# Research on Buffer Calculation Model of Critical Chain Based on Adjacency Information Entropy

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Calculation Method for Influencing Factors of Buffer Size

#### 2.1. A Method to Calculate the Influence of Multi-Objective Constraints on Buffer Size

_{ia}, the average duration T

_{im}and the longest duration T

_{ib}of process i, respectively [31,32,33].

#### 2.1.1. Cost-Time Model

_{i}is the total cost corresponding to process i under the duration T

_{i}; δ is the indirect cost rate; C

_{i}

_{min}is the minimum total cost corresponding to process i under the average duration T

_{im}; C

_{ia}is the total cost corresponding to process i under the minimum duration T

_{ia}; and C

_{ib}is the total cost corresponding to process i under the maximum duration T

_{ib}, a

_{i}is the duration corresponding to the minimum direct cost of process i, and b

_{i}is the minimum direct cost of process i.

#### 2.1.2. Quality-Time Model

_{i}is the quality level corresponding to process i under the duration T

_{i}; Q

_{ib}is the highest quality level corresponding to process i under the longest duration T

_{ib}; and Q

_{ia}is the lowest quality level corresponding to process i under the shortest duration T

_{ia}. g

_{i}is the growth rate of process quality level with duration, and f

_{i}is the adjustment coefficient of the relationship between process quality level and duration.

#### 2.1.3. Safety-Time Model

_{i}is the safety level of process i; ${\theta}_{i}$ is the influence coefficient of process i duration on safety level, 0 < ${\theta}_{i}$ ≤ 1; P

_{i}is the probability of safety accident occurring in process i; P

_{io}is the initial probability of safety accident occurring in process i; ∆P

_{io}is the change value of the initial probability P

_{io}of safety accident occurring in process i, ∆P

_{io}

_{min}is the minimum change value and ∆P

_{io}

_{max}is the maximum change value.

#### 2.1.4. Environment—Time Model

_{i}is the environmental influence value of process i; d

_{d}is the distance between the project site and the adjacent downtown area.

#### 2.1.5. Calculation of Process Safety Time

_{i}) is:

_{im}of process i; ${T}_{i}^{Q}$ is the duration corresponding to the target quality level Q

_{im}of process i; ${T}_{i}^{S}$ is the duration corresponding to the target safety level S

_{im}of process i; ${T}_{i}^{E}$ is the duration corresponding to the target environmental influence level E

_{im}of process i; k

_{C}, k

_{Q}, k

_{S}and k

_{E}are the weighting coefficients of cost, quality, safety and environment respectively. Let k

_{C}= k

_{Q}= k

_{S}= k

_{E}= 0.25.

#### 2.2. A Method to Calculate the Influence of Multi-Resource Constraints on Buffer Size

_{i}is the resource influence coefficient of process i; n is the total kind of resources required to complete process i; ${u}_{i}^{k}$ is the utilization coefficient of the kth resource in process i; w

^{k}is the restriction coefficient of the kth resource; ${r}_{i}^{k}$ is the demand of process i for the kth resource; ${\overline{r}}^{k}$ is the average demand of resource k for the process requiring the kth resource; R

^{k}is the limit of the kth resource. λ

_{i}is the influence coefficient of process i duration on resource constraints; T

_{i}is the duration of process i; T is the sum of the process duration on the critical chain. The larger ${u}_{i}^{k}$ is, indicating that process i is more influenced by the limitation of the kth resource. The larger w

^{k}is, indicating that the kth resource is more limited.

#### 2.3. A Method to Calculate the Influence of Process Relay Potential on Buffer Size

_{i}. When L

_{i}> 0, it indicates that process i has resource surplus after the immediate preceding process cooperation and resource allocation; When L

_{i}= 0, it indicates that there is cooperation and resource allocation between the immediately preceding processes in process i, but there is no need for resource replenishment; When L

_{i}< 0, it indicates that the process needs resource compensation. The per capita construction speed $\overline{v}$ for the project as planned is:

_{i}is the engineering quantity of process i; Y

_{ij}is the number of people in the jth professional title in process i.

_{i}of process i is obtained by considering the above effects [15,39]:

_{i}is the comprehensive capability index of process i; α

_{i}is the average quality of personnel of process i; β

_{i}is the efficiency coefficient of personnel-material-machine cooperation during cross construction of process i; γ

_{i}is the resource reserve coefficient of process i; D is the difficulty degree of the process; M and N are the equipping rate and utilization rate of equipment, respectively; B

_{ij}is the jth professional title weight of process i. The weight value and weight distribution of personnel professional titles are shown in Table 1.

#### 2.4. A Method to Calculate the Influence of Entropy of Process Adjacency Information on Buffer Size

_{i}is the probability that the system is in a certain state.

_{i}is the adjacency information entropy of process i; p

_{ij}is a probability function indicating the importance of process i in its neighbor process j; Γ

_{i}is the set of processes directly adjacent to process i; A

_{j}is the adjacency degree of process j; k

_{w}is the degree value of process w; ${k}_{w}^{\prime}$ is the number of immediately preceding processes directly adjacent to process w; ${k}_{w}^{\u2033}$ is the number of immediately following processes directly adjacent to process w.

## 3. Buffer Size Calculation Model

#### 3.1. Initial Buffer

_{l}is the initial buffer size of the lth line; ST

_{i}is the safety time of process i; L

_{i}is the relay potential of process i.

#### 3.2. Import Buffer

_{l}is the size of the lth non-critical line import buffer; FF

_{li}is the free time difference of the last process i of non-critical chain l; Ω

_{i}is the set of all immediately following processes j of process i; ES

_{j}is the earliest start time of immediately following process j of process i; EF

_{i}is the earliest end time of process i.

#### 3.3. Remaining Buffer

_{l}of the lth non-critical chain is:

#### 3.4. Project Buffer

## 4. Example Analysis

#### 4.1. Example Introduction

_{i}of each process on the environment shall not be greater than 1.20, and the safety level S

_{i}shall not be lower than 0.95. The parameters of each process are shown in Table 2. The project requires three kinds of resources, and each process requires at least one resource. The resource demand of each process, the limit of each resource, and the resource constrained parameters are shown in Table 3. Using Crystal Ball software for Monte Carlo simulation of each process, 5000 simulation results were extracted, and taking process B as an example, the frequency distribution of process B simulation results is shown in Figure 3. The green section represents the Beta PERT distribution probability density function with a minimum value equal to 4, a most likely value equal to 6, and a maximum value equal to 10 as the characteristic values. The blue section represents the results of 5000 simulations.

#### 4.2. Process Safety Time under Multi-Objective Constraints

_{B}= RMB 1,175,300), ${T}_{B}^{Q}$ for 7.34 d under the constraint of target quality level (Q

_{B}= 0.97), ${T}_{B}^{S}$ for 6.15 d under the constraint of target safety level (≥0.95), and ${T}_{B}^{E}$ for 6.50 d under the constraint of target environmental level (≤1.20), yielding Z

_{B}= 6.54 d and ST

_{B}= 1.74 d. The calculation of process safety time under multi-objective constraints is shown in Table 4. The network schedule based on multi-objective constraints is shown in Figure 4.

#### 4.3. Calculation of Resource Influence Coefficient

^{1}= 8, R

^{2}= 1, R

^{3}= 2, we get ${u}_{A}^{1}$ = 4/8 = 0.5, ${u}_{A}^{2}$ = 1/1 = 1, ${u}_{A}^{3}$ = 1/2 = 0.5. From R

^{1}= 8, there are nine processes using resource 1, we get ${\overline{r}}^{1}$ = 3.67, w

^{1}= 3.67/8 = 0.46. From T

_{A}= 2.15 d, T = 25.49 d, we get λ

_{A}= 2.15/25.49 = 0.26, R

_{A}= 0.15. The adjusted network schedule considering multi-resource constraints is shown in Figure 5 [27].

#### 4.4. Calculation of Process Relay Potential

_{D}=0.9 and β

_{G}= 0.85. The resource reserve coefficient is γ = 0.98. The difficulty of the processes are D

_{D}= 0.8 and D

_{G}= 0.9. The equipment allocation rate is M

_{D}= M

_{G}= 0.99. The equipment utilization rate is N

_{D}= N

_{G}= 0.95. The process durations are T

_{D}= 4.33 d, T

_{G}= 5.18 d. According to Equations (19–22), the average speed of the relay network of this project is $\overline{v}$ = 2.02, v

_{D}= 1.12, v

_{G}= 1.99. From “$\overline{v}$ > v

_{G}> v

_{D}, T

_{G}> T

_{D}, T

_{G}+ ($\overline{v}$ − v

_{G})T

_{G}/v

_{G}< T

_{D}+ (v − v

_{D})T

_{D}/v

_{D}” yields P

_{D}= −0.41, P

_{G}= −0.41. The network schedule considering the process relay potential is shown in Figure 6.

#### 4.5. Calculation of Process Adjacency Information Entropy

_{A}= 3, k

_{B}= 1, k

_{C}= 2, k

_{D}= 3 can be obtained from Equation (27); A

_{A}= 6, A

_{B}= 3, A

_{C}= 6, A

_{D}= 10 can be obtained from Equation (26); H

_{A}= 1.02 can be obtained from Equations (24) and (25); ${{\displaystyle H}}_{A}^{*}$ = 0.55 can be obtained from Equation (28).

#### 4.6. Calculation of Buffer Size

_{G}= 1.61 d and PB = 6.14 d can be obtained from Equations (32)–(34).

#### 4.7. Comparison and Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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Title | Professor | Associate Professor | Engineer | Assistant Engineer | Technicians and Below |
---|---|---|---|---|---|

Weight value | 9 | 7 | 5 | 3 | 1 |

Weight distribution | 0.36 | 0.28 | 0.2 | 0.12 | 0.04 |

Process | T_{ia} | T_{im} | T_{ib} | C_{ia} | C_{i}_{min} | C_{i} | Q_{ia} | Q_{ib} | Q_{i} | P_{io} | P_{io}_{max} | P_{io}_{min} |
---|---|---|---|---|---|---|---|---|---|---|---|---|

B | 5.48 | 6.25 | 8.28 | 140.73 | 117.28 | 117.53 | 0.95 | 0.98 | 0.97 | 0.04 | 0.95 | 0.15 |

A | 1.72 | 2.01 | 2.63 | 37.01 | 30.85 | 30.91 | 0.92 | 0.96 | 0.95 | 0.06 | 0.90 | 0.12 |

D | 3.47 | 4.11 | 5.75 | 88.11 | 73.42 | 73.60 | 0.92 | 0.95 | 0.94 | 0.06 | 0.89 | 0.10 |

E | 7.17 | 8.11 | 10.37 | 186.33 | 155.28 | 155.65 | 0.92 | 0.97 | 0.95 | 0.05 | 0.90 | 0.10 |

I | 3.18 | 3.88 | 5.33 | 82.27 | 68.56 | 68.72 | 0.94 | 0.98 | 0.97 | 0.06 | 0.95 | 0.12 |

C | 2.46 | 3.02 | 4.24 | 61.34 | 51.11 | 51.24 | 0.92 | 0.97 | 0.95 | 0.05 | 0.85 | 0.10 |

G | 4.12 | 4.98 | 6.87 | 109.09 | 90.91 | 91.13 | 0.93 | 0.96 | 0.95 | 0.06 | 0.95 | 0.12 |

F | 3.17 | 4.00 | 5.86 | 84.92 | 70.76 | 70.93 | 0.92 | 0.95 | 0.94 | 0.03 | 0.92 | 0.08 |

H | 3.44 | 4.12 | 5.77 | 88.26 | 73.55 | 73.72 | 0.93 | 0.98 | 0.96 | 0.05 | 0.85 | 0.05 |

Process | Resource | λ_{i} | R_{i} | ||
---|---|---|---|---|---|

1 | 2 | 3 | |||

A | 4 | 1 | 1 | 0.26 | 0.15 |

B | 6 | 0 | 1 | 0.08 | 0.13 |

C | 2 | 0 | 0 | 0.17 | 0.23 |

D | 2 | 1 | 1 | 0.33 | 0.51 |

E | 5 | 1 | 1 | 0.16 | 0.19 |

F | 3 | 0 | 1 | 0.12 | 0.01 |

G | 4 | 0 | 1 | 0.20 | 0.10 |

H | 4 | 0 | 1 | 0.16 | 0.07 |

I | 3 | 1 | 0 | 0.17 | 0.08 |

R^{k} | 8 | 1 | 2 | ||

w^{k} | 0.46 | 1.00 | 0.50 |

Process | ${\mathit{T}}_{\mathit{i}}^{\mathit{C}}$ | ${\mathit{T}}_{\mathit{i}}^{\mathit{Q}}$ | ${\mathit{T}}_{\mathit{i}}^{\mathit{S}}$ | ${\mathit{T}}_{\mathit{i}}^{\mathit{E}}$ | Z_{i} | ST_{i} |
---|---|---|---|---|---|---|

B | 6.17 | 7.34 | 6.15 | 6.50 | 6.54 | 1.74 |

A | 1.98 | 2.40 | 2.20 | 2.04 | 2.15 | 0.48 |

D | 4.04 | 4.98 | 4.20 | 4.11 | 4.33 | 1.42 |

E | 8.01 | 9.07 | 8.07 | 8.50 | 8.41 | 1.96 |

I | 3.80 | 4.78 | 3.90 | 3.77 | 4.06 | 1.27 |

C | 2.96 | 3.52 | 3.00 | 2.92 | 3.10 | 1.14 |

G | 4.89 | 5.94 | 5.00 | 4.88 | 5.18 | 1.69 |

F | 3.91 | 4.95 | 3.91 | 3.76 | 4.13 | 1.73 |

H | 4.05 | 4.82 | 4.30 | 4.08 | 4.31 | 1.46 |

Process Type (1) | Process (2) | Three-Time Estimate Distribution (3) | T_{95%}(4) | Z_{i}(5) | ST_{i}(6) | R_{i}(7) | ${\mathit{H}}_{\mathit{i}}^{*}$ (8) | L_{i}(9) | Buffer (10) | FF_{i}(11) | FB (12) | KB (13) | PB (14) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Critical path process | B | (4, 6, 10) | 8.28 | 6.54 | 1.74 | 0.15 | 0.066 | −0.33 | 4.16 | - | - | - | 5.75 |

A | (1, 2, 3) | 2.63 | 2.15 | 0.48 | 0.13 | 0.550 | - | - | - | - | |||

D | (2, 4, 7) | 5.75 | 4.33 | 1.42 | 0.23 | 0.991 | −0.41 | - | - | - | |||

E | (5, 8, 12) | 10.37 | 8.41 | 1.96 | 0.51 | 0.951 | 3.08 | - | - | - | |||

I | (1, 4, 6) | 5.33 | 4.06 | 1.27 | 0.19 | 0.518 | −0.70 | - | - | - | |||

Non-critical path process | C | (1, 3, 5) | 4.24 | 3.10 | 1.14 | 0.01 | 0.541 | 0.00 | 1.78 | 2.22 | 1.78 | 0 | |

G | (2, 5, 8) | 6.87 | 5.18 | 1.69 | 0.10 | 0.991 | −0.41 | 4.11 | 2.52 | 2.52 | 1.59 | ||

F | (1, 4, 7) | 5.86 | 4.13 | 1.73 | 0.07 | 1.003 | 0.00 | 3.70 | 4.28 | 3.70 | 0 | ||

H | (2, 4, 7) | 5.77 | 4.31 | 1.46 | 0.08 | 0.002 | −0.74 | 2.32 | - | 2.32 | 0 |

**Table 6.**Project schedule duration and critical chain buffer size under multiple buffer size calculation models.

Name of Methods | Factors Considered in the Method | Import Buffer (Day) | Project Buffer (Day) | Planned Project Duration (Day) | |||
---|---|---|---|---|---|---|---|

FB_{1}(Process C) | FB_{2}(Process G) | FB_{3}(Process F) | FB_{4}(Process H) | ||||

Method used in this article | Multi-objective, multi-resource constraints, relay potential and adjacency information entropy | 1.78 | 2.52 | 3.7 | 2.32 | 5.75 | 31.24 |

Hu Chen’s method | Project duration risks and multiple resource constraints | 1.54 | 1.15 | 2.87 | 2.65 | 6.44 | 36.36 |

Chu Chunchao’s method | Resource utilization, project complexity, risk preference of decision-makers | 2.06 | 4.11 | 4.52 | 4.11 | 11.60 | 35.60 |

Cut-and-paste | Safety time of activity | 1.00 | 1.5 | 1.5 | 1.50 | 7.00 | 31.00 |

Root variance method | Variance of activity | 2.00 | 3.00 | 3.00 | 2.00 | 6.78 | 30.78 |

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## Share and Cite

**MDPI and ACS Style**

Nie, X.; Li, M.; Lu, J.; Wang, B.
Research on Buffer Calculation Model of Critical Chain Based on Adjacency Information Entropy. *Buildings* **2023**, *13*, 942.
https://doi.org/10.3390/buildings13040942

**AMA Style**

Nie X, Li M, Lu J, Wang B.
Research on Buffer Calculation Model of Critical Chain Based on Adjacency Information Entropy. *Buildings*. 2023; 13(4):942.
https://doi.org/10.3390/buildings13040942

**Chicago/Turabian Style**

Nie, Xiangtian, Min Li, Jilan Lu, and Bo Wang.
2023. "Research on Buffer Calculation Model of Critical Chain Based on Adjacency Information Entropy" *Buildings* 13, no. 4: 942.
https://doi.org/10.3390/buildings13040942