Solving your first model with Hexaly Modeler

In this section, we show you how to model and solve your first problem: the optimization of the shape of a bucket. With a limited surface of material (S=π), we try to build a bucket that holds the largest volume.

This small example is more precisely described in our example tour. Here our main goal is to learn how to write and launch a model.

Writing the model

Below is a Hexaly program which models this non linear problem (see examples/optimal_bucket).

use io;

/* Declare the optimization model */
function model() {
    PI = 3.14159265359;

    // Numerical decisions
    R <- float(0, 1);
    r <- float(0, 1);
    h <- float(0, 1);

    // Surface must not exceed the surface of the plain disc
    surface <- PI * pow(r, 2) + PI * (R + r) * sqrt(pow(R - r, 2) + pow(h, 2));
    constraint surface <= PI;

    // Maximize the volume
    volume <- PI * h / 3 * (pow(R, 2) + R * r + pow(r, 2));
    maximize volume;

/* Parametrize the solver */
function param() {
    if (hxTimeLimit == nil) hxTimeLimit = 2;

/* Write the solution in a file with the following format:
 *  - surface and volume of the bucket
 *  - values of R, r and h */
function output() {
    if (solFileName == nil) return;
    local solFile = io.openWrite(solFileName);
    solFile.println(surface.value, "  ", volume.value);
    solFile.println(R.value, "  ", r.value, "  ", h.value);

All the variables of the model, called expressions, are declared using left arrows <-. Decision variables are introduced using the built-in function float() (or also bool(), int(), set(), list()). Intermediate expressions can be built upon these decision variables by using other operators or functions. For example, in the model above: power (pow), square root (sqrt), less than or equal to (<=). Many other mathematical operators are available, allowing you to model and solve highly-nonlinear combinatorial optimization problems. The keywords constraint or maximize are used for tagging expressions as constrained or maximized.

Launching the model

To solve this model, call Hexaly Optimizer with the HXM file as argument.

On Windows:

hexaly %HX_HOME%\examples\optimal_bucket\optimal_bucket.hxm

Note that on Windows, in a PowerShell window you would use the following line:

hexaly $env:HX_HOME\examples\optimal_bucket\optimal_bucket.hxm

On Linux or Mac OS the command line looks like this:

hexaly /opt/hexaly_|major|_|minor|/examples/optimal_bucket/optimal_bucket.hxm

Then, the following trace will appear in your console:

Hexaly Optimizer 12.5.20240516-Win64. All rights reserved.
Load optimal_bucket.hxm...
Run model...
Run param...
Run solver...

Model:  expressions = 26, decisions = 3, constraints = 1, objectives = 1
Param:  time limit = 2 sec, no iteration limit

[objective direction ]:     maximize

[  0 sec,       0 itr]:            0
[ optimality gap     ]:         100%
[  0 sec,   42898 itr]:      0.68709
[ optimality gap     ]:      < 0.01%

42898 iterations performed in 0 seconds

Optimal solution:
  obj    =      0.68709
  gap    =      < 0.01%
  bounds =     0.687189

If no time limit is set, the search will continue until optimality is proven (Optimal solution message) or until you force the stop of the program by pressing Ctrl+C. The trace in console starts with the key figures of the model: number of expressions, decisions, constraints and objectives.

Once the search is finished, the total number of iterations and the elapsed time are displayed, as well as the status and the value of the best solution found. The solution status can be Inconsistent, Infeasible, Feasible or Optimal.