matlab遺傳演算法解決多約束的mtsp問題？
有一道很經典的數學建模題，快遞公司送貨策略。那道題到底有無最正確的答案。怎麼用matlab遺傳演算法解決這個多約束的mtsp問題？
MTSP問題是肯定有最優解的，不過只有窮舉法這一種方法能100%獲得最優解。遺傳演算法只是有可能算出最優解，多數時候還只是次優的。
下面這是一份代碼，可以直接在Matlab運行，是定起點定終點的多旅行商代碼，其中的城市坐標數據是隨機的。當然你可以很容易的修改數據使之符合你的需要。
下面有運行結果展示，可以湊合看看。
% MTSPF_GA Fixed Multiple Traveling Salesmen Problem (M-TSP) Genetic Algorithm (GA)
% Finds a (near) optimal solution to a variation of the M-TSP by setting
% up a GA to search for the shortest route (least distance needed for
% each salesman to travel from the start location to individual cities
% and back to the original starting place)
%
% Summary:
% 1. Each salesman starts at the first point, and ends at the first
% point, but travels to a unique set of cities in between
% 2. Except for the first, each city is visited by exactly one salesman
%
% Note: The Fixed Start/End location is taken to be the first XY point
%
% Input:
% USERCONFIG (structure) with zero or more of the following fields:
% - XY (float) is an Nx2 matrix of city locations, where N is the number of cities
% - DMAT (float) is an NxN matrix of city-to-city distances or costs
% - NSALESMEN (scalar integer) is the number of salesmen to visit the cities
% - MINTOUR (scalar integer) is the minimum tour length for any of the
% salesmen, NOT including the start/end point
% - POPSIZE (scalar integer) is the size of the population (should be divisible by 8)
% - NUMITER (scalar integer) is the number of desired iterations for the algorithm to run
% - SHOWPROG (scalar logical) shows the GA progress if true
% - SHOWRESULT (scalar logical) shows the GA results if true
% - SHOWWAITBAR (scalar logical) shows a waitbar if true
%
% Input Notes:
% 1. Rather than passing in a structure containing these fields, any/all of
% these inputs can be passed in as parameter/value pairs in any order instead.
% 2. Field/parameter names are case insensitive but must match exactly otherwise.
%
% Output:
% RESULTSTRUCT (structure) with the following fields:
% (in addition to a record of the algorithm configuration)
% - OPTROUTE (integer array) is the best route found by the algorithm
% - OPTBREAK (integer array) is the list of route break points (these specify the indices
% into the route used to obtain the individual salesman routes)
% - MINDIST (scalar float) is the total distance traveled by the salesmen
%
% Route/Breakpoint Details:
% If there are 10 cities and 3 salesmen, a possible route/break
% combination might be: rte = [5 6 9 4 2 8 10 3 7], brks = [3 7]
% Taken together, these represent the solution [1 5 6 9 1][1 4 2 8 10 1][1 3 7 1],
% which designates the routes for the 3 salesmen as follows:
% . Salesman 1 travels from city 1 to 5 to 6 to 9 and back to 1
% . Salesman 2 travels from city 1 to 4 to 2 to 8 to 10 and back to 1
% . Salesman 3 travels from city 1 to 3 to 7 and back to 1
%
% Usage:
% mtspf_ga
% -or-
% mtspf_ga(userConfig)
% -or-
% resultStruct = mtspf_ga;
% -or-
% resultStruct = mtspf_ga(userConfig);
% -or-
% [...] = mtspf_ga("Param1",Value1,"Param2",Value2, ...);
%
% Example:
% % Let the function create an example problem to solve
% mtspf_ga;
%
% Example:
% % Request the output structure from the solver
% resultStruct = mtspf_ga;
%
% Example:
% % Pass a random set of user-defined XY points to the solver
% userConfig = struct("xy",10*rand(35,2));
% resultStruct = mtspf_ga(userConfig);
%
% Example:
% % Pass a more interesting set of XY points to the solver
% n = 50;
% phi = (sqrt(5)-1)/2;
% theta = 2*pi*phi*(0:n-1);
% rho = (1:n).^phi;
% [x,y] = pol2cart(theta(:),rho(:));
% xy = 10*([x y]-min([x;y]))/(max([x;y])-min([x;y]));
% userConfig = struct("xy",xy);
% resultStruct = mtspf_ga(userConfig);
%
% Example:
% % Pass a random set of 3D (XYZ) points to the solver
% xyz = 10*rand(35,3);
% userConfig = struct("xy",xyz);
% resultStruct = mtspf_ga(userConfig);
%
% Example:
% % Change the defaults for GA population size and number of iterations
% userConfig = struct("popSize",200,"numIter",1e4);
% resultStruct = mtspf_ga(userConfig);
%
% Example:
% % Turn off the plots but show a waitbar
% userConfig = struct("showProg",false,"showResult",false,"showWaitbar",true);
% resultStruct = mtspf_ga(userConfig);
%
% See also: mtsp_ga, mtspo_ga, mtspof_ga, mtspofs_ga, mtspv_ga, distmat
%
% Author: Joseph Kirk
% Email: jdkirk630@gmail.com
% Release: 2.0
% Release Date: 05/01/2014
function varargout = mtspf_ga(varargin)
% Initialize default configuration
defaultConfig.xy = 10*rand(40,2)
defaultConfig.dmat = []; % N*N距離矩陣
defaultConfig.nSalesmen = 8;
defaultConfig.minTour = 3;
defaultConfig.popSize = 80;
defaultConfig.numIter = 5e3;
defaultConfig.showProg = true;
defaultConfig.showResult = true;
defaultConfig.showWaitbar = false;
% Interpret user configuration inputs
if ~nargin
userConfig = struct();
elseif isstruct(varargin{1})
userConfig = varargin{1};
else
try
userConfig = struct(varargin{:});
catch
error("Expected inputs are either a structure or parameter/value pairs");
end
end
% Override default configuration with user inputs
configStruct = get_config(defaultConfig,userConfig);
% Extract configuration
xy = configStruct.xy;
dmat = configStruct.dmat;
nSalesmen = configStruct.nSalesmen;
minTour = configStruct.minTour;
popSize = configStruct.popSize;
numIter = configStruct.numIter;
showProg = configStruct.showProg;
showResult = configStruct.showResult;
showWaitbar = configStruct.showWaitbar;
if isempty(dmat)
nPoints = size(xy,1);
a = meshgrid(1:nPoints);
dmat = reshape(sqrt(sum((xy(a,:)-xy(a",:)).^2,2)),nPoints,nPoints);
end
% Verify Inputs 驗證輸入
[N,dims] = size(xy);
[nr,nc] = size(dmat);
if N ~= nr || N ~= nc
error("Invalid XY or DMAT inputs!")
end
n = N - 1; % Separate Start/End City
% Sanity Checks
nSalesmen = max(1,min(n,round(real(nSalesmen(1)))));
minTour = max(1,min(floor(n/nSalesmen),round(real(minTour(1)))));
popSize = max(8,8*ceil(popSize(1)/8));
numIter = max(1,round(real(numIter(1))));
showProg = logical(showProg(1));
showResult = logical(showResult(1));
showWaitbar = logical(showWaitbar(1));
% Initializations for Route Break Point Selection 路徑斷點選擇的初始化
nBreaks = nSalesmen-1;
dof = n - minTour*nSalesmen; % degrees of freedom
addto = ones(1,dof+1);
for k = 2:nBreaks
addto = cumsum(addto);
end
cumProb = cumsum(addto)/sum(addto);
% Initialize the Populations
popRoute = zeros(popSize,n); % population of routes
popBreak = zeros(popSize,nBreaks); % population of breaks
popRoute(1,:) = (1:n) + 1;
popBreak(1,:) = rand_breaks();
for k = 2:popSize
popRoute(k,:) = randperm(n) + 1;
popBreak(k,:) = rand_breaks();
end
% Select the Colors for the Plotted Routes 所畫路徑的顏色
pclr = ~get(0,"DefaultAxesColor");
clr = [1 0 0; 0 0 1; 0.67 0 1; 0 1 0; 1 0.5 0];
if nSalesmen &> 5
clr = hsv(nSalesmen);
end
% Run the GA
globalMin = Inf;
totalDist = zeros(1,popSize);
distHistory = zeros(1,numIter);
tmpPopRoute = zeros(8,n);
tmpPopBreak = zeros(8,nBreaks);
newPopRoute = zeros(popSize,n);
newPopBreak = zeros(popSize,nBreaks);
if showProg
figure("Name","MTSPF_GA | Current Best Solution","Numbertitle","off");
hAx = gca;
end
if showWaitbar
hWait = waitbar(0,"Searching for near-optimal solution ...");
end
for iter = 1:numIter
% Evaluate Members of the Population 人口評估
for p = 1:popSize
d = 0;
pRoute = popRoute(p,:);
pBreak = popBreak(p,:);
rng = [[1 pBreak+1];[pBreak n]]";
for s = 1:nSalesmen
d = d + dmat(1,pRoute(rng(s,1))); % Add Start Distance
for k = rng(s,1):rng(s,2)-1
d = d + dmat(pRoute(k),pRoute(k+1));
end
d = d + dmat(pRoute(rng(s,2)),1); % Add End Distance
end
totalDist(p) = d;
end
% Find the Best Route in the Population
[minDist,index] = min(totalDist);
distHistory(iter) = minDist;
if minDist &< globalMin
globalMin = minDist;
optRoute = popRoute(index,:);
optBreak = popBreak(index,:);
rng = [[1 optBreak+1];[optBreak n]]";
if showProg
% Plot the Best Route 實時展示最優路徑
for s = 1:nSalesmen
rte = [1 optRoute(rng(s,1):rng(s,2)) 1];
if dims &> 2, plot3(hAx,xy(rte,1),xy(rte,2),xy(rte,3),".-","Color",clr(s,:));
else plot(hAx,xy(rte,1),xy(rte,2),".-","Color",clr(s,:)); end
hold(hAx,"on");
end
if dims &> 2, plot3(hAx,xy(1,1),xy(1,2),xy(1,3),"o","Color",pclr);
else plot(hAx,xy(1,1),xy(1,2),"o","Color",pclr); end
title(hAx,sprintf("Total Distance = %1.4f, Iteration = %d",minDist,iter));
hold(hAx,"off");
drawnow;
end
end
% Genetic Algorithm Operators
randomOrder = randperm(popSize);
for p = 8:8:popSize
rtes = popRoute(randomOrder(p-7:p),:);
brks = popBreak(randomOrder(p-7:p),:);
dists = totalDist(randomOrder(p-7:p));
[ignore,idx] = min(dists); %#ok
bestOf8Route = rtes(idx,:);
bestOf8Break = brks(idx,:);
routeInsertionPoints = sort(ceil(n*rand(1,2)));
I = routeInsertionPoints(1);
J = routeInsertionPoints(2);
for k = 1:8 % Generate New Solutions
tmpPopRoute(k,:) = bestOf8Route;
tmpPopBreak(k,:) = bestOf8Break;
switch k
case 2 % Flip
tmpPopRoute(k,I:J) = tmpPopRoute(k,J:-1:I);
case 3 % Swap
tmpPopRoute(k,[I J]) = tmpPopRoute(k,[J I]);
case 4 % Slide
tmpPopRoute(k,I:J) = tmpPopRoute(k,[I+1:J I]);
case 5 % Modify Breaks
tmpPopBreak(k,:) = rand_breaks();
case 6 % Flip, Modify Breaks
tmpPopRoute(k,I:J) = tmpPopRoute(k,J:-1:I);
tmpPopBreak(k,:) = rand_breaks();
case 7 % Swap, Modify Breaks
tmpPopRoute(k,[I J]) = tmpPopRoute(k,[J I]);
tmpPopBreak(k,:) = rand_breaks();
case 8 % Slide, Modify Breaks
tmpPopRoute(k,I:J) = tmpPopRoute(k,[I+1:J I]);
tmpPopBreak(k,:) = rand_breaks();
otherwise % Do Nothing
end
end
newPopRoute(p-7:p,:) = tmpPopRoute;
newPopBreak(p-7:p,:) = tmpPopBreak;
end
popRoute = newPopRoute;
popBreak = newPopBreak;
% Update the waitbar
if showWaitbar ~mod(iter,ceil(numIter/325))
waitbar(iter/numIter,hWait);
end
end
if showWaitbar
close(hWait);
end
if showResult
% Plots 畫圖
figure("Name","MTSPF_GA | Results","Numbertitle","off");
subplot(2,2,1);
if dims &> 2, plot3(xy(:,1),xy(:,2),xy(:,3),".","Color",pclr);
else plot(xy(:,1),xy(:,2),".","Color",pclr); end
title("City Locations");
subplot(2,2,2);
imagesc(dmat([1 optRoute],[1 optRoute]));
title("Distance Matrix");
subplot(2,2,3);
rng = [[1 optBreak+1];[optBreak n]]";
for s = 1:nSalesmen
rte = [1 optRoute(rng(s,1):rng(s,2)) 1];
if dims &> 2, plot3(xy(rte,1),xy(rte,2),xy(rte,3),".-","Color",clr(s,:));
else plot(xy(rte,1),xy(rte,2),".-","Color",clr(s,:)); end
title(sprintf("Total Distance = %1.4f",minDist));
hold on;
end
if dims &> 2, plot3(xy(1,1),xy(1,2),xy(1,3),"o","Color",pclr);
else plot(xy(1,1),xy(1,2),"o","Color",pclr); end
subplot(2,2,4);
plot(distHistory,"b","LineWidth",2);
title("Best Solution History");
set(gca,"XLim",[0 numIter+1],"YLim",[0 1.1*max([1 distHistory])]);
end
% Return Output
if nargout
resultStruct = struct( ...
"xy", xy, ...
"dmat", dmat, ...
"nSalesmen", nSalesmen, ...
"minTour", minTour, ...
"popSize", popSize, ...
"numIter", numIter, ...
"showProg", showProg, ...
"showResult", showResult, ...
"showWaitbar", showWaitbar, ...
"optRoute", optRoute, ...
"optBreak", optBreak, ...
"minDist", minDist);
varargout = {resultStruct};
end
% Generate Random Set of Break Points
function breaks = rand_breaks()
if minTour == 1 % No Constraints on Breaks
tmpBreaks = randperm(n-1);
breaks = sort(tmpBreaks(1:nBreaks));
else % Force Breaks to be at Least the Minimum Tour Length
nAdjust = find(rand &< cumProb,1)-1;
spaces = ceil(nBreaks*rand(1,nAdjust));
adjust = zeros(1,nBreaks);
for kk = 1:nBreaks
adjust(kk) = sum(spaces == kk);
end
breaks = minTour*(1:nBreaks) + cumsum(adjust);
end
end
end
% Subfunction to override the default configuration with user inputs
% 將輸入初始化，什麼都不輸入，就用這個應該是
function config = get_config(defaultConfig,userConfig)
% Initialize the configuration structure as the default
config = defaultConfig;
% Extract the field names of the default configuration structure
defaultFields = fieldnames(defaultConfig);
% Extract the field names of the user configuration structure
userFields = fieldnames(userConfig);
nUserFields = length(userFields);
% Override any default configuration fields with user values
for i = 1:nUserFields
userField = userFields{i};
isField = strcmpi(defaultFields,userField);
if nnz(isField) == 1
thisField = defaultFields{isField};
config.(thisField) = userConfig.(userField);
end
end
end
求代碼
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