What is the mathematical definition of the weighted-sum scalarization for two objectives?

Conditions

• There are exactly two objectives being combined.
• The second weight is defined as $1-w_1$.
• The objective is to minimize or maximize the combined scalar $J$.

Reasoning, step by step

1. Identify the first objective function, $OBJ_1$.
2. Identify the second objective function, $OBJ_2$.
3. Assign a weight $w_1$ to the first objective.
4. Assign the complementary weight $1-w_1$ to the second objective.
5. Compute the aggregate objective $J$ as the sum of the weighted objectives.

Example

The board displays the boxed equation $J = (w_1 \cdot OBJ_1) + (1-w_1) \cdot OBJ_2$. The lecturer explains that each selected objective is multiplied by a weight representing its importance, and the second weight is typically one minus that weight.

Common misconceptions

• Assuming the weights can be arbitrary and do not need to sum to 1.
• Believing that $w_1$ must always be greater than 0.5.
• Thinking that the scalarization eliminates the need to track the individual objective values.

Watch the explanation

Explore next

• How does varying the weight $w_1$ affect the resulting solution?
• What is the purpose of the weight table in the procedure?
• Why is the bi-objective case used for visualization?