# Developing a Decision-Making Framework to Improve Healthcare Service Quality during a Pandemic
## 3 Method
### Multi-Objective Linear Programming  
Step 10: Development of an Initial Decision Matrix (IDM).  
In this stage, every stakeholder compares their criteria with all of the alternatives on a 10-point scale, where 1 is the least preferred and 10 is the most preferred. The main advantage of using SIMUS is its capability to group stakeholders' preference values into a single IDM in order to evaluate the alternatives with consensus. Since this study also wants to show the differences in the ranking of alternatives with and without consensus, the SMART evaluation values obtained in MAMCA were used as inputs for the IDM.  
Step 11: Solving the Multi-Objective Linear Programming (MOLP).  
This study considered nine criteria, with all being considered objective functions. For instance, if Criteria 1 was considered as the objective function, it was removed from the IDM, and the remaining criteria (2-9) were considered as constraints. Similarly, Criteria 2 would be removed from the IDM (as the objective function) in the second iteration, and Criteria 1 would be re-inserted and considered as a constant along with the rest of the criteria (Criteria 1, 3,..., 9). This process is repeated until SIMUS has considered every criterion as an objective function. Equation (5) shows a general representation of the first criteria optimization, where \(a_{n}\), \(b_{n}\), \(c_{n}\), \(d_{n}\), \(e_{n}\) are the variables that represent the stakeholders preference values, and \(Z_{n}\) represents the RHS threshold value. Equation (6) shows the ninth criteria optimization problem.  
\[\begin{array}{c}\max a_{1}x_{1}+b_{1}x_{2}+c_{1}x_{3}+d_{1}x_{4}+e_{1}x_{5} \\ s.t.\;\;a_{n}x_{1}+b_{n}x_{2}+c_{n}x_{3}+d_{n}x_{4}+e_{n}x_{5}\;\leq Z_{n},\; \forall n=2,\ldots 9\\ x_{j}\geq 0,\;\forall j=1,\ldots,5\end{array} \tag{5}\]  
\[\begin{array}{c}\max a_{9}x_{1}+b_{9}x_{2}+c_{9}x_{3}+d_{9}x_{4}+e_{9}x_{5} \\ s.t.\;\;a_{n}x_{1}+b_{n}x_{2}+c_{n}x_{3}+d_{n}x_{4}+e_{n}x_{5}\;\leq Z_{n},\; \forall n=1,\ldots 8\\ x_{j}\geq 0,\;\forall j=1,\ldots,5\end{array} \tag{6}\]  
Step 12: Forming an Efficient Result Matrix (ERM).  
In this step, the solutions are arranged to form a Pareto Efficient Matrix (PEM) and Efficient Result Matrix (ERM), since the obtained values were optimal [21]. The grouped values are then normalized, and the participating factor (PF) is determined based on how many objectives an alternative is able to satisfy. Next, the normalized participating factor (NPF) is calculated by dividing the PF by the number of criteria considered [21]. Once this has been done, the final results are achieved by multiplying SC \(\times\) PF, which enables the determination of the ERM alternative ranking.  
Step 13: Forming a Project Dominance Matrix (PDM).  
In this step, the alternatives are ranked to cross-verify the ERM results, as this helps the decision maker to compare or cross-verify the results obtained through SIMUS. The differences in value between each alternative and the other alternatives are grouped to form a PDM matrix. Next the final ranking is determined based on the difference between the dominant (rows) and subordinate (columns) values. Naturally, the alternative rankings produced by the ERM and PDM will be exactly matched with each other, and there will be no deviation in the result.  
Step 14: Result Analysis and Final Selection of Best Alternative with Consensus.  
Finally, the alternative rankings based on the PDM and ERM are compared with the BWM-MAMCA results. The DM can choose the best alternative by comparing their respective results. In the next section, the proposed framework is implemented in two different scenarios.