I am trying to solve the multi-objective function of my linear program. Are there another approaches other than the weighted sum approximation?
Yes. There are plenty of other approaches to handle multiple objectives. First of all, you need to figure out, what you consider an optimal solution (set) to your multi objective optimization problem. To name a few notions of optimality you might consider
- Pareto optimality
- Lexicographic optimality (as @Kuifje suggests in a comment to the question)
- Max-ordering optimality
- Combinations of some of the above
Next, you need to figure out a way to actually achieve an optimal solution (set). Here you also have quite a toolbox to choose from. I am most familiar with the methodology developed for finding the set of non-dominated outcome vectors (the image of the set of Pareto optimal solutions). I your problem is in fact a multi-objective linear program (MOLP) (no integer constrained variables) the following methods can be used to generate all non-dominated outcome vectors to your MOLP
- The perpendicular search method (a simple search strategy based on weighted sum scalarizations). This method is very easy to implement for two objectives, and not so easy to implement for more than two. A method for using this dichotomic search when having more than two objectives is described in this paper by A. Przybylski, X. Gandibleux, and M. Ehrgott.
- The $\varepsilon$-constraint method
- The weighted $\varepsilon$-method
- Benson's method. Details of a primal-dual version is described in this paper by A. Löhne and B. Weißing. The algorithm is implemented in BENSOLVE.
Generally, I would advice you to study chapter 2, 3, 4, and 5 in Multicriteria Optimization by Matthias Ehrgott, 2nd edition. It gives a thorough and easy to follow indtroduction to the different optimality concepts and to some of the methods listed above.