# Comparison of MOEAs in an Optimization-Decision Methodology for a Joint Order Batching and Picking System

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## Abstract

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## 1. Introduction

- The introduction of an integrated multi-criteria treatment of Order Batching and Order Picking problems. The decision criteria are the minimization of both operational cost and earliness in picking times.
- A more realistic analysis of the Order Batching and Order Picking problems using a multi-objective optimization model of nonlinear mixed-integer programming, by considering a multi-level storage system with an explicit inclusion of the due times of the requests and a zero tardiness policy.
- The incorporation into the model of the actual scheduling procedures for different pick-up teams with capacity restrictions used in real-world warehouses.
- The consideration of storage systems that combine multiple two and three-dimensional blocks.

## 2. Problem Description and Literature Review

## 3. The Model

#### 3.1. Sets and Parameters

- -
- $\mathcal{P}=\left\{1,\dots ,p,\dots ,\overline{\mathcal{P}}\right\}$ is the set of indexes of different articles required items in storage. ${\mathcal{P}}_{i}$ represents the subset of different articles requested by customer i. ${\mathcal{P}}_{b}$ represents the subset of articles grouped in batch b, and which can be requested by different customers.
- -
- $\mathcal{I}=\left\{1,\dots ,i,\dots ,\overline{\mathcal{I}}\right\}$ is the set of the index of customers and orders in the current wave. $\overline{\mathcal{I}}$ is the number of customers and orders, each customer places a single order with different articles.
- -
- $\mathcal{B}=\left\{1,\dots ,b,\dots ,\overline{\mathcal{B}}\right\}$ is the set of the index of batches to be picked up. ${\mathcal{B}}_{i}$ represents the subset of batches containing items of order i.
- -
- $\mathcal{W}=\left\{{w}_{1},\dots ,{w}_{p},\dots ,{w}_{\overline{\mathcal{P}}}\right\}$ is the set of weights, each article has a unit weight.
- -
- $\mathcal{D}=\left\{{d}_{1},\dots ,{d}_{i},\dots ,{d}_{\overline{\mathcal{I}}}\right\}$ is the set of customer order deadlines.
- -
- $\mathcal{L}=\left\{{\ell}_{0},{\ell}_{1},\dots ,{\ell}_{p},\dots ,{\ell}_{\overline{\mathcal{P}}}\right\}$ is the set of storage positions of the requested articles. ${\ell}_{0}$ represents the dispatch area. ${\ell}_{p}$ is the coordinates of the storage position of article p is given by $\left({x}_{p},{y}_{p},{z}_{p}\right)$.
- -
- $\mathcal{K}=\left\{1,\dots ,k,\dots ,\overline{\mathcal{K}}\right\}$ is the set of the index of pick-up teams. $\overline{\mathcal{K}}$ the total number of available teams.
- -
- $Cap$ is the total capacity of the pick-up teams.
- -
- ${\mathcal{S}}_{b}=\langle {s}_{1},\dots ,{s}_{u},\dots ,{s}_{{\overline{S}}_{b}}\rangle $ is the sequence of positions to be visited to conform batch b.
- -
- $\mathcal{Q}=\left\{1,\dots ,{q}_{i,p},\dots ,\overline{\mathcal{Q}}\right\}$ is the number of units of item p requested by customer i. ${\mathcal{Q}}_{i}={\sum}_{p\in {\mathcal{P}}_{i}}{q}_{i,p}$ is the total demand of articles by customer i. ${\mathcal{Q}}_{p}={\sum}_{i\in \mathcal{I}}{q}_{i,p}$ denotes the total number of requested units of p.

#### 3.2. Decision Variables

- -
- ${x}_{hlkb}$, which equals 1 if h is grabbed immediately before the article at the storage site l by the pick-up team k according to the sequencing of batch b, where $h,l\in \mathcal{V}$, $k\in \mathcal{K}$ and $b\in \mathcal{B}$. This means that ${x}_{hlkb}=1$ if k has to go through $edge(h,l)$ to pick up the goods in batch b.
- -
- ${y}_{hkb}$ equals 1 if k picks up the item in storage position h for batch b, where $h\in \mathcal{V}$, $k\in \mathcal{K}$ and $b\in \mathcal{B}$.

- -
- A time variable ${\mathfrak{g}}_{b}$ defines the starting time of batch b.
- -
- A time variable ${\mathfrak{f}}_{b}$ represents the end of the picking process of batch b.

#### 3.3. Bi-Objective OBP–OPP Model

## 4. Distribution Center Layout

## 5. Methodology

#### 5.1. Solution Methods at the MOO Stage

#### 5.1.1. NSGA-II

#### 5.1.2. SPEA2

#### 5.1.3. PESA-II

#### 5.2. The Decision Method at MCDM Stage

- A decision matrix is defined in which each element ${x}_{ij}=\phantom{\rule{4pt}{0ex}}{f}_{j}\left({a}_{i}\right)$ corresponds to the evaluation of the alternative i, according to criterion j.
- Weights are assigned to criteria, $w=\left\{{w}_{j},j=1,\phantom{\rule{4pt}{0ex}}2,\dots ,m\right\}$ associated with each objective function.
- The initial values are normalized according to a procedure in [57]:$${v}_{ij}=\frac{\left|{x}_{ij}-\underset{j}{min}{x}_{ij}\right|}{\left|\underset{j}{max}{x}_{ij}-\underset{j}{min}{x}_{ij}\right|}$$
- The ideal and anti-ideal solutions are identified. The ideal solution ${I}^{+}$ is the solution with the best possible values for each of the objective functions ${I}^{+}=\left({I}_{1}^{+},\dots ,{I}_{j}^{+},\dots ,{I}_{m}^{+}\right)$. The anti-ideal solution ${I}^{-}$ is the solution with the worst possible values for each of the objective functions ${I}^{-}=\left({I}_{1}^{-},\dots ,{I}_{j}^{-},\dots ,{I}_{m}^{-}\right)$.
- The distances of each action to its ideal and anti-ideal are measured by the metric ${L}_{p}$.$$\begin{array}{l}{L}_{p}^{{I}^{+}}={\left[{\sum}_{j=1}^{m}{w}_{j}^{p}\xb7{\left|{v}_{ij}-{I}^{+}\right|}^{p}\right]}^{\frac{1}{p}}with1\le p\le \infty \\ {L}_{p}^{{I}^{-}}={\left[{\sum}_{j=1}^{m}{w}_{j}^{p}\xb7{\left|{v}_{ij}-{I}^{-}\right|}^{p}\right]}^{\frac{1}{p}}with1\le p\le \infty \end{array}$$Here, ${v}_{ij}$ is the normalized value of action i for criterion j, and m is the number of criteria.
- The similarity ratios $S\left(v\right)$ are computed, yielding an ordering of actions:$$S\left(v\right)=\frac{{L}_{p}^{{I}^{-}}\left({v}^{i}\right)}{{L}_{p}^{{I}^{+}}\left({v}^{i}\right)+{L}_{p}^{{I}^{-}}\left({v}^{i}\right)}$$

#### 5.3. Measures of Performance

- Number of Pareto Solutions (NPS): This metric counts the number of non-dominated solutions found by each algorithm.
- Mean Ideal Distance (MID): It measures the closeness between the Pareto solution and the ideal point $(0,0)$ as:$$MID=\frac{{\sum}_{i=1}^{n}{c}_{i}}{n}$$
- Spread of Non-dominance Solution (SNS): It is a measure of the diversity of the Pareto front solutions. It is given by:$$SNS=\sqrt{\frac{{\sum}_{i=1}^{n}{\left(MID-{c}_{i}\right)}^{2}}{n-1}}$$
- Hypervolume (HV): It measures the size of the region dominated by the Pareto front $\left(P\right)$ and is limited by a point of reference dominated by the front. It takes into account both the convergence towards the Pareto front and the distribution of solutions:$$HV=\left|\left\{\bigcup _{i=1}^{n}A\left({x}_{i}\right):\forall {x}_{i}\in P\phantom{\rule{4pt}{0ex}}\right\}\right|$$

#### 5.4. Characterization of the MOEAs

## 6. Results

#### 6.1. Data Sets and Parameters Settings

#### 6.2. Numerical Experiments

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Ord. | S(v) | v1 | v2 |
---|---|---|---|

1° | 0.583 | 0.318 | 0.516 |

2° | 0.575 | 0.437 | 0.412 |

3° | 0.572 | 0.401 | 0.454 |

4° | 0.571 | 0.231 | 0.627 |

5° | 0.571 | 0.267 | 0.591 |

6° | 0.57 | 0.304 | 0.557 |

7° | 0.568 | 0.217 | 0.646 |

8° | 0.563 | 0.014 | 0.861 |

9° | 0.563 | 0.515 | 0.359 |

10° | 0.563 | 0.178 | 0.696 |

11° | 0.563 | 0.387 | 0.488 |

12° | 0.562 | 0.123 | 0.753 |

13° | 0.56 | 0.139 | 0.74 |

14° | 0.56 | 0.37 | 0.51 |

15° | 0.559 | 0.295 | 0.587 |

16° | 0.558 | 0.432 | 0.452 |

17° | 0.558 | 0.103 | 0.781 |

18° | 0.556 | 0.632 | 0.255 |

19° | 0.555 | 0.579 | 0.311 |

20° | 0.555 | 0.716 | 0.175 |

21° | 0.551 | 0.66 | 0.238 |

22° | 0.55 | 0.618 | 0.281 |

23° | 0.547 | 0.05 | 0.856 |

24° | 0.546 | 0.075 | 0.833 |

25° | 0.542 | 0.682 | 0.234 |

26° | 0.541 | 0.616 | 0.303 |

27° | 0.54 | 0.699 | 0.22 |

28° | 0.535 | 0.825 | 0.106 |

29° | 0.534 | 0.763 | 0.169 |

30° | 0.534 | 0.847 | 0.086 |

31° | 0.531 | 0.786 | 0.152 |

32° | 0.531 | 0.855 | 0.083 |

33° | 0.531 | 0.816 | 0.122 |

34° | 0.524 | 0.911 | 0.041 |

35° | 0.517 | 0.942 | 0.024 |

36° | 0.515 | 0.933 | 0.036 |

37° | 0.506 | 0.908 | 0.08 |

38° | 0.505 | 0.972 | 0.018 |

39° | 0.5 | 0 | 1 |

40° | 0.5 | 1 | 0 |

Ord. | S(v) | v1 | v2 |
---|---|---|---|

1° | 0.608 | 0.322 | 0.463 |

2° | 0.604 | 0.298 | 0.494 |

3° | 0.599 | 0.391 | 0.412 |

4° | 0.596 | 0.175 | 0.633 |

5° | 0.596 | 0.282 | 0.527 |

6° | 0.596 | 0.356 | 0.453 |

7° | 0.591 | 0.273 | 0.545 |

8° | 0.59 | 0.629 | 0.191 |

9° | 0.59 | 0.254 | 0.566 |

10° | 0.588 | 0.2 | 0.624 |

11° | 0.588 | 0.475 | 0.349 |

12° | 0.586 | 0.236 | 0.591 |

13° | 0.585 | 0.546 | 0.283 |

14° | 0.584 | 0.599 | 0.233 |

15° | 0.581 | 0.613 | 0.224 |

16° | 0.58 | 0.459 | 0.38 |

17° | 0.58 | 0.448 | 0.392 |

18° | 0.576 | 0.746 | 0.101 |

19° | 0.575 | 0.706 | 0.145 |

20° | 0.571 | 0.597 | 0.261 |

21° | 0.569 | 0.534 | 0.327 |

22° | 0.567 | 0.821 | 0.044 |

23° | 0.564 | 0.126 | 0.746 |

24° | 0.56 | 0.739 | 0.141 |

25° | 0.56 | 0.161 | 0.719 |

26° | 0.558 | 0.704 | 0.179 |

27° | 0.556 | 0.144 | 0.744 |

28° | 0.546 | 0.112 | 0.797 |

29° | 0.545 | 0.121 | 0.789 |

30° | 0.541 | 0.068 | 0.85 |

31° | 0.54 | 0.032 | 0.888 |

32° | 0.535 | 0.102 | 0.828 |

33° | 0.535 | 0.065 | 0.865 |

34° | 0.535 | 0.012 | 0.923 |

35° | 0.532 | 0.031 | 0.913 |

36° | 0.531 | 0.013 | 0.966 |

37° | 0.525 | 0.021 | 0.963 |

38° | 0.505 | 0.03 | 0.909 |

39° | 0.5 | 0 | 1 |

40° | 0.5 | 1 | 0 |

Ord. | S(v) | v1 | v2 |
---|---|---|---|

1° | 0.577 | 0.358 | 0.488 |

2° | 0.566 | 0.434 | 0.434 |

3° | 0.563 | 0.297 | 0.578 |

4° | 0.56 | 0.406 | 0.475 |

5° | 0.553 | 0.557 | 0.338 |

6° | 0.551 | 0.486 | 0.412 |

7° | 0.546 | 0.509 | 0.399 |

8° | 0.545 | 0.524 | 0.387 |

9° | 0.545 | 0.604 | 0.307 |

10° | 0.545 | 0.118 | 0.793 |

11° | 0.539 | 0.236 | 0.686 |

12° | 0.538 | 0.726 | 0.197 |

13° | 0.538 | 0.656 | 0.269 |

14° | 0.536 | 0.16 | 0.767 |

15° | 0.534 | 0.821 | 0.112 |

16° | 0.532 | 0.255 | 0.681 |

17° | 0.529 | 0.75 | 0.192 |

18° | 0.528 | 0.868 | 0.076 |

19° | 0.527 | 0.703 | 0.244 |

20° | 0.525 | 0.09 | 0.859 |

21° | 0.525 | 0.901 | 0.05 |

22° | 0.524 | 0.896 | 0.056 |

23° | 0.522 | 0.925 | 0.031 |

24° | 0.522 | 0.943 | 0.012 |

25° | 0.521 | 0.939 | 0.019 |

26° | 0.521 | 0.929 | 0.029 |

27° | 0.519 | 0.807 | 0.155 |

28° | 0.519 | 0.802 | 0.16 |

29° | 0.518 | 0.104 | 0.859 |

30° | 0.517 | 0.698 | 0.267 |

31° | 0.517 | 0.778 | 0.187 |

32° | 0.515 | 0.797 | 0.173 |

33° | 0.508 | 0.972 | 0.012 |

34° | 0.503 | 0.991 | 0.004 |

35° | 0.501 | 0.061 | 0.953 |

36° | 0.5 | 0.987 | 0.01 |

37° | 0.5 | 0.07 | 0.94 |

38° | 0.5 | 0.989 | 0.006 |

39° | 0.5 | 0 | 1 |

40° | 0.5 | 1 | 0 |

Ord. | S(v) | v1 | v2 |
---|---|---|---|

1° | 0.765 | 0.29 | 0.179 |

2° | 0.765 | 0.419 | 0.051 |

3° | 0.757 | 0.439 | 0.047 |

4° | 0.757 | 0.439 | 0.047 |

5° | 0.756 | 0.255 | 0.234 |

6° | 0.755 | 0.315 | 0.175 |

7° | 0.754 | 0.382 | 0.109 |

8° | 0.754 | 0.382 | 0.109 |

9° | 0.747 | 0.248 | 0.258 |

10° | 0.744 | 0.494 | 0.019 |

11° | 0.744 | 0.494 | 0.019 |

12° | 0.742 | 0.469 | 0.047 |

13° | 0.737 | 0.481 | 0.045 |

14° | 0.737 | 0.481 | 0.045 |

15° | 0.734 | 0.519 | 0.014 |

16° | 0.734 | 0.519 | 0.014 |

17° | 0.732 | 0.372 | 0.165 |

18° | 0.718 | 0.244 | 0.321 |

19° | 0.705 | 0.173 | 0.416 |

20° | 0.702 | 0.582 | 0.013 |

21° | 0.697 | 0.6 | 0.007 |

22° | 0.697 | 0.6 | 0.007 |

23° | 0.685 | 0.234 | 0.396 |

24° | 0.665 | 0.156 | 0.514 |

25° | 0.662 | 0.669 | 0.006 |

26° | 0.646 | 0.703 | 0.006 |

27° | 0.644 | 0.152 | 0.559 |

28° | 0.636 | 0.145 | 0.583 |

29° | 0.634 | 0.142 | 0.591 |

30° | 0.605 | 0.786 | 0.004 |

31° | 0.605 | 0.786 | 0.004 |

32° | 0.605 | 0.786 | 0.004 |

33° | 0.597 | 0.042 | 0.763 |

34° | 0.596 | 0.131 | 0.677 |

35° | 0.594 | 0.127 | 0.685 |

36° | 0.565 | 0.11 | 0.76 |

37° | 0.509 | 0.039 | 0.943 |

38° | 0.506 | 0.048 | 0.913 |

39° | 0.5 | 1 | 0 |

40° | 0.5 | 0 | 1 |

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**Figure 1.**Illustrations of a small layout representation with a position marker for a particular batch. (

**a**) Orders batch (highlighted) in warehouse layout. (

**b**) A graph-based representation of the underlying topology. (

**c**) A picking tour for a particular batch of orders.

**Table 1.**Pseudocode of NSGA-II (adapted from [18]).

1: | Input: $N,{P}_{c},{P}_{m},T,D$ |

2: | Output: ${P}^{*}$ |

3: | Begin: |

4: | P ← Initialize population(N) |

5: | Q ← ∅ |

6: | while $t<T$ do: |

7: | F ← Fitness Evaluation ($P,D$) |

8: | $(P,F)$ ← Non-Dominated Sorting $(P,F)$ |

9: | $(P,F)$ ← Crowding Distance $(P,F)$ |

10: | $(P,F)$ ← Sort Population $(P,F)$ |

11: | ${X}_{c}$ ← Selection parents $(P,{P}_{c})$ |

12: | ${Y}_{c}$ ← Crossover(${X}_{c},D$) |

13: | $({Y}_{c},{F}_{c})$ ← Fitness Evaluation $({Y}_{c},D)$ |

14: | ${X}_{m}$ ← Selection Individuals ($P,{P}_{m}$) |

15: | ${Y}_{m}$ ← Mutation($Xm,\mu ,D$) |

16: | $({Y}_{m},{F}_{m})$ ← Fitness Evaluation $\left({Y}_{m}\right)$ |

17: | R ← Join $P,{Y}_{c},{Y}_{m}$ |

18: | $(R,F)$ ← Non-Dominated Sorting $(R,F)$ |

19: | $(R,F)$ ← Crowding Distance $(R,F)$ |

20: | $(R,F)$ ← Sort Population $(R,F)$ |

21: | P ← Truncate $(R,N)$ |

22: | end-while |

23: | return: $P*$ ← P |

**Table 2.**Pseudocode of SPEA2 (based on [19]).

1: | Input: $N,T,D$ |

2: | Output: ${P}^{*}$ |

3: | Initialization: |

4: | P ← Initialize population(N) |

5: | E ← ∅ Create empty external set (archive) |

6: | While $t<T$ do: |

7: | ${F}_{E},{F}_{P}$ ← Fitness Evaluation $(P,E,D)$; |

8: | E ← Non-dominated individuals in P and E; |

9: | E ← If size(E) > N: truncate(E) otherwise fill E with dominated individuals in P; |

10: | if $t\ge T$ then |

11: | ${P}^{*}$ ← E |

12: | Stop |

13: | end-if |

14: | X ← Selection parents $\left(E\right)$ |

15: | P ← apply Crossover and Mutation operators to X |

16: | end-while |

17: | return: ${P}^{*}$ |

**Table 3.**Pseudocode of PESA-II (Based on [20]).

1: | Input: $T,N,{P}_{c},D$ |

2: | Output: ${P}^{*}$ |

3: | Initialization: |

4: | P ← Initialize population(N) |

5: | $({F}_{P},{F}_{E})$ ← Fitness Evaluation $(P,D)$; |

6: | E ← ∅ Create empty external set (archive) |

7: | While $t<T$ do: |

8: | E ← Non-dominated individuals in P |

9: | P ← ∅ |

10: | while size(P) = N do: |

11: | X ← Selection parents $\left(E\right)$ with ${P}_{c}$ |

12: | Y ← Crossover operator $\left(X\right)$ |

13: | P ← Mutation operator $\left(Y\right)$ |

14: | X ← Selection parents $\left(E\right)$ with $(1-{P}_{c})$ |

15: | P ← Mutation operator $\left({X}^{\prime}\right)$ |

16: | end-while |

17: | end-while |

18: | return: ${P}^{*}$ |

Articles | ||||||
---|---|---|---|---|---|---|

Orders | A1 | A2 | A3 | A4 | A5 | Due Dates |

O1 | 1 | 0 | 1 | 0 | 0 | 0.50 |

O2 | 0 | 1 | 2 | 2 | 1 | 0.35 |

O3 | 1 | 1 | 0 | 1 | 1 | 0.25 |

O4 | 2 | 0 | 0 | 1 | 0 | 0.75 |

Order Index (${D}_{I}$) | 1 | 2 | 3 | 4 |

Nº Art. diff. (${D}_{{q}_{ip}}$) | 2 | 4 | 4 | 2 |

Due Date (${D}_{dd}$) | 0.25 | 0.75 | 0.15 | 0.50 |

Weight (${D}_{W}$) | 10 | 12 | 11 | 12 |

Genome 1 (Batch) | 1 | 2 | 2 | 1 |

Order Index | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 |

Article Index | 1 | 3 | 2 | 3 | 4 | 5 | 1 | 2 | 4 | 5 | 1 | 4 |

Quantity | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 |

Genome 2 (sequence) | 1 | 2 | 1 | 3 | 4 | 2 | 5 | 1 | 4 | 2 | 1 | 3 |

Genome 1 | Genome 2 | |||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Chromosome | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 3 | 4 | 2 | 5 | 1 | 4 | 2 | 1 | 3 |

1: | Input: ${X}_{1},{X}_{2}$ and D. Parent solutions and data parametric information |

2: | Output: ${Y}_{1}$ and ${Y}_{2}$. Offspring solutions generated by the crossover operation |

3: | Initialization: Obtain the genomes of each parent. ${x}_{11},{x}_{12},{x}_{21}$, and ${x}_{22}$ |

4: | Generate k random integers, between 1 and $\overline{\mathcal{I}}$ |

5: | Swap k elements belonging to the first genome in the two parents and store in ${y}_{11}$ and ${y}_{21}$ |

6: | Identify batches with changes in genome 1 of each offspring |

7: | Apply the closest neighbor heuristic and store the modified batches in the second genomes of each offspring ${y}_{12}$ and ${y}_{22}$ |

9: | Termination: Join the pairs of genomes 1 and 2 of each offspring and save them in ${Y}_{1}$ and ${Y}_{2}$ |

1: | Input: ${X}_{3}$ and D. Parent solution and data parametric information |

2: | Output: ${Y}_{3}$. Offspring solution generated by the mutation operation |

3: | Initialization: Obtain the genomes of parent. ${x}_{31}$, and ${x}_{32}$ |

4: | Generate random integer, between 1 and $\overline{\mathcal{B}}$ |

5: | Insert the integer into a random position between 1 and $\overline{\mathcal{I}}$ in the first genome of the parent and store in ${y}_{31}$ |

6: | Identify batches with changes in genome 1 |

7: | Apply a lambda swap procedure to enhance the local sequence within the modified batches of genome 2 and save it in ${y}_{32}$ |

9: | Termination: Join the pairs of genomes 1 and 2 of the mutated child and save it in ${Y}_{3}$ |

Instance | DS0 | DS1 | DS2 | DS3 | DS4 | DS5 | DS6 |
---|---|---|---|---|---|---|---|

Problem size ^{1} | S | S | M | L | S | M | L |

Dimensions of storage system | 2D | 2D | 2D | 2D | 3D | 3D | 3D |

Number of orders in a wave | 25 | 40 | 80 | 200 | 40 | 100 | 250 |

Number of different articles in a wave of orders | 30 | 80 | 160 | 300 | 80 | 200 | 400 |

Maximum number of units per order | 89 | 117 | 138 | 132 | 116 | 116 | 124 |

Minimum number of different items per order | 18 | 18 | 23 | 22 | 21 | 19 | 27 |

Total number of units ordered per wave | 1459 | 2690 | 4589 | 11,889 | 2096 | 5805 | 15,080 |

Weight per order | 1512 | 1426 | 1403 | 1978 | 1200 | 1416 | 1886 |

^{1}Small; M: Medium; L: Large.

DS0 | DS1 | DS2 | DS3 | DS4 | DS5 | DS6 | |
---|---|---|---|---|---|---|---|

NPS | |||||||

NSGA-II | 39.875 | 39.852 | 39.764 | 39.687 | 39.624 | 39.625 | 39.545 |

PESA-II | 39.913 | 39.858 | 39.826 | 39.770 | 39.716 | 39.652 | 39.616 |

SPEA2 | 39.868 | 39.826 | 39.763 | 39.724 | 39.681 | 39.635 | 39.572 |

MID | |||||||

NSGA-II | 5.991 | 2.681 | 5.606 | 19.62 | 2.370 | 4.570 | 36.07 |

PESA-II | 5.860 | 2.469 | 5.897 | 21.81 | 2.579 | 5.211 | 38.69 |

SPEA2 | 6.393 | 2.693 | 5.888 | 21.85 | 2.440 | 4.724 | 36.53 |

SNS | |||||||

NSGA-II | 3.936 | 1.155 | 2.558 | 6.801 | 1.969 | 2.157 | 19.572 |

PESA-II | 4.049 | 1.161 | 2.946 | 8.936 | 2.268 | 2.681 | 21.083 |

SPEA2 | 3.669 | 1.032 | 2.667 | 7.437 | 1.916 | 2.239 | 19.922 |

HV | |||||||

NSGA-II | 1.170 | 1.166 | 1.320 | 0.371 | 1.178 | 1.264 | 9.737 |

PESA-II | 1.170 | 1.155 | 1.316 | 0.354 | 1.129 | 1.241 | 9.694 |

SPEA2 | 1.174 | 1.161 | 1.315 | 0.333 | 1.114 | 1.241 | 9.617 |

NPS | MID | SNS | HV | |||||
---|---|---|---|---|---|---|---|---|

DS0 | ${u}_{i}-{u}_{j}$ | $\mathit{p}$-value | ${u}_{i}-{u}_{j}$ | $\mathit{p}$-value | ${u}_{i}-{u}_{j}$ | $\mathit{p}$-value | ${u}_{i}-{u}_{j}$ | $\mathit{p}$-value |

NSGA-II vs. PESA-II | −0.038 | 0.025 | 0.131 | 0.026 | −0.112 | 0.000 | 0.000 | 0.000 |

NSGA-II vs. SPEA2 | 0.007 | 0.502 | −0.402 | 0.016 | 0.267 | 0.000 | 0.000 | 0.001 |

SPEA2 vs. PESA-II | −0.045 | 0.025 | 0.533 | 0.020 | −0.379 | 0.000 | 0.004 | 0.002 |

DS1 | ${u}_{i}-{u}_{j}$ | $\mathit{p}$-value | ${u}_{i}-{u}_{j}$ | $\mathit{p}$-value | ${u}_{i}-{u}_{j}$ | $\mathit{p}$-value | ${u}_{i}-{u}_{j}$ | $\mathit{p}$-value |

NSGA-II vs. PESA-II | −0.006 | 0.628 | 0.212 | 0.019 | −0.007 | 0.575 | 0.011 | 0.000 |

NSGA-II vs. SPEA2 | 0.026 | 0.056 | −0.013 | 0.738 | 0.122 | 0.000 | 0.005 | 0.000 |

SPEA2 vs. PESA-II | −0.032 | 0.032 | 0.225 | 0.025 | −0.129 | 0.000 | 0.007 | 0.000 |

DS2 | ${u}_{i}-{u}_{j}$ | $\mathit{p}$-value | ${u}_{i}-{u}_{j}$ | $\mathit{p}$-value | ${u}_{i}-{u}_{j}$ | $\mathit{p}$-value | ${u}_{i}-{u}_{j}$ | $\mathit{p}$-value |

NSGA-II vs. PESA-II | −0.062 | 0.025 | −0.292 | 0.000 | −0.380 | 0.000 | 0.004 | 0.002 |

NSGA-II vs. SPEA2 | 0.002 | 0.938 | −0.282 | 0.000 | 0.109 | 0.000 | 0.005 | 0.000 |

SPEA2 vs. PESA-II | −0.063 | 0.025 | −0.009 | 0.791 | −0.279 | 0.000 | 0.000 | 0.000 |

DS3 | ${u}_{i}-{u}_{j}$ | $\mathit{p}$-value | ${u}_{i}-{u}_{j}$ | $\mathit{p}$-value | ${u}_{i}-{u}_{j}$ | $\mathit{p}$-value | ${u}_{i}-{u}_{j}$ | $\mathit{p}$-value |

NSGA-II vs. PESA-II | −0.083 | 0.025 | −2.189 | 0.000 | −2.135 | 0.000 | 0.017 | 0.051 |

NSGA-II vs. SPEA2 | −0.037 | 0.085 | −2.233 | 0.000 | −0.636 | 0.000 | 0.038 | 0.000 |

SPEA2 vs. PESA-II | −0.046 | 0.034 | 0.044 | 0.858 | −1.499 | 0.000 | −0.020 | 0.015 |

DS4 | ${u}_{i}-{u}_{j}$ | $\mathit{p}$-value | ${u}_{i}-{u}_{j}$ | $\mathit{p}$-value | ${u}_{i}-{u}_{j}$ | $\mathit{p}$-value | ${u}_{i}-{u}_{j}$ | $\mathit{p}$-value |

NSGA-II vs. PESA-II | −0.092 | 0.025 | −0.209 | 0.000 | −0.299 | 0.000 | 0.048 | 0.000 |

NSGA-II vs. SPEA2 | −0.057 | 0.037 | −0.070 | 0.221 | 0.053 | 0.215 | 0.064 | 0.000 |

SPEA2 vs. PESA-II | −0.035 | 0.122 | −0.139 | 0.015 | −0.352 | 0.000 | −0.010 | 0.000 |

DS5 | ${u}_{i}-{u}_{j}$ | $\mathit{p}$-value | ${u}_{i}-{u}_{j}$ | $\mathit{p}$-value | ${u}_{i}-{u}_{j}$ | $\mathit{p}$-value | ${u}_{i}-{u}_{j}$ | $\mathit{p}$-value |

NSGA-II vs. PESA-II | −0.027 | 0.286 | −0.641 | 0.000 | −0.528 | 0.000 | 0.023 | 0.000 |

NSGA-II vs. SPEA2 | −0.010 | 0.702 | −0.154 | 0.001 | 0.082 | 0.009 | 0.027 | 0.000 |

SPEA2 vs. PESA-II | −0.017 | 0.502 | −0.487 | 0.000 | −0.442 | 0.000 | 0.000 | 0.000 |

DS6 | ${u}_{i}-{u}_{j}$ | $\mathit{p}$-value | ${u}_{i}-{u}_{j}$ | $\mathit{p}$-value | ${u}_{i}-{u}_{j}$ | $\mathit{p}$-value | ${u}_{i}-{u}_{j}$ | $\mathit{p}$-value |

NSGA-II vs. PESA-II | −0.071 | 0.036 | −2.622 | 0.000 | −1.511 | 0.000 | 0.043 | 0.000 |

NSGA-II vs. SPEA2 | −0.027 | 0.370 | −0.463 | 0.001 | 0.350 | 0.016 | 0.120 | 0.000 |

SPEA2 vs. PESA-II | −0.044 | 0.131 | −2.160 | 0.000 | −1.162 | 0.000 | −0.080 | 0.000 |

**Table 13.**MCDM performance and comparison with the benchmark [2].

M1 | TOPSIS | M1 | TOPSIS | M1 | TOPSIS | |||
---|---|---|---|---|---|---|---|---|

TOC (Small) | TOC (Medium) | TOC (Large) | ||||||

DS1 | 1092.60 | 1078.50 | DS2 | 3104.83 | 2914.24 | DS3 | 9207.13 | 8217.20 |

DS4 | 1322.20 | 1287.51 | DS5 | 4431.03 | 4098.40 | DS6 | 14,546.63 | 12,848.32 |

TOPSIS/M1 | TOPSIS/M1 | TOPSIS/M1 | |||
---|---|---|---|---|---|

Small | Medium | Large | |||

DS1 | 0.987 | DS2 | 0.939 | DS3 | 0.892 |

DS4 | 0.974 | DS5 | 0.925 | DS6 | 0.883 |

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**MDPI and ACS Style**

Miguel, F.M.; Frutos, M.; Méndez, M.; Tohmé, F.; González, B.
Comparison of MOEAs in an Optimization-Decision Methodology for a Joint Order Batching and Picking System. *Mathematics* **2024**, *12*, 1246.
https://doi.org/10.3390/math12081246

**AMA Style**

Miguel FM, Frutos M, Méndez M, Tohmé F, González B.
Comparison of MOEAs in an Optimization-Decision Methodology for a Joint Order Batching and Picking System. *Mathematics*. 2024; 12(8):1246.
https://doi.org/10.3390/math12081246

**Chicago/Turabian Style**

Miguel, Fabio Maximiliano, Mariano Frutos, Máximo Méndez, Fernando Tohmé, and Begoña González.
2024. "Comparison of MOEAs in an Optimization-Decision Methodology for a Joint Order Batching and Picking System" *Mathematics* 12, no. 8: 1246.
https://doi.org/10.3390/math12081246