## 4 Proposed Framework
### Layer 3: Multi Objective Optimization  
On the bases of these opinions (requirements/ suggestions/ factors) in the second layer, the past performance data of suppliers were collected, and an analysis was performed using the optimization algorithm in the third layer. This layer analyzed the capabilities of all the given suppliers against all requirements suggested by the managers. The past data of suppliers, i.e., values against factors from layer 2, were used to identify the most appropriate supplier. To make this framework more flexible, a change in objectives was allowed, such that any firm may change the objective according to its need. A combined objective function was formulated to evaluate the suppliers. Therefore, all the factors were used in this combined objective function. All vendors were evaluated individually on the basis of this objective function. Basically, the effectiveness of each supplier was assessed with respect to all these objectives. Metaheuristic algorithms used these values to output priority based on the best-suited suppliers, considering all objective values and constraints. The process began with initialization using Equation (5). This essentially acts like a worker bee that produces a change in position (solution) in memory depending on local information (visual information) and tests the amount of nectar (value) of the new source of fitness (new solution):  
\[x_{ij}=x_{\text{jmin}}+rand[1,0](x_{\text{jmax}}-x_{\text{jmin}}) \tag{5}\]  
Initially, random solutions were generated from the given data. These solutions were further used to determine the most appropriate solution. After the generation of initial random solutions, the probability of each solution was calculated to indicate the optimal solutions. Then, a probability check was performed using a probabilistic equation, i.e., Equation (6). This process acts like an onlooker bee, evaluating the information of nectar of all working bees and choosing a food source with a probability of a high quantity of nectar, where SN is the \(n^{\text{th}}\) solution:  
\[p_{i}=\frac{fit_{i}}{\sum_{n=1}^{SN}fit_{n}} \tag{6}\]  
Solutions in the previous steps were stored, and the more random solutions were generated as fitness values in the neighborhood of the optimum of the initially generated random values. Thisprocess helped to optimize the final solutions. After this probability check, random solutions are again generated using Equation (7):  
\[v_{ij}=x_{ij}+\varnothing_{ij}(x_{ij}-x_{kj}) \tag{7}\]  
where \(v_{ij}\) is the new solution, and \(x_{ij}\) is the old or previous solution. After finally using the objective function, each supplier was evaluated. After the random search and probability check, we used our constraints and objective function to obtain our final solution. The last step essentially followed the case of the scout bee, which produces a change in position in its memory and checks for an even greater quantity of nectar. With identification of a new maximum, this bee stores the new position and forgets the previous.  
In the developed framework, any enterprise can be used as a case study or problem, and unique data are then taken from that enterprise.