Most businesses want to maximize profits. But what about other priorities? Environmentally conscious firms, for example, may want to maximize profits and minimize greenhouse gas emissions.
Juggling multiple priorities can complicate decision making. Mathematical optimization supports leaders in navigating these tradeoffs.
What Does “Multiple Objectives” Mean in Mathematical Optimization?
Optimization problems with multiple objectives analyze multiple, competing objectives (which are your goals, or the things you want to optimize). First-year economics students may be familiar with several aspects of multiple objective problems, especially “opportunity costs” (i.e., improving performance on one objective inherently means diverting resources or attention from another) and “pareto optimum” (i.e., a state in which it is impossible to improve the firm’s performance on one metric without decreasing its performance on another).
Common multiple objective problems include:
- Profits vs. Greenhouse Gas Emissions: For firms that want to earn profit, but mitigate environmental and reputational costs.
- Cost vs. Quality: For firms that want to maintain brand and product integrity while meeting the needs of price-conscious customers.
- Labor Costs vs. Service Quality: For customer-facing firms that want to minimize labor costs while maintaining a high level of service to customers.
- Risk vs. Returns: For investment portfolios that aim to maximize returns while minimizing risks.
The challenge presented by multiple objective problems is navigating the tradeoffs between their multiple goals. How do you align profits with environmental impact? Or balance risk with potential returns? To make these decisions, it’s important to understand the measurable tradeoffs between the available options.
Can Gurobi Solve Multiple Objective Problems?
Yes. Gurobi enables you to solve multiple objective problems using a number of techniques, including:
- Blended Objectives: Optimizes a weighted combination of objectives. For example, your team may determine that lowering costs is twice as valuable as increasing quality, so your weighted combination favors cost-cutting at twice the value.
- Hierarchical Objectives (i.e., the Lexicographic Approach): Optimizes the highest priority objective, then the second highest, until all objectives have been optimized in priority order. In this example, your team would first prioritize cost cutting, and then optimize quality standards within the optimized cost boundary.
- Hybrid Approach: Optimizes a combination of blended and hierarchical objectives. For example, your team may maximize profits first, then optimize for a blend of cost cutting and quality.
For more information on techniques for solving multiple objective problems, check out our documentation on multiple objectives.
What are Best Practices for Optimizing Multiple Objectives?
There are a few best practices to follow if you’re looking to optimize for multiple objectives:
- Clearly Define and Prioritize Objectives: Take time to understand your objectives. Which ones are most critical? It may be tempting to optimize for everything, but try to focus on the most important metrics for the question at hand.
- Ensure Objectives are Comparable: Do not compare apples to oranges. If possible, the objectives’ units of measure should be the same. This is especially critical for weighted objectives.
- Understand Tradeoffs: Work with subject matter experts to understand the relationship between your objectives. For example, is there a level of cost cutting that is so extreme that it would no longer be possible to meet minimum quality standards? Your model should reflect that so that cost cutting does not go too far.
- Use Solver Logs and Diagnostics: As the Gurobi solver analyzes your problem, it develops logs, summaries, and solution attributes that can help your team understand how it navigated objectives and found an answer. If the answer surprises you, the logs and diagnostics may help you understand the outcome on a deeper level.
- Iterate and Refine Based on Results: You do not have to stick with the first solution the solver produces. Discuss the findings with your leadership and subject matter experts. The output and logs may reveal additional insights or tradeoffs that they had not anticipated. After analyzing the results, you might want to tweak your model and try again.
Making the Qualitative Quantitative
Optimizing multiple objectives is difficult. When you optimize one metric, everything is objective, easily measurable, and singularly focused. Navigating between two or more metrics can feel more qualitative than data scientists and optimization professionals are accustomed to.
By clearly defining objectives, consulting with leadership and subject matter experts, and standardizing units of measure when possible, you can convert qualitative tradeoffs into quantitative insights.
For more information on solving for multiple objectives and other advanced modeling features offered by Gurobi, check out this article on hidden Gurobi gems.
