# Knapsack Balancing via Multiobjectivization
## 5 Illustrative Examples  
In the examples, to solve bi-objective optimization problems, we use the general methodology of multiobjective optimization outlined in Section 4.1. Example problemsFrom the 7 th multiple constraint knapsack problem (in the format described under [[https://people.brunel.ac.uk/~%/Emastijb/jeb/orlib/mknapinfo.html](https://people.brunel.ac.uk/~%/Emastijb/jeb/orlib/mknapinfo.html)]([https://people.brunel.ac.uk/~%/Emastijb/jeb/orlib/mknapinfo.html](https://people.brunel.ac.uk/~%/Emastijb/jeb/orlib/mknapinfo.html)), accessed on 5 May 2024) (5 constraints and 50 items) in the [PERSON]'s OR-Library ([[https://people.brunel.ac.uk/~%Emastijb/jeb/orlib/files/mknap1.txt](https://people.brunel.ac.uk/~%Emastijb/jeb/orlib/files/mknap1.txt)]([https://people.brunel.ac.uk/~%Emastijb/jeb/orlib/files/mknap1.txt](https://people.brunel.ac.uk/~%Emastijb/jeb/orlib/files/mknap1.txt)) accessed on 5 May 2024), we derived 5 knapsack problems. In each such problem, denoted \(SP_{i}^{p}\), \(i=1,...,5\), the objective function and _ith_ constraint of the original multiple constraint knapsack problem serve as the objective function and the constraint, respectively. Next, each knapsack problem is reformulated as the corresponding \(\mathcal{L}n\)-\(\mathcal{S}um\)-\(\mathcal{P}rod^{p}\) KP problem (see (11)). Pareto front approximations  
To derive reasonably informative approximations of Pareto fronts to our example problems (11), we use the following set of \(\lambda\) vectors:  
\[\Lambda:=\{\lambda\in\mathbb{R}^{2}\mid\lambda_{1}=1-0.005j,\ \lambda_{2}=1- \lambda_{1},\ j\in\mathbb{N},\,j>0,\ \lambda_{1}>0\}. \tag{16}\]  
By this definition, \(|\Lambda|=199\), and \(\lambda_{1}+\lambda_{2}=1\) for all \(\lambda\in\Lambda\). We assume that \(\lambda\)'s are ordered due to the decreasing value of their first component, and each \(\lambda\) has a corresponding index, resulting from this order. So, \(\lambda^{1}=(0.995,0.005)\), \(\lambda^{2}=(0.990,0.010),\ldots,\ \lambda^{199}=(0.005,0.995)\). We also assume that \(\epsilon=\rho=0.001\). The method to compute Pareto optimal solutions to multiobjective optimization problems applied here can provide, in general, only subsets of Pareto optimal sets, there is no guarantee that all Pareto optimal solutions are derived. Here, wanting for the illustration to compute as many Pareto optimal solutions as possible, we take advantage of the fact that all instances of problem (11) are solved to optimality in a split second. Thus, the rather large number of \(\lambda\)'s used in the example (in practical terms one can even say extraagantly large) is not an issue. Solver  
To derive \(\hat{y}\) and Pareto optimal solutions for each \(\lambda\in\Lambda\), we use Gurobi (version 10.0.0) for Microsoft Windows (x64). Gurobi ([28]) is a general mixed-integer solver, this means it solves linear and quadratic programming problems with integer variables. It is a commercial product but licensed free for any academic user, with all functionalities available under paid licenses. In our case, the solver is installed on a laptop equipped with an Intel Core i7-7700 HQ CPU with 16 GB RAM. Results  
The results for \(SP_{1}^{p}\) and \(SP_{2}^{p}\) are reported in Tables 2 and 3. The results for \(SP_{i}^{p}\), \(i=3,4,5\), are reported in the Appendix A. In these tables, the row with \(j=0\) corresponds to the optimal solution of \(\mathcal{S}um\) KP, \(x^{opt\_sum}\), and the row with \(j=200\) corresponds to the optimal solution of \(\mathcal{P}rod^{p}\) KP, \(x^{opt\_prod}\). As observed in many multiobjective combinatorial problems (see, e.g., [29]), multiple \(\lambda\) vectors can correspond to the same Pareto optimal outcome. Furthermore, this is also the case for our test problems, e.g., for problem \(SP_{1}^{p}\) (Table 2) the outcomes of Pareto optimal solutions for \(\lambda^{2}\) to \(\lambda^{46}\) are the same as those corresponding to \(\lambda^{1}\). Repeated Pareto optimal outcomes for \(\lambda\in\Lambda\) are not reported in the tables.