Designing Compromise Solutions with Multi‑Objective Optimization

In real life, decisions often require considering multiple opinions and preferences to find the most feasible solution. Because different people assign different weights to factors, the same object can receive different evaluations.

Example: I tried to persuade my father to undergo a gastrointestinal endoscopy. He refused, citing his own reasons, leading to a conflict about whether to proceed with the examination.

We face two options: undergo the test or not, but the father and son evaluate them differently. How to resolve?

Typically, we "comprehensively consider" various factors and their relative importance to make a judgment. If parties have different weightings for the criteria, a consensus may be impossible, requiring a compromise.

How to compromise? This article discusses a mathematical model for designing compromise solutions.

Basic Idea

Assuming we accept each party's weights as reasonable and the resulting outcome as rational, the compromise solution is likely a "step back, broaden horizons" approach, meaning the new design will adjust some indicators. Determining which indicators and by how much is a computational problem.

Mathematical Model

Assume there are n decision makers, each with preferences over m indicators. Let the original solution have indicator values x₀ . We seek a new compromise solution with indicator values x .

Define variables and functions:

w_i: weight of the i‑th decision maker for the dimensions, with Σ_i w_i = 1.

f_i(x): satisfaction function of the i‑th decision maker for the compromise solution, defined as the weighted sum of squared deviations of each indicator.

The objective is to minimize the total dissatisfaction of all decision makers:

Minimize Σ_i w_i * f_i(x)

To reflect sensitivity to different indicators, we set an allowable adjustment range for each indicator, with lower and upper bounds l_j and u_j.

Thus the model can be expressed as a quadratic programming problem with constraints l_j ≤ x_j ≤ u_j.

Case Study

Consider two decision makers evaluating three indicators. The original indicator values are omitted for brevity. Their weights and dimension weights are as follows:

Decision maker 1, weight …

Decision maker 2, weight …

Allowable adjustment ranges are defined for each indicator.

Formulate the optimization model and solve the quadratic programming problem to obtain a compromise solution.

Conclusion

This article proposes a multi‑objective optimization model for generating compromise solutions when stakeholders have differing preferences. By adjusting indicator values within permissible ranges, a solution with higher overall satisfaction can be found. The method applies to corporate decisions, public policy, and any scenario requiring negotiation among multiple parties.

In personal reflection, I consider a compromise with my father, such as regular non‑invasive check‑ups combined with symptom monitoring, balancing his concerns with health monitoring needs.

Mathematical modeling can inspire, but its usefulness depends on whether the underlying assumptions hold, such as open communication and mutual respect for each other's weights.

Ultimately, designing compromise solutions involves not only calculations but also human communication, understanding, and respect; the model provides a rational tool, but the key lies in interpersonal dynamics.