Capacitated Vehicle Routing Problem

Note

Installation Required: This functionality requires MATLAB Support Package for Quantum Computing.

Capacitated vehicle routing is a combination of a knapsack problem and a traveling salesperson problem. The problem is for a vehicle (or set of vehicles) to visit a group of customers that are geographically distributed. The vehicle has a capacity constraint, where the capacity refers to a quantity that the vehicle delivers to each customer. The problem has a central depot, and the vehicle must return to the depot after each visit to a set of customers, or route. The problem is to visit the customers at minimal cost, where the cost is the total length of the route for visiting a group of customers.

The following figure shows four routes originating from a single point, the depot. These routes do not represent a minimal solution, because nodes 2 and 3 (at least) should be visited in the opposite order. The route containing nodes 2 and 3 has a self-intersection, which does not occur in an optimal tour.

Four tours starting from a central point, the depot

To solve a capacitated vehicle routing problem, follow the steps in Feld and coauthors [1]. While Feld gives several solution approaches, this example uses just one:

Create clusters that represent groups of customers visited by a vehicle in a single route. This step is a knapsack problem.
Solve the traveling salesperson problem for each cluster.

Solve the cluster creation problem using a classical algorithm. Solve the traveling salesperson problems by mapping them to QUBO problems. Solve the QUBO problems using the solve function, which internally uses the tabu search algorithm.

Problem Data and Variables

Create random integer locations for 24 customers, with random integer demands at the locations from 100 to 2500 in increments of 100. The depot, which is one more location, is at coordinates [0,0].

rng(1) % For reproducibility
numCustomers = 24; % Depot at [0 0] makes 25 locations
depot = [0 0]; % Depot at the origin
loc = [depot; randi([-50,50],numCustomers,2)]; % Integers from -50 to 50, 24-by-2
demand = 100*randi([1,25],numCustomers,1);
capacity = 6000;

Plot the locations and demands.

% Plot the locations with demands overlaid
figure;
scatter(loc(:,1),loc(:,2),'filled','SizeData',25);
hold on
text(loc(:,1),loc(:,2),["Depot"; num2str(demand)]);
title("Customer Locations and Demands");

24 random integer locations from -50 to 50 in each coordinate, with associated demands from 100 to 2500

Create the matrix of distances between customers from the coordinate vector. Use the Pythagorean rule and Euclidean distance.

numLocations = numCustomers + 1;
[X,Y] = meshgrid(1:numLocations);
dist = hypot(loc(X(:),1)- loc(Y(:),1),loc(X(:),2) - loc(Y(:),2));
dist = reshape(dist,numLocations,numLocations);

Set up variables for the remainder of the example by removing the depot, site 1, from the problem data.

customerCoords = loc(2:end,:);
costMatrix = dist;
vehicleCapacity = capacity;

Create Clusters

Feld [1] gives the following two-step approach for creating clusters:

Create initial clusters based on the problem coordinates and capacity limits.
Refine the initial clusters to obtain shorter traveling salesperson routes.

For this example, create initial clusters using the following iterative approach:

Start with the customer that is farthest from the depot.
Iteratively add customers one at a time, choosing each customer to add as the one closest to the current mean position of the customers in the cluster. That is, that the second customer added to the cluster is the one closest to the first customer, the third customer is the one closest to the average position of the first two, and so on.
Stop adding customers to the cluster when adding the next one would exceed the capacity constraint. At this event, return to the first step and start a new cluster.

Create the clusters as structures with the fields customers, center, and demand. Use the addToCluster helper function to add a customer to the cluster.

clusters = [];

% CLUSTER GENERATION

% Select the first node of the first cluster
[~,idx] = max(costMatrix(1,:));

currentCluster.Customers = idx;
currentCluster.Center = customerCoords(idx,:);
currentCluster.Demand = demand(idx);

unclustered.Coords = customerCoords;
unclustered.Customers = 1:numCustomers;
unclustered.Coords(idx,:) = [];
unclustered.Customers(idx) = [];

% While not all the nodes have beed added
while ~isempty(unclustered.Customers)
    % Find the next closest node to the cluster
    distance = sqrt(sum((currentCluster.Center - unclustered.Coords).^2,2));
    [~,nextIdx] = min(distance);
    next = unclustered.Customers(nextIdx);
    % Remove the node from the unclustered list
    unclustered.Coords(nextIdx,:) = [];
    unclustered.Customers(nextIdx) = [];

    % If the customer can be added without violating capacity constraints,
    % add the customer to the current cluster; otherwise, start a new cluster
    if currentCluster.Demand + demand(next) <= vehicleCapacity
        currentCluster = addToCluster(currentCluster,next,customerCoords,demand);
    else
        clusters = [clusters, currentCluster];
        currentCluster.Customers = next;
        currentCluster.Center = customerCoords(next,:);
        currentCluster.Demand = demand(next);
    end
end
clusters = [clusters, currentCluster];

After creating the initial clusters, you can improve them. To do so, reassign a customer to a different cluster if doing so places the customer closer to the mean position of the new cluster compared to the current cluster, without exceeding the capacity constraint of the new cluster. Continue to perform cluster improvement steps until no more steps are possible, or until 10 improvement steps are made.

Improve the clusters using the addToCluster and removeFromCluster helper functions.

% CLUSTER IMPROVEMENT
iterations = 0;
% For each cluster
while iterations < 10
    for i = 1:numel(clusters)
        % And each customer within the cluster
        for customer = clusters(i).Customers
            % Calculate the customer's distance from the center
            d_i = sqrt(sum((clusters(i).Center - customerCoords(customer,:)).^2));
            % For each alternative cluster
            for j = [1:i-1, i+1:numel(clusters)]
                % Calculate the customer's distance to the center of the
                % alternative cluster
                d_j = sqrt(sum((clusters(j).Center - customerCoords(customer,:)).^2));
                % Move the customer to the alternative cluster if it is closer and
                % capacity constraints are met
                if d_j < d_i && clusters(j).Demand + demand(customer) < vehicleCapacity
                    % Remove the customer from the original cluster
                    % and add the customer to the new cluster
                    clusters(i) = removeFromCluster(clusters(i), ...
                        customer, customerCoords, demand);
                    clusters(j) = addToCluster(clusters(j), ...
                        customer, customerCoords, demand);                    
                    break
                end
            end
        end
    end
    iterations = iterations + 1;
end

Adjust the customer labels so that they correspond to the original problem, with the depot being site 1 and the first customer being site 2.

nRoutes = numel(clusters);
for i = 1:nRoutes
    clusters(i).Customers = clusters(i).Customers + 1;
end

Create and Solve Traveling Salesperson Problems

Each cluster represents a group of customers visited in one vehicle route, which starts at the depot and returns to the depot. For each route, the vehicle should travel the smallest possible distance. This problem is the traveling salesperson problem (TSP) for each cluster.

For each cluster, create a distance matrix for the customers in that cluster. Collect the various TSPs and their solutions in structures. Each structure contains the customers associated with that TSP, the distances between customers, and the solution to the TSP, which is the route with the minimal distance.

Create the TSPs to be solved by conversion to QUBO problems.

TSPsolutions = cell(nRoutes,1);
Routes = cell(nRoutes,1);
customerCoords = loc; % Return depot to list

For each cluster of customers, compute the pairwise distances and formulate the corresponding TSPs. Convert each TSP to a QUBO problem using the tsp2qubo helper function. Solve the TSPs using the solvemyTSP helper function.

for rt = 1:nRoutes
    % Compute pairwise distance
    cluster = clusters(rt);
    coords = [depot; customerCoords(cluster.Customers,:)];
    M = height(coords);
    [X,Y] = meshgrid(1:M);
    dist = hypot(coords(X(:),1)- coords(Y(:),1),coords(X(:),2) - coords(Y(:),2));
    
    % Collect TSP data
    tsp.CostMatrix = reshape(dist,M,M);
    tsp.Customers = cluster.Customers;
    % Solve this TSP
    TSPsolutions{rt} = solvemyTSP(tsp);
    Routes{rt} = TSPsolutions{rt}.Route;
end

Plot the resulting routes using the plotClusters helper function.

plotClusters(clusters,loc,Routes)

Five colored line plots showing routes starting from and ending at depot.

Display the capacity needed for each route.

d = arrayfun(@(x)x.Demand,clusters)

d =

        5000        5800        5400        4600        5700

The demand in each route is less than the capacity limit of 6000.

Helper Functions

This code creates the addToCluster helper function. Note that this helper function uses the updateCluster helper function.

function cluster = addToCluster(cluster,next,customerCoordinates,demandVector)
    cluster.Customers = [cluster.Customers, next];
    cluster = updateCluster(cluster,next,customerCoordinates,demandVector,1);
end

This code creates the removeFromCluster helper function. Note that this helper function uses the updateCluster helper function.

function cluster = removeFromCluster(cluster,next,customerCoordinates,demandVector)
    cluster.Customers(cluster.Customers == next) = [];   
    cluster = updateCluster(cluster,next,customerCoordinates,demandVector,-1);
end

This code creates the updateCluster helper function.

function cluster = updateCluster(cluster,next,customerCoordinates,demandVector,s)
% Update the cluster as well as the lists of available coordinates and
% customers

currentN = numel(cluster.Customers);
previousN = currentN - s;

% Update center
newX = (cluster.Center(1)*previousN + s*customerCoordinates(next,1))/(currentN);
newY = (cluster.Center(2)*previousN + s*customerCoordinates(next,2))/(currentN);
cluster.Center = [newX,newY];

% Update demand
cluster.Demand = cluster.Demand + s*demandVector(next);
end

This code creates the plotClusters helper function.

function plotClusters(clusters,customerCoords,Routes)
    % Plot the result of the clustering algorithm in 2D space. Plot the
    % routes as well if they are provided. 

    % Plot the depot
    f = figure;

    scatter(customerCoords(1,1), customerCoords(1,2),"filled");
    ax = f.CurrentAxes;
    text(customerCoords(1,1), customerCoords(1,2),"Depot");
    hold on

    % Plot each cluster and label customers
    for i = 1:numel(clusters)
        customer = clusters(i).Customers;
        scatter(customerCoords(customer,1),...
                customerCoords(customer,2),"filled",SizeData=25); % Locations
        text(customerCoords(customer,1),...
               customerCoords(customer,2),num2str(customer')); % Labels
    end

    % If the routes are provided, plot them
    if nargin > 2
        % Reset color index - 1 is for the depot, so start at 2
        colorIdx = 1;
        for k = 1:numel(Routes)
            route = Routes{k};
            % Advance color index for the next route
            colorIdx = colorIdx + 1;
            % Append depot as the start of the route
            cu = [1 clusters(k).Customers]; 
            tr = cu(route); % Route in original indices
            for customer = 1:numel(tr) - 1
                ax.ColorOrderIndex = colorIdx; % Keep current route color
                plot(customerCoords(tr(customer:customer+1),1),...
                    customerCoords(tr(customer:customer+1),2));
            end
            ax.ColorOrderIndex = colorIdx;
            plot(customerCoords([tr(end),tr(1)],1),...
                customerCoords([tr(end),tr(1)],2)); 
        end
    end
    drawnow
    hold off
end

This code creates the tsp2qubo helper function.

function QP = tsp2qubo(dist)
%   QP = TSP2QUBO(DIST) returns a QUBO problem from the traveling salesperson
%   problem specified by the distance matrix DIST. DIST is an N-by-N
%   nonnegative matrix where DIST(i,j) is the distance between locations
%   i and j.

% Copyright 2023 The MathWorks, Inc.

N = size(dist,1);
% Create constraints on routes
A = eye(N);
B = ones(N);
Q0 = kron(A,B);
Q1 = kron(B,A);
% Create upper diagonal matrices of distances
v = ones(N-1,1);
A2 = diag(v,1);
Q2 = kron(B,A2); % Q2 has a diagonal just above the main diagonal in each block
C = kron(dist,B);
Q2 = Q2.*C; % Q2 has an upper diagonal dist(i,j)
% Create dist(j,i) in the upper-right corner of each block
E = zeros(N);
E(1,N) = 1;
Q3 = kron(B,E); % Q3 has a 1 in the upper-right corner of each block
CP = kron(dist',B); % dist' for D(j,i)
Q3 = Q3.*CP; % Q3 has dist(j,i) in the upper-right corner of each block
% Add the multipliers
M = max(max(dist));
QN = sparse(M*(Q0 + Q1)*N^2 + Q2 + Q3);
% Symmetrize
QN = (QN + QN.')/2;

% Include the constant and linear terms
c = -4*ones(N^2,1)*M*N^2;
d = 2*N*M*N^2;

QP = qubo(QN,c,d);

end

This code creates the solvemyTSP helper function. Note that this helper function uses the solveTSPwithTabu and convertSolutionToRoute helper functions.

function TSPsolution = solvemyTSP(tsp)
% solvemyTSP solves the TSP by first converting it to a QUBO problem 
% and then using tabu search to find a solution. 

% Inputs: 
%   tsp: Structure with fields
%       customers: the customers for the tsp
%       costMatrix: the cost matrix for the tsp
% Outputs: 
%   TSP_solution: Structure with fields
%       Route: The order of the customers in the best route found. For example,
%       if there are 5 customers in this tsp, the Route might be [2 3 1 5 4]
%       for customers [13 6 3 12 7]. 
%       customers: The customers in the current tsp, taken directly
%       from the tsp input
%       quboFval: The fval returned by the qubo algorithm

n = numel(tsp.Customers);

% If the number of customers is less than 3, the solution is trivial and
% there is no need to solve the QUBO. 
if n < 3
    TSPsolution.Route = 1:n;
    TSPsolution.Customers = tsp.Customers;
    TSPsolution.QuboFval = tsp.CostMatrix(1,2) + tsp.CostMatrix(2,1);
    return
end
    
% Solve with tabu search
TSPsol = solveTSPwithTabu(tsp.CostMatrix);

% Convert the QUBO solution to a route
TSPsolution.Route = convertSolutionToRoute(TSPsol);
TSPsolution.QuboFval = TSPsol.BestFunctionValue;
TSPsolution.Customers = tsp.Customers;
end

This code creates the solveTSPwithTabu helper function. Note that this helper function uses the tsp2qubo and checkTSPConstraints helper functions.

function solution = solveTSPwithTabu(costMatrix)
% Solves the TSP with tabu search. Convert the TSP to a QUBO
% according to the penalty term and return the best feasible solution. 
% Convert the TSP problem to QUBO
Q = tsp2qubo(costMatrix);
x = solve(Q);

validSolutions = checkTSPConstraints(x);
% If no valid solution is found yet, try again up to 10 times
if validSolutions == false
   it = 1;
   while (validSolutions == false) && it < 10
       x = solve(Q);
       validSolutions = checkTSPConstraints(x);
       it = it + 1;
   end
end

if validSolutions == false
   solution = [];
else
   solution = x;
end
end

This code creates the checkTSPConstraints helper function.

function validSolutions = checkTSPConstraints(x)
% The solutions are encoded as [x11, x12, x13, ... x21, x23, ...]'
% where xij indicates that customer i is visited in the jth position of the
% route. To establish that the constraints are satisfied, verify
% that each customer is visited only once and each position in the route
% has only one customer assigned to it.
N = size(x.BestX,1);
n = sqrt(N);
y = reshape(x.BestX,n,n);

% Ensure each customer has only one position assigned
r = all(sum(y,1) == 1);

% Ensure each route position (row) on has one customer assigned
s = all(sum(y,2) == 1);
validSolutions = r && s;
end

This code creates the convertSolutionToRoute helper function.

function Route = convertSolutionToRoute(sol)
% The solutions are encoded as [x11, x12, x13, ... x21, x23, ...]'
% where xij indicates that customer i is visited in the jth position of the
% route. Decode the solution into a route.
solution = sol.BestX;
n = sqrt(numel(solution));
selected_indices = find(solution);
order = mod(selected_indices - 1, n) + 1;
[~, node_order] = sort(order);
nodes = 1:n;

Route = nodes(node_order);
end

References

[1] Feld, Sebastian, Christoph Roch, Thomas Gabor, Christian Seidel, Florian Neukart, Isabella Galter, Wolfgang Mauerer, and Claudia Linnhoff-Popien. A Hybrid Solution Method for the Capacitated Vehicle Routing Problem Using a Quantum Annealer. Available at https://arxiv.org/abs/1811.07403.