# Semiconductor Fab Scheduling with Self-Supervised and Reinforcement
Learning††thanks: This work was funded by KWF project 28472, cms electronics
GmbH, FunderMax GmbH, Hirsch Armbänder GmbH, incubed IT GmbH, Infineon
Technologies Austria AG, Isovolta AG, Kostwein Holding GmbH, and
Privatstiftung Kärntner Sparkasse.

Pierre Tassel,1 Benjamin Kovács,1 Martin Gebser,12 Konstantin Schekotihin,1
Patrick Stöckermann,3 Georg Seidel3

###### Abstract

Semiconductor manufacturing is a notoriously complex and costly multi-step
process involving a long sequence of operations on expensive and quantity-
limited equipment. Recent chip shortages and their impacts have highlighted
the importance of semiconductors in the global supply chains and how reliant
on those our daily lives are. Due to the investment cost, environmental
impact, and time scale needed to build new factories, it is difficult to ramp
up production when demand spikes. This work introduces a method to
successfully learn to schedule a semiconductor manufacturing facility more
efficiently using deep reinforcement and self-supervised learning. We propose
the first adaptive scheduling approach to handle complex, continuous,
stochastic, dynamic, modern semiconductor manufacturing models. Our method
outperforms the traditional hierarchical dispatching strategies typically used
in semiconductor manufacturing plants, substantially reducing each order’s
tardiness and time until completion. As a result, our method yields a better
allocation of resources in the semiconductor manufacturing process.

## Introduction

Modern semiconductor manufacturing is one of the most complex industrial
processes to date (Iwai, Kakushima, and Wong 2005; Ravi 2021). Typical
production sequences involve a multitude of complex elementary operations
handled by highly specialized tools that require specific setups according to
variable product and process designs (Yugma et al. 2015). Over the years,
Moore’s Law has pushed the economic and manufacturing limits of industry by
approximately doubling the number of transistors per given area every two
years. On the other hand, the costs for research and development,
manufacturing, and testing become more and more elevated with each new chip
generation (Rupp and Selberherr 2010). Thus, increasing semiconductor
production capacities takes time, and it is impossible to ramp up chip output
overnight (Ravi 2021; Leslie 2022). Recent disruption in supply chains due to
the Coronavirus disease (COVID-19) pandemic and severe droughts distorted the
market for microchips, leading to a record of 22 weeks waiting time per order
in October 2021, rather than the around 12 weeks of late 2019, where experts
predict that supply will not catch up with demand for another year or more
(Leslie 2022). In this regard, optimization of production processes is an
important prerequisite for the mitigation of the current critical situation on
the markets. Beyond the economic and manufacturing challenges, increased
production facilities have a negative ecological impact, as the industry is
very resource-intensive, especially in terms of electricity, land, and water
consumption (Kuo, Kuo, and Chen 2022). In 2010, the semiconductor industry
accounted for $1.3$–$2.0\%$ of the total US electricity consumption in the
manufacturing sector (Gopalakrishnan, Mardikar, and Korakakis 2010). The chip-
making process is also water-intensive, and an average semiconductor factory
consumes around $20\,000$ tons of water a day with an estimate of $97.26\%$
used during manufacturing (Frost and Hua 2017). Increasing semiconductor fab
efficiency has thus been identified as one of the ways to achieve ecological
benefits while reducing the environmental impact of semiconductor
manufacturing (AG 2022).

Applications of machine learning techniques to combinatorial optimization
problems like planning and scheduling have recently emerged as a prospective
research area (Bengio, Lodi, and Prouvost 2021). As most of these problems are
$\mathit{NP}$-complete or even harder, exhaustive search methods do not scale
and quickly become inefficient. The same observation can be made for various
scheduling problems, most of which are $\mathit{NP}$-hard (Garey, Johnson, and
Sethi 1976; Sotskov and Shakhlevich 1995). Applying machine learning
techniques to scheduling problems has shown to be a promising approach to
generate good approximate solutions in reasonable time (Park et al. 2021;
Zhang et al. 2020; Stricker et al. 2018). Recently, Tassel et al. (2022)
suggested an RL-based method that can handle very large instances of the
resource constraint scheduling problem. However, previous work focuses on
scenarios that significantly simplify real-world problems (Stricker et al.
2018; Tassel, Gebser, and Schekotihin 2021). That is, considered problem
instances represent finite, deterministic, and static scheduling problems with
predefined sets of machines, jobs, and their operations, which remain constant
during the scheduling process. Therefore, our work is the first to consider
the _continuous_ , _stochastic_ , and _dynamic_ scheduling process of a
semiconductor fab, addressing a wide range of challenges encountered in large-
scale production environments.

##### Contributions.

In this paper, we propose a novel adaptive scheduling method to improve the
yield and reduce customer order delays in a realistic semiconductor
manufacturing environment. Using _Reinforcement Learning_ (RL) and _Self-
Supervised Learning_ (SSL), we train a single RL agent’s _Neural Network_
acting as a global dispatcher. Our contributions can be summarized as follows:

RL Training Approach. Due to the specificities of the semiconductor
manufacturing process and our model’s architecture, we have developed a
training loop using _Self-Supervised Learning_ to pre-train part of our
network and derivative-free _Evolution Strategies_ to train another part of
the network.

Feature Set. For each wafer lot, we have designed a feature set that can
efficiently represent the current status of the lot in the manufacturing
process.

Objective Function. We have designed an objective function to compare the
quality of different schedules. This function aims to reduce the average
tardiness and cycle-time, thus increasing the throughput of each lot type
while incorporating their priorities. The function also considers unfinished
lots (i.e., the WIP) in the computation of the score, correcting potential
bias in evaluating a continuous production environment.

Model Architecture. We introduce an order- and size-invariant neural network
architecture for a single-agent dispatcher using global self-attention on the
features of each wafer lot to be scheduled. Our architecture can efficiently
encode information such as the type of a required tool family without
increasing the dimensions of the input representation.

Empirical Evaluation. We evaluate our method on a large-scale, continuous,
stochastic and dynamic semiconductor simulation model from the literature
(Hassoun et al. 2019), using an open source event-based semiconductor
fabrication plant simulation (Kovacs et al. 2022). This testbed was designed
to help researchers to assess the performance of integrated dispatching
strategies and aims to credibly reflect the complexity of modern semiconductor
manufacturing. It includes two models, one representing a High-Volume/Low-Mix
(HV/LM) production scenario and the other a Low-Volume/High-Mix (LV/HM)
production plan. Our approach outperforms traditional dispatching heuristics
in both of these scenarios, achieving so far unmatched state-of-the-art
performance.

## Background

The problem of scheduling in semiconductor manufacturing can be modeled as a
variant of the Job-shop Scheduling Problem (JSP) (Gupta and Sivakumar 2006).
In classical JSP, given $n$ jobs and $m$ machines, each operation of a job
must be non-preemptively processed by exactly one machine within a minimal
makespan, i.e., the total completion time for all jobs. Although simple in
appearance, the problem is $\mathit{NP}$-complete, and unless $\mathit{NP}=P$,
it cannot be solved efficiently (Applegate and Cook 1991). Due to the limited
practical applicability of the classical JSP, multiple variants of this
problem, which better reflect real-world scheduling scenarios, have emerged
over the years (Potts and Strusevich 2009). For instance, in flow-shop
scheduling, each job has exactly $m$ operations and an operation $i$ is non-
preemptively executed on machine $i$. In a flexible JSP, for every operation,
there can be several machines able to process it. Complex JSP further extends
the flexible variant with batching machines, reentrant flows, sequence-
dependent setup times and release dates (Knopp, Dauzère-Pérès, and Yugma
2017). The term batching is used in the context of parallel processing and/or
in that of sequential processing (often referred to as cascading or serial
batching), denoting capacities for overlapping the execution times of multiple
jobs’ operations (Huang and Wu 2017).

However, a realistic model of the semiconductor manufacturing process cannot
be directly mapped to existing JSP variants. Therefore, (Hassoun et al. 2019)
presented SMT2020, a set of stochastic semiconductor fab simulation models,
aiming to credibly represent the complexity of modern semiconductor
manufacturing. These models go beyond previously proposed semiconductor
simulation testbeds such as MiniFab (Spier and Kempf 1995), MIMAC (Hassoun and
Kalir 2017), Harris (Kayton et al. 1997), and SEMATECH 300mm (Kiba et al.
2009) by including larger-scale datasets and more realistic characterizations
of manufacturing processes, incorporating the following features:

  1. 1.

Different Simulation Scenarios: The simulation includes models of Low-
Volume/High-Mix (LV/HM) and High-Volume/Low-Mix (HV/LM) wafer fabs, where mix
stands for high/low variety of products, and volume for small/large
quantities, respectively. The LV/HM scenario includes $10$ different products
with $10$ associated routes, reflecting make-to-order (MTO) environments for
supplying foundries, producing logic or memory devices. In the HV/LM setting,
$2$ high-volume products are considered to represent dedicated fabs, typically
producing logic devices in a make-to-stock (MTS) environment.

  2. 2.

Realistic Scale: Each product has a dedicated route with at least $500$ steps,
reflecting the complexity of modern wafer fabs. Similarly, both scenarios
contain $106$ tool groups organized in $12$ families, sometimes called machine
families or work centers. The LV/HM scenario includes a total of $1\,265$
machines, while HV/LM features $1\,071$ machines. At the beginning of the
scheduling horizon, there are $2\,700$ lots in progress for the LV/HM dataset
and $3\,635$ for the HV/LM scenario.

  3. 3.

Machine Availability: Simulation models include scheduled downtime due to
planned tool maintenance and unscheduled downtime due to machine breakdown.

  4. 4.

Batching and Cascading: SMT2020 scenarios involve batch and/or cascade
processing for certain tool groups.

  5. 5.

Loading and Transportation: Loading and unloading times on the tools and
transportation times between the different tool groups are considered in the
simulation.

  6. 6.

Machine Setup: Tools can have operation- and sequence-dependent
configurations, which must be set up on a machine before processing
operations. A configuration might imply a minimum/maximum number of operations
to be performed before it can/must be changed. Moreover, some operations
require reconfiguring a machine, even if the same configuration was in place
before.

  7. 7.

Tool Family Specificities: Certain functional areas of a fab have specific
constraints. For instance, steppers in the photolithography area have an lot-
to-lens dedication, enforcing the product to revisit the same tool in a future
step of the route. Metrology steps performing quality control can be skipped
by a production process with some probability. Dry etching steps have a
Critical Queue Time (CQT) constraint enforcing the next operation to be
performed with no more than a given amount of time delay.

  8. 8.

Lot Priorities: Lots can have different priorities: Super Hot Lots have the
highest and Hot Lots have a higher priority at all stages of processing than
regular lots.

  9. 9.

Variable Lot Size: The processing time of certain operations might depend on
the number of wafers in a lot, which can vary from one lot to another.

††footnotetext: Code and dataset available at:
https://github.com/ingambe/RL4SemiconductorFabSched

SMT2020 provides hierarchical dispatching heuristics, combining $4$ stages in
the order of significance as follows:

  1. 1.

Critical Queue Time: Lots with a CQT constraint in the previous operation have
the top priority.

  2. 2.

Lot Priorities: Lots must be scheduled according to their priorities, i.e,
Super Hot Lots scheduled first, then Hot Lots, and finally the regular lots.

  3. 3.

Setup Avoidance: Reconfiguration of a tool is time-consuming, and therefore,
it is preferable to schedule operations in a way that avoids switching the
setup.

  4. 4.

Generic Dispatching Heuristic: To break ties, the HV/LM and LV/HM scenarios
use First-Come First-Serve (FCFS) or Critical Ratio (CR) heuristics,
respectively, where CR considers the due date and the remaining processing
time required to complete a lot.

The resulting simulation models are substantially different from the static
and deterministic JSP, where all job and machine parameters are known in
advance and the environment is not changing during the computation and
execution of a schedule. In contrast, the SMT2020 scenarios are dynamic and
stochastic as changes in the environment, like the release of new lots with
different priorities or unscheduled downtimes of machines, might occur with
some probability at any point in time.

## Related Work

The Semiconductor Factory Scheduling Problem (SFSP) is often decomposed into
tool scheduling problems (single machine job-shop), work center scheduling
(parallel machine job-shop), or work area scheduling (flexible job-shop)
(Mönch et al. 2011). However, decomposition approaches merely approximate the
SFSP since combinatorial optimization methods, like _Constraint Programming_
(CP) or _Mixed Integer Programming_ (MIP), do not scale up to the full problem
due to the high implementation efforts and long computation times (Branke et
al. 2016). That is, a realistic instance of SFSP contains millions of
operations to schedule each year, thus going far beyond the capabilities of
modern optimization algorithms (Da Col and Teppan 2019; Kovács et al. 2021).
On the other hand, deterministic approximations of SFSP might reduce the
robustness of schedules in presence of unexpected events, like machine
breakdowns or reworks. Finally, SFSP is a dynamic problem where new customer
orders constantly arrive, requiring to recompute a schedule once the existing
one gets inconsistent with new data.

Combined methods try to mitigate scalability issues by applying combinatorial
optimization only for bottleneck machines (often lithographic tools), while
using dispatching heuristics for other operations (Govind et al. 2008; Ham and
Cho 2015; Klemmt et al. 2009). Often, such partial optimization approaches
only marginally improve over deterministic heuristic solutions, which have in
turn been shown to be less effective than meta-heuristics constructed for
stochastic SFSP-like settings (Nguyen et al. 2013).

The drawbacks of combinatorial optimization methods on large and stochastic
scheduling problems made dispatching heuristics a de-facto industry standard.
Respective scheduling systems only manage to approximate optimal solutions,
but they compute results in seconds and, therefore, can react to changes in
real time (Waschneck et al. 2016; Ham and Cho 2015). Any dispatching heuristic
is a handcrafted heuristic designed by experts taking experimental studies and
personal experience on typical scheduling scenarios into account. The quality
of different dispatching heuristics is evaluated using SFSP simulators with
various objectives like reducing mean cycle-time, average tardiness, and
delayed lots. Although algorithms based on dispatching heuristics have shown
good runtime results, there is no universal set of heuristics that work best
for all evaluation scenarios. For instance, their performance depends on the
current Work-In-Progress (WIP) (Nyhuis and Wiendahl 2012), selected key
performance indicators (Uzsoy et al. 1993), and types of machines used
(Fordyce et al. 2015).

Attempts to apply RL to semiconductor manufacturing aim to overcome the issues
of dispatching heuristics in complex SFSP environments. (Waschneck et al.
2018) propose a multi-agent RL approach in which each agent represents a work
center, i.e., a tool family. This work models a factory with $4$ work centers
and $48$ WIP lots to schedule, where transportation times and machine
breakdowns are disregarded. Agents are trained in two phases: (1) only one
agent (i.e., a work center) is trained while the others are controlled by a
heuristic, and (2) all agents controlling the $4$ work centers are trained
together. Results obtained with the trained agents are comparable to the
original dispatching strategy, but as the action space depends on the number
of WIP lots, the approach is not easy to generalize. Stricker et al. (2018)
propose a single-agent RL method for a testbed with $3$ work centers and a
total of $8$ machines, without setup changes between different kinds of
operations. The dispatching agent can select from a set of $12$ actions to
decide how to schedule a given lot. Since the action space depends on the
number of machines, a neural network trained on one fab cannot be reused for
others and it might be difficult to apply this approach to thousands of
machines as in SMT2020. However, the trained agent improves the initial
dispatching strategy significantly by $8\%$ better fab utilization.

## Learning Algorithm

This work aims at the development of a scheduler for the full SFSP, in which
the objective is to efficiently dispatch the lots to schedule on the machines,
so that metrics like mean cycle-time, average tardiness, and the number of
delayed lots are minimized. In this section, we formulate semiconductor fab
dispatching decision making as an RL problem, present the agent network
architecture, the self-supervision pre-training of the tool type encoding, and
the agent training settings.

### Approach Overview

Following the Markov Decision Process (MDP) formulation introduced by (Tassel
et al. 2022), the state-transition function of our deep RL agent considers
sequences of lots rather than a fixed number of actions assigning lots to
machines. In particular, at any timepoint when a set of lots must be
dispatched, the agent is trained to order lots in a sequence in which they are
assigned to machines using a set-up avoidance scheme. In the following, we
formulate dispatching decisions on lot operations to be scheduled in terms of
an MDP, including the state space $\mathcal{S}$, action space $\mathcal{A}$,
reward $r_{t}$ and transition $s_{t+1}\sim p(\;\cdot\mid s_{t},a_{t})$
functions.

##### State Space $\mathcal{S}$.

A state $s_{t}\in\mathcal{S}$ is given by the set $\mathcal{L}^{t}$ of legal
lots at timepoint $t$. A lot $l$ is _legal_ if its operation sequence
$O_{l}=(\tau_{1},\dots,\tau_{n})$ comprises an operation $\tau_{m}$ that can
be scheduled to free resources at $t$. Moreover, each lot $l$ is associated
with a set of features:  _(1)_ critical ratio, i.e., a ratio of the time
remaining till the due date $d_{l}$ to the expected processing time to finish
the job;  _(2)_ remaining time until the deadline $d_{l}$;  _(3)_ total
waiting time of the lot since its release;  _(4)_ waiting time of the lot
since the last operation;  _(5)_ number of lot-to-lens dedications until lot
completion, i.e., once a machine is used to perform a specific operation, it
will always be used for remaining operations $\tau_{i}\in O_{l}$;  _(6)_ lot
priority $w_{l}$;  _(7)_ sum of mean processing times for all remaining
operations of the lot;  _(8)_ minimum machine setup time to perform the
current operation of the lot;  _(9)_ mean processing time
$p_{\tau}\in\mathbb{N}^{+}$ of the current operation $\tau\in O_{l}$;  _(10)_
number of available machines with compatible setup;  _(11)_ minimum batch
size;  _(12)_ maximum batch size; and  _(13)_ the family of the required tool
group.

##### Action Space $\mathcal{A}$.

The action space $\mathcal{A}$ is composed of all legal lots at timepoint $t$,
i.e., $\mathcal{A}=\mathcal{L}^{t}$. Hence, the action space is discrete, and
the cardinality of $\mathcal{A}$ may vary from one timepoint to another. The
agent receives as input a set of dispatchable lots for the current timepoint
$t$, and provides as output the order in which to dispatch the given lots. For
each lot $l\in\mathcal{L}^{t}$, the network produces a scalar representing the
priority of $l$. The ordered set of lots to dispatch is then obtained by
sorting the lots according to their priority scores.

##### Reward $r_{t}$.

Usually, SFSP uses the average cycle-time, tardiness, and the number of
delayed lots for each lot type to compute the quality of a schedule.
Unfortunately, evaluations of these criteria provide meaningful results only
when computed for a complete schedule and might be misleading for incomplete
ones. Therefore, we have designed an objective function assigning scores for
any (in)complete schedule. This function penalizes tardiness of a lot and long
cycle-times while rewarding high throughputs. Moreover, the score for each lot
is weighted based on its priority.

Our objective function comprises two parts: one for the lots already finished
at the time of the evaluation and another for the lots still in the WIP. Both
parts are then summed together to obtain the final result. Considering lots in
the WIP for evaluating a schedule is particularly important in the case of a
continuous scheduling environment such as SFSP, as one potential bias of a
method is to only optimize metrics for the evaluated time windows but risk
bringing the fab into a catastrophic state in the future.

The first part, only considering lots finished at the evaluation time $t$, is
defined as:

$\displaystyle w_{l}*(p+(c_{l}-d_{l}))$ $\displaystyle\quad\text{if
}c_{l}>d_{l}$ $0$ otherwise

where $K$ the set of all lot types, $\lvert k_{i}\rvert$ the number of lots of
type $k_{i}$, $c_{l}$ the completion date and $d_{l}$ the deadline for lot
$l$. The constant term $p$ allows to penalize delayed lots independently of
whether the tardiness is large or small. The second part, focusing on lots in
the WIP at time $t$, is defined as:

$O_{p}=\sum_{k_{i}\in K}\;\frac{1}{\lvert k_{i}\rvert}\sum_{l\in
k_{i}}w_{l}*x_{i,l}$ $\displaystyle p+(t-(d_{l}-(a_{i}*e_{l})))$
$\displaystyle\quad\text{if }d_{l}-(a_{i}*e_{l})<t$ $0$ otherwise

where $a_{i}$ is the average cycle-time of lots of type $k_{i}$ computed on
the finished lots, and $e_{l}$ the sum of mean left-to-process operations’
processing time. The term $d_{l}-(a_{i}*e_{l})$ evaluates the deadline for
lots in the WIP by forecasting the completion date of a given lot based on the
average processing time for operations to be performed until completion. Thus,
the minimization of the objective function $O=O_{f}+O_{p}$ leads to the
reduction of tardiness, increase in throughput, and decrease in the average
cycle-time for each lot type, just as the original SFSP objective. Note that,
due to the specificities of our training approach, which employs a derivative-
free optimization method, we do not require a continuous and differentiable
reward function.

Figure 1: Policy network architecture with an embedding encoding, injecting
information about the tool family required for an operation. The network
outputs a priority score for each lot representation.

##### State Transition.

At each decision timepoint $t$, the simulator provides information about the
current state $s_{t}$ of a fab, and awaits an ordered list of lots to
dispatch. Similar to static dispatching heuristics, the ordered list is
processed using a hierarchical strategy, first giving higher priority to lots
with a CQT constraint, then lots’ priorities (i.e., _Hot Lots_), and finally
the original order provided by the agent. The simulator then allocates the
lots to their associated free machines in the given order until no free
resources remain or all lots are scheduled. Next, new events from the events
stack are handled and the timepoint is incremented. In _SMT2020_ , lots are
assigned to machines using the following heuristic strategy:

  1. _(1)_

If there is a lot-to-lens dedication constraint, assign to the mandatory
machine.

  2. _(2)_

If the current operation does not require a setup, assign to a machine without
setup, if any. Otherwise, assign to the machine with the highest waiting time.

  3. _(3)_

If the operation requires a setup, assign to a machine with this setup, if
available. Otherwise, assign to the machine with the lowest setup time.

This strategy is greedy as it merely determines advantageous allocations at a
timepoint $t$, does neither backtrack nor consider future incoming lots, i.e.,
no ”lookahead”. However, since _SMT2020_ supplies information about the legal
lots at timepoint $t$ only, the machine allocation strategy is also used by
our agent.

### Policy Architecture

The agent policy is a function $\pi_{\theta}(\mathcal{A}\mid s_{t})$ that maps
the current state of the simulator $s_{t}$ to the ordered set $\mathcal{A}$ of
legal actions (lots) to dispatch for the current timepoint $t$. $\pi_{\theta}$
is implemented as a deep neural network of the structure illustrated in Figure
1.

Before feeding the state representation to the policy neural network, we
rescale the features of the legal lots using a virtual batch normalization
from statistics over a $2$-months horizon collected before training. For each
lot representation $l\in s_{t}$, we inject information about the tool family
required to perform the current operation, using a learned encoding, and keep
the first $12$ features as the lot representation (i.e., the $13$-th feature
is the tool family index that is only used by the learned encoding). Encodings
are often used to inject positional information via absolute positional
embeddings (Vaswani et al. 2017) or relative bias factors (Shaw, Uszkoreit,
and Vaswani 2018), where a positional encoding is either fixed or learned
(Haviv et al. 2022). In our case, the injected information is the family of
the tool required to perform the current operation on a lot. We learn an
associate machine encoding for each of the $12$ tool families and add it to
the input lot representation. Moreover, we experimented with input embedding
before the machine encoding, similar to (Dosovitskiy et al. 2021; Vaswani et
al. 2017), but observed degradation of performance.

The transformed state representation $s^{\prime}_{t}$ is passed to a double-
head self-attention mechanism and a simple, position-wise, fully connected
Feed-Forward Network (FFN). We employ a residual connection with scaling
factor (He et al. 2016; Szegedy et al. 2017) around $s^{\prime}_{t}$ and the
FFN:

$\bm{y}=\alpha\bm{x}+\beta f(\alpha\bm{x})$

with $\alpha$ and $\beta$ as two learnable parameters. The multi-head
attention layer allows the network to determine the priority of each lot not
only based on the features of the given lot but taking all lots to schedule at
the current timepoint into account. It contains two attention heads, each of
dimension $d_{\mathit{head}}=6$. The FFN comprises three linear
transformations of dimension $16$ with a _tanh_ activation in between:

$\mathrm{FFN}(x)=\tanh(\tanh(xW_{1}+b_{1})W_{2}+b_{2})W_{3}+b_{3}$

Similar to the transformer architecture (Vaswani et al. 2017), we apply the
FFN to each lot representation separately. The inner layers have
dimensionality $d_{\mathit{FFN}}=16$, while the network outputs a scalar
representing the estimated lot priority.

### Training

With an average of $7\,254\,125$ allocated steps per year, training the policy
network is challenging since the search space is highly combinatorial. Also,
agents need to be trained with a long horizon since some products have a long
cycle-time with completion up to three months in the future, while disruptions
such as machine breakdown occur relatively rarely. Therefore, we use a
training horizon of $6$ months, as it provides a good balance between training
runtime and the diversity of events handled by an agent.

#### Self-Supervised Learning.

Figure 2: Self-supervised pretext task architecture. The lot representation is
extended with information about the required tool family, which then must be
correctly predicted by the network.

While the multi-head attention and FFN layers are trained together, the
machine encoding is trained separately using a self-supervised pretext task
(Doersch, Gupta, and Efros 2015; Gidaris, Singh, and Komodakis 2018). The goal
of the machine encoding is to inject information about the required tool
family into the lot representation. This information is very important as
different tool families have very different processes and constraints, for
example, lithographic tools are usually considered critical due to their
scarcity, etching steps often have a coupling constraint with their next
operation, and other tool families also have their specificities. To achieve
this goal, we propose to train a network as shown in Figure 2, having a
similar architecture as the policy network. The network receives a set of lot
representations as input and yields a probability distribution over all
possible tool families for each lot as output.

Therefore, given a set of $N$ training state representations
$D=\\{x_{i}\\}_{i=0}^{N}$ sampled from a $2$-months horizon simulation, the
self-supervised training objective that the model must learn to solve is:

$\min_{\theta}\frac{1}{N}\sum_{i=1}^{N}loss(x_{i},\theta)+\lambda\|\theta_{e}\|^{2}$

where the loss function $loss(.)$ is the cross-entropy between the predicated
and the ground-truth tool family, $\theta$ the parameters of the network,
$\theta_{e}$ the parameters of the machine encoding, and $\lambda$ the
regularization constant. This $L_{2}$ regularization is required to prevent
the encoding from taking too large values and, thus, destroying the
information about lot features. The $\lambda=0.2$ regularization constant is
tuned such that the performance on the downstream task (i.e., the fab
dispatching) is maximized.

The pretext task is trained until convergence using Stochastic Gradient
Descent (SGD) and backpropagation. The multi-head attention and the linear
layer are then discarded, and only the learned machine encoding weights are
kept frozen for the downstream task.

#### Policy Network Training.

Objective functions of scheduling problems and discrete sets of possible job
allocations result in learning environments with non-smooth objectives, which
are unsuitable for training by backpropagating gradients. We overcome this
issue by using a combination of two methods: estimation of a gradient with the
Natural Evolution Strategies (NES) algorithm (Salimans et al. 2017) and a
modern gradient-based optimization algorithm to update weights of the policy
network.

NES belongs to a broad family of Evolution Strategies (ES), which are black-
box, derivative-free methods inspired by natural evolution. When applied to
the training of a policy network $\pi_{\theta}$, NES methods use the parameter
vector $\theta$ of the policy to generate a population of parameter vectors
$\theta^{\prime}_{i}\sim\mathcal{N}(\theta,\sigma^{2}I)$ by adding noise
sampled from an isotropic multivariate Gaussian to the original parameter
vector $\theta$, thus obtaining a set of $n$ successor networks. Then, each of
the $n$ networks in the population is used to evaluate an objective function,
the episodic reward function in our case, to estimate the _natural gradient_
for the policy network:

$\nabla_{\theta}F(\pi_{\theta})\approx\frac{1}{\sigma_{t}*N}\sum_{i=1}^{N}F(\pi_{\theta_{i}^{\prime}})\epsilon_{i}$

where $\epsilon_{i}\sim\mathcal{N}(0,\mathbb{I})$ is sampled from a
multivariate Gaussian, $\theta_{i}^{\prime}=\theta+\sigma_{t}\epsilon_{i}$,
$F(\pi_{\theta_{i}^{\prime}})$ is estimated using a single trajectory, and
$\sigma_{t}>0$ is a constant defined as $\sigma_{t}=0.975^{t}\sigma$ with
$\sigma=0.005$.

The estimated natural gradient is provided to Adam optimizer (Kingma and Ba
2015) with $\beta_{1}=0.9$, $\beta_{2}=0.98$, and $\epsilon=10^{-4}$, which
updates the parameters $\theta$ of the policy network. Within the $i$-th run,
we decay the learning rate with a cosine annealing (Loshchilov and Hutter
2017) as follows:

$\eta_{i}=\frac{\eta_{max}}{2}\left(1+\cos\left(\frac{i}{i_{max}}\pi\right)\right)$

where $\eta_{max}=0.01$ is the maximum learning rate, and $i_{max}=40$ the
maximum number of iterations.

The training algorithm continues to repeat these steps till convergence.
Because NES works with function evaluations, there is no need to rely on a
continuous reward function. Therefore, we can use the previously defined
unbiased and episodic objective function.

Lot Type | Ours | Hierarchical CR | Hierarchical FIFO
---|---|---|---
Super Hot Lots | On-Time |  $\mathbf{42.50\%}$$\displaystyle\,\pm\,$ | $13.94$ |  $41.84\%$$\displaystyle\,\pm\,$ | $12.45$ |  $38.06\%$$\displaystyle\,\pm\,$ | $10.64$
Theoretical Cycle-Time: $\mathbf{24.75}$ | Cycle-Time |  $\mathbf{33.61}$$\displaystyle\,\pm\,$ | $0.51$ |  $33.89$$\displaystyle\,\pm\,$ | $0.43$ |  $33.99$$\displaystyle\,\pm\,$ | $0.44$
Hot Lots | On-Time |  $\mathbf{55.86\%}$$\displaystyle\,\pm\,$ | $3.58$ |  $46.76\%$$\displaystyle\,\pm\,$ | $3.77$ |  $47.66\%$$\displaystyle\,\pm\,$ | $3.74$
Theoretical Cycle-Time: $\mathbf{19.65}$ | Cycle-Time |  $\mathbf{26.54}$$\displaystyle\,\pm\,$ | $0.14$ |  $26.94$$\displaystyle\,\pm\,$ | $0.13$ |  $26.91$$\displaystyle\,\pm\,$ | $0.12$
Regular Lots | On-Time |  $\mathbf{44.91\%}$$\displaystyle\,\pm\,$ | $2.36$ |  $0.00\%$$\displaystyle\,\pm\,$ | $0.00$ |  $0.00\%$$\displaystyle\,\pm\,$ | $0.00$
Theoretical Cycle-Time: $\mathbf{17.31}$ | Cycle-Time |  $\mathbf{52.24}$$\displaystyle\,\pm\,$ | $1.80$ |  $70.80$$\displaystyle\,\pm\,$ | $0.60$ |  $72.57$$\displaystyle\,\pm\,$ | $0.61$
| Cost |  $\mathbf{33\,171.37}$$\displaystyle\,\pm\,$ | $1\,655.53$ |  $39\,813.37$$\displaystyle\,\pm\,$ | $899.89$ |  $38\,013.67$$\displaystyle\,\pm\,$ | $1\,327.73$

(a) High-Volume/Low-Mix (HV/LM)

Lot Type | Ours | Hierarchical CR | Hierarchical FIFO
---|---|---|---
Super Hot Lots | On-Time |  $\mathbf{97.38\%}$$\displaystyle\,\pm\,$ | $4.63$ |  $97.19\%$$\displaystyle\,\pm\,$ | $4.72$ |  $95.04\%$$\displaystyle\,\pm\,$ | $5.44$
Theoretical Cycle-Time: $\mathbf{24.75}$ | Cycle-Time |  $\mathbf{33.64}$$\displaystyle\,\pm\,$ | $0.44$ |  $33.73$$\displaystyle\,\pm\,$ | $0.42$ |  $33.86$$\displaystyle\,\pm\,$ | $0.45$
Hot Lots | On-Time |  $96.03\%$$\displaystyle\,\pm\,$ | $3.20$ |  $\mathbf{96.34\%}$$\displaystyle\,\pm\,$ | $3.04$ |  $95.38\%$$\displaystyle\,\pm\,$ | $3.35$
Theoretical Cycle-Time: $\mathbf{17.31}$ | Cycle-Time |  $\mathbf{23.67}$$\displaystyle\,\pm\,$ | $0.23$ |  $23.71$$\displaystyle\,\pm\,$ | $0.23$ |  $23.80$$\displaystyle\,\pm\,$ | $0.23$
Regular Lots | On-Time |  $\mathbf{56.47\%}$$\displaystyle\,\pm\,$ | $2.36$ |  $0.00\%$$\displaystyle\,\pm\,$ | $0.00$ |  $0.06\%$$\displaystyle\,\pm\,$ | $0.09$
Theoretical Cycle-Time: $\mathbf{17.31}$ | Cycle-Time |  $\mathbf{44.15}$$\displaystyle\,\pm\,$ | $1.80$ |  $52.25$$\displaystyle\,\pm\,$ | $0.30$ |  $52.38$$\displaystyle\,\pm\,$ | $0.63$
| Cost |  $\mathbf{94\,349.38}$$\displaystyle\,\pm\,$ | $2\,716.51$ |  $108\,855.67$$\displaystyle\,\pm\,$ | $2\,044.28$ |  $102\,170.71$$\displaystyle\,\pm\,$ | $2\,497.89$

(b) Low-Volume/High-Mix (LV/HM)

Table 1:  Comparison against the original dispatching heuristics on the
SMT2020 testbed for the HV/LM and LV/HM scenarios with $6$ months of
simulation. For each lot priority level, we report the aggregated mean and
standard deviation of the percentage of lots on-time and the cycle-time (see
the appendix for more detailed results). Our approach outperforms the original
ones in both scenarios, especially for regular lots, which represent the major
part of all lots in production.

## Empirical Evaluation

We evaluated our approach against the original _SMT2020_ hierarchical
dispatching heuristics described in the background, using a machine equipped
with AMD EPYC $7452$ CPU and $128$GB of RAM for benchmarking and training,
with the training iterations taking $3$ hours on average. As discussed in the
related work, to our knowledge, there are no CP- or MIP-based approaches in
the literature for handling such large problems of stochastic, dynamic, and
continuous nature. Therefore, as a baseline, we rely on proposed heuristic
methods, which are the de-facto standard in semiconductor industry (Govind et
al. 2008; Ham and Cho 2015).

For each dataset, we run all dispatching strategies with a horizon of $6$
months with $100$ different random seeds and report the percentage of lots
that meet the deadlines defined at release time along with the cycle-time,
i.e., the average number of days from release to completion per given lot
type. We provide the mean and the standard deviation of each metric collected
for agents trained in a considered scenario. The penalty constant was set to
$p=10$ encouraging the learner to prefer fewer delayed lots with a larger
delay over a larger number of delayed lots with a very short delay each. In
comparison, experimentation conducted with $p=5$ leads to fewer lots on time
but a shorter average cycle-time.

Our evaluation settings are realistic since the production scenario rarely
changes during the lifetime of a fab. Due to page constraints, Table 1 only
reports the aggregated metrics for each lot priority level. To get a full
picture of the performance of our approach, we invite the reader to refer to
the appendix, providing more fine-grained metrics per lot type.

As we can see in Table 1(a), for the HV/LM dataset, our approach outperforms
the state of the art, obtaining solutions of $16.68\%$ lower cost when
compared to the best-performing dispatching heuristic. Both the average
tardiness and the cycle-time of lots are reduced. While the absolute reduction
in solution cost is already significant, as the unknown “optimal” solution
(i.e., the optimal solution for each seed) has a cost greater than $0$, we
have an even higher relative improvement. Focusing on Regular Lots, the vast
majority of all lots, the original dispatching heuristics are not capable of
scheduling them to meet the deadline. In contrast, our approach meets the
deadline for $44.91\%$ of the lots and reduces the cycle-time by $26.21\%$
when compared to the CR dispatching heuristic, thus substantially decreasing
the average time from release to completion of a lot.

Aggregated results for the LV/HM dataset are reported in Table 1(b). As with
the HV/LM scenario, our method outperforms the state of the art, obtaining
solutions of $7.66\%$ lower cost when compared to the best-performing
dispatching heuristic. Although the metrics obtained for Hot Lots are very
similar, our approach is always capable of reducing the cycle-time and
achieving overall lower tardiness of lots. Also paralleling the HV/LM
scenario, the agent vastly outperforms static dispatching heuristics when
focusing on Regular Lots, meeting the deadline for $56.47\%$ of them. In
comparison, Regular Lots scheduled by standard heuristics met their deadlines
in almost none of the cases. Furthermore, our approach greatly reduces
tardiness and the cycle-time by about $15.5\%$, thus making better usage of
fab resources.

For extended evaluations, including results per lot type, comparison with
additional dispatching heuristics from the literature, and a $1$-year
evaluation horizon, we invite the reader to refer to the appendix. In all of
these scenarios, we observe similar results as described above. Our approach
outperforms other methods from the literature, improving almost all metrics
while only adding a negligible overhead on computation time with an, on
average, $1.02$ ms longer decision time per timepoint.

## Conclusion

We present a new scheduling method capable of improving the usage of
semiconductor fab resources. Unlike prior works focusing on toy datasets with
a handful of products, machines, or constraints, we aim at realistic
semiconductor manufacturing models in full complexity and scale. Using a
combination of self-supervised learning to learn an embedding encoding
information about available tool families, and reinforcement learning to train
an attention-based model capable of considering all lots to allocate, we
trained an agent to dispatch lots to machines efficiently. Extensive
evaluation against a wide range of dispatching heuristics in the literature,
including those proposed for the considered datasets, shows the ability of our
approach to substantially improve semiconductor scheduling capacities and
achieve better usage of a fab’s resources.

While these initial results are encouraging, many challenges remain. Mainly,
applying our approach in production first requires building trust of operators
responsible for the production. In fact, it can be challenging to convince
operators to follow decisions made by such a black-box system, especially if
some decisions might initially seem counterintuitive. Another challenge is to
provide a more global vision of the fab to the agent by including lots
currently processed by a tool in the state representation. It would allow the
agent to anticipate which lots are getting ready next and possibly decide to
leave a tool unallocated for a soon-to-be-available lot. In this regard, with
a WIP exceeding $3\,000$ lots, the squared complexity of the dot-product
attention mechanism used in our current architecture might, however, limit the
possibilities to incorporate currently processed lots.

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Lot Type | Ours | Hierarchical CR | Hierarchical FIFO
---|---|---|---
Super Hot Lots 3 | On-Time |  $\mathbf{42.50\%}$$\displaystyle\,\pm\,$ | $13.94$ |  $41.84\%$$\displaystyle\,\pm\,$ | $12.45$ |  $38.06\%$$\displaystyle\,\pm\,$ | $10.64$
Theoretical Cycle-Time: $\mathbf{24.75}$ | Cycle-Time |  $\mathbf{33.61}$$\displaystyle\,\pm\,$ | $0.51$ |  $33.89$$\displaystyle\,\pm\,$ | $0.43$ |  $33.99$$\displaystyle\,\pm\,$ | $0.44$
Hot Lots 3 | On-Time |  $\mathbf{51.77\%}$$\displaystyle\,\pm\,$ | $3.71$ |  $42.99\%$$\displaystyle\,\pm\,$ | $3.58$ |  $43.09\%$$\displaystyle\,\pm\,$ | $3.85$
Theoretical Cycle-Time: $\mathbf{24.75}$ | Cycle-Time |  $\mathbf{33.56}$$\displaystyle\,\pm\,$ | $0.17$ |  $34.02$$\displaystyle\,\pm\,$ | $0.14$ |  $34.02$$\displaystyle\,\pm\,$ | $0.13$
Hot Lots 4 | On-Time |  $\mathbf{59.94\%}$$\displaystyle\,\pm\,$ | $3.45$ |  $50.53\%$$\displaystyle\,\pm\,$ | $3.95$ |  $52.26\%$$\displaystyle\,\pm\,$ | $3.63$
Theoretical Cycle-Time: $\mathbf{14.54}$ | Cycle-Time |  $\mathbf{19.52}$$\displaystyle\,\pm\,$ | $0.11$ |  $19.86$$\displaystyle\,\pm\,$ | $0.12$ |  $19.80$$\displaystyle\,\pm\,$ | $0.10$
Regular Lots 3 | On-Time |  $\mathbf{51.72\%}$$\displaystyle\,\pm\,$ | $2.47$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$
Theoretical Cycle-Time: $\mathbf{24.75}$ | Cycle-Time |  $\mathbf{63.71}$$\displaystyle\,\pm\,$ | $1.90$ |  $90.77$$\displaystyle\,\pm\,$ | $0.53$ |  $93.61$$\displaystyle\,\pm\,$ | $0.71$
Regular Lots 4 | On-Time |  $\mathbf{38.1\%}$$\displaystyle\,\pm\,$ | $2.04$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$
Theoretical Cycle-Time: $\mathbf{14.54}$ | Cycle-Time |  $\mathbf{40.77}$$\displaystyle\,\pm\,$ | $1.7$ |  $50.83$$\displaystyle\,\pm\,$ | $0.66$ |  $51.52$$\displaystyle\,\pm\,$ | $0.5$
| Cost |  $\mathbf{33\,171.37}$$\displaystyle\,\pm\,$ | $1\,655.53$ |  $39\,813.37$$\displaystyle\,\pm\,$ | $899.89$ |  $38\,013.67$$\displaystyle\,\pm\,$ | $1\,327.73$

(a) High-Volume/Low-Mix (HV/LM)

Lot Type | Ours | Hierarchical CR | Hierarchical FIFO
---|---|---|---
Super Hot Lots 3 | On-Time |  $\mathbf{97.38\%}$$\displaystyle\,\pm\,$ | $4.63$ |  $97.19\%$$\displaystyle\,\pm\,$ | $4.72$ |  $95.04\%$$\displaystyle\,\pm\,$ | $5.44$
Theoretical Cycle-Time: $\mathbf{24.75}$ | Cycle-Time |  $\mathbf{33.64}$$\displaystyle\,\pm\,$ | $0.44$ |  $33.73$$\displaystyle\,\pm\,$ | $0.42$ |  $33.86$$\displaystyle\,\pm\,$ | $0.45$
Hot Lots 1 | On-Time |  $97.08\%$$\displaystyle\,\pm\,$ | $3.29$ |  $\mathbf{97.60\%}$$\displaystyle\,\pm\,$ | $2.77$ |  $96.4\%$$\displaystyle\,\pm\,$ | $3.42$
Theoretical Cycle-Time: $\mathbf{21.75}$ | Cycle-Time |  $\mathbf{29.81}$$\displaystyle\,\pm\,$ | $0.29$ |  $29.9$$\displaystyle\,\pm\,$ | $0.27$ |  $29.97$$\displaystyle\,\pm\,$ | $0.26$
Hot Lots 2 | On-Time |  $98.65\%$$\displaystyle\,\pm\,$ | $2.2$ |  $\mathbf{99.23\%}$$\displaystyle\,\pm\,$ | $1.8$ |  $98.15\%$$\displaystyle\,\pm\,$ | $2.59$
Theoretical Cycle-Time: $\mathbf{23.3}$ | Cycle-Time |  $\mathbf{31.5}$$\displaystyle\,\pm\,$ | $0.24$ |  $31.66$$\displaystyle\,\pm\,$ | $0.26$ |  $31.81$$\displaystyle\,\pm\,$ | $0.25$
Hot Lots 3 | On-Time |  $\mathbf{97.93\%}$$\displaystyle\,\pm\,$ | $2.55$ |  $97.90\%$$\displaystyle\,\pm\,$ | $2.87$ |  $96.93\%$$\displaystyle\,\pm\,$ | $3.38$
Theoretical Cycle-Time: $\mathbf{24.75}$ | Cycle-Time |  $\mathbf{33.56}$$\displaystyle\,\pm\,$ | $0.32$ |  $33.71$$\displaystyle\,\pm\,$ | $0.27$ |  $33.88$$\displaystyle\,\pm\,$ | $0.3$
Hot Lots 4 | On-Time |  $96.62\%$$\displaystyle\,\pm\,$ | $3.55$ |  $\mathbf{96.92\%}$$\displaystyle\,\pm\,$ | $3.42$ |  $95.91\%$$\displaystyle\,\pm\,$ | $3.67$
Theoretical Cycle-Time: $\mathbf{14.54}$ | Cycle-Time |  $\mathbf{19.87}$$\displaystyle\,\pm\,$ | $0.22$ |  $19.94$$\displaystyle\,\pm\,$ | $0.22$ |  $20$$\displaystyle\,\pm\,$ | $0.21$
Hot Lots 5 | On-Time |  $\mathbf{94.17\%}$$\displaystyle\,\pm\,$ | $3.03$ |  $94.05\%$$\displaystyle\,\pm\,$ | $3.06$ |  $94.20\%$$\displaystyle\,\pm\,$ | $2.88$
Theoretical Cycle-Time: $\mathbf{10.10}$ | Cycle-Time |  $\mathbf{14.09}$$\displaystyle\,\pm\,$ | $0.17$ |  $14.11$$\displaystyle\,\pm\,$ | $0.17$ |  $14.06$$\displaystyle\,\pm\,$ | $0.17$
Hot Lots 6 | On-Time |  $96.42\%$$\displaystyle\,\pm\,$ | $2.62$ |  $\mathbf{97.15\%}$$\displaystyle\,\pm\,$ | $2.47$ |  $96.64\%$$\displaystyle\,\pm\,$ | $2.48$
Theoretical Cycle-Time: $\mathbf{12.95}$ | Cycle-Time |  $\mathbf{17.47}$$\displaystyle\,\pm\,$ | $0.16$ |  $17.53$$\displaystyle\,\pm\,$ | $0.21$ |  $17.53$$\displaystyle\,\pm\,$ | $0.2$
Hot Lots 7 | On-Time |  $\mathbf{95.22\%}$$\displaystyle\,\pm\,$ | $3.68$ |  $94.82\%$$\displaystyle\,\pm\,$ | $3.54$ |  $94.28\%$$\displaystyle\,\pm\,$ | $3.87$
Theoretical Cycle-Time: $\mathbf{15.46}$ | Cycle-Time |  $\mathbf{21.02}$$\displaystyle\,\pm\,$ | $0.21$ |  $21.05$$\displaystyle\,\pm\,$ | $0.22$ |  $21.11$$\displaystyle\,\pm\,$ | $0.23$
Hot Lots 8 | On-Time |  $\mathbf{92.41\%}$$\displaystyle\,\pm\,$ | $3.71$ |  $92.39\%$$\displaystyle\,\pm\,$ | $3.77$ |  $90.32\%$$\displaystyle\,\pm\,$ | $4.01$
Theoretical Cycle-Time: $\mathbf{16.02}$ | Cycle-Time |  $\mathbf{22.21}$$\displaystyle\,\pm\,$ | $0.21$ |  $22.21$$\displaystyle\,\pm\,$ | $0.22$ |  $22.33$$\displaystyle\,\pm\,$ | $0.24$
Hot Lots 9 | On-Time |  $95.96\%$$\displaystyle\,\pm\,$ | $3.28$ |  $\mathbf{96.27\%}$$\displaystyle\,\pm\,$ | $3.37$ |  $95.03\%$$\displaystyle\,\pm\,$ | $3.54$
Theoretical Cycle-Time: $\mathbf{16.95}$ | Cycle-Time |  $23.16$$\displaystyle\,\pm\,$ | $0.2$ |  $\mathbf{23.05}$$\displaystyle\,\pm\,$ | $0.21$ |  $23.22$$\displaystyle\,\pm\,$ | $0.23$
Hot Lots 10 | On-Time |  $95.88\%$$\displaystyle\,\pm\,$ | $4.1$ |  $\mathbf{97.04\%}$$\displaystyle\,\pm\,$ | $3.33$ |  $95.96\%$$\displaystyle\,\pm\,$ | $3.64$
Theoretical Cycle-Time: $\mathbf{17.32}$ | Cycle-Time |  $24.01$$\displaystyle\,\pm\,$ | $0.28$ |  $\mathbf{23.96}$$\displaystyle\,\pm\,$ | $0.25$ |  $24.08$$\displaystyle\,\pm\,$ | $0.25$
Regular Lots 1 | On-Time |  $\mathbf{65.80\%}$$\displaystyle\,\pm\,$ | $1.77$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$
Theoretical Cycle-Time: $\mathbf{21.75}$ | Cycle-Time |  $\mathbf{50.77}$$\displaystyle\,\pm\,$ | $1.67$ |  $61.76$$\displaystyle\,\pm\,$ | $0.33$ |  $69.76$$\displaystyle\,\pm\,$ | $0.78$
Regular Lots 2 | On-Time |  $\mathbf{69.58\%}$$\displaystyle\,\pm\,$ | $1.8$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$
Theoretical Cycle-Time: $\mathbf{23.3}$ | Cycle-Time |  $\mathbf{49.31}$$\displaystyle\,\pm\,$ | $1.26$ |  $70.22$$\displaystyle\,\pm\,$ | $0.45$ |  $70.84$$\displaystyle\,\pm\,$ | $0.76$
Regular Lots 3 | On-Time |  $\mathbf{58.29\%}$$\displaystyle\,\pm\,$ | $2.42$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$
Theoretical Cycle-Time: $\mathbf{24.75}$ | Cycle-Time |  $\mathbf{61.85}$$\displaystyle\,\pm\,$ | $2.38$ |  $74.40$$\displaystyle\,\pm\,$ | $0.46$ |  $78.75$$\displaystyle\,\pm\,$ | $0.88$
Regular Lots 4 | On-Time |  $\mathbf{46.02\%}$$\displaystyle\,\pm\,$ | $3.7$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$
Theoretical Cycle-Time: $\mathbf{14.54}$ | Cycle-Time |  $\mathbf{40.85}$$\displaystyle\,\pm\,$ | $2.58$ |  $44.7$$\displaystyle\,\pm\,$ | $0.34$ |  $44.88$$\displaystyle\,\pm\,$ | $0.49$
Regular Lots 5 | On-Time |  $\mathbf{24.3\%}$$\displaystyle\,\pm\,$ | $2.12$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$
Theoretical Cycle-Time: $\mathbf{10.10}$ | Cycle-Time |  $41.57$$\displaystyle\,\pm\,$ | $2.48$ |  $\mathbf{31.28}$$\displaystyle\,\pm\,$ | $0.19$ |  $33.16$$\displaystyle\,\pm\,$ | $0.53$
Regular Lots 6 | On-Time |  $\mathbf{58.18\%}$$\displaystyle\,\pm\,$ | $2.70$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0.59\%$$\displaystyle\,\pm\,$ | $0.92$
Theoretical Cycle-Time: $\mathbf{12.95}$ | Cycle-Time |  $\mathbf{33.55}$$\displaystyle\,\pm\,$ | $1.57$ |  $43.54$$\displaystyle\,\pm\,$ | $0.35$ |  $39.03$$\displaystyle\,\pm\,$ | $0.59$
Regular Lots 7 | On-Time |  $\mathbf{65.29\%}$$\displaystyle\,\pm\,$ | $2.24$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$
Theoretical Cycle-Time: $\mathbf{15.46}$ | Cycle-Time |  $\mathbf{36.7}$$\displaystyle\,\pm\,$ | $1.42$ |  $45.08$$\displaystyle\,\pm\,$ | $0.31$ |  $47.73$$\displaystyle\,\pm\,$ | $0.52$
Regular Lots 8 | On-Time |  $\mathbf{67.04\%}$$\displaystyle\,\pm\,$ | $1.83$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$
Theoretical Cycle-Time: $\mathbf{16.02}$ | Cycle-Time |  $\mathbf{38.43}$$\displaystyle\,\pm\,$ | $1.50$ |  $52.13$$\displaystyle\,\pm\,$ | $0.36$ |  $51.81$$\displaystyle\,\pm\,$ | $0.65$
Regular Lots 9 | On-Time |  $\mathbf{53.25\%}$$\displaystyle\,\pm\,$ | $2.66$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$
Theoretical Cycle-Time: $\mathbf{16.95}$ | Cycle-Time |  $\mathbf{44.48}$$\displaystyle\,\pm\,$ | $1.60$ |  $48.24$$\displaystyle\,\pm\,$ | $0.25$ |  $33.86$$\displaystyle\,\pm\,$ | $0.45$
Regular Lots 10 | On-Time |  $\mathbf{56.97\%}$$\displaystyle\,\pm\,$ | $2.39$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$
Theoretical Cycle-Time: $\mathbf{17.32}$ | Cycle-Time |  $\mathbf{43.95}$$\displaystyle\,\pm\,$ | $1.67$ |  $51.13$$\displaystyle\,\pm\,$ | $0.29$ |  $54.01$$\displaystyle\,\pm\,$ | $0.63$
| Cost |  $\mathbf{94\,349.38}$$\displaystyle\,\pm\,$ | $2\,716.51$ |  $108\,855.67$$\displaystyle\,\pm\,$ | $2\,044.28$ |  $102\,170.71$$\displaystyle\,\pm\,$ | $2\,497.89$

(b) Low-Volume/High-Mix (LV/HM)

Table 2:  Detailed results of the comparison against the original dispatching
rules on the SMT2020 testbed for the HV/LM and LV/HM scenarios with $6$ months
of simulation, providing aggregated metrics for each lot priority level. For
each lot type, we report the mean and standard deviation of the percentage of
lots on-time and the cycle-time. Our approach outperforms the original one on
both datasets.

Lot Type | Ours | Hierarchical SPT | Hierarchical SRPT | Hierarchical EDD | Hierarchical LS
---|---|---|---|---|---
Super Hot Lots 3 | On-Time |  $42.50\%$$\displaystyle\,\pm\,$ | $13.94$ |  $41.76\%$$\displaystyle\,\pm\,$ | $11.48$ |  $37.82\%$$\displaystyle\,\pm\,$ | $12.76$ |  $37.08\%$$\displaystyle\,\pm\,$ | $15.20$ |  $\mathbf{43.37\%}$$\displaystyle\,\pm\,$ | $13.01$
Theoretical Cycle-Time: $\mathbf{24.75}$ | Cycle-Time |  $\mathbf{33.61}$$\displaystyle\,\pm\,$ | $0.51$ |  $33.81$$\displaystyle\,\pm\,$ | $0.45$ |  $33.95$$\displaystyle\,\pm\,$ | $0.42$ |  $33.93$$\displaystyle\,\pm\,$ | $0.53$ |  $33.74$$\displaystyle\,\pm\,$ | $0.45$
Hot Lots 3 | On-Time |  $\mathbf{51.77\%}$$\displaystyle\,\pm\,$ | $3.71$ |  $44.40\%$$\displaystyle\,\pm\,$ | $4.44$ |  $41.06\%$$\displaystyle\,\pm\,$ | $3.93$ |  $43.32\%$$\displaystyle\,\pm\,$ | $3.76$ |  $44.5\%$$\displaystyle\,\pm\,$ | $3.43$
Theoretical Cycle-Time: $\mathbf{24.75}$ | Cycle-Time |  $\mathbf{33.56}$$\displaystyle\,\pm\,$ | $0.17$ |  $33.86$$\displaystyle\,\pm\,$ | $0.19$ |  $34.08$$\displaystyle\,\pm\,$ | $0.16$ |  $33.94$$\displaystyle\,\pm\,$ | $0.15$ |  $33.87$$\displaystyle\,\pm\,$ | $0.15$
Hot Lots 4 | On-Time |  $\mathbf{59.94\%}$$\displaystyle\,\pm\,$ | $3.45$ |  $51.94\%$$\displaystyle\,\pm\,$ | $3.84$ |  $50.33\%$$\displaystyle\,\pm\,$ | $3.4$ |  $51.7\%$$\displaystyle\,\pm\,$ | $3.84$ |  $52.64\%$$\displaystyle\,\pm\,$ | $3.33$
Theoretical Cycle-Time: $\mathbf{14.54}$ | Cycle-Time |  $\mathbf{19.52}$$\displaystyle\,\pm\,$ | $0.11$ |  $19.79$$\displaystyle\,\pm\,$ | $0.12$ |  $19.86$$\displaystyle\,\pm\,$ | $0.11$ |  $19.82$$\displaystyle\,\pm\,$ | $0.13$ |  $19.77$$\displaystyle\,\pm\,$ | $0.10$
Regular Lots 3 | On-Time |  $\mathbf{51.72\%}$$\displaystyle\,\pm\,$ | $2.47$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$
Theoretical Cycle-Time: $\mathbf{24.75}$ | Cycle-Time |  $\mathbf{63.71}$$\displaystyle\,\pm\,$ | $1.90$ |  $106.91$$\displaystyle\,\pm\,$ | $1.59$ |  $109.9$$\displaystyle\,\pm\,$ | $0.97$ |  $87.58$$\displaystyle\,\pm\,$ | $0.72$ |  $86.24$$\displaystyle\,\pm\,$ | $0.55$
Regular Lots 4 | On-Time |  $38.1\%$$\displaystyle\,\pm\,$ | $2.04$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $\mathbf{69.14\%}$$\displaystyle\,\pm\,$ | $2.49$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$
Theoretical Cycle-Time: $\mathbf{14.54}$ | Cycle-Time |  $40.77$$\displaystyle\,\pm\,$ | $1.7$ |  $53.25$$\displaystyle\,\pm\,$ | $2.02$ |  $\mathbf{29.43}$$\displaystyle\,\pm\,$ | $0.25$ |  $51.6$$\displaystyle\,\pm\,$ | $1.15$ |  $52.52$$\displaystyle\,\pm\,$ | $0.93$
| Cost |  $\mathbf{33\,171.37}$$\displaystyle\,\pm\,$ | $1\,655.53$ |  $39\,813.37$$\displaystyle\,\pm\,$ | $899.89$ |  $40\,615.84$$\displaystyle\,\pm\,$ | $1\,660.51$ |  $39\,695.29$$\displaystyle\,\pm\,$ | $1\,341.17$ |  $39\,519.56$$\displaystyle\,\pm\,$ | $1\,484.84$

(a) High-Volume/Low-Mix (HV/LM)

Lot Type | Ours | Hierarchical SPT | Hierarchical SRPT | Hierarchical EDD | Hierarchical LS
---|---|---|---|---|---
Super Hot Lots 3 | On-Time |  $97.38\%$$\displaystyle\,\pm\,$ | $4.63$ |  $\mathbf{98.79\%}$$\displaystyle\,\pm\,$ | $2.45$ |  $97.48\%$$\displaystyle\,\pm\,$ | $4.81$ |  $96.9\%$$\displaystyle\,\pm\,$ | $5.23$ |  $97.46\%$$\displaystyle\,\pm\,$ | $4.63$
Theoretical Cycle-Time: $\mathbf{24.75}$ | Cycle-Time |  $33.64$$\displaystyle\,\pm\,$ | $0.44$ |  $\mathbf{33.32}$$\displaystyle\,\pm\,$ | $0.35$ |  $33.67$$\displaystyle\,\pm\,$ | $0.44$ |  $33.61$$\displaystyle\,\pm\,$ | $0.44$ |  $33.72$$\displaystyle\,\pm\,$ | $0.45$
Hot Lots 1 | On-Time |  $97.08\%$$\displaystyle\,\pm\,$ | $3.29$ |  $96.66\%$$\displaystyle\,\pm\,$ | $2.09$ |  $\mathbf{97.16\%}$$\displaystyle\,\pm\,$ | $2.5$ |  $97\%$$\displaystyle\,\pm\,$ | $3.27$ |  $96.76\%$$\displaystyle\,\pm\,$ | $3.65$
Theoretical Cycle-Time: $\mathbf{21.75}$ | Cycle-Time |  $29.81$$\displaystyle\,\pm\,$ | $0.29$ |  $\mathbf{29.65}$$\displaystyle\,\pm\,$ | $0.19$ |  $29.82$$\displaystyle\,\pm\,$ | $0.28$ |  $29.81$$\displaystyle\,\pm\,$ | $0.23$ |  $29.93$$\displaystyle\,\pm\,$ | $0.28$
Hot Lots 2 | On-Time |  $\mathbf{98.65\%}$$\displaystyle\,\pm\,$ | $2.2$ |  $97.83\%$$\displaystyle\,\pm\,$ | $2.01$ |  $98.81\%$$\displaystyle\,\pm\,$ | $1.86$ |  $98.38\%$$\displaystyle\,\pm\,$ | $2.26$ |  $98.77\%$$\displaystyle\,\pm\,$ | $2.03$
Theoretical Cycle-Time: $\mathbf{23.3}$ | Cycle-Time |  $\mathbf{31.5}$$\displaystyle\,\pm\,$ | $0.24$ |  $31.6$$\displaystyle\,\pm\,$ | $0.23$ |  $31.68$$\displaystyle\,\pm\,$ | $0.22$ |  $31.65$$\displaystyle\,\pm\,$ | $0.27$ |  $31.67$$\displaystyle\,\pm\,$ | $0.23$
Hot Lots 3 | On-Time |  $\mathbf{97.93\%}$$\displaystyle\,\pm\,$ | $2.55$ |  $96.83\%$$\displaystyle\,\pm\,$ | $2.04$ |  $98.18\%$$\displaystyle\,\pm\,$ | $2.53$ |  $97.67\%$$\displaystyle\,\pm\,$ | $2.94$ |  $97.99\%$$\displaystyle\,\pm\,$ | $2.47$
Theoretical Cycle-Time: $\mathbf{24.75}$ | Cycle-Time |  $\mathbf{33.56}$$\displaystyle\,\pm\,$ | $0.32$ |  $33.8$$\displaystyle\,\pm\,$ | $0.23$ |  $33.82$$\displaystyle\,\pm\,$ | $0.29$ |  $33.81$$\displaystyle\,\pm\,$ | $0.28$ |  $33.81$$\displaystyle\,\pm\,$ | $0.29$
Hot Lots 4 | On-Time |  $96.62\%$$\displaystyle\,\pm\,$ | $3.55$ |  $95.9\%$$\displaystyle\,\pm\,$ | $2.43$ |  $96.4\%$$\displaystyle\,\pm\,$ | $3.53$ |  $\mathbf{97.81\%}$$\displaystyle\,\pm\,$ | $2.88$ |  $97.18\%$$\displaystyle\,\pm\,$ | $3.26$
Theoretical Cycle-Time: $\mathbf{14.54}$ | Cycle-Time |  $19.87$$\displaystyle\,\pm\,$ | $0.22$ |  $\mathbf{19.79}$$\displaystyle\,\pm\,$ | $0.16$ |  $19.9$$\displaystyle\,\pm\,$ | $0.23$ |  $19.8$$\displaystyle\,\pm\,$ | $0.19$ |  $19.83$$\displaystyle\,\pm\,$ | $0.24$
Hot Lots 5 | On-Time |  $94.17\%$$\displaystyle\,\pm\,$ | $3.03$ |  $94.02\%$$\displaystyle\,\pm\,$ | $2.42$ |  $93.86\%$$\displaystyle\,\pm\,$ | $3.14$ |  $94.26\%$$\displaystyle\,\pm\,$ | $3.14$ |  $\mathbf{94.92\%}$$\displaystyle\,\pm\,$ | $3.32$
Theoretical Cycle-Time: $\mathbf{10.10}$ | Cycle-Time |  $14.09$$\displaystyle\,\pm\,$ | $0.17$ |  $\mathbf{14}$$\displaystyle\,\pm\,$ | $0.15$ |  $\mathbf{14}$$\displaystyle\,\pm\,$ | $0.17$ |  $14$$\displaystyle\,\pm\,$ | $0.16$ |  $13.96$$\displaystyle\,\pm\,$ | $0.17$
Hot Lots 6 | On-Time |  $96.42\%$$\displaystyle\,\pm\,$ | $2.62$ |  $96.63\%$$\displaystyle\,\pm\,$ | $1.91$ |  $96.92\%$$\displaystyle\,\pm\,$ | $2.49$ |  $\mathbf{97.23\%}$$\displaystyle\,\pm\,$ | $2.88$ |  $\mathbf{97.23\%}$$\displaystyle\,\pm\,$ | $2.38$
Theoretical Cycle-Time: $\mathbf{12.95}$ | Cycle-Time |  $17.47$$\displaystyle\,\pm\,$ | $0.16$ |  $17.46$$\displaystyle\,\pm\,$ | $0.13$ |  $17.45$$\displaystyle\,\pm\,$ | $0.17$ |  $17.47$$\displaystyle\,\pm\,$ | $0.19$ |  $\mathbf{17.42}$$\displaystyle\,\pm\,$ | $0.19$
Hot Lots 7 | On-Time |  $95.22\%$$\displaystyle\,\pm\,$ | $3.68$ |  $\mathbf{95.71\%}$$\displaystyle\,\pm\,$ | $2.61$ |  $95.85\%$$\displaystyle\,\pm\,$ | $3.52$ |  $94.49\%$$\displaystyle\,\pm\,$ | $3.39$ |  $94.68\%$$\displaystyle\,\pm\,$ | $3.57$
Theoretical Cycle-Time: $\mathbf{15.46}$ | Cycle-Time |  $21.02$$\displaystyle\,\pm\,$ | $0.21$ |  $\mathbf{20.9}$$\displaystyle\,\pm\,$ | $0.16$ |  $20.91$$\displaystyle\,\pm\,$ | $0.25$ |  $20.99$$\displaystyle\,\pm\,$ | $0.19$ |  $21$$\displaystyle\,\pm\,$ | $0.21$
Hot Lots 8 | On-Time |  $92.41\%$$\displaystyle\,\pm\,$ | $3.71$ |  $\mathbf{93.7\%}$$\displaystyle\,\pm\,$ | $2.6$ |  $93.23\%$$\displaystyle\,\pm\,$ | $2.86$ |  $93.01\%$$\displaystyle\,\pm\,$ | $3.64$ |  $93.31\%$$\displaystyle\,\pm\,$ | $3.08$
Theoretical Cycle-Time: $\mathbf{16.02}$ | Cycle-Time |  $22.21$$\displaystyle\,\pm\,$ | $0.21$ |  $\mathbf{20.9}$$\displaystyle\,\pm\,$ | $0.16$ |  $22.09$$\displaystyle\,\pm\,$ | $0.22$ |  $22.14$$\displaystyle\,\pm\,$ | $0.23$ |  $22.05$$\displaystyle\,\pm\,$ | $0.2$
Hot Lots 9 | On-Time |  $95.96\%$$\displaystyle\,\pm\,$ | $3.28$ |  $95.96\%$$\displaystyle\,\pm\,$ | $2.26$ |  $\mathbf{96.96\%}$$\displaystyle\,\pm\,$ | $3$ |  $96.65\%$$\displaystyle\,\pm\,$ | $3.01$ |  $96.27\%$$\displaystyle\,\pm\,$ | $3.05$
Theoretical Cycle-Time: $\mathbf{16.95}$ | Cycle-Time |  $23.16$$\displaystyle\,\pm\,$ | $0.2$ |  $\mathbf{22.92}$$\displaystyle\,\pm\,$ | $0.14$ |  $23$$\displaystyle\,\pm\,$ | $0.23$ |  $23.07$$\displaystyle\,\pm\,$ | $0.21$ |  $23.04$$\displaystyle\,\pm\,$ | $0.19$
Hot Lots 10 | On-Time |  $95.88\%$$\displaystyle\,\pm\,$ | $4.1$ |  $96.1\%$$\displaystyle\,\pm\,$ | $1.92$ |  $97.16\%$$\displaystyle\,\pm\,$ | $3.41$ |  $96.8\%$$\displaystyle\,\pm\,$ | $3.25$ |  $\mathbf{97.76\%}$$\displaystyle\,\pm\,$ | $2.67$
Theoretical Cycle-Time: $\mathbf{17.32}$ | Cycle-Time |  $24.01$$\displaystyle\,\pm\,$ | $0.28$ |  $23.92$$\displaystyle\,\pm\,$ | $0.2$ |  $23.92$$\displaystyle\,\pm\,$ | $0.21$ |  $\mathbf{23.91}$$\displaystyle\,\pm\,$ | $0.23$ |  $\mathbf{23.91}$$\displaystyle\,\pm\,$ | $0.23$
Regular Lots 1 | On-Time |  $\mathbf{65.80\%}$$\displaystyle\,\pm\,$ | $1.77$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$
Theoretical Cycle-Time: $\mathbf{21.75}$ | Cycle-Time |  $\mathbf{50.77}$$\displaystyle\,\pm\,$ | $1.67$ |  $92.04$$\displaystyle\,\pm\,$ | $1.88$ |  $70.87$$\displaystyle\,\pm\,$ | $1.46$ |  $62.59$$\displaystyle\,\pm\,$ | $0.39$ |  $62.79$$\displaystyle\,\pm\,$ | $0.42$
Regular Lots 2 | On-Time |  $\mathbf{69.58\%}$$\displaystyle\,\pm\,$ | $1.80$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$
Theoretical Cycle-Time: $\mathbf{23.3}$ | Cycle-Time |  $\mathbf{49.31}$$\displaystyle\,\pm\,$ | $1.26$ |  $96.8$$\displaystyle\,\pm\,$ | $1.78$ |  $107.63$$\displaystyle\,\pm\,$ | $2.4$ |  $67.27$$\displaystyle\,\pm\,$ | $0.42$ |  $65.96$$\displaystyle\,\pm\,$ | $0.43$
Regular Lots 3 | On-Time |  $\mathbf{58.29\%}$$\displaystyle\,\pm\,$ | $2.42$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$
Theoretical Cycle-Time: $\mathbf{24.75}$ | Cycle-Time |  $\mathbf{61.85}$$\displaystyle\,\pm\,$ | $2.38$ |  $101.23$$\displaystyle\,\pm\,$ | $1.86$ |  $122.26$$\displaystyle\,\pm\,$ | $4.63$ |  $71.34$$\displaystyle\,\pm\,$ | $0.42$ |  $69.87$$\displaystyle\,\pm\,$ | $0.44$
Regular Lots 4 | On-Time |  $46.02\%$$\displaystyle\,\pm\,$ | $3.7$ |  $4.32\%$$\displaystyle\,\pm\,$ | $3.87$ |  $\mathbf{74.31\%}$$\displaystyle\,\pm\,$ | $4.11$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$
Theoretical Cycle-Time: $\mathbf{14.54}$ | Cycle-Time |  $40.85$$\displaystyle\,\pm\,$ | $2.58$ |  $47$$\displaystyle\,\pm\,$ | $3.37$ |  $\mathbf{30.43}$$\displaystyle\,\pm\,$ | $0.53$ |  $45.57$$\displaystyle\,\pm\,$ | $0.48$ |  $45.4$$\displaystyle\,\pm\,$ | $0.45$
Regular Lots 5 | On-Time |  $24.3\%$$\displaystyle\,\pm\,$ | $2.12$ |  $0.18\%$$\displaystyle\,\pm\,$ | $0.41$ |  $\mathbf{90.2\%}$$\displaystyle\,\pm\,$ | $0.49$ |  $0.28\%$$\displaystyle\,\pm\,$ | $0.69$ |  $0\%$$\displaystyle\,\pm\,$ | $0$
Theoretical Cycle-Time: $\mathbf{10.10}$ | Cycle-Time |  $41.57$$\displaystyle\,\pm\,$ | $2.48$ |  $58.7$$\displaystyle\,\pm\,$ | $2.14$ |  $\mathbf{16.4}$$\displaystyle\,\pm\,$ | $0.09$ |  $33.75$$\displaystyle\,\pm\,$ | $0.54$ |  $35.7$$\displaystyle\,\pm\,$ | $0.34$
Regular Lots 6 | On-Time |  $58.18\%$$\displaystyle\,\pm\,$ | $2.70$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $\mathbf{69.13\%}$$\displaystyle\,\pm\,$ | $5.29$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$
Theoretical Cycle-Time: $\mathbf{12.95}$ | Cycle-Time |  $33.55$$\displaystyle\,\pm\,$ | $1.57$ |  $71.1$$\displaystyle\,\pm\,$ | $1.91$ |  $\mathbf{29.08}$$\displaystyle\,\pm\,$ | $0.57$ |  $43.48$$\displaystyle\,\pm\,$ | $0.42$ |  $43.27$$\displaystyle\,\pm\,$ | $0.43$
Regular Lots 7 | On-Time |  $65.29\%$$\displaystyle\,\pm\,$ | $2.24$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $\mathbf{81.55\%}$$\displaystyle\,\pm\,$ | $1.28$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$
Theoretical Cycle-Time: $\mathbf{15.46}$ | Cycle-Time |  $36.7$$\displaystyle\,\pm\,$ | $1.42$ |  $74.96$$\displaystyle\,\pm\,$ | $1.87$ |  $\mathbf{30.02}$$\displaystyle\,\pm\,$ | $0.34$ |  $47.02$$\displaystyle\,\pm\,$ | $0.46$ |  $47.65$$\displaystyle\,\pm\,$ | $0.46$
Regular Lots 8 | On-Time |  $\mathbf{67.04\%}$$\displaystyle\,\pm\,$ | $1.83$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $59.89\%$$\displaystyle\,\pm\,$ | $3.36$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$
Theoretical Cycle-Time: $\mathbf{16.02}$ | Cycle-Time |  $38.43$$\displaystyle\,\pm\,$ | $1.50$ |  $77.84$$\displaystyle\,\pm\,$ | $1.86$ |  $\mathbf{37.59}$$\displaystyle\,\pm\,$ | $0.56$ |  $50.74$$\displaystyle\,\pm\,$ | $0.29$ |  $50.78$$\displaystyle\,\pm\,$ | $0.31$
Regular Lots 9 | On-Time |  $53.25\%$$\displaystyle\,\pm\,$ | $2.66$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $\mathbf{68.75\%}$$\displaystyle\,\pm\,$ | $2.16$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$
Theoretical Cycle-Time: $\mathbf{16.95}$ | Cycle-Time |  $44.48$$\displaystyle\,\pm\,$ | $1.60$ |  $79.7$$\displaystyle\,\pm\,$ | $1.89$ |  $\mathbf{36.07}$$\displaystyle\,\pm\,$ | $0.48$ |  $51.24$$\displaystyle\,\pm\,$ | $0.41$ |  $52$$\displaystyle\,\pm\,$ | $0.43$
Regular Lots 10 | On-Time |  $\mathbf{56.97\%}$$\displaystyle\,\pm\,$ | $2.39$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $53.87\%$$\displaystyle\,\pm\,$ | $5.15$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$
Theoretical Cycle-Time: $\mathbf{17.32}$ | Cycle-Time |  $43.95$$\displaystyle\,\pm\,$ | $1.67$ |  $82.14$$\displaystyle\,\pm\,$ | $1.79$ |  $\mathbf{39.93}$$\displaystyle\,\pm\,$ | $0.68$ |  $53.09$$\displaystyle\,\pm\,$ | $0.42$ |  $53.23$$\displaystyle\,\pm\,$ | $0.43$
| Cost |  $\mathbf{94\,349.38}$$\displaystyle\,\pm\,$ | $2\,716.51$ |  $141\,538.20$$\displaystyle\,\pm\,$ | $2\,390.37$ |  $115\,084.5$$\displaystyle\,\pm\,$ | $2\,961.95$ |  $111\,275.17$$\displaystyle\,\pm\,$ | $2\,067.71$ |  $109\,030.26$$\displaystyle\,\pm\,$ | $2\,290.72$

(b) Low-Volume/High-Mix (LV/HM)

Table 3: Detailed results of additional dispatching rules on the SMT2020
testbed for the HV/LM and LV/HM scenarios with $6$ months of simulation. In
addition to the original dispatching heuristics previously reported, we
compare against the Shortest Processing Time (SPT), Shortest Remaining Process
Time (SRPT), Earliest Due Date (EDD), and Least Slack (LS) dispatching rules.
We observe similar results as with the original dispatching rules.

Lot Type | Ours | Hierarchical CR | Hierarchical FIFO
---|---|---|---
Super Hot Lots 3 | On-Time |  $44.58\%$$\displaystyle\,\pm\,$ | $9.54$ |  $\mathbf{46.21\%}$$\displaystyle\,\pm\,$ | $9.46$ |  $40.21\%$$\displaystyle\,\pm\,$ | $8.03$
Theoretical Cycle-Time: $\mathbf{24.75}$ | Cycle-Time |  $\mathbf{33.73}$$\displaystyle\,\pm\,$ | $0.38$ |  $33.77$$\displaystyle\,\pm\,$ | $0.37$ |  $33.98$$\displaystyle\,\pm\,$ | $0.11$
Hot Lots 3 | On-Time |  $\mathbf{48.68\%}$$\displaystyle\,\pm\,$ | $2.77$ |  $43.99\%$$\displaystyle\,\pm\,$ | $2.86$ |  $41.22\%$$\displaystyle\,\pm\,$ | $3.09$
Theoretical Cycle-Time: $\mathbf{24.75}$ | Cycle-Time |  $\mathbf{33.75}$$\displaystyle\,\pm\,$ | $0.12$ |  $33.91$$\displaystyle\,\pm\,$ | $0.12$ |  $34.12$$\displaystyle\,\pm\,$ | $0.13$
Hot Lots 4 | On-Time |  $\mathbf{57.04\%}$$\displaystyle\,\pm\,$ | $2.68$ |  $50.77\%$$\displaystyle\,\pm\,$ | $2.97$ |  $50.47\%$$\displaystyle\,\pm\,$ | $2.57$
Theoretical Cycle-Time: $\mathbf{14.54}$ | Cycle-Time |  $\mathbf{19.65}$$\displaystyle\,\pm\,$ | $0.07$ |  $19.84$$\displaystyle\,\pm\,$ | $0.08$ |  $19.86$$\displaystyle\,\pm\,$ | $0.08$
Regular Lots 3 | On-Time |  $\mathbf{49.35\%}$$\displaystyle\,\pm\,$ | $1.61$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$
Theoretical Cycle-Time: $\mathbf{24.75}$ | Cycle-Time |  $\mathbf{79.91}$$\displaystyle\,\pm\,$ | $4.89$ |  $100.52$$\displaystyle\,\pm\,$ | $1.06$ |  $98.27$$\displaystyle\,\pm\,$ | $1.13$
Regular Lots 4 | On-Time |  $\mathbf{38.72\%}$$\displaystyle\,\pm\,$ | $1.69$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$
Theoretical Cycle-Time: $\mathbf{14.54}$ | Cycle-Time |  $\mathbf{47.8}$$\displaystyle\,\pm\,$ | $2.26$ |  $53.16$$\displaystyle\,\pm\,$ | $1.05$ |  $55.67$$\displaystyle\,\pm\,$ | $0.68$
| Cost |  $\mathbf{39\,775.41}$$\displaystyle\,\pm\,$ | $1\,036.54$ |  $46\,402.56$$\displaystyle\,\pm\,$ | $636.07$ |  $41\,079.57$$\displaystyle\,\pm\,$ | $743.56$

(a) High-Volume/Low-Mix (HV/LM)

Lot Type | Ours | Hierarchical CR | Hierarchical FIFO
---|---|---|---
Super Hot Lots 3 | On-Time |  $\mathbf{98.16\%}$$\displaystyle\,\pm\,$ | $2.92$ |  $97.9\%$$\displaystyle\,\pm\,$ | $3.24$ |  $96.69\%$$\displaystyle\,\pm\,$ | $3.46$
Theoretical Cycle-Time: $\mathbf{24.75}$ | Cycle-Time |  $\mathbf{33.57}$$\displaystyle\,\pm\,$ | $0.33$ |  $33.73$$\displaystyle\,\pm\,$ | $0.42$ |  $33.75$$\displaystyle\,\pm\,$ | $0.37$
Hot Lots 1 | On-Time |  $96.01\%$$\displaystyle\,\pm\,$ | $2.52$ |  $\mathbf{96.43\%}$$\displaystyle\,\pm\,$ | $2.26$ |  $95.06\%$$\displaystyle\,\pm\,$ | $2.43$
Theoretical Cycle-Time: $\mathbf{21.75}$ | Cycle-Time |  $\mathbf{29.92}$$\displaystyle\,\pm\,$ | $0.21$ |  $29.9$$\displaystyle\,\pm\,$ | $0.27$ |  $30.05$$\displaystyle\,\pm\,$ | $0.2$
Hot Lots 2 | On-Time |  $\mathbf{98.88\%}$$\displaystyle\,\pm\,$ | $1.87$ |  $98.49\%$$\displaystyle\,\pm\,$ | $1.56$ |  $96.9\%$$\displaystyle\,\pm\,$ | $2.44$
Theoretical Cycle-Time: $\mathbf{23.3}$ | Cycle-Time |  $\mathbf{31.66}$$\displaystyle\,\pm\,$ | $0.18$ |  $31.66$$\displaystyle\,\pm\,$ | $0.26$ |  $31.93$$\displaystyle\,\pm\,$ | $0.21$
Hot Lots 3 | On-Time |  $\mathbf{96.88\%}$$\displaystyle\,\pm\,$ | $2.50$ |  $97.60\%$$\displaystyle\,\pm\,$ | $1.83$ |  $95.63\%$$\displaystyle\,\pm\,$ | $2.52$
Theoretical Cycle-Time: $\mathbf{24.75}$ | Cycle-Time |  $\mathbf{33.83}$$\displaystyle\,\pm\,$ | $0.24$ |  $33.8$$\displaystyle\,\pm\,$ | $0.18$ |  $34.14$$\displaystyle\,\pm\,$ | $0.21$
Hot Lots 4 | On-Time |  $94.86\%$$\displaystyle\,\pm\,$ | $2.65$ |  $\mathbf{96.38\%}$$\displaystyle\,\pm\,$ | $2.47$ |  $94.65\%$$\displaystyle\,\pm\,$ | $2.65$
Theoretical Cycle-Time: $\mathbf{14.54}$ | Cycle-Time |  $\mathbf{19.93}$$\displaystyle\,\pm\,$ | $0.16$ |  $19.92$$\displaystyle\,\pm\,$ | $0.16$ |  $20.03$$\displaystyle\,\pm\,$ | $0.14$
Hot Lots 5 | On-Time |  $\mathbf{93.79\%}$$\displaystyle\,\pm\,$ | $2.19$ |  $94.21\%$$\displaystyle\,\pm\,$ | $2.1$ |  $93.98\%$$\displaystyle\,\pm\,$ | $2.43$
Theoretical Cycle-Time: $\mathbf{10.10}$ | Cycle-Time |  $\mathbf{14.14}$$\displaystyle\,\pm\,$ | $0.14$ |  $14.13$$\displaystyle\,\pm\,$ | $0.13$ |  $14.10$$\displaystyle\,\pm\,$ | $0.12$
Hot Lots 6 | On-Time |  $95.95\%$$\displaystyle\,\pm\,$ | $2.07$ |  $\mathbf{97.03\%}$$\displaystyle\,\pm\,$ | $1.86$ |  $96.04\%$$\displaystyle\,\pm\,$ | $2.28$
Theoretical Cycle-Time: $\mathbf{12.95}$ | Cycle-Time |  $\mathbf{17.56}$$\displaystyle\,\pm\,$ | $0.13$ |  $17.58$$\displaystyle\,\pm\,$ | $0.15$ |  $17.58$$\displaystyle\,\pm\,$ | $0.14$
Hot Lots 7 | On-Time |  $\mathbf{94\%}$$\displaystyle\,\pm\,$ | $2.28$ |  $93.73\%$$\displaystyle\,\pm\,$ | $2.33$ |  $93.4\%$$\displaystyle\,\pm\,$ | $2.51$
Theoretical Cycle-Time: $\mathbf{15.46}$ | Cycle-Time |  $\mathbf{21.15}$$\displaystyle\,\pm\,$ | $0.16$ |  $21.12$$\displaystyle\,\pm\,$ | $0.16$ |  $21.21$$\displaystyle\,\pm\,$ | $0.14$
Hot Lots 8 | On-Time |  $\mathbf{92.5\%}$$\displaystyle\,\pm\,$ | $2.87$ |  $92.89\%$$\displaystyle\,\pm\,$ | $2.71$ |  $91.47\%$$\displaystyle\,\pm\,$ | $3.1$
Theoretical Cycle-Time: $\mathbf{16.02}$ | Cycle-Time |  $\mathbf{22.24}$$\displaystyle\,\pm\,$ | $0.15$ |  $22.23$$\displaystyle\,\pm\,$ | $0.15$ |  $22.36$$\displaystyle\,\pm\,$ | $0.18$
Hot Lots 9 | On-Time |  $95.25\%$$\displaystyle\,\pm\,$ | $2.53$ |  $\mathbf{96.02\%}$$\displaystyle\,\pm\,$ | $2.46$ |  $94.83\%$$\displaystyle\,\pm\,$ | $2.4$
Theoretical Cycle-Time: $\mathbf{16.95}$ | Cycle-Time |  $23.1$$\displaystyle\,\pm\,$ | $0.15$ |  $\mathbf{23}$$\displaystyle\,\pm\,$ | $0.16$ |  $23.2$$\displaystyle\,\pm\,$ | $0.16$
Hot Lots 10 | On-Time |  $95.53\%$$\displaystyle\,\pm\,$ | $2.78$ |  $\mathbf{96.89\%}$$\displaystyle\,\pm\,$ | $2.38$ |  $95.19\%$$\displaystyle\,\pm\,$ | $2.3$
Theoretical Cycle-Time: $\mathbf{17.32}$ | Cycle-Time |  $24.04$$\displaystyle\,\pm\,$ | $0.19$ |  $\mathbf{23.96}$$\displaystyle\,\pm\,$ | $0.19$ |  $24.12$$\displaystyle\,\pm\,$ | $0.18$
Regular Lots 1 | On-Time |  $\mathbf{64.61\%}$$\displaystyle\,\pm\,$ | $1.62$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$
Theoretical Cycle-Time: $\mathbf{21.75}$ | Cycle-Time |  $\mathbf{59.61}$$\displaystyle\,\pm\,$ | $3.06$ |  $65.97$$\displaystyle\,\pm\,$ | $0.48$ |  $71.88$$\displaystyle\,\pm\,$ | $0.87$
Regular Lots 2 | On-Time |  $\mathbf{68.92\%}$$\displaystyle\,\pm\,$ | $1.8$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$
Theoretical Cycle-Time: $\mathbf{23.3}$ | Cycle-Time |  $\mathbf{59.17}$$\displaystyle\,\pm\,$ | $2.74$ |  $76.14$$\displaystyle\,\pm\,$ | $0.66$ |  $73.9$$\displaystyle\,\pm\,$ | $0.87$
Regular Lots 3 | On-Time |  $\mathbf{56.42\%}$$\displaystyle\,\pm\,$ | $1.88$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$
Theoretical Cycle-Time: $\mathbf{24.75}$ | Cycle-Time |  $\mathbf{73.06}$$\displaystyle\,\pm\,$ | $3.08$ |  $80.45$$\displaystyle\,\pm\,$ | $0.67$ |  $81.77$$\displaystyle\,\pm\,$ | $0.92$
Regular Lots 4 | On-Time |  $\mathbf{40.24\%}$$\displaystyle\,\pm\,$ | $2.34$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$
Theoretical Cycle-Time: $\mathbf{14.54}$ | Cycle-Time |  $\mathbf{62.05}$$\displaystyle\,\pm\,$ | $3.03$ |  $48.62$$\displaystyle\,\pm\,$ | $0.45$ |  $46.7$$\displaystyle\,\pm\,$ | $0.59$
Regular Lots 5 | On-Time |  $\mathbf{29.11\%}$$\displaystyle\,\pm\,$ | $2$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$
Theoretical Cycle-Time: $\mathbf{10.10}$ | Cycle-Time |  $50.06$$\displaystyle\,\pm\,$ | $2$ |  $\mathbf{33.68}$$\displaystyle\,\pm\,$ | $0.27$ |  $34.2$$\displaystyle\,\pm\,$ | $0.5$
Regular Lots 6 | On-Time |  $\mathbf{67.51\%}$$\displaystyle\,\pm\,$ | $2.08$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0.29\%$$\displaystyle\,\pm\,$ | $0.45$
Theoretical Cycle-Time: $\mathbf{12.95}$ | Cycle-Time |  $\mathbf{32.13}$$\displaystyle\,\pm\,$ | $1.57$ |  $47.9$$\displaystyle\,\pm\,$ | $0.5$ |  $40.64$$\displaystyle\,\pm\,$ | $0.54$
Regular Lots 7 | On-Time |  $\mathbf{65.05\%}$$\displaystyle\,\pm\,$ | $1.71$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$
Theoretical Cycle-Time: $\mathbf{15.46}$ | Cycle-Time |  $\mathbf{42.96}$$\displaystyle\,\pm\,$ | $1.77$ |  $48.66$$\displaystyle\,\pm\,$ | $0.44$ |  $49.08$$\displaystyle\,\pm\,$ | $0.61$
Regular Lots 8 | On-Time |  $\mathbf{70.46\%}$$\displaystyle\,\pm\,$ | $1.49$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$
Theoretical Cycle-Time: $\mathbf{16.02}$ | Cycle-Time |  $\mathbf{41.92}$$\displaystyle\,\pm\,$ | $1.66$ |  $56.55$$\displaystyle\,\pm\,$ | $0.51$ |  $52.9$$\displaystyle\,\pm\,$ | $0.64$
Regular Lots 9 | On-Time |  $\mathbf{54.84\%}$$\displaystyle\,\pm\,$ | $2.66$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$
Theoretical Cycle-Time: $\mathbf{16.95}$ | Cycle-Time |  $\mathbf{49.16}$$\displaystyle\,\pm\,$ | $1.56$ |  $51.5$$\displaystyle\,\pm\,$ | $0.37$ |  $53.47$$\displaystyle\,\pm\,$ | $0.64$
Regular Lots 10 | On-Time |  $\mathbf{54.66\%}$$\displaystyle\,\pm\,$ | $1.95$ |  $0\%$$\displaystyle\,\pm\,$ | $0$ |  $0\%$$\displaystyle\,\pm\,$ | $0$
Theoretical Cycle-Time: $\mathbf{17.32}$ | Cycle-Time |  $\mathbf{53.85}$$\displaystyle\,\pm\,$ | $1.93$ |  $54.96$$\displaystyle\,\pm\,$ | $0.44$ |  $55.78$$\displaystyle\,\pm\,$ | $0.69$
| Cost |  $\mathbf{109\,643.74}$$\displaystyle\,\pm\,$ | $1\,848.16$ |  $123\,895.88$$\displaystyle\,\pm\,$ | $1\,840.75$ |  $111\,977.53$$\displaystyle\,\pm\,$ | $1\,780.09$

(b) Low-Volume/High-Mix (LV/HM)

Table 4: Detailed results on the SMT2020 testbed for the HV/LM and LV/HM
scenarios with a $1$ year simulation horizon. The dispatching agents, which
were trained over a shorter horizon of $6$ months, are the same as in Tables
1–3.