Solving your first business problem with Hexaly Modeler¶

Reading input data¶

Data files for this problem are available in folder hexaly/examples/carsequencing and more files can be downloaded from http://www.csplib.org (problem number 001). The format of these files is:

• 1st line: number of cars; number of options; number of classes.

• 2nd line: for each option, the maximum number of cars with this option in a block (parameter P).

• 3rd line: for each option, the block size to which the maximum number refers (parameter Q).

• Then for each class: index of the class; number of cars in this class; for each option, whether or not this class requires it (0 or 1).

Below is the code of the input() function to read the input data. Note that undefined variables are equal to nil. This allows checking the existence of a variable (here, we throw an error if inFileName is not defined). Then, we simply read integers one after the other in the file (regardless of line breaks).

To simplify the modeling, we also define an initial sequence of cars, starting with all the cars belonging to the first class, followed by all the cars belonging to the second class, and so on:

use io;

/* Read instance data */
function input() {
    local usage = "Usage: hexaly car_sequencing.hxm inFileName=inputFile"
            + "[solFileName=outputFile] [hxTimeLimit=timeLimit]";

    if (inFileName == nil) throw usage;
    local inFile = io.openRead(inFileName);
    nbPositions = inFile.readInt();
    nbOptions = inFile.readInt();
    nbClasses = inFile.readInt();

    maxCarsPerWindow[o in 0...nbOptions] = inFile.readInt();
    windowSize[o in 0...nbOptions] = inFile.readInt();

    local pos = 0;
    for [c in 0...nbClasses] {
        inFile.readInt(); // Note: index of class is read but not used
        nbCars[c] = inFile.readInt();
        options[c][o in 0...nbOptions] = inFile.readInt();
        initialSequence[p in pos...pos+nbCars[c]] = c;
        pos += nbCars[c];
    }
}

Modeling the problem¶

One of the most important steps when modeling an optimization problem with Hexaly is to define the decision variables. These decisions are such that instantiating them allows evaluating all other expressions in your model (in particular, the constraints and objectives). Hexaly Optimizer will try to find the best possible decisions with respect to the given objectives and constraints. Here, we want to decide the order of cars in the production line. The best way to model an order with Hexaly is to define a list decision variable. Here, the list represents the sequence of cars: the i-th element of the list corresponds to the index of the i-th car to build. In other words, if the i-th element in the list is equal to j, then the i-th car to build is the one described by initialSequence[j]:

sequence <- list(nbPositions);

Defining the constraints of your model is another important modeling task. Generally speaking, constraints should be limited to strictly necessary requirements. In the car sequencing problem, it is physically impossible to have two cars at the same position. On the contrary, satisfying all P/Q ratios is not always possible, which is why we define it as a first-priority objective instead. Limiting the use of constraints is generally a good practice, as Hexaly Optimizer tends to benefit from denser search spaces. Therefore, symmetry-breaking constraints and other types of non-necessary constraints should also be avoided.

To satisfy the assignment requirements, each car to be produced must be present exactly once on the assembly line. Using the partition operator, we ensure that every element between 0 and the total number of positions (excluded) is present exactly once in the list, meaning that every car appears exactly once in the sequence:

constraint partition(sequence);

Before writing the objective function, we introduce some intermediate expressions. From the value of the sequence, we can compute, for each option and each position in the line, the number of cars with this option in the window starting at this position:

nbCarsWindows[o in 0...nbOptions][j in 0...nbPositions-windowSize[o]+1]
        <- sum[k in 0...windowSize[o]](options[initialSequence[sequence[j + k]]][o]);

Indeed, for each position p, the index of the car at position p is sequence[p]. This car is described by the expression initialSequence[sequence[p]]. To know whether this car includes any option o, we check the value of options[initialSequence[sequence[p]]][o]. This ability to define intermediate expressions makes the model more readable and more efficient (if these intermediate expressions are used in several other expressions).

We can then deduce the number of violations on overcapacities for each option and each window:

nbViolationsWindows[o in 0...nbOptions][p in 0...nbPositions-windowSize[o]+1]
        <- max(nbCarsWindows[o][p] - maxCarsPerWindow[o], 0);

Note that we are using here a different way to express a sum. Indeed, you can define a sum with the sum operator, but you can also use arithmetic operators + and -. Ultimately, the last step consists in defining the objective function, which is the sum of all violations:

totalViolations <- sum[o in 0...nbOptions][p in 0...nbPositions-windowSize[o]+1](
        nbViolationsWindows[o][p]);
minimize totalViolations;

Parameterizing the optimizer¶

Some parameters can be defined in the param() function in order to control the search:

function param() {
    if (hxTimeLimit == nil) hxTimeLimit = 60;
}

The control parameters can also be set in command line, instead of being defined in the HXM file. For instance, the following command line launches the resolution for 30 seconds on the instance hexaly/examples/carsequencing/instances/carseq_100_8_20_86.in.

> hexaly car_sequencing.hxm inFileName=instances/carseq_100_8_20_86.in hxTimeLimit=30

All control parameters can be found in the section Built-in variables and functions.

Setting an initial solution¶

The param function can also be used to set an initial solution. Having an initial solution is not necessary for Hexaly Optimizer to function properly, but it can sometimes be useful if you wish to start the optimization from an existing solution, or for debugging purposes.

To initialize the value of a list decision variable, you must first make sure it is empty, then add its elements one by one:

sequence.value.clear();
for [p in 0...nbPositions]
    sequence.value.add(p);

Launching the resolution¶

Having launched Hexaly Optimizer with the above command line, you should observe the following output:

Hexaly Optimizer 14.0.20250724-Linux64. All rights reserved.
Load car_sequencing.hxm...
Run input...
Run model...
Run param...
Run optimizer...

Model:  expressions = 10929, decisions = 1, constraints = 1, objectives = 1
Param:  time limit = 60 sec, no iteration limit

[objective direction ]:     minimize

[  0 sec,       0 itr]:          403
[ optimality gap     ]:      100.00%
[  1 sec,   32710 itr]:            6
[  2 sec,   97525 itr]:            2
[  3 sec,  167931 itr]:            1
[  4 sec,  241127 itr]:            1
[  4 sec,  241144 itr]:            0
[ optimality gap     ]:           0%

241144 iterations performed in 4 seconds

Optimal solution:
obj    =            0
gap    =           0%
bounds =            0

Run output...

Each second (in accordance with parameter hxTimeBetweenDisplays, of default value 1), a brief report of the current state of the search is displayed: the number of elapsed seconds and the number of iterations performed, and the values of the objective functions given in lexicographic order (separated with |).

You can disable all displays by setting parameter hxVerbosity to 0.

Writing solutions¶

You can define an output() function that will be called by Hexaly at the end of the search. Here, we define a function writing the solution of the problem in a file whose name must be assigned (for instance in the command line) to some variable solFileName. It writes the solution in the following format:

1. First line: the objective value

2. Second line: for each position p, index of the class at position p

You can retrieve the value of any expression (decisions, intermediate expressions) thanks to the value attribute. Writing in a file is done with the usual print and println functions:

function output() {
    if (solFileName == nil) return;
    local solFile = io.openWrite(solFileName);
    solFile.println(totalViolations.value);
    local listSolution = sequence.value;
    for [p in 0...nbPositions]
        solFile.print(initialSequence[listSolution[p]], " ");
    solFile.println();
}

Complete code¶

Here is the complete Hexaly Modeler code for the Car Sequencing Problem:

use io;

/* Read instance data */
function input() {
    local usage = "Usage: hexaly car_sequencing.hxm "
            + "inFileName=inputFile [solFileName=outputFile] [hxTimeLimit=timeLimit]";

    if (inFileName == nil) throw usage;
    local inFile = io.openRead(inFileName);
    nbPositions = inFile.readInt();
    nbOptions = inFile.readInt();
    nbClasses = inFile.readInt();

    maxCarsPerWindow[o in 0...nbOptions] = inFile.readInt();
    windowSize[o in 0...nbOptions] = inFile.readInt();

    local pos = 0;
    for [c in 0...nbClasses] {
        inFile.readInt(); // Note: index of class is read but not used
        nbCars[c] = inFile.readInt();
        options[c][o in 0...nbOptions] = inFile.readInt();
        initialSequence[p in pos...pos+nbCars[c]] = c;
        pos += nbCars[c];
    }
}

/* Declare the optimization model */
function model() {
    // sequence[i] = j if class initially placed on position j is produced on position i
    sequence <- list(nbPositions);

    // sequence is a permutation of the initial production plan, all indexes must
    // appear exactly once
    constraint partition(sequence);

    // Number of cars with option o in each window
    nbCarsWindows[o in 0...nbOptions][j in 0...nbPositions-windowSize[o]+1]
            <- sum[k in 0...windowSize[o]](options[initialSequence[sequence[j + k]]][o]);

    // Number of violations of option o capacity in each window
    nbViolationsWindows[o in 0...nbOptions][p in 0...nbPositions-windowSize[o]+1]
            <- max(nbCarsWindows[o][p] - maxCarsPerWindow[o], 0);

    // Minimize the sum of violations for all options and all windows
    totalViolations <- sum[o in 0...nbOptions][p in 0...nbPositions-windowSize[o]+1](
            nbViolationsWindows[o][p]);
    minimize totalViolations;
}

/* Parametrize the solver */
function param() {
    // Set the initial solution
    sequence.value.clear();
    for [p in 0...nbPositions]
        sequence.value.add(p);
    if (hxTimeLimit == nil) hxTimeLimit = 60;
}

/* Write the solution in a file with the following format:
* - 1st line: value of the objective;
* - 2nd line: for each position p, index of class at positions p. */
function output() {
    if (solFileName == nil) return;
    local solFile = io.openWrite(solFileName);
    solFile.println(totalViolations.value);
    local listSolution = sequence.value;
    for [p in 0...nbPositions]
        solFile.print(initialSequence[listSolution[p]], " ");
    solFile.println();
}