1.4. Other MCDM Methods

1.4.1. LoPM Method

Farag (2020) describes the Limits on Property method as a material selection technique in which performance requirements are defined as lower limits, upper limits, and target values for relevant properties.

Candidate materials (alternatives) are first screened to ensure that all properties satisfy the specified bounds. The remaining materials are then ranked using a merit parameter that incorporates weighting factors reflecting the relative importance of each property. The merit parameter is defined as:

\[m = \sum_{i=1}^{n_l} \alpha_i \frac{Y_i}{X_i} + \sum_{j=1}^{n_u} \alpha_j \frac{X_j}{Y_j} + \sum_{k=1}^{n_t} \alpha_k \left| \frac{X_k}{Y_k} - 1 \right|\]

where \(l\), \(u\), and \(t\) denote lower-limit, upper-limit, and target-value properties, respectively; \(\alpha\) are weighting factors; \(X\) are candidate material properties; and \(Y\) are specified limits or target values. A lower value of \(m\) indicates a more suitable material.

Cost may be included either as an upper-limit property or as a modifier to the merit parameter:

\[m' = m \frac{C_X}{C_Y}\]

where \(C_X\) is the candidate material cost and \(C_Y\) is the specified cost limit.

Reference

Farag, M. M. (2020). Materials and process selection for engineering design. CRC Press.

1.4.2. LMAW Method

Pamučar et al. (2021) propose the Logarithm Methodology of Additive Weights (LMAW) as a multi-criteria decision-making (MCDM) method that combines logarithmic normalization, additive aggregation, and Bonferroni operator to provide stable and reliable rankings of alternatives. A distinctive feature of LMAW is its logarithmic framework for determining criteria weights and aggregating performance values, which contributes to strong resistance to the rank reversal problem in dynamic decision environments.

Assume a set of \(m\) alternatives \(A_i \ (i=1,\dots,m)\) evaluated with respect to \(n\) criteria \(C_j \ (j=1,\dots,n)\). Criteria weights are denoted by \(w_j\), satisfying \(\sum_{j=1}^n w_j = 1\). Criteria may be of benefit (maximization) or cost (minimization) type. Evaluations can be quantitative or qualitative and may involve multiple experts.

1.4.2.1. Steps of the LMAW Method

Step 1: Form the initial decision matrix

Alternatives are evaluated against all criteria, producing the initial decision matrix \(X = [\vartheta_{ij}]\). When multiple experts are involved, individual matrices are aggregated using the Bonferroni mean:

\[\begin{split}\vartheta_{ij} = \left( \frac{1}{k(k-1)} \sum_{x=1}^{k} (\vartheta_{ij}^{(x)})^{p} \sum_{\substack{y=1 \\ y \neq x}}^{k} (\vartheta_{ij}^{(y)})^{q} \right)^{\frac{1}{p+q}}\end{split}\]

where \(p,q \ge 0\) are Bonferroni stabilization parameters and \(k\) is the number of experts.

Step 2: Standardize the decision matrix

The elements of the decision matrix are standardized to obtain \(\tilde{X} = [\tilde{x}_{ij}]\):

• For benefit criteria:

\[\tilde{\vartheta}_{ij} = \frac{\vartheta_{ij} + \max_i \vartheta_{ij}}{\max_i \vartheta_{ij}}\]

• For cost criteria:

\[\tilde{\vartheta}_{ij} = \frac{\vartheta_{ij} + \min_i \vartheta_{ij}}{\vartheta_{ij}}\]

This ensures comparability of criteria with different units and scales.

Step 3: Determination of criteria weights

The criteria weights in the LMAW method are obtained using a logarithmic priority-based procedure, which allows experts to express the relative importance of criteria on a predefined linguistic scale.

First, each expert \(e \ (e=1,\dots,k)\) assigns priority values to the criteria, forming a priority vector:

\[\mathbf{P}^e = (\gamma_1^e, \gamma_2^e, \dots, \gamma_n^e)\]

where \(\gamma_j^e\) denotes the priority of criterion \(C_j\) given by expert \(e\). Higher values indicate higher importance.

Step 3.1: Definition of the absolute anti-ideal point

An absolute anti-ideal point \(\gamma_{\text{AIP}}\) is introduced to ensure logarithmic stability. It must be strictly smaller than the smallest priority value in the priority vector:

\[\gamma_{\text{AIP}} = \frac{\min_j \gamma_j^e}{s}\]

where \(s\) is a constant greater than the logarithm base (for natural logarithms, \(s = 3\) is recommended, in this case \(\gamma_{\text{AIP}} = 0.5\)).

Step 3.2: Construction of the relation vector

For each criterion, the relation between its priority value and the absolute anti-ideal point is calculated as:

\[\eta_j^e = \frac{\gamma_j^e}{\gamma_{\text{AIP}}}\]

This yields the relation vector:

\[\mathbf{R}^e = (\eta_1^e, \eta_2^e, \dots, \eta_n^e)\]

Step 3.3: Calculation of criteria weights

The weight of criterion \(C_j\) for expert \(e\) is computed using a logarithmic transformation:

\[w_j^e = \frac{\log(\eta_j^e)}{\log\!\left(\prod_{j=1}^{n} \eta_j^e\right)}\]

When multiple experts participate in the evaluation, the final aggregated criteria weights \(w_j\) are obtained by applying the Bonferroni aggregator to the individual expert weights \(w_j^e\):

\[\begin{split}w_j = \left( \frac{1}{k(k-1)} \sum_{x=1}^{k} \left(w_j^{(x)}\right)^p \sum_{\substack{y=1 \\ y \neq x}}^{k} \left(w_j^{(y)}\right)^q \right)^{\frac{1}{p+q}}\end{split}\]

Step 4: Construct the weighted matrix

First, logarithmic normalization of standardized values is performed:

\[\varphi_{ij} = \frac{\ln(\tilde{\vartheta}_{ij})} {\ln\!\left(\prod_{i=1}^{m} \tilde{\vartheta}_{ij}\right)}\]

The weighted matrix \(N = [\xi_{ij}]\) is then computed as:

\[\xi_{ij} = \frac{2 \varphi_{ij}^{w_j}}{(2 - \varphi_{ij})^{w_j} + \varphi_{ij}^{w_j}}\]

Step 5: Compute the final performance score

The overall performance index of each alternative is obtained by additive aggregation:

\[Q_i = \sum_{j=1}^{n} \xi_{ij}\]

Alternatives are ranked in descending order of \(Q_i\). A higher value of \(Q_i\) indicates a more preferable alternative. Due to its logarithmic weighting scheme and aggregation structure, the LMAW method demonstrates strong robustness and resistance to rank reversal.

Reference

Pamučar, D., Žižović, M., Biswas, S., & Božanić, D. (2021). A new logarithm methodology of additive weights (LMAW) for multi-criteria decision-making: Application in logistics. Facta Universitatis, Series: Mechanical Engineering, 19(3), 361–380.