Complexity Assessment of Assembly Supply Chains from the Sustainability Viewpoint

Testing and Comparison of ASC Complexity Indicators

Testing of C#5

C#5 is proposing the following statement:

If i, j, and k are constant and the number of echelons is increasing, then the complexity of ASC(i,j,k) ˂ ASC(i,j,k) with an increasing number of echelons.

Let us test ASC₁(3,4,3) and ASC₂(3,4,3), shown in Figure 7. While in Figure 7a, ASC₁ 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, ASC₂ contains three machines (i = 3), four operations (k = 4), the number of parts equals four (j = 4), and the number of echelons equals three.

Applying Equation (4), according to complexity indicator I_vd, we obtained the following results for I_vd1 and I_vd2:

I_vd1 = 3 × log₂3 + 3 × log₂3 + 2 × log₂2 = 11.51 bits

I_vd2 = 3 × log₂3 + 3 × log₂3 + 2 × log₂2 = 11.51 bits

Applying Equation (5), according to complexity indicator PCI, we obtained the following results for PCI₁ and PCI₂:

PCI₁ = −(0.5 × log₂0.5 + 0.5 × log₂0.5 + 0.5 × log₂0.5 + 0.5 × log₂0.5) = 4 bits

PCI₂ = −(0.33 × log₂0.33 + 0.33 × log₂0.33 + 0.33 × log₂0.33 + 0.33 × log₂0.33 + 0.33 × log₂0.33 + 0.33 × log₂0.33 + 0.5 × log₂0.5 + 0.5 × log₂0.5 + 1 × log₂1) = 4.17 bits

Applying Equation (9), according to complexity indicator SDC, we obtained the following results for SDC₁ and SDC₂:

SDC₁ = 4 × ln4 + 2 × ln2 + 2 × ln2 = 8.32 nats

SDC₂ = 4 × ln4 + 3 × ln3 + 2 × ln2 = 10.23 nats

Applying Equation (10), according to complexity indicator MFC, we obtained the following results for MFC₁ and MFC₂:

MFC₁ = 0 × 3 + 1 × 7 + 1 × 6 = 13

MFC₂ = 0 × 4 + 1 × 7 + 1 × 6 = 13

According to the three complexity indicators PCI, SDC, and MFC, it was proved that the complexity of ASC₁ ˂ the complexity of ASC₂, then ASC(i,j,k) ˂ ASC(i,j,k) with an increasing number of echelons.