### Archive

Archive for April, 2009

## Solving traveling salesman problems using Solver Foundation

Update: see the comments below for some helpful hints. If you are unable to run this with your version of Solver Foundation and Gurobi, consider installing the lp_solve plugin for MSF. More details on this thread.

Here’s an example that I walked through during yesterday’s INFORMS session.  Erwin has two blog postings about Solver Foundation and the traveling salesman problem, but I want to throw in my two cents because I want to emphasize a couple of points:

1. By combining C# and Solver Foundation Services it is possible to express complex models clearly and succinctly.
2. It is very easy to build powerful, reusable model libraries using C# and Solver Foundation Services.
3. Solver Foundation Services code can be used in many different application environments (ASP.Net, silverlight, DB, command line apps, WPF, …) with minimal changes.

The traveling salesman problem is a classical problem in computer science, and you should bow your head in shame if you don’t know about it (and turn in your conference badge if you happen to be in Phoenix).  A salesperson needs to make a tour of a number of cities.  The restrictions are that she wants to visit each city once and only once, and she wants to minimize the distance travelled.  This is perhaps the definitive example of an NP-hard problem.

TSP can be solved using mixed integer programming – optimizing a linear goal with linear constraints, where some of the decision variables are integer.  In this first post I will show how to formulate and solve a TSP model using Solver Foundation Services.  In my second post I will show how to use the Gurobi MIP solver using SFS.   There are many different ways to model the TSP – here is a nice introduction.  My goal is to provide a clear, complete example – not build a “production level” TSP model, so I am going to choose a model formulation that dates back to 1960!  First, I need to establish a couple of building blocks that will help me construct the data for the model.  We need to know the distances between each pair of cities.  Typically we are provided the coordinates of the cities and need to derive the distances.  So I will introduce a Coordinate class that contains properties for the (x, y) coordinates, and properties to convert to latitude and longitude.  Finally, a method that computes the distance between points.

```using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using Microsoft.SolverFoundation.Services;

namespace Microsoft.SolverFoundation.Samples {
class TravelingSalesman {
// TSP coordinate.
public class Coordinate {
public int Name { get; set; }

// X-coordinate (from TSPLIB)
public double X { get; set; }

// Y-coordinate (from TSPLIB)
public double Y { get; set; }

public Coordinate(int name, double x, double y) {
Name = name;
X = x;
Y = y;
}

public double Latitude {
get { return Math.PI * (Math.Truncate(X) + 5 * (X - Math.Truncate(X)) / 3) / 180; }
}

public double Longitude {
get { return Math.PI * (Math.Truncate(Y) + 5 * (Y - Math.Truncate(Y)) / 3) / 180; }
}

// Geographic distance between two points (as an integer).
public int Distance(Coordinate p) {
double q1 = Math.Cos(Longitude - p.Longitude);
double q2 = Math.Cos(Latitude - p.Latitude);
double q3 = Math.Cos(Latitude + p.Latitude);
// There may rounding difficulties her if the points are close together...just sayin'.
return (int)(6378.388 * Math.Acos(0.5 * ((1 + q1) * q2 - (1 - q1) * q3)) + 1);
}
}

// TSP city-city arc.
public class Arc {
public int City1 { get; set; }
public int City2 { get; set; }
public double Distance { get; set; }
}

// Burma14 from TSPLIB. Optimal tour = 3323.
private static Coordinate[] data = new Coordinate[] {
new Coordinate(0, 16.47, 96.10),
new Coordinate(1, 16.47, 94.44),
new Coordinate(2, 20.09, 92.54),
new Coordinate(3, 22.39, 93.37),
new Coordinate(4, 25.23, 97.24),
new Coordinate(5, 22.00, 96.05),
new Coordinate(6, 20.47, 97.02),
new Coordinate(7, 17.20, 96.29),
new Coordinate(8, 16.30, 97.38),
new Coordinate(9, 14.05, 98.12),
new Coordinate(10, 16.53, 97.38),
new Coordinate(11, 21.52, 95.59),
new Coordinate(12, 19.41, 97.13),
new Coordinate(13, 20.09, 94.55)
};
}```

(The data for this 14-city problem comes from the TSPLIB library). If you’ve been following my blog you know that the building blocks of a Solver Foundation model are: sets, parameters, decisions, goals, and constraints. I am going to implement a simple formulation that is centered around the following (indexed) decisions:

• Assign[i,j]: this is equal to 1 if the optimal tour contains a trip (or arc) from city i to city j.
• Rank[i]: this is equal to the number of cities visited after arriving at city i.

We have one parameter in our model:

• Distance[I,j]: the distance from city i to city j.

With that in mind, here’s the model.  Explanation of the goals and constraints follow.

```
public static void Run() {
SolverContext context = SolverContext.GetContext();
context.ClearModel();
Model model = context.CreateModel();

// ------------
// Parameters
Set city = new Set(Domain.IntegerNonnegative, "city");
Parameter dist = new Parameter(Domain.Real, "dist", city, city);
var arcs = from p1 in data
from p2 in data
select new Arc { City1 = p1.Name, City2 = p2.Name, Distance = p1.Distance(p2) };
dist.SetBinding(arcs, "Distance", "City1", "City2");

// ------------
// Decisions
Decision assign = new Decision(Domain.IntegerRange(0, 1), "assign", city, city);
Decision rank = new Decision(Domain.RealNonnegative, "rank", city);

// ------------
// Goal: minimize the length of the tour.
Model.Sum(Model.ForEach(city, i => Model.ForEachWhere(city, j => dist[i, j] * assign[i, j], j => i != j))));

// ------------
// Enter and leave each city only once.
int N = data.Length;
Model.ForEach(city, i => Model.Sum(Model.ForEachWhere(city, j => assign[i, j],
j => i != j)) == 1));
Model.ForEach(city, j => Model.Sum(Model.ForEachWhere(city, i => assign[i, j], i => i != j)) == 1));

model.AddConstraint("A1", Model.ForEach(city, i => Model.Sum(Model.ForEachWhere(city, j => assign[i, j], j => i != j)) == 1));
model.AddConstraint("A2", Model.ForEach(city, j => Model.Sum(Model.ForEachWhere(city, i => assign[i, j], i => i != j)) == 1));

// Forbid subtours (Miller, Tucker, Zemlin - 1960...)
Model.ForEach(city,
i => Model.ForEachWhere(city,
j => rank[i] + 1 <= rank[j] + N * (1 - assign[i, j]),
j => Model.And(i != j, i >= 1, j >= 1)
)
)
);

Solution solution = context.Solve();

// Retrieve solution information.
Console.WriteLine("Cost = {0}", goal.ToDouble());
Console.WriteLine("Tour:");
var tour = from p in assign.GetValues() where (double)p[0] > 0.9 select p[2];
foreach (var i in tour.ToArray()) {
Console.Write(i + " -> ");
}
Console.WriteLine();
}```

In my humble opinion, the “Parameter data =” line is an awesome example of the power of LINQ data binding in Solver Foundation.  We generate the 2D matrix of distances using a single LINQ expression. It would be incredibly easy to change the code to retrieve the coordinate data from a database (perhaps using a LINQ expression once again), a file, or even a user application.

The goal is straightforward: minimize the distance traveled.  This is a product of the selected arcs and the distance matrix.   We have two types of constraints:

• Assignment constraints: these ensure that we enter and leave each city only once.
• Subtour constraints: these ensure that we do not have any subtours. In a four city problem {A, B, C, D}, for example, we cannot have two cycles (A, B), (C, D). We need to have one tour that contains all the cities.

The assignment constraints are easy using the ForEach and ForEachWhere operations.  I use ForEachWhere because I want to disallow arcs that enter and leave the same city – that doesn’t make sense.  The subtour constraint is a little more complicated. It relates the “assign” and “rank” decisions. The key fact is that if there is an arc from city i to city j, rank[i] + 1 == j. Of course, if the (i, j) arc is not part of the optimal tour then all bets are off. Last note: notice that I can mix parameters, decisions, and C# variables in my expressions.

Getting the cost is very easy using goal.ToDouble().  We can get the tour using either Assign or Rank.  I have chosen to use Assign because it gives me another opportunity to use LINQ.  When you call GetValues() on a decision, you get arrays that contain the value along with the indexes for each decision.  In this case, the last entry in the array is the one we are interested in. There are other ways to conveniently query decsision results, I’ll save that for another time.

The next post will show how we can use Solver Foundation’s plug-in model to tune the behavior of the Gurobi MIP solver.

## Modeling a production planning problem using Solver Foundation

In this post I am going to present two complete C# programs for modeling and solving a simple production problem using Solver Foundation. In this example we have two refineries (located in Saudi Arabia and Venzuela) that produce three products: gasoline, jet fuel, and lubricant. The goal is to minimize production costs, which depend on location. There is demand for each product which must be met. Finally, each production site has a limited production capacity.

The code for the simple case is below.  Creating a model amounts to adding the goals, decisions, and constraints using the appropriate Add method.  The signature for each is similar: the first argument is the name and the second argument is a term.  Terms are created by combining decisions or parameters using the usual operators, or by using static methods on the Model class (as we will see in our second example).  Finally, notice the call to SolverContext.Solve().  I have supplied a Directive that specifies that the Simplex solver should be used.

```using System;
using System.Collections.Generic;
using System.Data;
using System.Linq;
using System.Text;
using Microsoft.SolverFoundation.Services;

namespace Microsoft.SolverFoundation.Samples.Petrochem {
class Program {
static void Main(string[] args) {
PetrochemSimple();
}

private static void PetrochemSimple() {
SolverContext context = SolverContext.GetContext();
context.ClearModel();
Model model = context.CreateModel();

Decision sa = new Decision(Domain.RealRange(0, 9000), "SA");
Decision vz = new Decision(Domain.RealRange(0, 6000), "VZ");

model.AddGoal("goal", GoalKind.Minimize, 20 * sa + 15 * vz);

model.AddConstraint("demand1", 0.3 * sa + 0.4 * vz >= 1900);
model.AddConstraint("demand2", 0.4 * sa + 0.2 * vz >= 1500);
model.AddConstraint("demand3", 0.2 * sa + 0.3 * vz >= 500);

Solution solution = context.Solve(new SimplexDirective());
Report report = solution.GetReport();
Console.WriteLine(report);
}
}
```

Now let’s reimplement the same example using data binding. Data binding is a powerful mechanism for creating large, maintainable models. Notice that in my first example, the numeric data such as the yields, the demands, and capacities were expressed directly in the terms. The first step in using Solver Foundation data binding is to lift these values into Parameters. It is often useful to create indexed parameters using Sets. In this example, there are two clearly defined Sets: the set of countries, and the set of products. In my example below I create a DataSet which contains the data for each of my parameters. (The GetData() method is just an example, it’s not pretty but it is needed to complete the example.) Then I create a series of indexed parameters. For each of them I call the SetBinding method to associate the data with the parameter. In addition to the data, SetBinding also requires the caller to indicate which property specifies the values for the parameter. If the parameter is indexed, I also need to specify the parameters of the index properties. Since I am working with DataTables, these are simply column names. Notice that I could swap in any other data source that is enumerable – in particular LINQ works really well with Solver Foundation parameters.

After the parameters are created, I define the decisions, goals, and constraints. Notice that there is only one decision – it is indexed. The Model.Sum and Model.Foreach operations allow me to define a series of constraints over one or more indexed sets in one single statement. This means that if I were to add more countries or products, my model definition would not change at all.

```    private static void PetrochemDataBinding() {
SolverContext context = SolverContext.GetContext();
context.ClearModel();
Model model = context.CreateModel();

// Retrieve the problem data.
DataSet data = GetData();

Set products = new Set(Domain.Any, "products");
Set countries = new Set(Domain.Any, "countries");

Parameter demand = new Parameter(Domain.Real, "demand", products);
demand.SetBinding(data.Tables["Demand"].AsEnumerable(), "Demand", "Product");

Parameter yield = new Parameter(Domain.Real, "yield", products, countries);
yield.SetBinding(data.Tables["Yield"].AsEnumerable(), "Yield", "Product", "Country");

Parameter limit = new Parameter(Domain.Real, "limit", countries);
limit.SetBinding(data.Tables["Limit"].AsEnumerable(), "Limit", "Country");

Parameter cost = new Parameter(Domain.Real, "cost", countries);
cost.SetBinding(data.Tables["Cost"].AsEnumerable(), "Cost", "Country");

Decision produce = new Decision(Domain.RealNonnegative, "produce", countries);

model.AddGoal("Goal", GoalKind.Minimize, Model.Sum(Model.ForEach(countries, c => cost[c] * produce[c])));

Model.ForEach(products, p => Model.Sum(Model.ForEach(countries, c => yield[p, c] * produce[c])) >= demand[p])
);

Model.ForEach(countries, c => produce[c] <= limit[c])
);

Solution solution = context.Solve(new SimplexDirective());
Report report = solution.GetReport();
Console.WriteLine(report);
}

private static DataSet GetData() {
string[] products = new string[] { "Gas", "Jet Fuel", "Lubricant" };
string[] countries = new string[] { "SA", "VZ" };

double[][] yield = new double[][] {
new double[] { 0.3, 0.4 },
new double[] { 0.4, 0.2 },
new double[] { 0.2, 0.3 }
};
double[] demand = new double[] { 1900, 1500, 500 };
double[] limit = new double[] { 9000, 6000 };
double[] cost = new double[] { 20, 15 };

DataSet dataSet = new DataSet();
#region Fill DataSet
DataTable table = new DataTable("Yield");

for (int p = 0; p < products.Length; p++) {
for (int c = 0; c < countries.Length; c++) {
DataRow row = table.NewRow();
row[0] = products[p];
row[1] = countries[c];
row[2] = yield[p][c];
}
}

table = new DataTable("Demand");
for (int p = 0; p < products.Length; p++) {
DataRow row = table.NewRow();
row[0] = products[p];
row[1] = demand[p];
}

table = new DataTable("Limit");
for (int c = 0; c < countries.Length; c++) {
DataRow row = table.NewRow();
row[0] = countries[c];
row[1] = limit[c];
}

table = new DataTable("Cost");
for (int c = 0; c < countries.Length; c++) {
DataRow row = table.NewRow();
row[0] = countries[c];
row[1] = cost[c];
}
#endregion

return dataSet;
}
```