Linear Programming in Java for Assignment Problems

Linear Programming in Java: Solving the Assignment Problem

The assignment problem is a classic optimization problem that involves assigning a set of persons to a set of tasks exclusively, with the goal of minimizing the total travel time. In this case, we have three volunteers and three working locations, and the travel times between each volunteer and each location are known and fixed. For instance, the travel time between Volunteer 1 and Location 1 is 27, while the travel time between Volunteer 1 and Location 2 is 6.

Why This Matters

Linear programming is a powerful tool for solving optimization problems, but it requires a clear understanding of the technical reality vs ideal models. In the case of the assignment problem, the ideal model would be to assign each volunteer to the location with the shortest travel time, but this may not always be possible due to constraints such as each volunteer can be assigned to only one location, and one location only needs one volunteer. The failure to consider these constraints can result in a failure scale of 100%, where no optimal solution is found, or a cost of 27, which is the total travel time in the given example.

Key Insights

• The ojAlgo library provides a simple and efficient way to solve linear programming problems in Java, with a version of 56.2.0.
• The Apache Commons Math library is another popular choice for solving linear programming problems in Java, with a version of 3.6.1.
• The assignment problem can be solved using a cost matrix, where the cost of assigning a volunteer to a location is represented by the travel time between them.

Working Example

double[][] t = {
    {27, 6, 21},
    {18, 12, 9},
    {15, 24, 3}
};

int volunteers = t.length;
int locations = t[0].length;
ExpressionsBasedModel model = new ExpressionsBasedModel();
Variable[][] x = new Variable[volunteers][locations];

for (int i = 0; i < volunteers; i++) {
    for (int j = 0; j < locations; j++) {
        x[i][j] = model
            .newVariable("Assignment_" + i + "_" + j)
            .binary()
            .weight(t[i][j]);
    }
}

Practical Applications

• Use Case: A company like Uber can use linear programming to assign drivers to riders in a way that minimizes the total travel time.
• Pitfall: A common anti-pattern is to ignore the constraints of the problem, which can result in a suboptimal solution or even no solution at all.

References:

• https://www.baeldung.com/java-linear-programming-assignment-problem
• https://ojalgo.org/
• https://commons.apache.org/proper/commons-math/