Ordinal priority approach

The OPA method[edit]

The OPA model is a linear programming model, which can be solved using a simplex algorithm. The steps of this method are as follows:^[11]


Step 1: Identifying the experts and determining the preference of experts based on their working experience, educational qualification, etc.


Step 2: identifying the criteria and determining the preference of the criteria by each expert.


Step 3: identifying the alternatives and determining the preference of the alternatives in each criterion by each expert.


Step 4: Constructing the following linear programming model and solving it by an appropriate optimization software such as LINGO, GAMS, MATLAB, etc.


${\textstyle {\begin{aligned}&MaxZ\\&S.t.\\&Z\leq r_{i}{\bigg (}r_{j}{\big (}r_{k}(w_{ijk}^{r_{k}}-w_{ijk}^{{r_{k}}+1}){\big )}{\bigg )}\;\;\;\;\forall i,j\;and\;r_{k}\\&Z\leq r_{i}r_{j}r_{m}w_{ijk}^{r_{m}}\;\;\;\forall i,j\;and\;r_{m}\\&\sum _{i=1}^{p}\sum _{j=1}^{n}\sum _{k=1}^{m}w_{ijk}=1\\&w_{ijk}\geq 0\;\;\;\forall i,j\;and\;k\\&Z:Unrestricted\;in\;sign\\\end{aligned}}}$


In the above model, ${\displaystyle r_{i}(i=1,...,p)}$ represents the rank of expert ${\displaystyle i}$, ${\displaystyle r_{j}(j=1...,n)}$ represents the rank of criterion ${\displaystyle j}$, ${\displaystyle r_{k}(k=1...,m)}$ represents the rank of alternative ${\displaystyle k}$, and ${\displaystyle w_{ijk}}$ represents the weight of alternative ${\displaystyle k}$ in criterion ${\displaystyle j}$ by expert ${\displaystyle i}$. After solving the OPA linear programming model, the weight of each alternative is calculated by the following equation:


${\displaystyle {\begin{aligned}&w_{k}=\sum _{i=1}^{p}\sum _{j=1}^{n}w_{ijk}\;\;\;\;\forall k\\\end{aligned}}}$


The weight of each criterion is calculated by the following equation:


${\displaystyle {\begin{aligned}&w_{j}=\sum _{i=1}^{p}\sum _{k=1}^{m}w_{ijk}\;\;\;\;\forall j\\\end{aligned}}}$


And the weight of each expert is calculated by the following equation:


${\displaystyle {\begin{aligned}&w_{i}=\sum _{j=1}^{n}\sum _{k=1}^{m}w_{ijk}\;\;\;\;\forall i\\\end{aligned}}}$