# Collaborative Planning in Non-Hierarchical Networks--An Intelligent Negotiation-Based Framework
## 3 Framework Proposal
### Multi-Objective Optimization Module  
This module aims to seek optimal solutions in each phase of the negotiation, taking into account multiple assessment criteria such as cost, due date compliance, and partner KPIs. In each phase, each combination of partner proposals corresponds to an alternative plan. The selection of the alternative plans considering multiple conflicting criteria can be formulated as a multi-objective optimization problem. The multi-objective mathematical formulation of the partner's selection problem considering the cost, the collaboration degree, and the quality certification as criteria can be expressed as follows:  
\[\text{Min}f_{1}(x)=\sum_{i=1}^{m}\sum_{j=1}^{m}\sum_{k=1}^{l}c_{ijk}x_{ijk} \text{ COST}, \tag{1}\]  
\[\text{max}f_{2}(x)=\sum_{i=1}^{m}\sum_{j=1}^{m}f_{ij}\sum_{k=1}^{l}x_{ijk} \text{ COLLABORATION}, \tag{2}\]  
\[\text{max}\ f_{3}(x)=\sum_{i=1}^{m}\sum_{j=1}^{m}q_{ij}\sum_{k=1}^{l}x_{ijk} \text{ QUALITY}, \tag{3}\]  
where \(n\) is the number of partners, \(m\) is the number of phases, and \(l\) is the number of proposals of each partner in the stage. There are three objectives to optimize: minimization of the cost (\(f_{1}(x)\)), maximization of the collaboration degree (\(f_{2}(x)\)), and maximization of the quality certification (\(f_{3}(x)\)). In this model, \(c_{ijk}\) denotes the costs of proposal \(k\) in phase \(j\) and partner \(i\); \(f_{ij}\) and \(q_{ij}\) are, respectively, the collaboration degree and quality certification of partner \(i\) in phase \(j\). The binary decision variable \(x_{ijk}\) indicates if proposal \(k\) in phase \(j\) and partner \(i\) is selected. The constraints of the model are as follows:  
* To impose that just one proposal is selected for each phase, \[\sum_{i=1}^{n}\sum_{k=1}^{l}x_{ijk}=1,\ \ \forall j\in\{1,\dots,m\};\] (4)
* To guarantee that the proposal selected in each phase does not overlap in time with the selected proposal of the next phase (precedence constraint), \[\sum_{i=1}^{n}\sum_{k=1}^{l}\Big{(}t_{ijk}+d_{ijk}\Big{)}x_{ijk}\leq\sum_{i=1 }^{n}\sum_{k=1}^{l}t_{ij+1k}x_{ij+1k}\,\ \forall j\in\{1,\dots,m-1\},\] (5) where \(t_{ijk}\) and \(d_{ijk}\) are, respectively, the initial date and duration of the proposal \(k\) in phase \(j\) and partner \(i\);
* To guarantee that the due date of the project (\(T\)) is satisfied,\[\sum_{i=1}^{n}\sum_{k=1}^{l}(t_{imk}+d_{imk})x_{imk}\leq T; \tag{6}\]
* To define the decision variables as binary values, where 1 means that the proposal is selected and 0 that the proposal is not selected, \[x_{ijk}\in\{0,1\}\ \forall i=1,\ldots,n\ \forall j=1,\ldots,m\ \forall k=1,\ldots,l.\] (7)  
This problem has a combinatorial nature, and the aim is to identify, from the set of all feasible alternative plans, those that are efficient. The multiple conflicting criteria of the problem give rise to the existence of a set of efficient solutions known as the Pareto-optimal set. A solution is Pareto-optimal if there is no other solution that dominates it, i.e., none of the objectives can be improved without deteriorating at least one of the other objectives. Pareto-optimal solutions are incomparable and define a Pareto-optimal frontier in the objective space representing different tradeoffs among objectives. The multi-objective optimization problem can be reformulated as a single objective optimization problem using scalarization methods by introducing some weights. Different weight combinations allow computing an approximation to the Pareto-optimal set using a single objective optimization method [32]. In order to search for efficient plans, the weighted sum method with different combinations of normalized weights was implemented. Moreover, since the objectives have different magnitude scales, normalization was performed for each objective.