Multiobjective Optimization
Multiobjective Optimization · Balancing goals that conflict
So far we have mostly looked for the best way to achieve a single goal. In real decisions, though, you often have to consider several goals at once. Raising the quality of a new product requires better parts or process improvements, and that raises costs. A company cannot simply raise quality or simply cut costs; it has to weigh both goals together.
Why we need to consider several goals at once
| Decision situation | Goals to weigh together |
|---|---|
| New product development | Minimize cost, maximize quality |
| Logistics route planning | Minimize shipping cost, minimize delivery time |
| Building an investment portfolio | Maximize return, minimize risk |
| Hospital resource allocation | Maximize treatment effect, minimize cost |
| Public policy making | Cut budget, improve policy effect and equity |
Finding suitable alternatives while considering several goals at once like this is called multiobjective optimization. The point is not to list goals, but to compare which alternatives are better when the goals conflict, and to find the options that fit the decision.
Single-objective versus multiobjective optimization
Minimize only the total cost C(x) of production plan x. The single plan with the lowest cost within the constraints is the answer.
We want to cut cost while also raising quality Q(x). If raising quality costs money, the two goals conflict, and there may be no alternative that is best on every goal at once.
That is why, in multiobjective optimization, what matters is not a single right answer but finding and comparing the efficient alternatives, each with its own strengths and weaknesses.
Pareto dominance and Pareto optimality
If one alternative is as good as or better than another on every goal, and strictly better on at least one, we say it Pareto dominates the other. Let's compare five product development plans (lower cost is better, higher quality is better).
| Plan | Cost | Quality score | Verdict |
|---|---|---|---|
| A | 40 | 70 | Not dominated → Pareto optimal |
| B | 50 | 85 | Not dominated → Pareto optimal |
| C | 60 | 90 | Not dominated → Pareto optimal |
| D | 45 | 65 | Dominated by A (A is cheaper and higher quality) |
| E | 55 | 80 | Dominated by B (B is cheaper and higher quality) |
There is no reason to choose D or E. A, B, and C, on the other hand, do not dominate one another. A is the cheapest but has low quality, C has the highest quality but is the most expensive, and B is in between. A calculation comparing every pair also confirmed that A, B, and C are the only plans that are not dominated.
Pareto optimal means a state in which no goal can be improved further without making performance on another goal worse. It does not mean the alternative is best on every goal; it means it is an option that no other alternative beats outright.
The Pareto frontier
The set of Pareto optimal alternatives is called the Pareto frontier. When the two goals are plotted, it is the boundary formed by alternatives where cutting cost further means lowering quality, and raising quality further means spending more.
In Figure 1, D lies in A's shaded area and E in B's, so they are excluded from the frontier. The frontier lets the decision maker focus on the alternatives worth choosing. Which of A, B, and C to pick, however, is a separate question that requires the decision maker's preferences, such as how much they are willing to pay for better quality. Moving along the frontier from A to B buys 15 points of quality for 10 more in cost (1.5 per unit), while going from B to C buys 5 points of quality for 10 in cost (0.5 per unit). These exchange rates become the basis for the choice.
Main methods of multiobjective optimization
5.1 Weighted sum method
Qn is the normalized quality score, and Cn is a cost preference score transformed so that it rises as cost falls. If cost is measured in thousands of dollars and quality on a 100-point scale, you cannot simply add them, so you have to put them on a common scale with normalization or a utility function. Here we used min–max normalization (quality 65–90, cost 40–60).
As Figure 2 shows, A ranks first when the quality weight wQ is below 0.45, B when it is 0.45–0.71, and C when it is above 0.71. This is why you should check whether the choice changes when the weights shift a little. The weighted sum method also cannot find solutions in parts of the frontier that dip inward (concave parts). Suppose that instead of B we had only B′(50, 78). B′ is a Pareto optimal solution, not dominated by A or C, but because it lies below the straight line joining A and C, it never ranks first at any weight (the gray line in Figure 2).
5.2 ε-constraint method
Optimize one goal and set acceptable thresholds for the others. When there are clear requirements that must be met, such as a budget or a minimum quality, you can set up the problem without choosing weights arbitrarily.
In Figure 3, setting the cost cap to 40, 50, and 60 selects A, B, and C respectively. Repeating with different caps yields the frontier points one by one, and unlike the weighted sum method, it can also find solutions in the concave part (B′).
5.3 Goal programming
Set target values and reduce how far you deviate from them (the deviations). Not every deviation needs to be reduced, so you distinguish them to suit the problem, for example penalizing only excess for cost and only shortfall for quality. Evaluating the three plans against the goals of cost 50 or less and quality 85 or more gives:
| Plan | Cost excess d⁺ (goal 50) | Quality shortfall d⁻ (goal 85) | Total deviation (weight 1) |
|---|---|---|---|
| A (40, 70) | 0 | 15 | 15 |
| B (50, 85) | 0 | 0 | 0 → both goals met |
| C (60, 90) | 10 | 0 | 10 |
Adding a goal such as delivery within 10 days adds deviation terms in the same way. If the deviations are in different units, you have to align them with weights or normalization.
5.4 Pareto-based evolutionary algorithms
When there are several goals and the variables are complex or the objective functions are nonlinear, evolutionary algorithms are used to search for Pareto optimal alternatives. A well-known method, NSGA-II, generates and evaluates many candidate solutions, keeps the ones that do not dominate each other, and keeps them spread evenly along the frontier, approximating the whole frontier. The results are generally approximations, so you need to check that the search range and computational settings are appropriate.
Which method should you choose?
| Situation | Suitable approach | Why |
|---|---|---|
| You can set the importance of each goal | Weighted sum method | Compares several goals as a single score |
| There are requirements that must be met | Epsilon-constraint method | Optimizes one goal while limiting the others |
| Targets and allowed deviations are clear | Goal programming | Manages the shortfall or excess for each goal |
| Exploring many alternatives in a complex problem | Pareto-based evolutionary algorithms | Approximately searches for efficient solutions across several objectives |
When choosing a method, look first at what the decision maker knows rather than at computational convenience. Methods can also be combined: you can set the budget and minimum quality as constraints and then examine the trade-offs among the remaining goals with a frontier.
Applications of multiobjective optimization
Weighs cost, quality, durability, and development time together to eliminate inferior designs and present the designs worth choosing.
Compares the trade-off between expected return and risk with a frontier. In practice, estimation error, liquidity, and transaction costs are also considered.
Lays out the balance among different values such as cost, treatment effect, access, budget efficiency, policy effect, and equity.
Comparing five product development plans on cost and quality, D and E were excluded as Pareto dominated by A and B respectively, and the Pareto optimal plans were A(40, 70), B(50, 85), and C(60, 90). Applying the weighted sum method after min–max normalization, B was selected for quality weights in the range 0.45–0.71, and B was also selected by the epsilon-constraint method with a cost cap of 50 and by goal programming (cost 50 or less, quality 85 or more).
First remove the alternatives that are beaten, then choose from the remaining frontier by preference.
- Pareto dominance: being as good or better on every goal and better on at least one.
- Pareto frontier: the set of non-dominated alternatives; the final choice requires the decision maker's preferences.
- Weighted sum method: put goals on a common scale, then combine them with weights. It is sensitive to the weights and cannot find solutions in concave parts.
- Epsilon-constraint method: optimize one goal and cap the rest. Changing the caps lets you sweep the frontier.
- Goal programming · evolutionary algorithms: manage deviations from targets, or approximate the frontier in complex problems.
Next is 09 Robust and stochastic optimization. We look at how to design decisions when the inputs themselves, such as demand or costs, are uncertain.