## Abstract

## Keywords

## 1. Introduction

- Nishimura A.
- Nakajima R.
- Takagi R.
- Zhou G.
- Suzuki D.
- Kiyama M.
- Nozaki T.
- Owaki T.
- Takahara T.
- Nagai S.
- Nakamura T.
- Sugaya M.
- Terada K.
- Igarashi Y.
- Hanzawa H.
- Okano T.
- Shimizu T.
- Yamato M.
- Takeda S.

*J Tissue Eng Regen Med.*2019; 13: 2246-2255

- Yachie N.
- Takahashi K.
- Katayama T.
- Sakurada T.
- Kanda G.N.
- Takagi E.
- Hirose T.
- Katsura T.
- Moriya T.
- Kitano H.
- Tsujii J.
- Shiraki T.
- Kariyazaki H.
- Kamei M.
- Abe N.
- Fukuda T.
- Sawada Y.
- Hashiguchi Y.
- Matsukuma K.
- Murai S.
- Sasaki N.
- Ipposhi T.
- Urabe H.
- Kudo T.
- Umeno M.
- Ono S.
- Miyauchi K.
- Nakamura M.
- Kizaki T.
- Suyama T.
- Hatta T.
- Natsume T.
- Ohta T.
- Ozawa Y.
- Ihara S.
- Tamaki S.
- Antezana E.
- Garcia-Castro A.
- Perret J.-L.
- Ishiguro S.
- Mori H.
- Evans-Yamamoto D.
- Masuyama N.
- Tomita M.
- Katayama T.
- Matsumoto M.
- Nakayama H.
- Shirasawa A.
- Shimbo K.
- Yamada N.
- Nakayama K.I.
- Shimizu T.
- Saya H.
- Yamashita S.
- Matsushima T.
- Asahara H.
- Eguchi H.
- Mikamori M.
- Mori M.
- Natsume T.

*Nat Biotechnol.*2017; 35: 310-312

## 2. Materials and methods

### 2.1 S-LAB problem

*laboratory configuration*) and jobs that process multiple operations in a certain order (

*job definitions*) (Mathematical formulation of S-LAB problems as mixed integer problems is provided in Supplemental Material Section 2 of [

#### 2.1.1 Laboratory configuration ($\mathcal{L}$)

- •There are $M$ instruments in a laboratory, each of which has an instrument type ${T}_{m}(1\le m\le M,1\le {T}_{m}\le K)$, where $K$ is the number of different instrument types.
- •Each instrument has parking positions through which an instrument passes samples to transporters or receives them from transporters. Here, we assume that each instrument has a sufficient number of parking positions so that samples can wait if the instrument is being used by another operation.

#### 2.1.2 Job definition ($\mathcal{J}$)

*jobs, operations*, and

*constraints*.

- •There are $J$ jobs. The $j$th ($1\le j\le J$) job is composed of ${N}_{j}$ operations. In total, there are $N={\sum}_{j=1}^{J}{N}_{j}$ operations.

- •The $a$-th ($1\le a\le N$) operation ${O}_{a}$ has a compatible instrument type ${C}_{a}$ ($1\le {C}_{a}\le K$); that is, the $a$-th operation needs to be processed by an instrument $m$ with the instrument type ${T}_{m}={C}_{a}$.
- •The $a$-th operation has process time ${\tau}_{a}$, which is intrinsically determined by the combination of the nature of the operation and the instrument type ${C}_{a}$ to process it.
- •There can be a dependency between one operation ${O}_{a}$ and another ${O}_{b}$ $(a<b)$ in the same job; that is, ${O}_{b}$ must start after ${O}_{a}$ ends. The operation dependency graph $G$ is a directed acyclic graph wherein each node represents an operation and each directed edge represents the dependency between a pair of operations.${G}_{a,b}=\{\begin{array}{cc}1,& \mathrm{if}\phantom{\rule{0.33em}{0ex}}{O}_{b}\phantom{\rule{0.16em}{0ex}}\mathrm{must}\phantom{\rule{0.33em}{0ex}}\text{start}\phantom{\rule{4.pt}{0ex}}\text{after}\phantom{\rule{4.pt}{0ex}}{O}_{a}\phantom{\rule{0.16em}{0ex}}\mathrm{ends}\\ 0,& \mathrm{otherwise}\end{array}$

Note that such $G$ can be obtained easily by topological sort of operation indices [[36]]. We also define $Lis{t}_{G}^{\top}$ as the adjacency list of the transpose graph of $G$: $Lis{t}_{G}^{\top}\left[b\right]=\{a,...\}$ ($a<b$). For example, if $Lis{t}_{G}^{\top}\left[5\right]=\{1,2,4\}$, then ${O}_{5}$ must start after ${O}_{1}$, ${O}_{2}$ and ${O}_{4}$ end. - •Let $Lis{t}_{TCMB}\left[b\right]$ be the list of TCMBs between ${O}_{a}$ $(a<b)$ and ${O}_{b}$. A TCMB sets the upper limit $\alpha $ of the maximum tolerable difference between the start or end time of each of a pair of operations ${O}_{a}$ and ${O}_{b}$ [24,31]. Each TCMB is represented as a 4-tuple $(From,a,To,\alpha )$. $\alpha $ can vary among different TCMBs. $From$ and $To$ indicate either “Start” or “End” of an operation. There are four pairs of $From$ and $To$:
- 1.Start - Start: The absolute difference between the start time of operation $b$ and the start time of operation $a$ must be less than or equal to $\alpha $.
- 2.End - Start: The absolute difference between the end time of operation $b$ and the start time of operation $a$ must be less than or equal to $\alpha $.
- 3.Start - End: The absolute difference between the start time of operation $b$ and the end time of operation $a$ must be less than or equal to $\alpha $.
- 4.End - End: The absolute difference between the end time of operation $b$ and the end time of operation $a$ must be less than or equal to $\alpha $.

For example, “$Lis{t}_{TCMB}\left[2\right]=\{Start,1,End,5\}$” means that ${O}_{2}$ must start within 5 min before and after ${O}_{1}$ ends. - 1.

- •Constraint 1: When multiple instruments are allocated to multiple operations, one instrument can process at most one operation at a time.
- •Constraint 2: The order of operations defined by $G$ needs to be maintained.
- •Constraint 3: The start and end time of all operations must satisfy $TCMB$.
- •Constraint 4: There must be a buffer time between a pair of operations consecutively processed on the same instrument. The length of the buffer time $\beta $ ($\beta >0$) is determined by the user.

#### 2.1.3 Scheduling solutions

- •The start time for all operations is designated as $S=\left\{{S}_{a}\right\}\phantom{\rule{0.33em}{0ex}}(1\le a\le N)$.
- •An
*instrument allocation*is $E=\left\{{E}_{a}\right\}\phantom{\rule{0.33em}{0ex}}(1\le a\le N,1\le {E}_{a}\le M)$, where ${E}_{a}$ is the instrument that processes ${O}_{a}$, chosen from the set of compatible instruments whose instrument type ${T}_{{E}_{a}}={C}_{a}$. $E$ is computed based on the laboratory configuration $\mathcal{L}$, job definition $\mathcal{J}$, the buffer time $\beta $, process time $\tau $ and the start time ${S}_{a}$. See Algorithm 5. - •A
*schedule*is defined by determining the start time ${S}_{a}$ and the processor ${E}_{a}$ for each ${O}_{a}$. - •A scheduling solution that meets all the constraints is designated as a
*feasible*solution. A feasible solution with the smallest objective function value (see 2.1.4) is designated as an*optimal*solution. A schedule that does not meet all the constraints is designated as an*failed*solution.

#### 2.1.4 Objective

#### 2.1.5 Mathematical formulation

${F}_{a}^{\left(m\right)}$ is the Boolean equivalent of the $m$-th instrument which process ${O}_{a}$. ${Q}_{a,b}^{\left(m\right)}$ is the Boolean equivalent of the $m$-th instrument which process ${O}_{a}$ and ${O}_{b}$. If ${O}_{a}$ and ${O}_{b}$ are processed with the $m$-th instrument, ${Q}_{a,b}^{\left(m\right)}=1$. Using these terms, all constraints is rewritten as follows:

- •Constraints 1 and 4$\sum _{m=1}^{M}{F}_{a}^{\left(m\right)}=1\phantom{\rule{2.em}{0ex}}\forall a.$${S}_{a}+{\tau}_{a}+\beta \le {S}_{b}\phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{4.pt}{0ex}}{Q}_{a,b}^{\left(m\right)}=1\phantom{\rule{2.em}{0ex}}\forall a,b,m$
- •Constraint 2${S}_{a}+{\tau}_{a}\le {S}_{b}\phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{4.pt}{0ex}}{G}_{a,b}=1\phantom{\rule{2.em}{0ex}}\forall a,b$
- •Constraint 3 (TCMB)For each TCMB$(From,a,To,\alpha )$ of $Lis{t}_{TCMB}\left[b\right]$ (see 2.1.2)
- 1.$|{S}_{a}-{S}_{b}|\le \alpha $
- 2.$|{S}_{a}-({S}_{b}+{\tau}_{b})|\le \alpha $
- 3.$|({S}_{a}+{\tau}_{a})-\left({S}_{b}\right)|\le \alpha $
- 4.$|({S}_{a}+{\tau}_{a})-({S}_{b}+{\tau}_{b})|\le \alpha $

- 1.

### 2.2 SAGAS algorithm

#### 2.2.1 Overview of SAGAS algorithm

#### 2.2.2 Simulated annealing step (SA)

*temperature*(min), which is gradually reduced from a higher temperature to a lower temperature. Starting with the generation of an initial solution (schedule), the next solution ${S}_{next}$ is generated from the current solution ${S}_{current}$ at each temperature and accepted or rejected based on temperature-dependent transition probabilities $p({S}_{current},{S}_{next})$ (described below). Finally, SA outputs ${S}_{SA}$, the start time for all operations with the shortest execution time ever found during the simulated annealing calculation, and $Scor{e}_{SA}$, the corresponding objective function value. Note that ${S}_{SA}$ is not guaranteed to meet all the constraints; it can be either a feasible or failed solution.

When $Temp$ becomes $Tem{p}_{L}$ or smaller, the calculation of SA stops and outputs ${S}_{SA}$. To obtain a feasible solution at a time sufficient for real-world use, we introduced the user-provided runtime threshold of SA ($SA\_msec$), in which the time $Temp$ decreases to $Tem{p}_{L}$. Strictly, using elapsed time ($Elapsed\_time$) of SA and $SA\_msec$, $Temp$ is calculated as the following equation:

#### 2.2.3 Final modification step (Mod)

- 1.First step
- a.At the first step, it sets a criterion time ${t}_{criteria}$ as 1.
- b.After that, it decreases the start time of all operations, which is started after ${t}_{criteria}$ by 1 and updates the schedule if the score improves.
- c.After then, it increases ${t}_{criteria}$ by 1.
- d.If ${t}_{criteria}$ is equal to the start time of the lastly processed operation, it sets a criterion time ${t}_{criteria}$ as 1 again.
- e.It repeats (b) and (d) until the score finishes improving.

- a.
- 2.Second step
- a.At the second step, it sets criterion time ${t}_{criteria}$ as 1.
- b.After that, it decreases the start time of all the operations of the $j$-th ($1\le j\le J$) job, which is started after ${t}_{criteria}$ by 1, and updates the schedule if the score improves.
- c.After then, it increases ${t}_{criteria}$ by 1 after checking the all jobs.
- d.if ${t}_{criteria}$ is equal to the start time of the lastly processed operation, it sets a criterion time ${t}_{criteria}$ as 1 again.
- e.It repeats (b) and (d) until the score finishes improving.

- a.
- 3.Third step
- a.At the third step, it updates the schedule if the score improves every time it decreases the start time of each operation by 1.
- b.It repeats (a) until the score finishes improving.

- a.

#### 2.2.4 Greedy algorithm step (Greedy)

- 1.Let ${R}^{-}$ and ${R}^{+}$ be the minimum and maximum values for $S\left[b\right]$, respectively, that satisfy Constraint 2 and 3. ${R}^{-}$ and ${R}^{+}$ are calculated by Algorithm 4.
- 2.Starting from ${R}^{-}$ to ${R}^{+}$, Greedy assigns a tentative value to $S\left[b\right]$ and test whether the value of $S\left[b\right]$ meets all the constraints. When the value of $S\left[b\right]$ meets all the constraints, do Step 1 for the next operation ${O}_{b+1}$.
- 3.If any value of $S\left[b\right]$ does not meet all of the constraints, Greedy tries shifting the start time of $\left({O}_{v}\right)$, the operation specifying ${R}^{+}$, by 1.
- 4.Then, if the value of $S\left[v\right]$ exceeds ${w}_{p}$ defined in SA, the calculation exits with no solution; otherwise, do the Step 1 for ${O}_{v}$.

#### 2.2.5 Instrument allocation

### 2.3 Implementation and code availability

### 2.4 Performance evaluation

### 2.5 S-LAB problems for simulated experiments

#### 2.5.1 Gu et al. 2016 (Gu2016)

#### 2.5.2 RT-qPCR (qPCR)

- Nagura-Ikeda M.
- Imai K.
- Tabata S.
- Miyoshi K.
- Murahara N.
- Mizuno T.
- Horiuchi M.
- Kato K.
- Imoto Y.
- Iwata M.
- Mimura S.
- Ito T.
- Tamura K.
- Kato Y.

*J Clin Microbiol.*2020; 58

Takara Bio Inc.. Direct one-step RT-qPCR mix for SARS-CoV-2 protocol-at-a-glance. https://www.takarabio.com/resourcedocument/x225225, Accessed: 2022-7-7.

#### 2.5.3 Library preparation for RNA sequencing (RNAseq)

- Yachie N.
- Takahashi K.
- Katayama T.
- Sakurada T.
- Kanda G.N.
- Takagi E.
- Hirose T.
- Katsura T.
- Moriya T.
- Kitano H.
- Tsujii J.
- Shiraki T.
- Kariyazaki H.
- Kamei M.
- Abe N.
- Fukuda T.
- Sawada Y.
- Hashiguchi Y.
- Matsukuma K.
- Murai S.
- Sasaki N.
- Ipposhi T.
- Urabe H.
- Kudo T.
- Umeno M.
- Ono S.
- Miyauchi K.
- Nakamura M.
- Kizaki T.
- Suyama T.
- Hatta T.
- Natsume T.
- Ohta T.
- Ozawa Y.
- Ihara S.
- Tamaki S.
- Antezana E.
- Garcia-Castro A.
- Perret J.-L.
- Ishiguro S.
- Mori H.
- Evans-Yamamoto D.
- Masuyama N.
- Tomita M.
- Katayama T.
- Matsumoto M.
- Nakayama H.
- Shirasawa A.
- Shimbo K.
- Yamada N.
- Nakayama K.I.
- Shimizu T.
- Saya H.
- Yamashita S.
- Matsushima T.
- Asahara H.
- Eguchi H.
- Mikamori M.
- Mori M.
- Natsume T.

*Nat Biotechnol.*2017; 35: 310-312

Qiagen N.V.. RNeasy kits, QIAGEN. https://www.qiagen.com/us/products/discovery-and-translational-research/dna-rna-purification/rna-purification/total-rna/rneasy-kits/, Accessed: 2022-7-7.

Illumina, Inc.. TruSeq® RNA sample preparation v2 guide. https://support.illumina.com/content/dam/illumina-support/documents/documentation/chemistry_documentation/samplepreps_truseq/truseqrna/truseq-rna-sample-prep-v2-guide-15026495-f.pdf, Accessed: 2022-7-7.

#### 2.5.4 Composite of qPCR and RNAseq

## 3. Results

### 3.1 SAGAS can find feasible solutions and optimal solutions for a simple S-LAB problem

### 3.2 SAGAS is effective for complex S-LAB problems

### 3.3 SAGAS enables the simulation-based design of laboratory configuration

### 3.4 SAGAS depends on computation time and complexity of problem

**Supplementary Figure 1**).

## 4. Discussion

S-LAB problem | Scheduler | Minimum execution time (min) | Avarge execution time (min) | Avarage computation time (min) |
---|---|---|---|---|

Gu $\times $1 | Greedy | 90 (1) | 90 (1) | 0.000 (1) |

$N=17$ | SA | 111 (4) | 118 (4) | 0.025 (5) |

SAGAS | 87 (5) | 87 (5) | 0.025 (5) | |

SA-Mod | 87 (4) | 88 (4) | 0.025 (5) | |

Gu $\times $5 | Greedy | 386 (1) | 386 (1) | 0.513 (1) |

$N=17\times 5$ | SA | NA(0) | NA(0) | 3.000 (5) |

SAGAS | 386 (5) | 386 (5) | 3.518 (5) | |

SA-Mod | 389 (1) | 389 (1) | 3.028 (5) | |

qPCR $\times $5 | Greedy | 152 (1) | 152 (1) | 0.001 (1) |

$N=16\times 5$ | SA | 236 (5) | 256 (5) | 3.000 (5) |

SAGAS | 152 (5) | 152 (5) | 3.026 (5) | |

SA-Mod | 192 (5) | 200 (5) | 3.012 (5) | |

RNAseq $\times $5 | Greedy | 1682 (1) | 1682 (1) | 0.014 (1) |

$N=28\times 5$ | SA | 2417 (5) | 2810 (5) | 3.000 (5) |

SAGAS | 1090 (5) | 1118 (5) | 3.320 (5) | |

SA-Mod | 1053 (5) | 1152 (5) | 3.675 (5) | |

qPCR $\times $5 | Greedy | 1885 (1) | 1885 (1) | 0.050 (1) |

RNAseq $\times $5 | SA | 3163 (5) | 3303 (5) | 3.000 (5) |

$N=16\times 5+28\times 5$ | SAGAS | 1145 (5) | 1187 (5) | 9.890 (5) |

SA-Mod | 1160 (5) | 1261 (5) | 7.322 (5) |

- Yachie N.
- Takahashi K.
- Katayama T.
- Sakurada T.
- Kanda G.N.
- Takagi E.
- Hirose T.
- Katsura T.
- Moriya T.
- Kitano H.
- Tsujii J.
- Shiraki T.
- Kariyazaki H.
- Kamei M.
- Abe N.
- Fukuda T.
- Sawada Y.
- Hashiguchi Y.
- Matsukuma K.
- Murai S.
- Sasaki N.
- Ipposhi T.
- Urabe H.
- Kudo T.
- Umeno M.
- Ono S.
- Miyauchi K.
- Nakamura M.
- Kizaki T.
- Suyama T.
- Hatta T.
- Natsume T.
- Ohta T.
- Ozawa Y.
- Ihara S.
- Tamaki S.
- Antezana E.
- Garcia-Castro A.
- Perret J.-L.
- Ishiguro S.
- Mori H.
- Evans-Yamamoto D.
- Masuyama N.
- Tomita M.
- Katayama T.
- Matsumoto M.
- Nakayama H.
- Shirasawa A.
- Shimbo K.
- Yamada N.
- Nakayama K.I.
- Shimizu T.
- Saya H.
- Yamashita S.
- Matsushima T.
- Asahara H.
- Eguchi H.
- Mikamori M.
- Mori M.
- Natsume T.

*Nat Biotechnol.*2017; 35: 310-312

**Supplementary Figure 2**, Suppl. Table S21). Systematic evaluation based on simulation experiments for a wide variety of S-LAB problems of various sizes will be necessary to understand the relationship between the size of S-LAB problems and the computation performance of SAGAS. Fourth, dynamic scheduling is not considered in this study. Dynamic scheduling is essential when accidents occur in an automated laboratory and experimental procedures cannot be executed according to a predetermined schedule. To achieve dynamic scheduling, it is worth considering extending SAGAS to allow the rescheduling of failed operations or jobs while taking into account operations that have already finished or are in progress.

## Funding

## Data availability

## Declaration of Competing Interest

## Acknowledgments

## Appendix A. Supplementary materials

- Supplementary Data S1
Supplementary Raw Research Data. This is open data under the CC BY license http://creativecommons.org/licenses/by/4.0/

- Supplementary Data S2
Supplementary Raw Research Data. This is open data under the CC BY license http://creativecommons.org/licenses/by/4.0/

- Supplementary Data S3
Supplementary Table 1: List of operations of one job in the simulated experiment Gu2016 $\times $1.

Supplementary Table 2: List of instruments used in the simulated experiment Gu2016 $\times $1.

Supplementary Table 3: List of the dependencies among the operations of one job in the simulated experiment Gu2016 $\times $1.

Supplementary Table 4: List of the TCMBs among the operations of one job in the simulated experiment Gu2016 $\times $1.

Supplementary Table 5: List of operations of one job in the simulated experiment Gu2016 $\times $5.

Supplementary Table 6: List of instruments used in the simulated experiment Gu2016 $\times $5.

Supplementary Table 7: List of the dependencies among the operations of one job in the simulated experiment Gu2016 $\times $5.

Supplementary Table 8: List of the TCMBs among the operations of one job in the simulated experiment Gu2016 $\times $5.

Supplementary Table 9: List of operations of one job in the simulated experiment qPCR $\times $5.

Supplementary Table 10: List of instruments used in the simulated experiment qPCR $\times $5.

Supplementary Table 11: List of the dependencies among the operations of one job in the simulated experiment qPCR $\times $5.

Supplementary Table 12: List of the TCMBs among the operations of one job in the simulated experiment qPCR $\times $5.

Supplementary Table 13: List of operations of one job in the simulated experiment RNAseq $\times $5.

Supplementary Table 14: List of instruments used in the simulated experiment RNAseq $\times $5.

Supplementary Table 15: List of the dependencies among the operations of one job in the simulated experiment RNAseq $\times $5.

Supplementary Table 16: List of the TCMBs among the operations of one job in the simulated experiment RNAseq $\times $5.

Supplementary Table 17: List of operations of one job in the simulated experiment qPCR $\times $5 RNAseq $\times $5.

Supplementary Table 18: List of instruments used in the simulated experiment qPCR $\times $5 RNAseq $\times $5.

Supplementary Table 19: List of the dependencies among the operations of one job in the simulated experiment qPCR $\times $5 RNAseq $\times $5.

Supplementary Table 20: List of the TCMBs among the operations of one job in the simulated experiment qPCR $\times $5 RNAseq $\times $5.

Supplementary Table 21: List of the number of times the Greedy step and SA-Mod step were chosen for SAGAS output, operations, dependencies, and TCMBs for the different S-LAB problems.

Supplementary Raw Research Data. This is open data under the CC BY license http://creativecommons.org/licenses/by/4.0/

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