# Multi-Phase and Integrated Multi-Objective Cyclic Operating Room Scheduling Based on an Improved NSGA-II Approach

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Multiphase and Integrated Multi-Objective Linear Programming Model

#### 2.1. Problem Description

#### 2.2. Operating Room Resource Allocation Model (Phase 1)

#### 2.3. Multiobjective Cyclic Master Surgery Schedule (MSS) Model (Phase 2)

**Step 1.**Because only elective surgeries are considered here, the SSs in a hospital get all of the surgeries that need to be conducted in the next period (three days), and the surgeons mark the patients’ deadlines for the surgeries and the standard duration of the surgeries through patient severity and the results of the examinations (so that the initial surgery pool-L

_{1}in the schedule is obtained).

**Step 2.**First of all, the scheme needs to pick out the surgeries from the initial surgery pool-L

_{1}on the first day and then arrange a suitable OR and surgical sequence for them. The process uses an integrated multi-objective model with four specific indicators selected: (1) the number of surgeries arranged every day, (2) the total OR overtime, (3) the total OR idle time, and (4) the equilibrium rate between ORs in order to select the surgeries and make the specific surgery schedule.

**Step 3.**When surgeries are scheduled on the first day, they are removed from the surgery pool-L

_{1}, and the deadlines for unscheduled surgeries are reduced by one day, $d{d}_{i}^{m}=d{d}_{i}^{m}-1$. The unselected surgeries form a new surgery pool-L

_{2}. The next two days in the period cycle through the above steps. Through the above three steps in the daily cycle, one can get a cyclic master surgery schedule.

## 3. Solution Approaches

#### 3.1. Goal Programming Approach (GPA)

#### 3.1.1. Determining the Priority of the Objectives via AHP

_{ij}represents the relative importance scale between the two indicators. The relative importance scales are mainly based on 1 to 9 and their reciprocals as a scale to reflect their importance. The method of scaling is shown in Table 3.

_{i}of these four indicators is obtained.

_{1}> P

_{2}> P

_{3}> P

_{4}.

#### 3.1.2. Transforming Models via GPA

#### 3.2. Improved Nondominated Sorting Genetic Algorithm II (NSGA-II)

_{1}and f

_{2}, we define the solution x

_{1}to dominate x

_{2}as follows: $\left[{f}_{1}\left({x}_{1}\right)\ge {f}_{1}\left({x}_{2}\right)\wedge {f}_{2}\left({x}_{1}\right)>{f}_{2}\left({x}_{2}\right)\right]\vee \left[{f}_{1}\left({x}_{1}\right)>{f}_{1}\left({x}_{2}\right)\wedge {f}_{2}\left({x}_{1}\right)\ge {f}_{2}\left({x}_{2}\right)\right]$. If all solutions are not dominated by any other solutions in X, then this set constitutes the Pareto optimal front. However, in general, it can only be local Pareto optimality, and there is no guarantee that global Pareto optimality will be obtained. We use the following definitions to describe this more clearly:

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

#### 3.2.1. Initial Population

#### 3.2.2. Non-Dominant Sorting and Crowding Distance Calculation

#### 3.2.3. Selection, Crossover, and Mutation

#### 3.2.4. Generating New Populations (Elite Strategy)

#### 3.2.5. Evaluation Function Stage

## 4. Case Study and Computational Results

#### 4.1. Case 1: The Exact Algorithm (GPA)

#### 4.1.1. Computational Results of Phases 1 and 2

**Phase 1.**According to the data, for the next 3 days, the available time of 15 ORs in the hospital was 360 h, and the planned surgery time of 10 SSs was 381.32 h, exceeding the 360 h that the ORs could provide. The total minimum demand was 360 h, which was equal to the total OR time of 360 h, which could meet the minimum demand of the SSs, so these data were feasible.

**Phase 2.**In our case study, 10 instances were generated corresponding to 10 different SSs based on Phase 1. As in the above, OR time allocation was completed in Phase 1. On this basis, the 10 instances are listed in Table 8.

_{1}(patient list) and schedule them on the first day. Then the remaining surgeries constituted a new surgical pool L

_{2}. After that, we selected surgeries from L

_{2}and arranged them on the second day according to the first day’s schedule, and so on, so as to realize the essence of the cyclic schedule. Our scheduling period was 3 days. All surgical cases we considered were elective and the surgical duration was predicted by surgeons, so this schedule would take a three-day cycle.

#### 4.1.2. Comparison and Discussion of GPA Solutions with Real Schedules

#### 4.2. Case 2: The Heuristic Algorithm (Improved NSGA-II)

#### 4.2.1. Data Description, Experiment Design, and Model Coding

#### 4.2.2. Experiment Results and Analysis

#### 4.2.3. Comparison and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Scheduling Gantt chart in Instance 10 via Goal Programming Approach (GPA) (2 allocated ORs).

**Figure 12.**Comparison of the three objective values of the five non-dominated solutions of (8, 100, 1).

Symbol | Description |
---|---|

M | set of surgical specialties |

m | index for surgical specialties in M |

T | scheduling period |

K | set of ORs |

H | normal open hours per OR per day |

Max_{k} | maximum overtime allowed per day per OR |

RT | total time required by each SS during the scheduling period |

rt_{m} | time required by the SS m during the scheduling period |

profit_{m} | surgical benefit per unit time of the SS m |

cost_{m} | the overtime cost of the SS m |

alloc_{m} | the number of ORs assigned to the SS m |

Min_{m} | the minimum OR time requirement for the SS m during the scheduling period |

Symbol | Description |
---|---|

I^{m} | set of surgeries in SS m |

i^{m} | index for surgery in I^{m} |

K | set of ORs |

k | index for OR in K |

N^{k} | the number of surgeries scheduled in OR k |

n^{k} | the nth surgery in OR k |

$tim{e}_{i}^{m}$ | the duration of surgery i^{m} |

$d{d}_{i}^{m}$ | the deadline of surgery i^{m} |

T_{normal} | the normal working hours in the OR |

Topen_{k} | the open hours in OR k for one day |

Tover | the overtime in all ORs during the day |

Tover_{k} | the overtime of OR k |

Num | the total number of surgeries scheduled during the day |

Scale | Meaning |
---|---|

1 | the same importance compared to two factors |

3 | one is slightly more important than the other |

5 | one is significantly more important than the other |

7 | one is obviously strongly more important than the other |

9 | one is extremely more important than the other |

2, 4, 6, 8 | the median value of the above two adjacent judgments |

1, 1/3, 1/5, etc. | the judgment comparing the factors i with j is a_{ij}, the judgment of the factor j, and i is a_{ji} = 1/a_{ij} |

Objectives | (1) | (2) | (3) | (4) | Relative Importance |
---|---|---|---|---|---|

the number of surgeries scheduled per day | 1 | 3 | 5 | 6 | 0.5761 |

OR overtime | 1/3 | 1 | 2 | 3 | 0.2224 |

OR idle time | 1/5 | 1/2 | 1 | 2 | 0.1251 |

OR equalization rate | 1/6 | 1/3 | 1/2 | 1 | 0.0764 |

Algorithm: Construction of IniPop |

Data: N |

Result: IniPop of N individuals |

i: =0; |

IniPop = Φ; |

While i! = N do |

/* imposed the constraints */ |

/* find the chromosomes randomly */ |

Find a chromosome_{i} as a feasible solution (or suboptimal) satisfying the constraints; |

IniPop = IniPop ∪ chromosome_{i}; |

i: = i+1; |

end |

Surgical Specialty | Surgery Time | Income per Unit | Overtime Cost | Number of ORs (New) | |
---|---|---|---|---|---|

Total Time (h) | Minimum Time (h) | ||||

Gynecology | 36.11 | 33 | 978 | 600 | 1 |

Otorhinolaryngology | 26.41 | 22 | 547 | 342 | 1 |

Hepatobiliary | 43.29 | 38 | 324 | 212 | 1 |

Orthopedics | 27.73 | 23 | 923 | 322 | 1 |

Stomatology | 30.11 | 26 | 567 | 122 | 1 |

Urology | 29.01 | 28 | 1200 | 860 | 1 |

General surgery | 30.78 | 29 | 300 | 120 | 1 |

Neurosurgery | 56.63 | 54 | 250 | 100 | 3 |

Pediatrics | 48.26 | 46 | 344 | 200 | 2 |

Oncology | 52.99 | 50 | 134 | 100 | 3 |

Surgical Specialty | Surgery Time | Income per Unit | Overtime Cost | Number of ORs (New) | |
---|---|---|---|---|---|

Total Time (h) | Minimum Time (h) | ||||

Gynecology | 36.11 | 33 | 978 | 600 | 2 |

Otorhinolaryngology | 26.41 | 22 | 547 | 342 | 1 |

Hepatobiliary | 43.29 | 38 | 324 | 212 | 2 |

Orthopedics | 27.73 | 23 | 923 | 322 | 1 |

Stomatology | 30.11 | 26 | 567 | 122 | 1 |

Urology | 29.01 | 28 | 1200 | 860 | 1 |

General surgery | 30.78 | 29 | 300 | 120 | 1 |

Neurosurgery | 56.63 | 54 | 250 | 100 | 2 |

Pediatrics | 48.26 | 46 | 344 | 200 | 2 |

Oncology | 52.99 | 50 | 134 | 100 | 2 |

Instance | OR | OR Time (h)*3 days | Patient List | Total Surgery Time (h) | Mean Surgery Time (h) |
---|---|---|---|---|---|

I1 | 2 | 16*3 | 9 | 36.11 | 4.01 |

I2 | 1 | 8*3 | 6 | 26.41 | 4.40 |

I3 | 2 | 16*3 | 9 | 43.29 | 4.81 |

I4 | 1 | 8*3 | 6 | 27.73 | 4.62 |

I5 | 1 | 8*3 | 9 | 30.11 | 3.35 |

I6 | 1 | 8*3 | 5 | 29.01 | 5.80 |

I7 | 1 | 8*3 | 5 | 30.78 | 6.16 |

I8 | 2 | 16*3 | 15 | 56.63 | 3.78 |

I9 | 2 | 16*3 | 8 | 48.26 | 6.03 |

I10 | 2 | 16*3 | 13 | 52.99 | 4.08 |

Day in Period | OR | Surgery Serial Number | Duration of Surgery (h) | Total Surgery Time (h) |
---|---|---|---|---|

1 | 1 | 1 | 3.57 | 8.33 |

2 | 1.75 | |||

3 | 1.00 | |||

4 | 2.01 | |||

2 | 5 | 4.58 | 4.58 | |

3 | 6 | 9.87 | 9.87 | |

2 | 1 | 7 | 4.74 | 8.32 |

8 | 1.25 | |||

9 | 2.33 | |||

2 | 10 | 4.33 | 4.33 | |

3 | 11 | 8.00 | 8.00 | |

3 | 1 | 12 | 3.89 | 9.56 |

13 | 5.67 | |||

2 and 3 | unscheduled | - | - |

Day in Period | OR | Surgery Serial Number | Duration of Surgery (h) | Total Surgery Time (h) |
---|---|---|---|---|

1 | 1 | 10 | 4.33 | 8.22 |

12 | 3.89 | |||

2 | 1 | 3.57 | 9.16 | |

8 | 1.25 | |||

4 | 2.01 | |||

9 | 2.33 | |||

2 | 1 | 5 | 4.58 | 9.32 |

7 | 4.74 | |||

2 | 6 | 9.87 | 9.87 | |

3 | 1 | 11 | 8.00 | 8.00 |

2 | 13 | 5.67 | 8.42 | |

3 | 1.00 | |||

2 | 1.75 |

OR | 1 | 2 | 3 | |
---|---|---|---|---|

Day 1 | Original | 0.33 | −3.42 | 1.87 |

New | 0.22 | 1.16 | - | |

Day 2 | Original | 0.32 | −3.67 | - |

New | 1.32 | 1.87 | 0.00 | |

Day 3 | Original | 1.56 | −8.00 | −8.00 |

New | 0.00 | 0.42 | - |

State | N | Minimum Value | Maximum Value | Mean Value | Variance |
---|---|---|---|---|---|

Original | 3 | 0.00 | 9.87 | 5.89 | 13.24 |

New | 2 | 8.00 | 9.87 | 8.83 | 0.44 |

Parameter Values of the Improved NSGA-II |
---|

pop = 200 |

gen = 5000 |

Tour = 2 |

mu = 20 |

mum = 20 |

crossover rate = 0.9 |

mutation rate = 1/k |

Instance (SS Size, Waiting Lists, One SS) | Number of Constraints | Length of Chromosomes (m) | Max Surgery Time (h) | Min Surgery Time (h) | Mean Surgery Time (h) |
---|---|---|---|---|---|

(2, 10, 1) | 40 | 200 | 8.35 | 1.92 | 4.02 |

(8, 10, 1) | 100 | 800 | 6.75 | 2.12 | 4.41 |

(16, 10, 1) | 180 | 1,600 | 10.58 | 2.45 | 4.81 |

(2, 100, 1) | 400 | 40,000 | 14.33 | 1.80 | 4.62 |

(8, 100, 1) | 1,000 | 80,000 | 7.83 | 1.74 | 3.35 |

(16, 100, 1) | 1,800 | 160,000 | 9.27 | 4.59 | 5.82 |

(2, 1000, 1) | 4,000 | 4,000,000 | 9.25 | 3.50 | 6.16 |

(8, 1000, 1) | 10,000 | 8,000,000 | 7.23 | 0.41 | 3.78 |

(16, 1000, 1) | 18,000 | 16,000,000 | 11.33 | 1.24 | 6.03 |

Objective Function | Schedules | Min | Max | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |||

(max) Objective 1 | 22 | 35 | 16 | 16 | 25 | 34 | 37 | 37 | 33 | 39 | 16 | 39 |

(min) Objective 2 | 7.67 | 22.43 | 6.04 | 5.34 | 14.54 | 19.93 | 21.13 | 23.43 | 20.06 | 31.56 | 5.34 | 31.56 |

(min) Objective 3 | 0.296 | 1.161 | 0.474 | 0.238 | 2.471 | 2.523 | 2.519 | 2.633 | 2.419 | 2.903 | 0.238 | 2.903 |

**Table 16.**Computational results: minimum and maximum values of the three objective functions and CPU time (in min).

Instance (SS Size, Waiting Lists, One SS) | (Max) Objective 1 | (Min) Objective 2 | (Min) Objective 3 | CPU Time/Min | |||
---|---|---|---|---|---|---|---|

Min | Max | Min | Max | Min | Max | ||

(2, 10, 1) | 8 | 10 | 3.88 | 5.78 | 0.083 | 1.472 | 6 |

(8, 10, 1) | 8 | 10 | 98.50 | 113.01 | 0.079 | 0.991 | 7 |

(16, 10, 1) | 9 | 10 | 267.30 | 301.69 | 0.071 | 1.688 | 11 |

(2, 100, 1) | 13 | 16 | 2.07 | 12.41 | 0.112 | 4.654 | 21 |

(8, 100, 1) | 50 | 56 | 10.05 | 30.08 | 0.097 | 1.066 | 32 |

(16, 100, 1) | 33 | 39 | 5.34 | 23.55 | 0.238 | 2.903 | 50 |

(2, 1000, 1) | 14 | 15 | 13.55 | 24.368 | 0.076 | 5.966 | 98 |

(8, 1000, 1) | 57 | 62 | 3.18 | 10.12 | 0.107 | 3.994 | 150 |

(16, 1000, 1) | 98 | 102 | 4.72 | 34.51 | 0.067 | 1.554 | 230 |

Instance (SS Size, Waiting Lists, One SS) | GPA | Improved NSGA-II | ||||||
---|---|---|---|---|---|---|---|---|

Objective 1 | Objective 2 | Objective 3 | CPU Time | (Max) Objective 1 | (Min) Objective 2 | (Min) Objective 3 | CPU Time | |

(2, 10, 1) | 10 | 7.56 | 2.764 | 0.566 | 10 | 3.88 | 0.083 | 6 |

(8, 10, 1) | 10 | 132.70 | 1.063 | 0.578 | 10 | 98.50 | 0.079 | 7 |

(16, 10, 1) | 10 | 312.40 | 1.759 | 0.822 | 10 | 267.30 | 0.071 | 11 |

(2, 100, 1) | 12 | 3.93 | 2.663 | 2 | 16 | 2.07 | 0.112 | 21 |

(8, 100, 1) | 45 | 12.10 | 2.731 | 9 | 56 | 10.05 | 0.097 | 32 |

(16, 100, 1) | 89 | 8.71 | 4.564 | 12 | 39 | 5.34 | 0.161 | 50 |

(2, 1000, 1) | - | - | - | Unsolved | 15 | 13.55 | 0.076 | 98 |

(8, 1000, 1) | - | - | - | Unsolved | 62 | 3.18 | 0.107 | 150 |

(16, 1000, 1) | - | - | - | Unsolved | 102 | 4.72 | 0.067 | 230 |

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**MDPI and ACS Style**

Lu, Q.; Zhu, X.; Wei, D.; Bai, K.; Gao, J.; Zhang, R.
Multi-Phase and Integrated Multi-Objective Cyclic Operating Room Scheduling Based on an Improved NSGA-II Approach. *Symmetry* **2019**, *11*, 599.
https://doi.org/10.3390/sym11050599

**AMA Style**

Lu Q, Zhu X, Wei D, Bai K, Gao J, Zhang R.
Multi-Phase and Integrated Multi-Objective Cyclic Operating Room Scheduling Based on an Improved NSGA-II Approach. *Symmetry*. 2019; 11(5):599.
https://doi.org/10.3390/sym11050599

**Chicago/Turabian Style**

Lu, Qian, Xiaomin Zhu, Dong Wei, Kaiyuan Bai, Jinsheng Gao, and Runtong Zhang.
2019. "Multi-Phase and Integrated Multi-Objective Cyclic Operating Room Scheduling Based on an Improved NSGA-II Approach" *Symmetry* 11, no. 5: 599.
https://doi.org/10.3390/sym11050599