CPLEX-求解VRPTW模型

问题定义：
VRPTW，即带时间窗约束的车辆路径问题，是在经典的VRP问题上加入了一个时间窗约束。
数学模型：


求解：
代码结构：

其中Data类表示定义参数，读取数据和初始化参数

package com.chb;

import java.io.BufferedReader;
import java.io.FileNotFoundException;
import java.io.FileReader;
import java.util.Scanner;

public class Data {
	public static int pointnum=102;
	public static int carnum;
	public static int cap;
	public static double E;
	public static double L;
	public static int point[][]=new int[pointnum][2];
	public static int s[]=new int[pointnum];
	public static double a[]=new double[pointnum];
	public static double b[]=new double[pointnum];
	public static int demand[]=new int[pointnum];
	public static int car[]=new int[pointnum];
	public static double dist[][]=new double[pointnum][pointnum];
	public static double arcs[][]=new double[pointnum][pointnum];
	
	//截断小数3.26434-->3.2
	public static double double_truncate(double v){
		int iv = (int) v;
		if(iv+1 - v <= 0.000000000001)
			return iv+1;
		double dv = (v - iv) * 10;
		int idv = (int) dv;
		double rv = iv + idv / 10.0;
		return rv;
	}
	
	//读取数据并且初始化参数
	public static void read_data(String path,Data data) throws Exception {
		String line=null;
		String substr[]=null;
		Scanner cin=new Scanner(new BufferedReader(new FileReader(path)));
		for (int i = 0; i < 4; i++) {
			line=cin.nextLine();
		}
		line=cin.nextLine();
		line.trim();
		substr=line.split("\\s+");
		carnum=Integer.parseInt(substr[1]);
		cap=Integer.parseInt(substr[2]);
		for (int i = 0; i < 4; i++) {
			line=cin.nextLine();
		}
		for (int i = 0; i < pointnum-1; i++) {
			line=cin.nextLine();
			line.trim();
			substr=line.split("\\s+");
			point[i][0]=Integer.parseInt(substr[2]);
			point[i][1]=Integer.parseInt(substr[3]);
			demand[i]=Integer.parseInt(substr[4]);
			a[i]=Integer.parseInt(substr[5]);
			b[i]=Integer.parseInt(substr[6]);
			s[i]=Integer.parseInt(substr[7]);
		}
		cin.close();
		//终点数据初始化
		point[pointnum-1]=point[0];
		demand[pointnum-1]=demand[0];
		a[pointnum-1]=a[0];
		b[pointnum-1]=b[0];
		s[pointnum-1]=s[0];
		E=a[0];
		L=b[0];
		double min1=1e15;
		double min2=1e15;
		
		//距离矩阵初始化
		for (int i = 0; i < pointnum; i++) {
			for (int j = 0; j < pointnum; j++) {
				if(i==j) {
					dist[i][j]=0;
					continue;
				}else {
					dist[i][j]=Math.sqrt(Math.pow(point[i][0]-point[j][0], 2)+Math.pow(point[i][1]-point[j][1], 2));
					dist[i][j]=double_truncate(dist[i][j]);
				}
			}
		}
		dist[pointnum-1][0]=0;
		dist[0][pointnum-1]=0;
		
		//距离矩阵满足三角关系
		for (int k = 0; k < pointnum; k++) {
			for (int i = 0; i < pointnum; i++) {
				for (int j = 0; j < pointnum; j++) {
					if(dist[i][j]>dist[i][k]+dist[k][j]) {
						dist[i][j]=dist[i][k]+dist[k][j];
					}
				}
			}
		}
		
		//初始化为完全图
		for (int i = 0; i <  pointnum; i++) {
			for (int j = 0; j <  pointnum; j++) {
				if(i==j) {
					arcs[i][j]=0;
				}else {
					arcs[i][j]=1;
				}
			}
		}
		
		//除去不符合时间窗和容量约束的边
		for (int i = 0; i < pointnum; i++) {
			for (int j = 0; j < pointnum; j++) {
				if(i==j) {
					continue;
				}
				if(a[i]+s[i]+dist[i][j]>b[j]||demand[i]+demand[j]>cap) {
					arcs[i][j]=0;
				}
				if(a[0]+s[i]+dist[0][i]+dist[i][pointnum-1]>b[pointnum-1]) {
					System.out.println("计算错误");
				}
			}
		}
		for (int i = 1; i < pointnum-1; i++) {
			if(b[i]-dist[0][i]min1||L

Solution类是对解的可行性进行判断

package com.chb;

import java.util.ArrayList;
import ilog.concert.IloException;

public class Solution {
	double epsilon = 0.0001;
	Data data = new Data();
	ArrayList> routes = new ArrayList<>();
	ArrayList> servetimes = new ArrayList<>();
	public Solution(Data data, ArrayList> routes, ArrayList> servetimes) {
		super();
		this.data = data;
		this.routes = routes;
		this.servetimes = servetimes;
	}
	//函数功能：比较两个数的大小
	public int double_compare(double v1,double v2) {
		if (v1 < v2 - epsilon) {
			return -1;
		}
		if (v1 > v2 + epsilon) {
			return 1;
		}
		return 0;
	}
	//函数功能：解的可行性判断
	public void fesible() throws IloException {
		//车辆数量可行性判断
		if (routes.size() > data.carnum) {
			System.out.println("error: carnum!!!");
			System.exit(0);
		}
		//车辆载荷可行性判断
		for (int k = 0; k < routes.size(); k++) {
			ArrayList route = routes.get(k);
			double capasity = 0;
			for (int i = 0; i < route.size(); i++) {
				capasity += data.demand[route.get(i)];
			}
			if (capasity > data.cap) {
				System.out.println("error: cap!!!");
				System.exit(0);
			}
		}
		//时间窗、车容量可行性判断
		for (int k = 0; k < routes.size(); k++) {
			ArrayList route = routes.get(k);
			ArrayList servertime = servetimes.get(k);
			double capasity = 0;
			for (int i = 0; i < route.size()-1; i++) {
				int origin = route.get(i);
				int destination = route.get(i+1);
				double si = servertime.get(i);
				double sj = servertime.get(i+1);
				if (si < data.a[origin] && si >  data.b[origin]) {
					System.out.println("error: servertime!");
					System.exit(0);
				}
				if (double_compare(si + data.dist[origin][destination],data.b[destination]) > 0) {
					System.out.println(origin + ": [" + data.a[origin] + ","+data.b[origin]+"]"+ " "+ si);
					System.out.println(destination + ": [" + data.a[destination] + ","+data.b[destination]+"]"+ " "+ sj);
					System.out.println(data.dist[origin][destination]);
					System.out.println(destination + ":" );
					System.out.println("error: forward servertime!");
					System.exit(0);
				}
				if (double_compare(sj - data.dist[origin][destination],data.a[origin]) < 0) {
					System.out.println(origin + ": [" + data.a[origin] + ","+data.b[origin]+"]"+ " "+ si);
					System.out.println(destination + ": [" + data.a[destination] + ","+data.b[destination]+"]"+ " "+ sj);
					System.out.println(data.dist[origin][destination]);
					System.out.println(destination + ":" );
					System.out.println("error: backward servertime!");
					System.exit(0);
				}
			}
			if (capasity > data.cap) {
				System.out.println("error: cap!!!");
				System.exit(0);
			}
		}
	}
}

类 Vrptw 是建立模型并求解模型

package com.chb;

import java.util.ArrayList;

import ilog.concert.IloException;
import ilog.concert.IloNumExpr;
import ilog.concert.IloNumVar;
import ilog.concert.IloNumVarType;
import ilog.cplex.IloCplex;

public class Vrptw {
	    Data data;					//定义类Data的对象
		IloCplex cplex;				//定义cplex内部类的对象		
		public IloNumVar[][][] x;	//x[i][j][k]表示弧arcs[i][j]被车辆k访问
		public IloNumVar[][] w;		//车辆访问所有点的时间矩阵
		double cost;				//目标值object
		Solution solution;			
		public Vrptw(Data data) {
			this.data = data;
		}
		//函数功能：解模型，并生成车辆路径和得到目标值
		public void solve() throws IloException {
			ArrayList> routes = new ArrayList<>();		//定义车辆路径链表
			ArrayList> servetimes = new ArrayList<>();	//定义花费时间链表
			//初始化车辆路径和花费时间链表，链表长度为车辆数k
			for (int k = 0; k < data.carnum; k++) {
				ArrayList r = new ArrayList<>();	//定义一个对象为int型的链表
				ArrayList t = new ArrayList<>();	//定义一个对象为double型的链表
				routes.add(r);								//将上述定义的链表加入到链表routes中
				servetimes.add(t);							//同上
			}
			//判断建立的模型是否可解
			if(cplex.solve() == false){
				//模型不可解
				System.out.println("problem should not solve false!!!");
				return;										//若不可解，则直接跳出solve函数
			}
			else{
				//模型可解，生成车辆路径
				for(int k = 0; k < data.carnum; k++){
					boolean terminate = true;
					int i = 0;
					routes.get(k).add(0);		
					servetimes.get(k).add(0.0);
					while(terminate){
						for (int j = 0; j < data.pointnum; j++) {
							if (data.arcs[i][j]>=0.5 && cplex.getValue(x[i][j][k])>=0.5) {
								routes.get(k).add(j);
								servetimes.get(k).add(cplex.getValue(w[j][k]));
								i = j;
								break;
							}
						}
						if (i == data.pointnum-1) {
							terminate = false;
						}
					}
				}
			}
			solution = new Solution(data,routes,servetimes);
			cost = cplex.getObjValue();
			System.out.println("routes="+solution.routes);
		}
		//函数功能：根据VRPTW数学模型建立VRPTW的cplex模型
		private void build_cplex() throws IloException {
			//model
			cplex = new IloCplex();
			cplex.setOut(null);
			//variables
			x = new IloNumVar[data.pointnum][data.pointnum][data.carnum];
			w = new IloNumVar[data.pointnum][data.carnum];				//车辆访问点的时间
			//定义cplex变量x和w的数据类型及取值范围
			for (int i = 0; i < data.pointnum; i++) {
				for (int k = 0; k < data.carnum; k++) {
					w[i][k] = cplex.numVar(0, 1e15, IloNumVarType.Float, "w" + i + "," + k);
				}
				for (int j = 0; j < data.pointnum; j++) {
					if (data.arcs[i][j]==0) {
						x[i][j] = null;
					}
					else{
						//Xijk,公式(10)-(11)
						for (int k = 0; k < data.carnum; k++) {
							x[i][j][k] = cplex.numVar(0, 1, IloNumVarType.Int, "x" + i + "," + j + "," + k);
						}
					}
				}
			}
			//加入目标函数
			//公式(1)
			IloNumExpr obj = cplex.numExpr();
			for(int i = 0; i < data.pointnum; i++){
				for(int j = 0; j < data.pointnum; j++){
					if (data.arcs[i][j]==0) {
						continue;
					}
					for(int k = 0; k < data.carnum; k++){
						obj = cplex.sum(obj, cplex.prod(data.dist[i][j], x[i][j][k]));
					}
				}
			}
			cplex.addMinimize(obj);
			//加入约束
			//公式(2)
			for(int i= 1; i < data.pointnum-1;i++){
				IloNumExpr expr1 = cplex.numExpr();
				for (int k = 0; k < data.carnum; k++) {
					for (int j = 1; j < data.pointnum; j++) {
						if (data.arcs[i][j]==1) {
							expr1 = cplex.sum(expr1, x[i][j][k]);
						}
					}
				}
				cplex.addEq(expr1, 1);
			}
			//公式(3)
			for (int k = 0; k < data.carnum; k++) {
				IloNumExpr expr2 = cplex.numExpr();
				for (int j = 1; j < data.pointnum; j++) {
					if (data.arcs[0][j]==1) {
						expr2 = cplex.sum(expr2, x[0][j][k]);
					}
				}
				cplex.addEq(expr2, 1);
			}
			//公式(4)
			for (int k = 0; k < data.carnum; k++) {
				for (int j = 1; j < data.pointnum-1; j++) {
					IloNumExpr expr3 = cplex.numExpr();
					IloNumExpr subExpr1 = cplex.numExpr();
					IloNumExpr subExpr2 = cplex.numExpr();
					for (int i = 0; i < data.pointnum; i++) {
						if (data.arcs[i][j]==1) {
							subExpr1 = cplex.sum(subExpr1,x[i][j][k]);
						}
						if (data.arcs[j][i]==1) {
							subExpr2 = cplex.sum(subExpr2,x[j][i][k]);
						}
					}
					expr3 = cplex.sum(subExpr1,cplex.prod(-1, subExpr2));
					cplex.addEq(expr3, 0);
				}
			}
			//公式(5)
			for (int k = 0; k < data.carnum; k++) {
				IloNumExpr expr4 = cplex.numExpr();
				for (int i = 0; i < data.pointnum-1; i++) {
					if (data.arcs[i][data.pointnum-1]==1) {
						expr4 = cplex.sum(expr4,x[i][data.pointnum-1][k]);
					}
				}
				cplex.addEq(expr4, 1);
			}
			//公式(6)
			double M = 1e5;
			for (int k = 0; k < data.carnum; k++) {
				for (int i = 0; i < data.pointnum; i++) {
					for (int j = 0; j < data.pointnum; j++) {
						if (data.arcs[i][j] == 1) {
							IloNumExpr expr5 = cplex.numExpr();
							IloNumExpr expr6 = cplex.numExpr();
							expr5 = cplex.sum(w[i][k], data.s[i]+data.dist[i][j]);
							expr5 = cplex.sum(expr5,cplex.prod(-1, w[j][k]));
							expr6 = cplex.prod(M,cplex.sum(1,cplex.prod(-1, x[i][j][k])));
							cplex.addLe(expr5, expr6);
						}
					}
				}
			}
			//公式(7)
			for (int k = 0; k < data.carnum; k++) {
				for (int i = 1; i < data.pointnum-1; i++) {
					IloNumExpr expr7 = cplex.numExpr();
					for (int j = 0; j < data.pointnum; j++) {
						if (data.arcs[i][j] == 1) {
							expr7 = cplex.sum(expr7,x[i][j][k]);
						}
					}
					cplex.addLe(cplex.prod(data.a[i], expr7), w[i][k]);
					cplex.addLe(w[i][k], cplex.prod(data.b[i], expr7));
				}
			}
			//公式(8)
			for (int k = 0; k < data.carnum; k++) {
				cplex.addLe(data.E, w[0][k]);
				cplex.addLe(data.E, w[data.pointnum-1][k]);
				cplex.addLe(w[0][k], data.L);
				cplex.addLe(w[data.pointnum-1][k], data.L);
			}
			//公式(9)
			for (int k = 0; k < data.carnum; k++) {
				IloNumExpr expr8 = cplex.numExpr();
				for (int i = 1; i < data.pointnum-1; i++) {
					IloNumExpr expr9 = cplex.numExpr();
					for (int j = 0; j < data.pointnum; j++) {
						if (data.arcs[i][j] == 1) {
							expr9=cplex.sum(expr9,x[i][j][k]);
						}
					}
					expr8 = cplex.sum(expr8,cplex.prod(data.demand[i],expr9));
				}
				cplex.addLe(expr8, data.cap);
			}
	}
	public static void main(String[] args) throws Exception {
		Data data=new Data();
		String path="data/c102.txt";
		data.read_data(path, data);
		System.out.println("输入成功");
		System.out.println("程序执行中**************************");
		Vrptw cplex=new Vrptw(data);
		cplex.build_cplex();
		double cplex_time1=System.nanoTime();
		cplex.solve();
		cplex.solution.fesible();
		double cplex_time2=System.nanoTime();
		double cplex_time=( cplex_time2- cplex_time1)/1e9;
		System.out.println("cplex_time:"+cplex_time);
		System.out.println("bestcost:"+cplex.cost);
	}
}

Data文件夹中是测试数据
运行结果截图：

本文转载、提炼来源于：https://mp.weixin.qq.com/s/OdzX_b9sLJ2PZ4WoDM5NnQ