new Double(object.toString());

But it seems weird to me that you're going from an Object to a Double. You should have a better idea what class of object you're starting with before attempting a conversion. You might have a bit of a code quality problem there.

Note that this is a conversion, not casting.

This is called Autoboxing which is a feature that was added in Java 5. It will automatically convert between primitive types and the wrapper types such as double (the primitive) and java.lang.Double (the object wrapper). The java compiler automatically transforms the line:

Double d = 5.1;

into:

Double d = Double.valueOf(5.1);

You need to create a class for that that will hold exactly these two values:

import java.util.*;

class MyValuePair {
    private int intValue;
    private double doubleValue;

    MyValuePair(double doubleValue, int intValue) {
        this.intValue = intValue;
        this.doubleValue = doubleValue;
    }

    //Getter and Setter as well as constructor(s) as needed
    public String toString() {
        return "(" + doubleValue + ", " + intValue + ")";
    }

}

public class Dijkstra {

    static Map<Integer, MyValuePair> map = new HashMap<>();

    public static void main(String[] args) {
        MyValuePair pair = new MyValuePair(4.0, 5);
        map.put(1, pair);
        System.out.println(map);
    }

}

So your value pairs go in MyValuePair and the objects of that class will go in the Map. You can create a new public class for it if you need (= in a new file with public keyword) or leave it in there as internal class.

Anything else with Lists and whatever you need will go in the Dijkstra class.

After the comment:

As said it is possible! But you do not want to do that. However this would be a code that would do the same where the Dijkstra class holds the values as well:

public class Dijkstra {

    static Map<Integer, Dijkstra> map = new HashMap<>();

    private double doubleValue;
    private int intValue;

    public Dijkstra(int intValue, double doubleValue) {
        this.intValue = intValue;
        this.doubleValue = doubleValue;
    }

    public String toString() {
        return "(" + doubleValue + ", " + intValue + ")";
    }

    public static void main(String[] args) {
        Dijkstra pair = new Dijkstra(4.0, 5);
        map.put(1, pair);
        System.out.println(map);
    }

}