# Complexity Assessment of Assembly Supply Chains from the Sustainability Viewpoint

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Works

## 3. Methodology Framework

## 4. Description of Possible ASC Structural Complexity Indicators

#### 4.1. Index of Vertex Degree

_{1}, N

_{2}, …, N

_{k}symbols. Then, entropy of information H (α) is calculated by the formula:

_{i}specifies the probability of the occurrence of the elements of the ith group.

_{i}as p

_{i}= w

_{i}/∑w

_{i}with ∑w

_{i}= w, and ∑p

_{i}= 1.

_{i}can be expressed by the formula:

_{max}− H, where H

_{max}is the maximum entropy that can exist in a system with the same number of elements, the information entropy of a graph with a total weight W and vertex weights w

_{i}can be expressed in the form of the equation:

_{i}= 1, then H

_{max}= Wlog

_{2}W, and by substituting W = ∑deg(v)

_{i}and w

_{i}= deg(v)

_{i}, the information content of the vertex degree distribution of a network, called the vertex degree index (I

_{vd}), is expressed as follows [28]:

#### 4.2. Process Complexity Indcator

_{ijk}means the probability that part j is being proceeded by operation k by individual machine i based on the scheduling order; O is the number of operations according to parts production; P is the number of parts produced in the manufacturing process; and M is the number of all machines of all types in the manufacturing process.

_{ijk}equals 1/M

_{s}, where M

_{s}presents the number of machines organized in serial. In the case when a part is processed on machines in a parallel manner, then p

_{ijk}= 1/M

_{p}, where M

_{p}represents the number of machines organized in parallel. In the case where we have a serial/parallel arrangement of machines and a part is processed on one of the parallel machines, then p

_{ijk}equals 1/M

_{s}.M

_{p}.

#### 4.3. System Design Complexity

_{1}, N

_{2}, …, N

_{k}as the numbers of interactions per each design parameter (DP) of the same matrix. Then, the so-called degree of disorder W can be expressed by the formula:

_{j}was considered by Boltzmann. For each magnitude φ

_{j}, its interval of admitted values is divided into small intervals of equal length ∆

_{j}. Then, the n-dimensional space, also known as µ-space or module space, can be divided into a system of cells of equal volume: υ

^{µ}= ∆

_{i},…, ∆

_{n}. K is the number of these cells in the total range of admitted values, R

^{µ}; then: υ

^{µ}= V

^{µ}/K, where V

^{µ}is the volume of R

^{µ}. The µ-cells are analogous to the cells Q

_{j}(j = 1, …, K) in the classification system. f

_{j}means the density in Q

_{j}, i.e., the number of molecules per unit of µ-volume: f

_{j}= N

_{j}/υ

^{µ}. The function defined by Boltzmann for a statistical description is:

_{j}= N

_{j}/υ

^{µ}, where the volume is assumed equal to unity, the following formula for complexity measure can be derived:

_{j}is interpreted as the number of interactions per single DP.

#### 4.4. Modified Flow Complexity

## 5. Definition of Testing Rules for ASC Complexity Indicators

- Rule#1:
- Static complexity should increase with the number of parts and the number of machines and operations required to process the part mix.
- Rule#2:
- Static complexity should increase with increases in sequence flexibility for the parts in the production batch.
- Rule#3:
- Static complexity should increase as the sharing of resources by parts increases.
- Rule#4:
- If the original part mix is split into two or more groups, then the complexity of processing should remain constant.

- Rule#1:
- Additional independent element: The element has no structure itself, so it has no complexity of its own. Because it is independent of the rest of the system, the complexity should not change.
- Rule#2:
- Union of two independent systems: Because there are no dependencies between the two systems, the complexity of the union should be simply the sum of the complexities of the subsystems.
- Rule#3:
- Union of two identical copies: Because there is no need for additional information to describe the second system, one could argue that the complexity should be equal to the complexity of one system. One has, however, to include the fact in the description that the second system is a copy of the first one. At least this part should not be extensive with respect to the system size.

**C#1:**- Static complexity should increase with the number of parts required to process the part mix.
**C#2:**- Static complexity should increase with the number of machines required to process the part mix.
**C#3:**- Static complexity should increase with the number of operations required to process the part mix.
**C#4:**- Static complexity should increase with increases in sequence flexibility for the parts in the production batch.
**C#5:**- Static complexity should increase with the number of echelons while the number of parts, machines, and operations is constant.

## 6. Analysis of Testing Criteria from Sustainability Viewpoint

- (Direct) energy costs;
- (Direct) material costs;
- (Direct) organizational costs.

## 7. Testing and Comparison of ASC Complexity Indicators

_{1}(i,j,k) and ASC

_{2}(i,j,k). One of them is more complex according to the defined criteria (C#1–C#5). P represents parts, j is the number of parts, O represents operations, k is the number of operations, M represents machines, and i is the number of machines.

#### 7.1. Testing of C#1

_{1}(3;4;3) and ASC

_{2}(3;5;3), shown in Figure 3. While in Figure 3a, ASC

_{1}consists of three machines and operations (i = k = 3), and the number of parts equals four (j = 4), in Figure 3b, ASC

_{2}contains three machines and operations (i = k = 3), and the number of parts equals five (j = 5).

_{vd}, we obtained the following results for static complexity using I

_{vd1}(for ASC

_{1}) and I

_{vd2}(for ASC

_{2}):

_{vd1}= 3 × log

_{2}3 + 3 × log

_{2}3 + 2 × log

_{2}2 = 11.51 bits

_{vd2}= 3 × log

_{2}3 + 4 × log

_{2}4 + 2 × log

_{2}2 = 14.76 bits

_{1}and PCI

_{2}:

_{1}= −(0.5 × log

_{2}0.5 + 0.5 × log

_{2}0.5 + 0.5 × log

_{2}0.5 + 0.5 × log

_{2}0.5) = 4 bits

_{2}= −(0.5 × log

_{2}0.5 + 0.5 × log

_{2}0.5 + 0.5 × log

_{2}0.5 + 0.5 × log

_{2}0.5 + 0.5 × log

_{2}0.5 + 0.5 × log

_{2}0.5) = 5 bits

_{1}and SDC

_{2}:

_{1}= 4 × ln4 + 2 × ln2 + 2 × ln2 = 8.32 nats

_{2}= 5 × ln5 + 2 × ln2 + 3 × ln3 = 12.73 nats

_{1}and MFC

_{2}:

_{1}= 0 × 3 + 1 × 7 + 1 × 6 = 13

_{2}= 0 × 3 + 1 × 8 + 1 × 7 = 15

_{1}(3,4,3) < ASC

_{2}(3,5,3), then ASC(i,j−1,k) < ASC(i,j,k).

#### 7.2. Testing of C#2

_{1}(3,4,3) and ASC

_{2}(4,4,4), shown in Figure 4. While in Figure 4a, ASC

_{1}consists of three machines and operations (i = k = 3), and the number of parts equals four (j = 4), in Figure 4b, ASC

_{2}contains four machines and operations (i = k = 4), and the number of parts equals four (j = 4).

_{vd}, we obtained the following results for I

_{vd1}and I

_{vd2}:

_{vd1}= 3 × log

_{2}3 + 3 × log

_{2}3 + 2 × log

_{2}2 = 11.51 bits

_{vd2}= 3 × log

_{2}3 + 3 × log

_{2}3 + 3 × log

_{2}3 + 3 × log

_{2}3 = 19.02 bits

_{1}and PCI

_{2}:

_{1}= −(0.5 × log

_{2}0.5 + 0.5 × log

_{2}0.5 + 0.5 × log

_{2}0.5 + 0.5 × log

_{2}0.5) = 4 bits

_{2}= −(0.5 × log

_{2}0.5 + 0.5 × log

_{2}0.5 + 0.25 × log

_{2}0.25 + 0.25 × log

_{2}0.25 + 0.5 × log

_{2}0.5 + 0.25 × log

_{2}0.25 + 0.25 × log

_{2}0.25 + 0.5 × log

_{2}0.5 + 0.5 × log

_{2}0.5 + 0.5 × log

_{2}0.5) = 5 bits

_{1}and SDC

_{2}:

_{1}= 4 × ln4 + 2 × ln2 + 2 × ln2 = 8.32 nats

_{1}and MFC

_{2}:

_{1}= 0 × 3 + 1 × 7 + 1 × 6 = 13

_{2}= 0 × 3 + 1 × 8 + 1 × 9 = 17

_{1}(3,4,3) < ASC

_{2}(4,4,4), then ASC(i−1,j,k) ˂ ASC(i,j,k).

#### 7.3. Testing of C#3

_{1}(3,4,3) and ASC

_{2}(3,4,4), shown in Figure 5. While in Figure 5a, ASC

_{1}consists of three machines and operations (i = k = 3), and the number of parts equals four (j = 4), in Figure 5b, ASC

_{2}contains three machines (i = 3), four operations (k = 4), and the number of parts equals four (j = 4).

_{vd}, we obtained the following results for I

_{vd1}and I

_{vd2}:

_{vd1}= 3 × log

_{2}3 + 3 × log

_{2}3 + 2 × log

_{2}2 = 11.51 bits

_{vd2}= 5 × log

_{2}5 + 3 × log

_{2}3 + 2 × log

_{2}2 = 18.37 bits

_{1}and PCI

_{2}:

_{1}= −(0.5 × log

_{2}0.5 + 0.5 × log

_{2}0.5 + 0.5 × log

_{2}0.5 + 0.5 × log

_{2}0.5) = 4 bits

_{2}= −(0.25 × log

_{2}0.25 + 0.25 × log

_{2}0.25 + 0.5 × log

_{2}0.5 + 0.25 × log

_{2}0.25 + 0.25 × log

_{2}0.25 + 0.5 × log

_{2}0.5 + 0.5 × log

_{2}0.5 + 0.5 × log

_{2}0.5 + 0.5 × log

_{2}0.5 + 0.5 × log

_{2}0.5) = 5 bits

_{1}and SDC

_{2}:

_{1}= 4 × ln4 + 2 × ln2 + 2 × ln2 = 8.32 nats

_{2}= 4 × ln4 + 2 × ln2 + 2 × ln2 = 8.32 nats

_{1}and MFC

_{2}:

_{1}= 0 × 3 + 1 × 7 + 1 × 6 = 13

_{2}= 0 × 3 + 1 × 7 + 1 × 7 = 14

_{vd}, PCI, and MFC, it was proved that the structural complexity of ASC

_{1}(3,4,3) < ASC

_{2}(3,4,4), then ASC(i,j,k−1) ˂ ASC(i,j,k).

#### 7.4. Testing of C#4

_{1}(3,4,3) and ASC

_{2}(3,4,3), shown in Figure 6. While in Figure 6a, ASC

_{1}consists of three machines and operations (i = k = 3), and the number of parts equals four (j = 4), in Figure 6b, ASC

_{2}consists of three machines and operations (i = k = 3), and the number of parts equals four (j = 4), but P

_{2}and P

_{3}can be processed on M

_{1}or M

_{2}.

_{vd}, we obtained the following results for I

_{vd1}and I

_{vd2}:

_{vd1}= 3 × log

_{2}3 + 3 × log

_{2}3 + 2 × log

_{2}2 = 11.51 bits

_{vd2}= 4 × log

_{2}4 + 4 × log

_{2}4 + 2 × log

_{2}2 = 18 bits

_{1}and PCI

_{2}:

_{1}= −(0.5 × log

_{2}0.5 + 0.5 × log

_{2}0.5 + 0.5 × log

_{2}0.5 + 0.5 × log

_{2}0.5) = 4 bits

_{2}= −(0.5 × log

_{2}0.5 + 0.5 × log

_{2}0.5 + 0.25 × log

_{2}0.25 + 0.25 × log

_{2}0.25 + 0.5 × log

_{2}0.5 + 0.25 × log

_{2}0.25 + 0.25 × log

_{2}0.25 + 0.5 × log

_{2}0.5 + 0.5 × log

_{2}0.5 + 0.5 × log

_{2}0.5) = 5 bits

_{1}and SDC

_{2}:

_{1}= 4 × ln4 + 2 × ln2 + 2 × ln2 = 8.32 nats

_{2}= 4 × ln4 + 3 × ln3 + 3 × ln3 = 12.14 nats

_{1}and MFC

_{2}:

_{1}= 0 × 3 + 1 × 7 + 1 × 6 = 13

_{2}= 0 × 3 + 1 × 7 + 1 × 8 = 15

_{1}˂ ASC

_{2}, then ASC(i,j,k) ˂ ASC(i,j,k) with an increasing number of routings.

#### 7.5. Testing of C#5

_{1}(3,4,3) and ASC

_{2}(3,4,3), shown in Figure 7. While in Figure 7a, ASC

_{1}consists of three machines and operations (i = k = 3), the number of parts equals four (j = 4), and the number of echelons equals two, in Figure 7b, ASC

_{2}contains three machines (i = 3), four operations (k = 4), the number of parts equals four (j = 4), and the number of echelons equals three.

_{vd}, we obtained the following results for I

_{vd1}and I

_{vd2}:

_{vd1}= 3 × log

_{2}3 + 3 × log

_{2}3 + 2 × log

_{2}2 = 11.51 bits

_{vd2}= 3 × log

_{2}3 + 3 × log

_{2}3 + 2 × log

_{2}2 = 11.51 bits

_{1}and PCI

_{2}:

_{1}= −(0.5 × log

_{2}0.5 + 0.5 × log

_{2}0.5 + 0.5 × log

_{2}0.5 + 0.5 × log

_{2}0.5) = 4 bits

_{2}= −(0.33 × log

_{2}0.33 + 0.33 × log

_{2}0.33 + 0.33 × log

_{2}0.33 + 0.33 × log

_{2}0.33 + 0.33 × log

_{2}0.33 + 0.33 × log

_{2}0.33 + 0.5 × log

_{2}0.5 + 0.5 × log

_{2}0.5 + 1 × log

_{2}1) = 4.17 bits

_{1}and SDC

_{2}:

_{1}= 4 × ln4 + 2 × ln2 + 2 × ln2 = 8.32 nats

_{2}= 4 × ln4 + 3 × ln3 + 2 × ln2 = 10.23 nats

_{1}and MFC

_{2}:

_{1}= 0 × 3 + 1 × 7 + 1 × 6 = 13

_{2}= 0 × 4 + 1 × 7 + 1 × 6 = 13

_{1}˂ the complexity of ASC

_{2}, then ASC(i,j,k) ˂ ASC(i,j,k) with an increasing number of echelons.

#### 7.6. Comparison and Selection of ASC Complexity Measure

## 8. Discussion of Results, Implications and Limitations

- (i)
- All described indicators sufficiently reflect organizational aspects of ASCs.
- (ii)
- Three of the complexity indicators, namely, I
_{vd}, MFC, and PCI, can be effectively used to measure ASC complexity in order to identify how ASC structural variants are influencing organizational costs, as well as energy costs. - (iii)
- The PCI complexity measure reflects all of the three cost items and covers the two crucial dimensions of sustainability, economic and environmental.

## 9. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**The sequence of the methodological procedures for the selection of the most suitable complexity indicator.

**Figure 2.**The possible assembly supply chain (ASC) network with four input components and the corresponding structural alternatives.

**Figure 3.**ASC consisting of: (

**a**) three machines, three operations, and four parts; (

**b**) three machines, three operations, and five parts.

**Figure 4.**ASC consisting of: (

**a**) three machine, three operations, and four parts; (

**b**) four machines, four operations, and four parts.

**Figure 5.**ASC consisting of: (

**a**) three machines, three operations, and four parts; (

**b**) three machines, four operations, and three parts.

**Figure 6.**ASC consisting of: (

**a**) three machines, three operations, and four parts; (

**b**) three machines, three operations, and four parts.

**Figure 7.**Manufacturing system consisting of: (

**a**) three machines, three operations, four parts, and two echelons; (

**b**) three machines, three operations, four parts, and three echelons.

Graph | (a) | (b) | (c) | (d) | (e) |
---|---|---|---|---|---|

I_{vd} | 8 bits | 10 bits | 9.51 bits | 11.51 bits | 11.51 bits |

Graph | (a) | (b) | (c) | (d) | (e) |
---|---|---|---|---|---|

PCI | 0 bits | 3 bits | 2 bits | 4 bits | 4.17 bits |

Graph | (a) | (b) | (c) | (d) | (e) |
---|---|---|---|---|---|

SDC | 5.55 nats | 8.84 nats | 6.93 nats | 8.32 nats | 10.23 nats |

Graph | (a) | (b) | (c) | (d) | (e) |
---|---|---|---|---|---|

MFC | 9 | 11 | 11 | 13 | 13 |

Testing Criteria | Material Costs | Energy Costs | Organizational Costs |
---|---|---|---|

C#2 | ✔ | ✔ | ✔ |

C#3 | - | ✔ | ✔ |

C#1 | - | - | ✔ |

C#4 | - | - | ✔ |

C#5 | - | - | ✔ |

Criteria | I_{vd} | SDC | MFC | PCI |
---|---|---|---|---|

C#2 | ✔ | ✔ | ✔ | ✔ |

C#3 | ✔ | X | ✔ | ✔ |

C#1 | ✔ | ✔ | ✔ | ✔ |

C#4 | ✔ | ✔ | ✔ | ✔ |

C#5 | X | ✔ | X | ✔ |

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## Share and Cite

**MDPI and ACS Style**

Modrak, V.; Soltysova, Z.; Onofrejova, D.
Complexity Assessment of Assembly Supply Chains from the Sustainability Viewpoint. *Sustainability* **2019**, *11*, 7156.
https://doi.org/10.3390/su11247156

**AMA Style**

Modrak V, Soltysova Z, Onofrejova D.
Complexity Assessment of Assembly Supply Chains from the Sustainability Viewpoint. *Sustainability*. 2019; 11(24):7156.
https://doi.org/10.3390/su11247156

**Chicago/Turabian Style**

Modrak, Vladimir, Zuzana Soltysova, and Daniela Onofrejova.
2019. "Complexity Assessment of Assembly Supply Chains from the Sustainability Viewpoint" *Sustainability* 11, no. 24: 7156.
https://doi.org/10.3390/su11247156