Parametric Biobjective Linear Programming

TL;DR

This work introduces parameterized biobjective linear programming ($PBLP$) with a single nonnegative parameter and develops a theoretical bridge to a corresponding triobjective problem through weighted-sum scalarization and weight-set decomposition. It establishes a two-case framework ($PBLP^1$ and $PBLP^2$) and proves a one-to-one correspondence between parametric solutions and triobjective solutions under specific weight mappings, characterizing the evolution of Pareto sets via line segments with calculable slopes. The paper proves that the number of breakpoints is finite and ties breakpoints to changes in extreme nondominated images, providing structural insights and several illustrative examples. Finally, it offers two algorithms—the Breakpoint Enumeration Algorithm and an Adapted Weight-Set Decomposition—to compute breakpoints and minimal solution sets, leveraging existing MOLP techniques and the geometry of weight sets. The results advance sensitivity analysis for parametric multi-objective problems and enable efficient tracking of Pareto-optimal solutions as the parameter varies, with clear pathways for practical implementation.

Abstract

We consider parametric linear programming problems with multiple objective functions depending linearly on some parameter. Both parametric (single-objective) linear programming and (non-parametric) multi-objective linear programming are well-researched topics. However, literature on the combination of both, parametric linear programming with multiple objectives, is scarce. This research gap encourages our work in this field. Our main focus is on biobjective linear programs with a single parameter. We establish a connection of this problem to non-parametric multi-objective problems. Using the so-called weight set decomposition, we are able to explain the behavior of parametric biobjective linear programs when the parameter value is variated. We investigate two special cases of parametric biobjective linear programs: In the first, there is only one parametric objective and, in the second, the parametric dependency is the same for both objectives. We prove that there is a one-to-one correspondence between the solution of the parametric program and the solution of the triobjective program using the weighted sum scalarization. We provide structural insights to the solution of the parametric biobjective linear program with respect to extreme weights of the weight set of the triobjective linear program and develop solution strategies for the parametric program.

Figures (11)

• Figure 1: An illustration of an optimal value function $P(\lambda)$ with three breakpoints.
• Figure 2: An illustration of $\mathcal{L}_{\mathcal{W}(\mathrm{PBLP}^1(\lambda))}$ for different values of $\lambda$ in the weight set of TOLP.
• Figure 3: An illustration of $\mathcal{L}_{\mathcal{W}(\mathrm{PBLP}^2(\lambda))}$ for different values of $\lambda$ in the weight set of TOLP.
• Figure 4: Weight set of TOLP with four weight set components.
• Figure 5: Illustration of line segments of $\mathrm{PBLP}^j$ for parameter values, $\lambda=0,1,2$ in the weight set of TOLP. (\ref{['fig5:a']}) Line segments $\mathcal{L}_{\mathcal{W}(\mathrm{PBLP}^1 (\lambda)}$ of $\mathrm{PBLP}^1$ and (\ref{['fig5:b']}) Line segments $\mathcal{L}_{\mathcal{W}(\mathrm{PBLP}^1 (\lambda)}$ of $\mathrm{PBLP}^2$

Theorems & Definitions (40)

• Definition 1: Linear Parametric Linear Program
• Definition 2
• Definition 3: Multi-objective Linear Program
• Definition 4: Efficiency and nondominance
• Definition 5
• Definition 6
• Theorem 7: Isermann Isermann1974
• Definition 8
• Proposition 9: Przybylski et al. Przybylski2010
• Definition 10: Parametric Biobjective Linear Program