Specifying Multiple Objectives

Let us first discuss the interface for managing multiple objectives. An empty model starts with one objective function, which is initially just 0.0. We'll refer to this as the primary objective. You can modify it in a few ways: you can set the Obj attribute, or you can use the setObjective method from your language API (e.g., Model.setObjective in Python). Note that for a multi-objective model, all objective functions must be linear, including the primary objective. In general, attributes and methods that aren't specific to multi-objective optimization will work with the primary objective function.

If you'd like to input additional objectives, your first step is to tell Gurobi how many objectives you'd like to have in your model. You do this by setting the NumObj attribute on the Gurobi model. Objectives are numbered 0 through NumObj-1. Once you've done this, you then modify or query objective <span>$</span>n<span>$</span> in two steps. The first is to set parameter ObjNumber to the desired objective number; the second is to use the ObjN variable attribute to access the corresponding objective entry. The primary objective is always objective <span>$</span>0<span>$</span>, so ObjN and Obj are equivalent when ObjNumber is 0. Additional objectives are always linear.

To give an example, after you set NumObj to 3, then ObjNumber can be set to <span>$</span>0<span>$</span>, <span>$</span>1<span>$</span>, or <span>$</span>2<span>$</span>. This parameter setting will determine the effect of setting attribute ObjN on a variable. Thus, for example, if you set ObjN to 5 for a variable x when ObjNumber is 2, then the objective coefficient in objective 2 for variable x will be set to 5.

You can also set an objective constant for each objective, using ObjNCon. Again, ObjNCon and ObjCon are equivalent when ObjNumber is <span>$</span>0<span>$</span>. You can also give each objective a name through ObjNName.

Note that a model has a single objective sense (controlled by the ModelSense attribute). This means that you can't maximize the first objective and minimize the second. However, you can achieve the same result with a simple trick. Each objective has a weight, and these weights are allowed to be negative. Minimizing an objective function is equivalent to maximizing the negation of that function.

You can change the number of objectives in your model as many times as you like. When you increase the objective count, the new objectives and their associated attributes are set to 0. When you decrease the count, objectives beyond the new count are discarded. If you set the number of objectives to zero, the model becomes a pure feasibility problem.

We have extended the LP and MPS file formats, so writing a model with multiple objectives to a file will capture those objectives. Similarly, if you read a model file that contains multiple objectives, then NumObj and ObjN will capture the objectives stored in the file. See the file format section for details.

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