如何在 ojAlgo ExpressionBasedModel 中鼓励平衡解决方案

Question

我有以下型号：

        ExpressionsBasedModel model = new ExpressionsBasedModel();

        Variable va = model.addVariable("va")
                .lower(0);

        Variable vb = model.addVariable("vb")
                .lower(0);

        Variable vc = model.addVariable("vc")
                .lower(0);

        Variable vd = model.addVariable("vd")
                .lower(0);

        Variable ve = model.addVariable("ve")
                .lower(0);

        Expression ef = model.addExpression("ef")
                .lower(0);

        Expression eg = model.addExpression("eg")
                .lower(0);

        Expression eh = model.addExpression("eh")
                .lower(-240);

        Expression ei = model.addExpression("ei")
                .lower(0);

        Expression ej = model.addExpression("ej")
                .lower(0);

        Expression ek = model.addExpression("ek")
                .lower(0);

        Expression el = model.addExpression("el")
                .lower(-2000);

        ef.set(va, -50);
        el.set(va, -100);
        eg.set(va, 100);

        eh.set(vb, -30);
        ef.set(vb, 40);
        ej.set(vb, 20);

        ek.set(vc, -30);
        eg.set(vc, -30);
        ei.set(vc, 60);

        ei.set(vd, -30);
        eg.set(vd, -30);
        ek.set(vd, 60);

        ej.set(ve, -40);
        el.set(ve, -40);
        ek.set(ve, 20);

        ei.weight(1);
        ek.weight(1);

        /*
         * These next 2 lines required to get balanced solution.
         * Without them ei = 720 and ek = 0.
         */
//        ek.lower(360);
//        ei.lower(360);

        model.maximise();

        BasicLogger.debug(model);

虽然ei和ek权重相等，但是我得到的解是：

############################################
0 <= va: 6.4 <= 6.4
0 <= vb: 8 <= 8
0 <= vc: 15.111111 (300)
0 <= vd: 6.222222 (300)
0 <= ve: 4 (200) <= 4
0 <= ef: 0.0
0 <= eg: 0.0
-240 <= eh: -240.0
0 <= ei: 720.0
0 <= ej: 0.0
0 <= ek: -0.0
-2000 <= el: -800.0
############################################

ei == 720, ek == 0.

我更喜欢 ei 和 ek 尽可能平衡的解决方案，(ei == 360, ek == 360)。是否有某种方法可以将“尽可能平衡”的要求编码为加权表达式？

在这个具体示例中，我注释掉了实际导致我想要的行为的较低值约束。在现实生活中，模型是动态的，我可能有 5 个同等权重的表达式。我不知道正确的值以设置较低的约束。

Answer 1

对于未来的读者，我能够通过首先将 ei 和 ek 表示的值转换为新变量，使用现有表达式 ei 和 [=16] 来做到这一点=] 将新变量的值限制为这些表达式以前具有的值。然后我添加了一个新变量，给它很大的权重，并限制它小于每个新变量。

        Expression ei = model.addExpression("ei")
                .lower(0);

        Expression ek = model.addExpression("ek")
                .lower(0);

变成了：

        Expression ei = model.addExpression("ei")
                .level(0);
        Variable vei = model.addVariable("vei")
                .lower(0);
        ei.set(vei, -1);

        Expression ek = model.addExpression("ek")
                .level(0);
        Variable vek = model.addVariable("vek")
                .lower(0);
        ek.set(vek, -1);

权重变为：

        vei.weight(1);
        vek.weight(1);

并添加了余额变量和约束：

        Variable balance = model.addVariable("balance")
                .weight(100);

        model.addExpression("balance constraint for vei")
                .set(vei, 1)
                .set(balance, -1)
                .lower(0);

        model.addExpression("balance constraint for vek")
                .set(vek, 1)
                .set(balance, -1)
                .lower(0);

这会产生一个尽可能平衡的解决方案:

############################################
0 <= va: 6.4 <= 6.4
0 <= vb: 8 <= 8
0 <= vc: 11.111111
0 <= vd: 10.222222
0 <= ve: 4 <= 4
0 <= vei: 360 (1)
0 <= vek: 360 (1)
balance: 360 (100)
0 <= ef: 0.0
0 <= eg: 0.0
-240 <= eh: -240.0
0 <= ei: 0.0 <= 0
0 <= ej: 0.0
0 <= ek: -0.0 <= 0
-2000 <= el: -800.0
0 <= balance constraint for vei: 0.0
0 <= balance constraint for vek: 0.0
############################################

如何在 ojAlgo ExpressionBasedModel 中鼓励平衡解决方案

How to encourage a balanced solution in ojAlgo ExpressionBasedModel

linear-programming

ojalgo