# Artificial Bee Colony Algorithm with Nelder–Mead Method to Solve Nurse Scheduling Problem

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## Abstract

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## 1. Introduction

- A hybrid meta-heuristic algorithm, namely artificial bee colony optimization with Nelder–Mead Method, is proposed.
- The search capability of ABC is enriched with the aid of the Nelder–Mead method, which consists of search strategies such as midpoint, reflection, expansion, contraction, and shrinkage processes. These search strategies enhance the balance between exploration and exploitation.
- NM-ABC is implemented and tested on the nurse scheduling problem (NSPLib).
- The performance of NM-ABC is compared with that of some classical optimization algorithms.

## 2. Nurse Scheduling Problem

## 3. Proposed Algorithm

#### 3.1. Artificial Bee Colony Algorithm

#### 3.1.1. Initialization

#### 3.1.2. Employed Bee Phase

#### 3.1.3. Probability Calculation

Algorithm 1: Probability calculation | |

1: | Fori = 1, 2, …, FS, do |

2: | Calculate probability values ${P}_{ij}$ for the solution ${v}_{i,j}$ |

3: | ${P}_{i}=\frac{fi{t}_{i}}{{{\displaystyle \sum}}_{j=1}^{FP}fi{t}_{j}}$ |

4: | $fi{t}_{i}=\{\begin{array}{c}\frac{1}{1+{f}_{i}},{f}_{i}\ge 0\\ 1+abs\left({f}_{i}\right),{f}_{i}0\end{array}$ |

5: | End For |

#### 3.1.4. Onlooker Bee Phase

#### 3.1.5. Scout Bee Phase

#### 3.2. Nelder–Mead Method

#### 3.2.1. Midpoint (M)

#### 3.2.2. Reflection (R)

#### 3.2.3. Expansion (E)

#### 3.2.4. Contraction (C)

#### 3.2.5. Shrinkage (S)

Algorithm 2: Nelder–Mead Method | |

1: | Produce new food source ${v}_{i}$ using modified Nelder–Mead Method |

2: | Let ${v}_{i}$ denote list of vertices |

3: | $\mathrm{\u027d}$, μ, λ, and ζ are the constants of likeness, extension, shrinkage, and contraction |

4: | f is the fitness method to be reduced |

5: | For i = 1, 2, …, n + 1 vertices, do |

6: | Order the vertices from deepest fitness function f(v_1) to maximum fitness function f(〖v〗_(n + 1)) |

7: | $f\left({v}_{1}\right)\le f\left({v}_{2}\right)\le \cdots \le f\left({v}_{n+1}\right)$ |

8: | Calculate midpoint for best two vertices |

9: | ${v}_{m}={\displaystyle \sum}\frac{{v}_{i}}{n}$, where i = 1, 2, …, n |

10: | Calculate reflection point v_r |

11: | ${v}_{r}={v}_{m}+\mathrm{\u027d}\left({v}_{m}-{v}_{n+1}\right)$ |

12: | if $f\left({v}_{1}\right)\le f\left({v}_{r}\right)\le f\left({v}_{n}\right)$ then |

13: | ${v}_{n}\leftarrow {v}_{r}$ and go to stopping criteria |

14: | End if |

15: | Calculate expansion point ${v}_{e}$ |

16: | if $f\left({v}_{r}\right)\le f\left({v}_{1}\right)$ then |

17: | ${v}_{n}\leftarrow {v}_{e}$ and go to stopping criteria |

18: | End if |

19: | if $f\left({v}_{e}\right)<f\left({v}_{r}\right)$ then |

20: | ${v}_{n}\leftarrow {v}_{e}$ and go to stopping criteria |

21: | else |

22: | ${v}_{n}\leftarrow {v}_{r}$ and go to stopping criteria |

23: | End if |

24: | Calculate contraction point ${v}_{c}$ |

25: | if $f\left({v}_{n}\right)\le f\left({v}_{r}\right)\le f\left({v}_{n+1}\right)$ then |

26: | Compute outside contraction |

27: | ${v}_{c}=\mathsf{\lambda}{v}_{n+1}+\left(1-\mathsf{\lambda}\right){v}_{m}$. |

28: | End if |

29: | if $f\left({v}_{r}\right)\ge f\left({v}_{n+1}\right)$ then |

30: | Compute inside contraction |

31: | ${v}_{c}=\mathsf{\lambda}{v}_{n+1}+\left(1-\mathsf{\lambda}\right){v}_{m}$. |

32: | End if |

33: | if $f\left({v}_{r}\right)\ge f\left({v}_{n}\right)$ then |

34: | Shrinkage is done between ${v}_{m}$ and the best vertex among ${v}_{r}$ and ${v}_{n+1}$. |

35: | End if |

36: | if $f\left({v}_{c}\right)<f\left({v}_{r}\right)$ then |

37: | ${v}_{n}\leftarrow {v}_{c}$ and go to Stopping criteria |

38: | else go to Shrinkage phase |

39: | End if |

40: | if $f\left({v}_{c}\right)\ge f\left({v}_{n+1}\right)$ then |

41: | ${v}_{n+1}\leftarrow {v}_{c}$ and go to Stopping criteria |

42: | else go to the Shrinkage phase |

43: | End if |

44: | Calculate Shrinkage |

45: | Shrink close the best individual with new apices |

46: | ${v}_{i}=\mathsf{\zeta}{v}_{i}+{v}_{1}\left(1-\mathsf{\zeta}\right)$, where i = 2, …, n + 1 |

47: | End for |

48: | Determine the new vertices of the simplex thus formed based on their fitness and continue with the process of the reflection phase |

#### 3.3. Nelder–Mead Method-Based ABC (NM-ABC)

Algorithm 3: NM-ABC | |

1: | Initialize the population |

2: | For i = 1, 2, …, FS, do |

3: | For j = 1, 2, …, S, do |

4: | Generate ${x}_{i,j}$ solution |

5: | ${x}_{i,j}={x}_{min,j}\pm rand\left(0,1\right)\ast \left({x}_{max,j}-{x}_{min,j}\right)$ |

6: | Where ${x}_{min,j}$ and ${x}_{max,j}$ are the min and max limit of the dimension $j$ |

7: | End for |

8: | Compute the objective of the population |

9: | $iter=1$ |

10: | Repeat |

11: | { |

12: | Employed Bee Phase |

13: | For each food source $i$ do |

14: | Generate candidate solution ${v}_{i}$ using Equation (14) |

15: | Select between ${v}_{i}$ and ${x}_{i}$ |

16: | End For |

17: | Onlooker Bee Phase |

18: | Set r = 0 |

19: | While (r <= FS) |

20: | If rand(0,1) < ${P}_{i}$ using Algorithm 3 |

21: | Generate candidate solution ${v}_{i}$ by Algorithm 2 |

22: | Select between ${v}_{i}$ and ${x}_{i}$ |

23: | r = r + 1 |

24: | End if |

25: | End while |

26: | Scout Bee Phase |

27: | Abandon the food source ${x}_{i}$, which cannot improve further using Equation (13) |

28: | Remember the best individual obtained so far |

29: | iter = iter + 1 |

30: | } |

31: | End for |

32: | Until iter = max FEs |

## 4. Experimental Results

#### 4.1. Experimental Setup

#### 4.2. Performance Metrics

#### 4.2.1. Average Best Time (ABT)

#### 4.2.2. Standard Deviation

#### 4.2.3. Least Error Rate

#### 4.2.4. Success Percentage

#### 4.2.5. Cost Reduction

#### 4.2.6. Gap

#### 4.2.7. #Both Feasible Solution

#### 4.2.8. #Feasible Solution

#### 4.3. Experimental Result Analysis

## 5. Discussions

^{6}for all the test cases.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Case | Type | Instances | Nurse | Day | Shift |
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1 | N25 | 1, 7, 12, 19, 25 | 25 | 7 | 4 |

2 | N25 | 2, 5, 9, 15, 27 | 25 | 7 | 4 |

3 | N25 | 1, 3, 16, 27, 35 | 25 | 7 | 4 |

4 | N25 | 5, 10, 25, 38, 41 | 25 | 7 | 4 |

5 | N25 | 7, 11, 30, 42, 47 | 25 | 7 | 4 |

6 | N50 | 1, 4, 12, 26, 29 | 50 | 7 | 4 |

7 | N50 | 3, 6,12, 26, 36 | 50 | 7 | 4 |

8 | N50 | 4, 9, 15, 40, 47 | 50 | 7 | 4 |

9 | N50 | 5, 10, 23, 29, 40 | 50 | 7 | 4 |

10 | N50 | 6,14, 20, 32, 41 | 50 | 7 | 4 |

11 | N60 | 2, 8, 14, 20, 32 | 60 | 28 | 4 |

12 | N60 | 3, 12, 19, 23, 34 | 60 | 28 | 4 |

13 | N60 | 1, 4, 19, 29, 40 | 60 | 28 | 4 |

14 | N60 | 5, 9, 15, 30, 43 | 60 | 28 | 4 |

15 | N60 | 6, 15, 26, 35, 44 | 60 | 28 | 4 |

Type | Method | Parameters and Values |
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M_{1} | Multi-Assignment Problem-based Algorithm (MAPA) [3] | Number of Iterations—1000, Penalty Violation Value—100 |

M_{2} | Hybrid Artificial Bee Colony Algorithm (HABC) [2] | Limit—100, Spin Track—10, Number of Population—100, Number of Iterations—100 |

M_{3} | Bee Colony Optimization Algorithm (BCO) [27] | Knowledge Base (b)—2 Number of Population—100, Number of Iterations—100 |

M_{4} | Hybrid Elitist–Ant System (HEAS) [12] | Population Size—100, Number of Iterations—1000, Pheromone Initial Values—0.01 Evaporation Rate—0.25 |

M_{5} | Harmony Search-Based Hyper-Heuristic Algorithm (HSHH) [1] | Number of Population—100, Number of Iterations—100 |

Proposed | Artificial Bee Colony with Nelder–Mead (NM-ABC) | Number of Bees—100, Number of Iterations—1000, Maximum Runs—20, Reflection Coefficient—α > 0 Expansion Coefficient—γ > 1 Contraction Coefficient—0 > β > 1 Shrinkage Coefficient—0 < δ < 1 |

**Table 3.**Summary of the Best result obtained by proposed Artificial bee colony with Nelder Mead method (NM-ABC) and competitor methods.

Case | Type | Instance | Optimal Value | NM-ABC | M_{1} | M_{2} | M_{3} | M_{4} | M_{5} |
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C-1 | N25 | 1 | 307 | 307 | 307 | 307 | 307 | 306 | 307 |

7 | 291 | 287 | 290 | 292 | 290 | 292 | 292 | ||

12 | 296 | 296 | 297 | 296 | 296 | 296 | 296 | ||

19 | 302 | 302 | 302 | 302 | 303 | 302 | 302 | ||

25 | 308 | 307 | 307 | 306 | 307 | 308 | 308 | ||

C-2 | N25 | 2 | 274 | 269 | 277 | 279 | 276 | 273 | 273 |

5 | 303 | 303 | 303 | 302 | 302 | 302 | 301 | ||

9 | 276 | 276 | 276 | 276 | 276 | 276 | 275 | ||

15 | 296 | 289 | 299 | 300 | 299 | 296 | 300 | ||

27 | 293 | 293 | 293 | 293 | 293 | 293 | 293 | ||

C-3 | N25 | 1 | 333 | 331 | 333 | 339 | 337 | 332 | 332 |

3 | 315 | 315 | 315 | 315 | 315 | 317 | 315 | ||

16 | 323 | 325 | 322 | 323 | 326 | 326 | 323 | ||

27 | 318 | 316 | 318 | 318 | 319 | 318 | 317 | ||

35 | 333 | 327 | 330 | 330 | 331 | 333 | 333 | ||

C-4 | N25 | 5 | 313 | 285 | 301 | 319 | 310 | 312 | 316 |

10 | 284 | 273 | 275 | 288 | 277 | 281 | 279 | ||

25 | 308 | 294 | 303 | 300 | 305 | 306 | 312 | ||

38 | 294 | 294 | 294 | 294 | 294 | 293 | 299 | ||

41 | 296 | 296 | 296 | 296 | 296 | 299 | 296 | ||

C-5 | N25 | 7 | 293 | 289 | 299 | 299 | 293 | 293 | 296 |

11 | 299 | 299 | 299 | 299 | 299 | 298 | 299 | ||

30 | 311 | 309 | 310 | 319 | 315 | 311 | 310 | ||

42 | 283 | 283 | 287 | 283 | 283 | 283 | 283 | ||

47 | 310 | 309 | 309 | 315 | 309 | 309 | 309 | ||

C-6 | N50 | 1 | 575 | 569 | 579 | 577 | 573 | 575 | 577 |

4 | 641 | 640 | 641 | 640 | 641 | 640 | 649 | ||

12 | 575 | 571 | 578 | 580 | 575 | 575 | 575 | ||

26 | 566 | 566 | 569 | 566 | 566 | 566 | 565 | ||

29 | 575 | 575 | 572 | 575 | 575 | 573 | 577 | ||

C-7 | N50 | 3 | 590 | 512 | 590 | 597 | 595 | 601 | 599 |

6 | 571 | 571 | 571 | 571 | 571 | 571 | 571 | ||

12 | 606 | 600 | 609 | 603 | 608 | 606 | 612 | ||

26 | 579 | 574 | 578 | 578 | 579 | 580 | 581 | ||

36 | 630 | 630 | 630 | 630 | 630 | 630 | 630 | ||

C-8 | N50 | 4 | 644 | 640 | 642 | 645 | 642 | 648 | 645 |

9 | 571 | 571 | 571 | 571 | 571 | 571 | 571 | ||

15 | 580 | 573 | 577 | 589 | 580 | 583 | 585 | ||

40 | 562 | 562 | 565 | 562 | 562 | 562 | 561 | ||

47 | 562 | 562 | 572 | 562 | 561 | 565 | 562 | ||

C-9 | N60 | 5 | 3362 | 3299 | 3370 | 3362 | 3362 | 3368 | 3372 |

10 | 3114 | 3107 | 3112 | 3119 | 3117 | 3114 | 3123 | ||

23 | 3476 | 3450 | 3479 | 3480 | 3475 | 3475 | 3477 | ||

29 | 3061 | 3025 | 3069 | 3061 | 3069 | 3061 | 3060 | ||

40 | 2786 | 2786 | 2786 | 2786 | 2786 | 2785 | 2786 | ||

C-10 | N60 | 6 | 2756 | 2756 | 2756 | 2756 | 2756 | 2756 | 2756 |

14 | 3394 | 3390 | 3393 | 3399 | 3399 | 3394 | 3400 | ||

20 | 3441 | 3441 | 3441 | 3441 | 3441 | 3440 | 3441 | ||

32 | 3398 | 3398 | 3397 | 3400 | 3398 | 3398 | 3401 | ||

41 | 3514 | 3504 | 3510 | 3520 | 3513 | 3514 | 3520 | ||

C-11 | N60 | 2 | 3870 | 3870 | 3870 | 3870 | 3870 | 3870 | 3870 |

8 | 3598 | 3598 | 3598 | 3598 | 3598 | 3598 | 3598 | ||

14 | 3703 | 3700 | 3705 | 3706 | 3708 | 3711 | 3705 | ||

20 | 3646 | 3646 | 3646 | 3646 | 3649 | 3646 | 3646 | ||

32 | 3642 | 3639 | 3639 | 3641 | 3645 | 3652 | 3641 | ||

C-12 | N60 | 3 | 2721 | 2720 | 2730 | 2721 | 2725 | 2729 | 2725 |

12 | 2988 | 2976 | 2988 | 2988 | 2990 | 2998 | 2992 | ||

19 | 2988 | 2988 | 2988 | 2987 | 2988 | 2998 | 2988 | ||

23 | 3432 | 3427 | 3431 | 3432 | 3432 | 3432 | 3431 | ||

34 | 3197 | 3197 | 3196 | 3197 | 3197 | 3197 | 3197 | ||

C-13 | N60 | 1 | 3244 | 3243 | 3244 | 3244 | 3244 | 3249 | 3249 |

4 | 2988 | 2969 | 2989 | 2985 | 2996 | 2987 | 3001 | ||

19 | 3136 | 3125 | 3141 | 3135 | 3139 | 3136 | 3139 | ||

29 | 3103 | 3100 | 3105 | 3103 | 3107 | 3102 | 3103 | ||

40 | 2834 | 2834 | 2834 | 2834 | 2834 | 2834 | 2834 | ||

C-14 | N60 | 5 | 3293 | 3275 | 3291 | 3299 | 3299 | 3300 | 3299 |

9 | 2959 | 2945 | 2959 | 2959 | 2969 | 2969 | 2959 | ||

15 | 3063 | 2972 | 3063 | 3056 | 3058 | 3069 | 3060 | ||

30 | 2935 | 2873 | 2928 | 2934 | 2934 | 2934 | 2934 | ||

43 | 2963 | 2965 | 2962 | 2963 | 2963 | 2963 | 2963 | ||

C-15 | N60 | 6 | 3031 | 2997 | 3031 | 3040 | 3030 | 3035 | 3035 |

15 | 3383 | 3383 | 3383 | 3383 | 3383 | 3383 | 3383 | ||

26 | 3969 | 3875 | 3978 | 3979 | 3969 | 3969 | 3979 | ||

35 | 3496 | 3492 | 3499 | 3496 | 3496 | 3500 | 3500 | ||

44 | 3475 | 3328 | 3481 | 3479 | 3472 | 3475 | 3473 |

**Table 4.**Summary of standard deviation, least error, and best time obtained by proposed Artificial bee colony with Nelder Mead method (NM-ABC) and competitor methods.

Case | Type | Instance | NM-ABC | M_{1} | M_{2} | M_{3} | M_{4} | M_{5} | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

SD | LER | Best Time | SD | LER | Best Time | SD | LER | Best Time | SD | LER | Best Time | SD | LER | Best Time | SD | LER | Best Time | |||

C-1 | N25 | 1 | 1.36 | 0 | 13.14 | 1.28 | 0 | 17.71 | 1.17 | 0 | 23.14 | 1.17 | 0 | 20.64 | 1.25 | 1 | 32 | 1.18 | 0 | 24.87 |

C-1 | N25 | 7 | 2.40 | 4 | 11.09 | 1.27 | 1 | 13.82 | 1.45 | 1 | 21.09 | 1.45 | 1 | 26.09 | 1.85 | 1 | 19.14 | 1.37 | 1 | 19.91 |

C-1 | N25 | 12 | 1.9 | 0 | 0.01 | 0.00 | 1 | 10.05 | 0.00 | 0 | 10.01 | 0.00 | 0 | 20.51 | 0.00 | 0 | 0.27 | 0.00 | 0 | 2.11 |

C-1 | N25 | 19 | 1.80 | 0 | 0.5 | 1.47 | 0 | 10.71 | 1.96 | 0 | 0.5 | 1.96 | 1 | 12 | 1.79 | 0 | 1.5 | 1.85 | 0 | 3.70 |

C-1 | N25 | 25 | 1.90 | 1 | 0.9 | 1.96 | 1 | 11.01 | 2.44 | 2 | 30.9 | 2.44 | 1 | 11.05 | 1.47 | 0 | 0.76 | 1.73 | 0 | 2.83 |

C-2 | N25 | 2 | 3.70 | 5 | 19.36 | 2.36 | 3 | 21.94 | 2.32 | 5 | 22.5 | 2.32 | 2 | 23.18 | 2.04 | 1 | 24.09 | 1.91 | 1 | 5.28 |

C-2 | N25 | 5 | 1.76 | 0 | 1.54 | 1.96 | 0 | 12.33 | 2.15 | 1 | 23.44 | 2.24 | 1 | 34.21 | 2.16 | 1 | 35.55 | 2.86 | 2 | 6.50 |

C-2 | N25 | 9 | 1.85 | 0 | 0.05 | 1.99 | 0 | 12.19 | 1.87 | 0 | 30 | 1.87 | 0 | 20.03 | 1.95 | 0 | 15.01 | 1.81 | 1 | 13.63 |

C-2 | N25 | 15 | 4.43 | 7 | 21.2 | 2.71 | 3 | 21.71 | 2.49 | 4 | 29.2 | 2.34 | 3 | 12.8 | 2.10 | 0 | 23.6 | 2.14 | 4 | 4.84 |

C-2 | N25 | 27 | 1.55 | 0 | 0.04 | 1.92 | 0 | 0.36 | 1.57 | 0 | 25 | 1.57 | 0 | 31.52 | 1.96 | 0 | 32.26 | 2.12 | 0 | 3.42 |

C-3 | N25 | 1 | 2.52 | 2 | 1.2 | 1.70 | 0 | 1.67 | 1.91 | 6 | 12 | 2.14 | 4 | 22.6 | 2.14 | 1 | 23.3 | 1.89 | 1 | 4.60 |

C-3 | N25 | 3 | 1.60 | 0 | 0.34 | 1.62 | 0 | 0.43 | 1.87 | 0 | 10.34 | 1.87 | 0 | 10.51 | 1.45 | 2 | 10.6 | 2.10 | 0 | 2.18 |

C-3 | N25 | 16 | 1.79 | 2 | 23.07 | 1.91 | 1 | 30.32 | 2.06 | 0 | 28 | 1.91 | 3 | 21.54 | 1.62 | 3 | 50.77 | 1.92 | 0 | 49.14 |

C-3 | N25 | 27 | 2.84 | 2 | 15.73 | 1.64 | 0 | 20.58 | 1.99 | 0 | 20 | 2.10 | 1 | 27.87 | 1.89 | 0 | 33.93 | 2.50 | 1 | 33.47 |

C-3 | N25 | 35 | 4.35 | 6 | 3.28 | 2.69 | 3 | 4.32 | 2.95 | 3 | 34.3 | 2.79 | 2 | 25.94 | 1.73 | 0 | 7.27 | 1.56 | 0 | 8.40 |

C-4 | N25 | 5 | 9.48 | 28 | 32.13 | 4.86 | 12 | 40.56 | 5.42 | 6 | 34 | 4.32 | 3 | 50.07 | 2.33 | 1 | 59.03 | 2.94 | 3 | 58.47 |

C-4 | N25 | 10 | 5.61 | 11 | 12.67 | 4.57 | 9 | 19.34 | 4.27 | 4 | 29 | 3.82 | 7 | 35.34 | 2.24 | 3 | 46.67 | 3.53 | 5 | 42.74 |

C-4 | N25 | 25 | 5.82 | 14 | 22.5 | 3.01 | 5 | 30.5 | 3.52 | 8 | 33.6 | 3.35 | 3 | 44.85 | 1.96 | 2 | 56.03 | 2.69 | 4 | 53.17 |

C-4 | N25 | 38 | 1.74 | 0 | 17.29 | 1.95 | 0 | 22.05 | 1.95 | 0 | 19.34 | 1.95 | 0 | 27.99 | 2.07 | 1 | 33.33 | 2.44 | 5 | 33.45 |

C-4 | N25 | 41 | 1.89 | 0 | 1.54 | 1.67 | 0 | 2.38 | 1.66 | 0 | 3.67 | 1.66 | 0 | 4.44 | 1.66 | 3 | 5.89 | 1.79 | 0 | 6.77 |

C-5 | N25 | 7 | 3.12 | 4 | 17.32 | 2.45 | 6 | 10.64 | 2.50 | 6 | 14.26 | 2.27 | 0 | 17.92 | 1.83 | 0 | 23.22 | 1.85 | 3 | 22.37 |

C-5 | N25 | 11 | 1.64 | 0 | 0.07 | 2.11 | 0 | 9.69 | 1.90 | 0 | 2.89 | 1.90 | 0 | 2.93 | 2.19 | 1 | 4.35 | 2.15 | 0 | 5.10 |

C-5 | N25 | 30 | 2.30 | 2 | 15.3 | 2.24 | 1 | 23.64 | 2.24 | 8 | 4.3 | 2.24 | 4 | 6.95 | 1.72 | 0 | 2.36 | 1.83 | 1 | 9.41 |

C-5 | N25 | 42 | 1.97 | 0 | 3.29 | 1.95 | 4 | 8.32 | 2.18 | 0 | 4.26 | 2.18 | 0 | 5.91 | 2.24 | 0 | 7.21 | 1.85 | 0 | 8.36 |

C-5 | N25 | 47 | 2.34 | 1 | 14.2 | 2.20 | 1 | 18.28 | 2.62 | 5 | 65 | 2.62 | 1 | 67.1 | 2.33 | 1 | 98.55 | 2.02 | 1 | 81.70 |

C-6 | N50 | 1 | 6.14 | 6 | 29.4 | 6.46 | 4 | 15.95 | 4.82 | 2 | 29 | 4.65 | 2 | 33.7 | 4.58 | 0 | 45.85 | 5.23 | 2 | 41.10 |

C-6 | N50 | 4 | 4.49 | 1 | 16.1 | 4.68 | 0 | 9.31 | 3.88 | 1 | 33.97 | 3.88 | 0 | 27.02 | 5.17 | 1 | 22.48 | 3.69 | 8 | 21.41 |

C-6 | N50 | 12 | 5.06 | 4 | 18.45 | 4.85 | 3 | 10.92 | 3.99 | 5 | 25.3 | 3.56 | 0 | 29.53 | 4.24 | 0 | 17.29 | 5.29 | 0 | 24.19 |

C-6 | N50 | 26 | 5.06 | 0 | 1.56 | 4.08 | 3 | 85.62 | 4.26 | 0 | 23.92 | 4.26 | 0 | 32.78 | 3.72 | 0 | 18.39 | 3.83 | 1 | 17.78 |

C-6 | N50 | 29 | 3.77 | 0 | 1.43 | 4.76 | 3 | 9.01 | 3.31 | 0 | 31.78 | 3.31 | 0 | 22.5 | 5.06 | 2 | 53.03 | 4.21 | 2 | 4.45 |

C-7 | N50 | 3 | 30.48 | 78 | 34.67 | 4.35 | 0 | 44.55 | 3.10 | 7 | 40.32 | 3.10 | 5 | 57.66 | 3.74 | 11 | 69.15 | 4.74 | 9 | 67.32 |

C-7 | N50 | 6 | 4.10 | 0 | 3.33 | 4.07 | 0 | 9.35 | 3.26 | 0 | 34.2 | 3.26 | 0 | 35.87 | 3.87 | 0 | 37.13 | 3.52 | 0 | 8.31 |

C-7 | N50 | 12 | 5.76 | 6 | 59.02 | 5.15 | 3 | 72.39 | 4.58 | 3 | 24.2 | 4.45 | 2 | 18.71 | 3.52 | 0 | 23.56 | 5.82 | 6 | 23.15 |

C-7 | N50 | 26 | 5.44 | 5 | 18.45 | 4.47 | 1 | 30.48 | 4.84 | 1 | 38.05 | 4.84 | 0 | 12.28 | 4.27 | 1 | 14.19 | 4.38 | 2 | 15.49 |

C-7 | N50 | 36 | 4.88 | 0 | 1.36 | 4.37 | 0 | 11.84 | 4.45 | 0 | 22 | 4.45 | 0 | 42.68 | 4.15 | 0 | 33.34 | 4.73 | 0 | 4.68 |

C-8 | N50 | 4 | 5.16 | 4 | 33.9 | 4.75 | 2 | 44.66 | 5.28 | 1 | 12.9 | 5.28 | 2 | 14.85 | 5.36 | 4 | 45.33 | 4.96 | 1 | 7.03 |

C-8 | N50 | 9 | 3.96 | 0 | 21.28 | 4.67 | 0 | 32.31 | 3.88 | 0 | 34.6 | 3.88 | 0 | 15.24 | 4.22 | 0 | 37.22 | 4.58 | 0 | 7.76 |

C-8 | N50 | 15 | 7.28 | 7 | 32.4 | 4.87 | 3 | 35.82 | 4.56 | 9 | 23.9 | 4.04 | 0 | 20.1 | 4.36 | 3 | 23.95 | 3.71 | 5 | 24.48 |

C-8 | N50 | 40 | 5.07 | 0 | 12.25 | 4.87 | 3 | 23.92 | 4.47 | 0 | 27.4 | 4.47 | 0 | 28.53 | 4.47 | 0 | 61.66 | 5.06 | 1 | 11.61 |

C-8 | N50 | 47 | 4.27 | 0 | 20.65 | 4.05 | 10 | 25.33 | 4.96 | 0 | 37.23 | 5.08 | 1 | 89.06 | 4.04 | 3 | 41.76 | 4.64 | 0 | 12.10 |

C-9 | N60 | 5 | 28.31 | 63 | 32.43 | 8.94 | 8 | 43.29 | 7.79 | 0 | 45.29 | 7.01 | 0 | 61.51 | 5.75 | 6 | 76.04 | 5.76 | 10 | 72.16 |

C-9 | N60 | 10 | 9.70 | 7 | 12.34 | 6.14 | 2 | 16.29 | 8.53 | 5 | 26.36 | 8.63 | 3 | 22.53 | 6.12 | 0 | 27.63 | 7.25 | 9 | 27.40 |

C-9 | N60 | 23 | 16.40 | 26 | 23.54 | 7.26 | 3 | 31.13 | 5.80 | 4 | 31.5 | 5.69 | 1 | 43.27 | 7.34 | 1 | 53.14 | 5.54 | 1 | 51.17 |

C-9 | N60 | 29 | 22.01 | 36 | 43.67 | 8.98 | 8 | 53.8 | 7.19 | 0 | 39.98 | 6.04 | 8 | 61.82 | 6.44 | 0 | 70.89 | 6.53 | 1 | 71.41 |

C-9 | N60 | 40 | 5.47 | 0 | 0.37 | 5.57 | 0 | 10.68 | 5.15 | 0 | 21.4 | 5.15 | 0 | 11.59 | 6.15 | 1 | 12.19 | 6.41 | 0 | 3.47 |

C-10 | N60 | 6 | 6.09 | 0 | 20.11 | 5.75 | 0 | 23.49 | 5.51 | 0 | 32.4 | 5.51 | 0 | 42.46 | 5.73 | 0 | 23.63 | 6.33 | 0 | 4.54 |

C-10 | N60 | 14 | 7.71 | 4 | 42.03 | 5.08 | 1 | 43.13 | 6.50 | 5 | 24.78 | 6.50 | 5 | 35.8 | 6.46 | 0 | 47.68 | 7.28 | 6 | 8.35 |

C-10 | N60 | 20 | 6.40 | 0 | 24.65 | 6.62 | 0 | 30.24 | 5.64 | 0 | 42 | 5.64 | 0 | 54.33 | 6.05 | 1 | 39.16 | 6.89 | 0 | 18.33 |

C-10 | N60 | 32 | 7.49 | 0 | 20.04 | 6.89 | 1 | 31.31 | 6.96 | 2 | 65.9 | 6.83 | 0 | 25.92 | 6.33 | 0 | 48.86 | 6.87 | 3 | 8.70 |

C-10 | N60 | 41 | 9.67 | 10 | 55.09 | 8.22 | 4 | 79.36 | 7.54 | 6 | 37.43 | 7.17 | 1 | 44.98 | 5.77 | 0 | 59.92 | 5.55 | 6 | 30.06 |

C-11 | N60 | 2 | 6.96 | 0 | 35.07 | 6.24 | 0 | 46.6 | 6.85 | 0 | 56.3 | 6.85 | 0 | 58.84 | 7.07 | 0 | 30.72 | 6.08 | 0 | 11.70 |

C-11 | N60 | 8 | 5.34 | 0 | 27.3 | 6.10 | 0 | 39.55 | 6.28 | 0 | 69.26 | 6.28 | 0 | 52.91 | 6.17 | 0 | 55.72 | 6.60 | 0 | 16.36 |

C-11 | N60 | 14 | 8.05 | 3 | 49.76 | 6.29 | 2 | 51.94 | 5.70 | 3 | 48.53 | 6.18 | 5 | 53.41 | 7.17 | 8 | 75.24 | 6.95 | 2 | 16.72 |

C-11 | N60 | 20 | 6.17 | 0 | 26.2 | 6.56 | 0 | 39.19 | 6.98 | 0 | 32.94 | 6.98 | 3 | 46.04 | 5.99 | 0 | 90.96 | 7.15 | 0 | 20.23 |

C-11 | N60 | 32 | 7.03 | 3 | 51.87 | 8.60 | 3 | 65.44 | 7.83 | 1 | 70.43 | 7.69 | 3 | 76.37 | 6.00 | 10 | 74.99 | 7.79 | 1 | 32.05 |

C-12 | N60 | 3 | 5.66 | 1 | 20.23 | 6.26 | 9 | 32.29 | 6.28 | 0 | 32.9 | 6.28 | 4 | 63.02 | 7.16 | 8 | 44.41 | 6.91 | 4 | 5.20 |

C-12 | N60 | 12 | 12.67 | 12 | 49.22 | 5.88 | 0 | 52.38 | 6.20 | 0 | 43.2 | 6.20 | 2 | 37.81 | 6.55 | 10 | 52.11 | 6.93 | 4 | 22.05 |

C-12 | N60 | 19 | 8.26 | 0 | 21.28 | 6.87 | 0 | 42.37 | 8.26 | 1 | 54.89 | 8.26 | 0 | 35.53 | 6.90 | 10 | 57.66 | 5.59 | 0 | 8.11 |

C-12 | N60 | 23 | 8.72 | 5 | 19.9 | 7.42 | 1 | 34.68 | 6.53 | 0 | 25.82 | 6.37 | 0 | 51.27 | 7.02 | 0 | 56.46 | 7.16 | 1 | 26.03 |

C-12 | N60 | 34 | 6.58 | 0 | 39 | 7.54 | 1 | 49.39 | 5.68 | 0 | 72 | 5.50 | 0 | 61.5 | 7.19 | 0 | 72.75 | 7.34 | 0 | 71.50 |

C-13 | N60 | 1 | 7.73 | 1 | 33.11 | 6.36 | 0 | 44.33 | 6.13 | 0 | 45.19 | 6.13 | 0 | 56.75 | 6.08 | 5 | 58.56 | 6.96 | 5 | 9.38 |

C-13 | N60 | 4 | 12.77 | 19 | 47.93 | 9.10 | 1 | 52.95 | 7.89 | 3 | 50.45 | 7.61 | 8 | 49.42 | 7.12 | 1 | 35.16 | 7.11 | 13 | 35.11 |

C-13 | N60 | 19 | 9.93 | 11 | 44.43 | 9.06 | 5 | 49.08 | 7.30 | 1 | 39.29 | 6.00 | 3 | 56.51 | 6.00 | 0 | 62.54 | 8.27 | 3 | 31.96 |

C-13 | N60 | 29 | 8.06 | 3 | 35.9 | 7.35 | 2 | 38.85 | 7.09 | 0 | 61.45 | 6.78 | 4 | 64.4 | 7.44 | 1 | 40.65 | 6.58 | 0 | 18.29 |

C-13 | N60 | 40 | 6.54 | 0 | 32.71 | 7.26 | 0 | 44.96 | 5.95 | 0 | 65.23 | 5.95 | 0 | 57.09 | 7.49 | 0 | 78.77 | 5.73 | 0 | 9.73 |

C-14 | N60 | 5 | 13.00 | 18 | 18.43 | 7.87 | 2 | 23.5 | 6.45 | 6 | 43.9 | 6.80 | 6 | 53.12 | 8.35 | 7 | 35.46 | 7.08 | 6 | 39.50 |

C-14 | N60 | 9 | 11.52 | 14 | 10.05 | 7.49 | 0 | 20.6 | 7.43 | 0 | 52.56 | 7.43 | 10 | 42.59 | 7.15 | 10 | 23.85 | 6.85 | 0 | 4.70 |

C-14 | N60 | 15 | 28.15 | 91 | 49.02 | 7.63 | 0 | 60.4 | 6.61 | 7 | 53 | 7.44 | 5 | 77.51 | 7.57 | 6 | 79.63 | 7.21 | 3 | 89.71 |

C-14 | N60 | 30 | 22.13 | 62 | 47.34 | 10.30 | 7 | 60.3 | 6.05 | 1 | 52.6 | 5.08 | 1 | 76.27 | 6.05 | 1 | 90.74 | 6.38 | 1 | 88.39 |

C-14 | N60 | 43 | 6.02 | 2 | 37.1 | 6.56 | 1 | 34 | 6.90 | 0 | 49 | 6.73 | 0 | 28.55 | 5.76 | 0 | 34.28 | 5.22 | 0 | 34.15 |

C-15 | N60 | 6 | 17.40 | 34 | 20.47 | 13.08 | 0 | 25.47 | 10.86 | 9 | 59.9 | 7.59 | 1 | 30.14 | 5.71 | 4 | 34.97 | 7.24 | 4 | 35.72 |

C-15 | N60 | 15 | 5.85 | 0 | 24.23 | 6.17 | 0 | 35.86 | 5.09 | 0 | 36.9 | 5.09 | 0 | 39.02 | 5.81 | 0 | 51.41 | 6.51 | 0 | 12.00 |

C-15 | N60 | 26 | 43.30 | 94 | 79.38 | 12.15 | 9 | 100.5 | 9.61 | 10 | 85.34 | 6.57 | 0 | 125.03 | 5.18 | 0 | 147.86 | 6.73 | 10 | 143.70 |

C-15 | N60 | 35 | 7.13 | 4 | 51.28 | 7.05 | 3 | 64.02 | 6.14 | 0 | 59.25 | 6.14 | 0 | 79.89 | 4.77 | 4 | 89.2 | 6.18 | 4 | 13.34 |

C-15 | N60 | 44 | 57.34 | 147 | 84.3 | 6.45 | 6 | 96.95 | 7.47 | 4 | 87 | 7.89 | 3 | 87.15 | 6.81 | 0 | 88.58 | 6.16 | 2 | 97.75 |

**Table 5.**Comparison of the number of #both feasible and #feasible solutions obtained by the proposed Artificial bee colony with Nelder Mead method (NM-ABC) and competitor’s methods.

Case | Type | Instances | NM-ABC | M_{1} | M_{2} | M_{3} | M_{4} | M_{5} | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

#BFS | #FS | #BFS | #FS | #BFS | #FS | #BFS | #FS | #BFS | #FS | #BFS | #FS | |||

1 | N25 | 1, 7, 12, 19, 25 | 3 | 5 | 2 | 4 | 3 | 4 | 2 | 4 | 2 | 4 | 4 | 4 |

2 | N25 | 2, 5, 9, 15, 27 | 3 | 5 | 3 | 3 | 2 | 3 | 2 | 3 | 3 | 5 | 1 | 4 |

3 | N25 | 1, 3, 16, 27, 35 | 1 | 4 | 3 | 5 | 3 | 4 | 1 | 2 | 1 | 3 | 3 | 5 |

4 | N25 | 5, 10, 25, 38, 41 | 2 | 5 | 2 | 5 | 2 | 3 | 2 | 5 | 0 | 4 | 1 | 3 |

5 | N25 | 7, 11, 30, 42, 47 | 2 | 5 | 1 | 3 | 2 | 2 | 3 | 4 | 3 | 5 | 2 | 4 |

6 | N50 | 1, 4, 12, 26, 29 | 2 | 5 | 0 | 2 | 2 | 3 | 4 | 5 | 3 | 5 | 1 | 2 |

7 | N50 | 3, 6,12, 26, 36 | 2 | 5 | 3 | 4 | 2 | 4 | 2 | 2 | 3 | 3 | 2 | 2 |

8 | N50 | 4, 9, 15, 40, 47 | 3 | 5 | 1 | 3 | 3 | 3 | 3 | 5 | 2 | 2 | 2 | 3 |

9 | N50 | 5, 10, 23, 29, 40 | 1 | 5 | 1 | 2 | 3 | 3 | 2 | 3 | 2 | 4 | 1 | 2 |

10 | N50 | 6,14, 20, 32, 41 | 3 | 5 | 3 | 5 | 2 | 2 | 3 | 4 | 4 | 5 | 2 | 2 |

11 | N60 | 2, 8, 14, 20, 32 | 3 | 5 | 3 | 4 | 3 | 4 | 2 | 2 | 3 | 3 | 3 | 4 |

12 | N60 | 3, 12, 19, 23, 34 | 2 | 5 | 2 | 4 | 4 | 5 | 3 | 3 | 2 | 2 | 2 | 3 |

13 | N60 | 1, 4, 19, 29, 40 | 1 | 5 | 2 | 2 | 3 | 5 | 2 | 2 | 2 | 4 | 2 | 2 |

14 | N60 | 5, 9, 15, 30, 43 | 0 | 4 | 2 | 5 | 2 | 4 | 1 | 3 | 1 | 2 | 2 | 4 |

15 | N60 | 6, 15, 26, 35, 44 | 1 | 5 | 2 | 2 | 4 | 3 | 2 | 4 | 2 | 3 | 2 | 2 |

**Table 6.**Comparison and assessment of ABT, ASD, and ASP obtained by proposed algorithm NM-ABC and competitor methods.

Case | Type | Instances | NM-ABC | M_{1} | M_{2} | M_{3} | M_{4} | M_{5} | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

ABT | ASD | ASP | ABT | ASD | ASP | ABT | ASD | ASP | ABT | ASD | ASP | ABT | ASD | ASP | ABT | ASD | ASP | |||

1 | N25 | 1, 7, 12, 19, 25 | 5.13 | 1.87 | 100 | 12.66 | 1.2 | 80 | 17.13 | 1.4 | 80 | 18.06 | 1.4 | 80 | 10.73 | 1.27 | 80 | 18.68 | 1.23 | 80 |

2 | N25 | 2, 5, 9, 15, 27 | 8.44 | 2.66 | 100 | 13.71 | 2.19 | 60 | 26.03 | 2.08 | 60 | 24.35 | 2.07 | 60 | 26.1 | 2.04 | 100 | 24.73 | 2.17 | 80 |

3 | N25 | 1, 3, 16, 27, 35 | 8.72 | 2.62 | 80 | 11.46 | 1.91 | 100 | 20.93 | 2.16 | 80 | 21.69 | 2.16 | 40 | 25.17 | 1.77 | 60 | 31.56 | 1.99 | 100 |

4 | N25 | 5, 10, 25, 38, 41 | 17.23 | 4.91 | 100 | 22.97 | 3.21 | 100 | 23.92 | 3.36 | 60 | 32.54 | 3.02 | 100 | 40.19 | 2.05 | 80 | 38.92 | 2.68 | 60 |

5 | N25 | 7, 11, 30, 42, 47 | 10.04 | 2.27 | 100 | 14.11 | 2.19 | 60 | 18.14 | 2.29 | 40 | 20.16 | 2.24 | 80 | 27.14 | 2.06 | 100 | 31.39 | 1.94 | 80 |

6 | N50 | 1, 4, 12, 26, 29 | 13.39 | 4.9 | 100 | 26.16 | 4.97 | 40 | 28.79 | 4.05 | 60 | 29.11 | 3.93 | 100 | 31.41 | 4.55 | 100 | 33.19 | 4.45 | 40 |

7 | N50 | 3, 6,12, 26, 36 | 23.37 | 10.13 | 100 | 33.72 | 4.48 | 80 | 31.75 | 4.05 | 80 | 33.44 | 4.02 | 40 | 35.47 | 3.91 | 60 | 37.79 | 4.64 | 40 |

8 | N50 | 4, 9, 15, 40, 47 | 24.1 | 5.15 | 100 | 32.41 | 4.64 | 60 | 27.21 | 4.63 | 60 | 33.56 | 4.55 | 100 | 41.98 | 4.49 | 40 | 44.6 | 4.59 | 60 |

9 | N50 | 5, 10, 23, 29, 40 | 22.47 | 16.38 | 100 | 31.04 | 7.38 | 40 | 32.91 | 6.89 | 60 | 40.14 | 6.5 | 60 | 47.98 | 6.36 | 80 | 45.12 | 6.3 | 40 |

10 | N50 | 6,14, 20, 32, 41 | 32.38 | 7.47 | 100 | 41.51 | 6.51 | 100 | 40.5 | 6.43 | 40 | 40.7 | 6.33 | 80 | 43.85 | 6.07 | 100 | 45.2 | 6.58 | 40 |

11 | N60 | 2, 8, 14, 20, 32 | 38.04 | 6.71 | 100 | 48.54 | 6.76 | 80 | 55.49 | 6.73 | 80 | 57.51 | 6.8 | 40 | 65.53 | 6.48 | 60 | 59.41 | 6.91 | 80 |

12 | N60 | 3, 12, 19, 23, 34 | 29.93 | 8.38 | 100 | 42.22 | 6.79 | 80 | 45.76 | 6.59 | 100 | 49.83 | 6.52 | 60 | 56.68 | 6.96 | 40 | 60.58 | 6.79 | 60 |

13 | N60 | 1, 4, 19, 29, 40 | 38.82 | 9.01 | 100 | 46.03 | 7.83 | 40 | 52.32 | 6.87 | 100 | 56.83 | 6.49 | 40 | 55.14 | 6.83 | 80 | 58.89 | 6.93 | 40 |

14 | N60 | 5, 9, 15, 30, 43 | 32.39 | 16.16 | 80 | 39.76 | 7.97 | 100 | 50.21 | 6.69 | 80 | 55.61 | 6.7 | 60 | 52.79 | 6.98 | 40 | 61.29 | 6.55 | 80 |

15 | N60 | 6, 15, 26, 35, 44 | 51.93 | 26.2 | 100 | 64.56 | 8.98 | 40 | 65.68 | 7.83 | 60 | 72.25 | 6.66 | 80 | 82.4 | 5.66 | 60 | 99.3 | 6.56 | 40 |

**Table 7.**Comparison and assessment of ACR and AGap obtained by proposed algorithm NM-ABC and competitor methods.

Case | Type | Instances | NM-ABC | M_{1} | M_{2} | M_{3} | M_{4} | M_{5} | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

ACR | AGap | ACR | AGap | ACR | AGap | ACR | AGap | ACR | AGap | ACR | AGap | |||

1 | N25 | 1, 7, 12, 19, 25 | 1.00 | 0.33 | 0.20 | 0.07 | 0.20 | 0.07 | 0.20 | 0.07 | 0.00 | 0.00 | −0.20 | −0.07 |

2 | N25 | 2, 5, 9, 15, 27 | 2.40 | 0.83 | −1.20 | −0.42 | −1.60 | −0.55 | −0.80 | −0.28 | 0.40 | 0.14 | 0.00 | 0.00 |

3 | N25 | 1, 3, 16, 27, 35 | 1.60 | 0.49 | 0.80 | 0.25 | −0.60 | −0.18 | −1.20 | −0.37 | −0.80 | −0.25 | 0.40 | 0.12 |

4 | N25 | 5, 10, 25, 38, 41 | 10.60 | 3.55 | 5.20 | 1.74 | −0.40 | −0.13 | 2.60 | 0.87 | 0.80 | 0.27 | −1.40 | −0.47 |

5 | N25 | 7, 11, 30, 42, 47 | 1.40 | 0.47 | −1.60 | −0.53 | −3.80 | −1.27 | −0.60 | −0.20 | 0.40 | 0.13 | −0.20 | −0.07 |

6 | N50 | 1, 4, 12, 26, 29 | 2.20 | 0.38 | −1.40 | −0.24 | −1.20 | −0.20 | 0.40 | 0.07 | 0.60 | 0.10 | −2.20 | −0.38 |

7 | N50 | 3, 6,12, 26, 36 | 17.80 | 2.99 | −0.40 | −0.07 | −0.60 | −0.10 | −1.40 | −0.24 | −2.40 | −0.40 | −3.40 | −0.57 |

8 | N50 | 4, 9, 15, 40, 47 | 2.20 | 0.38 | −1.60 | −0.27 | −2.00 | −0.34 | 0.60 | 0.10 | −2.00 | −0.34 | −1.00 | −0.17 |

9 | N50 | 5, 10, 23, 29, 40 | 26.40 | 0.84 | −3.40 | −0.11 | −1.80 | −0.06 | −2.00 | −0.06 | −0.80 | −0.03 | −3.80 | −0.12 |

10 | N50 | 6,14, 20, 32, 41 | 2.80 | 0.08 | 1.20 | 0.04 | −2.60 | −0.08 | −0.80 | −0.02 | 0.20 | 0.01 | −3.00 | −0.09 |

11 | N60 | 2, 8, 14, 20, 32 | 1.20 | 0.03 | 0.20 | 0.01 | −0.40 | −0.01 | −2.20 | −0.06 | −3.60 | −0.10 | −0.20 | −0.01 |

12 | N60 | 3, 12, 19, 23, 34 | 3.60 | 0.12 | −1.40 | −0.05 | 0.20 | 0.01 | −1.20 | −0.04 | −5.60 | −0.18 | −1.40 | −0.05 |

13 | N60 | 1, 4, 19, 29, 40 | 6.80 | 0.22 | −1.60 | −0.05 | 0.80 | 0.03 | −3.00 | −0.10 | −0.60 | −0.02 | −4.20 | −0.14 |

14 | N60 | 5, 9, 15, 30, 43 | 36.60 | 1.20 | 2.00 | 0.07 | 0.40 | 0.01 | −2.00 | −0.07 | −4.40 | −0.14 | −0.40 | −0.01 |

15 | N60 | 6, 15, 26, 35, 44 | 55.80 | 1.61 | −3.60 | −0.10 | −4.60 | −0.13 | 0.80 | 0.02 | −1.60 | −0.05 | −3.20 | −0.09 |

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**MDPI and ACS Style**

Muniyan, R.; Ramalingam, R.; Alshamrani, S.S.; Gangodkar, D.; Dumka, A.; Singh, R.; Gehlot, A.; Rashid, M. Artificial Bee Colony Algorithm with Nelder–Mead Method to Solve Nurse Scheduling Problem. *Mathematics* **2022**, *10*, 2576.
https://doi.org/10.3390/math10152576

**AMA Style**

Muniyan R, Ramalingam R, Alshamrani SS, Gangodkar D, Dumka A, Singh R, Gehlot A, Rashid M. Artificial Bee Colony Algorithm with Nelder–Mead Method to Solve Nurse Scheduling Problem. *Mathematics*. 2022; 10(15):2576.
https://doi.org/10.3390/math10152576

**Chicago/Turabian Style**

Muniyan, Rajeswari, Rajakumar Ramalingam, Sultan S. Alshamrani, Durgaprasad Gangodkar, Ankur Dumka, Rajesh Singh, Anita Gehlot, and Mamoon Rashid. 2022. "Artificial Bee Colony Algorithm with Nelder–Mead Method to Solve Nurse Scheduling Problem" *Mathematics* 10, no. 15: 2576.
https://doi.org/10.3390/math10152576