# Planning the Schedule for the Disposal of the Spent Nuclear Fuel with Interactive Multiobjective Optimization

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

#### 2.1. Parameters

N | be a total number of disposal periods |

Z | be a total number of removals from the reactor. |

a | the last removal before the first disposal period |

b | the disposal period when the last removal is done. |

${M}_{i}$ | number of assemblies belonging to the removal $i\in \mathcal{Z}$ |

Q | length of one disposal tunnel [m] |

${A}_{i,j}$ | storage time of an assembly belonging to the removal $i\in \mathcal{Z}$ |

in the period $j\in \mathcal{N}$ [period] | |

${P}_{i,j}$ | decay heat power of an assembly belonging to the removal $i\in \mathcal{Z}$ |

in the period $j\in \mathcal{N}$ [W]. |

${C}_{AS}$ | storage cost per one assembly per period [€] |

${C}_{IS}$ | costs related to the interim storage per period [€] |

${C}_{SP}$ | cost of a storage place per one assembly [€] |

${C}_{CA}$ | cost of one canister [€] |

${C}_{EF}$ | costs related to operating the encapsulation facility per period [€] |

${C}_{DT}$ | cost of a disposal tunnel per meter [€] |

${C}_{CT}$ | cost of a central tunnel per meter [€]. |

R | minimum storage time of an assembly [period] |

K | maximum capacity of a canister |

T | minimum number of canisters disposed in one period |

U | maximum number of canisters disposed in one period |

${p}_{max}^{low}$, ${p}_{max}^{up}$ | lower and upper bound for the maximum average power of |

a canister [W] | |

${d}_{CA}^{low}$, ${d}_{CA}^{up}$ | lower and upper bound for the distance between canisters [m] |

${d}_{DT}^{low},$${d}_{DT}^{up}$ | lower and upper bound for the distance between disposal |

tunnels [m]. |

#### 2.2. Continuous Variables

${x}_{i,j}$ | number of assemblies belonging to the removal $i\in \mathcal{Z}$ disposed during |

the period $j\in \mathcal{N}$ | |

${y}_{j}$ | number of canisters disposed during the period $j\in \mathcal{N}$ |

${z}_{i,j}$ | number of assemblies belonging to the removal $i\in \mathcal{Z}$ being in storage |

at the end of the period $j\in \mathcal{N}$ | |

${p}_{max}$ | maximum average power of a canister |

${d}_{DT}$ | distance between two adjacent disposal tunnels |

${d}_{CA}$ | distance between two adjacent canisters in a disposal tunnel. |

#### 2.3. Binary Variables

${e}_{ON}^{j}$ | encapsulation starts in the beginning of the period $j\in \mathcal{N}$ |

${e}_{OFF}^{j}$ | encapsulation ends in the beginning of the period $j\in \mathcal{N}$ |

${e}_{j}$ | encapsulation facility is in operation during the period $j\in \mathcal{N}$ |

${s}_{i,j}$ | assemblies belonging to the removal $i\in \mathcal{Z}$ take off from disposal |

at the beginning of the period $j\in \mathcal{N}$ | |

${r}_{i,j}$ | indicates that assemblies belonging to the removal $i\in \mathcal{Z}$ |

can be disposed during the period $j\in \mathcal{N}$. |

#### 2.4. Objectives

#### 2.5. Constraints—Interim Storage

#### 2.6. Constraints—Encapsulation Facility

#### 2.7. Constraints—Disposal Facility

## 3. Multiobjective Optimization Approach

#### 3.1. Mathematical Background

#### 3.2. Two-Slope Parameterized ASFs

**Theorem**

**1**

**Proof.**

#### 3.3. Multiobjective Interactive Method Utilizing the Two-Slope Parameterized ASFs

- Step 0.
- Give the ideal vector ${\mathit{f}}^{id}$, the nadir vector ${\mathit{f}}^{nad}$, and/or some Pareto optimal solution ${\mathit{f}}_{0}$ to the decision maker in order to illustrate the Pareto set.
- Step 1.
- Set the iteration counter $h=1$ and select the maximum number of iterations ${h}_{max}$. Ask the decision maker to provide the reference point ${\mathit{f}}_{h}^{R}$ and the number of solutions $s\in \{1,\dots ,k\}$ presented for each reference point. Initialize the positive coefficients ${\mathit{\lambda}}^{U}$ and ${\mathit{\lambda}}^{A}$.
- Step 2.
- Step 3.
- Present s solutions to the decision maker and ask the decision maker to select the most preferable solution among them as the current solution ${\mathit{f}}_{h}$ and go to Step 5 or if more solutions for the current reference point ${\mathit{f}}_{h}^{R}$ are needed go to Step 4.
- Step 4.
- Present supplementary solutions to the decision maker. Ask the decision maker to select the most preferable solution among the previous s solutions and the supplementary solutions as the current solution ${\mathit{f}}_{h}$ and go to Step 5.
- Step 5.
- If $h={h}_{max}$ or the decision maker is satisfied with the current solution ${\mathit{f}}_{h}$, stop with the current solution as the final solution ${\mathit{f}}^{*}$. Otherwise, ask the decision maker to specify the new reference point ${\mathit{f}}_{h+1}^{R}$ as the current reference point, set $h=h+1$, and go to Step 2.

## 4. Case Study: The Disposal in Finland

- Interim storage versus disposal facility: The interim storage-related goals all imply transferring the spent nuclear fuel from the interim storage as rapidly as possible. However, in order to minimize the disposal facility-related goals, the cooling times should be maximized.
- Encapsulation facility versus interim storage: By delaying the start of disposal, it is possible to shorten the operation time of the encapsulation facility, and thus, decrease the operating costs. Again, the delay at the start of the encapsulation can cause an increase of the inventories in the interim storage.
- Encapsulation facility versus disposal facility: The disposal should be started and ended as soon as possible. Both of these aims have a tendency to increase the canister heat load, and hence, affect the disposal facility goals. To minimize the operation time of the encapsulation facility, empty assembly positions can be used. However, the price to pay is the increased number of canisters. In addition, a larger number of canisters necessitates an increase in the disposal facility area.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Parameters of the Case Study

${M}_{i}=360,$$i\in \{1,3,5,7,9,11\}$ | $R=4$ | ${p}_{max}^{low}=1300$ |

${M}_{i}=240,$$i\in \{2,4,6,8,10\}$ | $U=500$ | ${d}_{DT}^{up}=50$ |

$a=5$ | $T=50$ | ${d}_{DT}^{low}=25$ |

$b=6$ | $Q=350$ | ${d}_{CA}^{up}=15$ |

$K=4$ | ${p}_{max}^{up}=1830$ | ${d}_{CA}^{low}=6$ |

$\mathit{j}=1$ | $\mathit{j}=2$ | $\mathit{j}=3$ | $\mathit{j}=4$ | $\mathit{j}=5$ | $\mathit{j}=6$ | $\mathit{j}=7$ | $\mathit{j}=8$ | $\mathit{j}=9$ | $\mathit{j}=10$ | $\mathit{j}=11$ | $\mathit{j}=12$ | $\mathit{j}=13$ | $\mathit{j}=14$ | $\mathit{j}=15$ | $\mathit{j}=16$ | $\mathit{j}=17$ | $\mathit{j}=18$ | $\mathit{j}=19$ | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$i=1$ | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 |

$i=2$ | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 |

$i=3$ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

$i=4$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 |

$i=5$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |

$i=6$ | $-1$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 |

$i=7$ | $-2$ | $-1$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |

$i=8$ | $-3$ | $-2$ | $-1$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |

$i=9$ | $-4$ | $-3$ | $-2$ | $-1$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |

$i=10$ | $-5$ | $-4$ | $-3$ | $-2$ | $-1$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |

$i=11$ | $-6$ | $-5$ | $-4$ | $-3$ | $-2$ | $-1$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |

$\mathit{j}=1$ | $\mathit{j}=2$ | $\mathit{j}=3$ | $\mathit{j}=4$ | $\mathit{j}=5$ | $\mathit{j}=6$ | $\mathit{j}=7$ | $\mathit{j}=8$ | $\mathit{j}=9$ | $\mathit{j}=10$ | $\mathit{j}=11$ | $\mathit{j}=12$ | $\mathit{j}=13$ | $\mathit{j}=14$ | $\mathit{j}=15$ | $\mathit{j}=16$ | $\mathit{j}=17$ | $\mathit{j}=18$ | $\mathit{j}=19$ | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$i=1$ | 695 | 632 | 578 | 531 | 489 | 451 | 418 | 388 | 361 | 338 | 316 | 297 | 280 | 264 | 251 | 238 | 227 | 216 | 207 |

$i=2$ | inf | 695 | 632 | 578 | 531 | 489 | 451 | 418 | 388 | 361 | 338 | 316 | 297 | 280 | 264 | 251 | 238 | 227 | 216 |

$i=3$ | inf | inf | 695 | 632 | 578 | 531 | 489 | 451 | 418 | 388 | 361 | 338 | 316 | 297 | 280 | 264 | 251 | 238 | 227 |

$i=4$ | inf | inf | inf | 695 | 632 | 578 | 531 | 489 | 451 | 418 | 388 | 361 | 338 | 316 | 297 | 280 | 264 | 251 | 238 |

$i=5$ | inf | inf | inf | inf | 695 | 632 | 578 | 531 | 489 | 451 | 418 | 388 | 361 | 338 | 316 | 297 | 280 | 264 | 251 |

$i=6$ | inf | inf | inf | inf | inf | 695 | 632 | 578 | 531 | 489 | 451 | 418 | 388 | 361 | 338 | 316 | 297 | 280 | 264 |

$i=7$ | inf | inf | inf | inf | inf | inf | 695 | 632 | 578 | 531 | 489 | 451 | 418 | 388 | 361 | 338 | 316 | 297 | 280 |

$i=8$ | inf | inf | inf | inf | inf | inf | inf | 695 | 632 | 578 | 531 | 489 | 451 | 418 | 388 | 361 | 338 | 316 | 297 |

$i=9$ | inf | inf | inf | inf | inf | inf | inf | inf | 695 | 632 | 578 | 531 | 489 | 451 | 418 | 388 | 361 | 338 | 316 |

$i=10$ | inf | inf | inf | inf | inf | inf | inf | inf | inf | 695 | 632 | 578 | 531 | 489 | 451 | 418 | 388 | 361 | 338 |

$i=11$ | inf | inf | inf | inf | inf | inf | inf | inf | inf | inf | 695 | 632 | 578 | 531 | 489 | 451 | 418 | 388 | 361 |

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**Figure 5.**The objective values corresponding the selected solutions $q=2$ and $q=8$ for the modified reference point 3 with objectives (1)–(8).

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**MDPI and ACS Style**

Montonen, O.; Ranta, T.; Mäkelä, M.M. Planning the Schedule for the Disposal of the Spent Nuclear Fuel with Interactive Multiobjective Optimization. *Algorithms* **2019**, *12*, 252.
https://doi.org/10.3390/a12120252

**AMA Style**

Montonen O, Ranta T, Mäkelä MM. Planning the Schedule for the Disposal of the Spent Nuclear Fuel with Interactive Multiobjective Optimization. *Algorithms*. 2019; 12(12):252.
https://doi.org/10.3390/a12120252

**Chicago/Turabian Style**

Montonen, Outi, Timo Ranta, and Marko M. Mäkelä. 2019. "Planning the Schedule for the Disposal of the Spent Nuclear Fuel with Interactive Multiobjective Optimization" *Algorithms* 12, no. 12: 252.
https://doi.org/10.3390/a12120252