Complexity Assessment of Assembly Supply Chains from the Sustainability Viewpoint

Testing and Comparison of ASC Complexity Indicators

Testing of C#2

C#2 is proposing the following statement:

If j is constant and i is increasing (i = k), then the structural complexity of ASC(i−1,j,k) ˂ ASC(i,j,k).

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

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 + 3 × log₂3 + 3 × log₂3 = 19.02 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 = −(0.5 × log₂0.5 + 0.5 × log₂0.5 + 0.25 × log₂0.25 + 0.25 × log₂0.25 + 0.5 × log₂0.5 + 0.25 × log₂0.25 + 0.25 × log₂0.25 + 0.5 × log₂0.5 + 0.5 × log₂0.5 + 0.5 × log₂0.5) = 5 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 + 2 × ln2 + 2 × ln2 + 2 × ln2 = 9.7 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 × 3 + 1 × 8 + 1 × 9 = 17

According to these complexity indicators, it was proved that ASC₁(3,4,3) < ASC₂(4,4,4), then ASC(i−1,j,k) ˂ ASC(i,j,k).