BUS606: Operations and Supply Chain Management

In order to identify reliable and consistent complexity indicator(s), four relevant complexity methods were described and mutually compared, namely index of vertex degree, modified flow complexity, system design complexity, and process complexity indicator.

Index of Vertex Degree

This indicator was originally developed by Bonchev to quantify the structural complexity of general networks. He adopted Shannon's information theory by applying the entropy of information H (α) in describing a message of N symbols. Further it is assumed that such symbols are distributed according to some equivalence criterion α into k groups of N₁, N₂, …, N_k symbols. Then, entropy of information H (α) is calculated by the formula:

$H(α)=−∑^{k}_{i=1}p_ilog_2p_i=−∑^{k}_{i=1} \dfrac{N_i}{N}log_2\dfrac{N_i}{N}$, (1)

where $p_i$ specifies the probability of the occurrence of the elements of the ith group.

Further, he substituted symbols or system elements for the vertices, and defined the probability for a randomly chosen system element i to have the weight w_i as p_i = w_i/∑w_i with ∑w_i = w, and ∑p_i = 1.

Then, the probability for a randomly chosen vertex i in the complete graph of V vertices to have a certain degree deg (v)i can be expressed by the formula:

$p_i=\dfrac{deg(v)_i}{∑^{V}_{i=1}deg(v)_i}$ (2)

Assuming that the information can be defined, according Shannon, as I = H_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:

$H(W)=Wlog_2W−∑_{i=1}^{V}w_ilog_2w_i$ (3)

Since the maximum entropy is when all w_i = 1, then H_max = Wlog₂W, 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:

$I_{vd}=∑_{i=1}^{V}deg(v)_ilog_2deg(v)_i$ (4)

The vertex degree index was subsequently applied to measure the structural complexity of assembly supply chains, and compared with other existing complexity measures.

The selected complexity indicator was applied on the following examples of ASCs by Hamta et al. (see Figure 2). All the five ASCs had the same number of input components, but they differed in the number of operations and the number of machines.

Figure 2. The possible assembly supply chain (ASC) network with four input components and the corresponding structural alternatives.
The complexity values obtained by using Equation (4) are shown in Table 1.

Table 1. Obtained complexity values by I_vd.

Process Complexity Indcator

An additional ASC structural complexity measure that was considered was the so-called process complexity indicator (PCI), which was introduced for the purpose of enumerating the operational complexity of manufacturing processes. Its expression is as follows:

$PCI=−∑_{i=1}^{M}∑_{j=1}^{P}∑_{k=1}^{O}p_{ijk}⋅log_2p_{ijk}$, (5)

where p_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.
It is assumed that machines in a given manufacturing process are organized in a serial and/or parallel manner in Equation (5). The probability that part j is being processed due to operation k on an individual machine i is calculated in the following way. In the case when a part is processed on machines in a serial manner, then pijk 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.
The complexity values obtained by using Equation (5) are shown in Table 2.

Table 2. Obtained complexity values by process complexity indicator (PCI).

System Design Complexity

Guenov proposed three indicators for architectural design complexity measurement based on Boltzmann's entropy theory and axiomatic design theory. Those indicators can be principally applied also as ASC complexity measures. However, only one of them has been treated as a potential indicator to measure ASC structural complexity. Its description is as follows. Let us denote N as the number of interactions within a design matrix, and N₁, N₂, …, 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:

$Ω=C^{N_1}_{N} * C^{N_2}_{N−N_1} * C^{N_3}_{N−N_1−N_2}…*…C^{N_K}_{N−N_1}−…−N_{K−1}=\dfrac{N!}{N_1!N_2!…N_K!,}$ (6)

where:

$C^{N_1}{N}=\dfrac{N!}{(N−N_1)!N1!}$ (7)

The multiplicity Ω in Equation (6) is often called the degree of disorder.
The state of gas body g at a given time t where the gas body consists of N molecules, each characterized by n magnitudes φj was considered by Boltzmann. For each magnitude φj, 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:

$H=∑_{j=1}^{K} [f_jlnf_j]υ^µ$ (8)

According to Equation (8) and $f_j = N_j/υ^µ$, where the volume is assumed equal to unity, the following formula for complexity measure can be derived:

$∑N_jl_nN_j$ (9)

where Nj is interpreted as the number of interactions per single DP.
Its application to measure the structural complexity of ASC was provided by Modrak and Soltysova, where the model of assembly process is structured with two groups of objects, while the first group is denoted as input components (ICs) and the second group as workstations. The input components are assembled at these workstations. Then, for the purpose of measuring process complexity, the following transformation is proposed: ICs are substituted by DPs, according to relation DPs = B. PVs (where PVs mean process variables and B is the design matrix that defines the characteristics of the process design) and workstations are considered as PVs. Subsequently, the assembly process structure is transformed into design matrix (DM) with DP–PV relations and finally, the structural complexity of ASC can be enumerated.

The complexity values obtained by using Equation (9) are shown in Table 3.

Table 3. Obtained complexity values by system design complexity (SDC).

Modified Flow Complexity

The next possible ASC complexity indicators were developed by Crippa. The most suitable indicator of these is considered to be the so-called modified flow complexity (MFC) indicator. The MFC indicator enumerates all tiers (including Tier 0), nodes, and links and adds all these counts, weighted with determined α, β, and γ and coefficients. Nodes and links are counted only once, even if they are repeated in the graph. Node and link repetition is included in coefficients. The MFC indicator can be enumerated by the following equations:

$MFC=α.T+ β.N+γ.L$ (10)

$α=MTI=\dfrac{TN−N}{(T−1).N}$ (11)

$β=MTR=\dfrac{TN}{N}$  (12)

$γ=MLR=\dfrac{LK}{L}$  (13)

where MTI is multi-tier index; MTR is multi-tier ratio; MLR is multi-link ratio; N is the number of nodes; TN is the number of nodes per ith tier level; L is number of links; LK is number of links per ith tier level; T is the number of tiers.

The complexity values obtained by using Equation (10) are shown in Table 4.

Table 4. Obtained values enumerated by modified flow complexity (MFC).