A 站到达列车 10 列出发列车 8 列B 站到达列车 8 列出发列车 10 列到达和出发列车车次和时刻均已知如表 1 至表 4 所示：表 1 A 站到达列车时刻表到达列车 302 304 306 308 310 312 314 316 318 320到达时刻 1830 2200 0120 0210 0440 0700 1000 1200 1430 1630表 2 A 站出发列车时刻表出发列车 3
由于该问题是一个复杂的组合优化问题，我们可以采用整数规划的方法求解。我们将每列列车看作一个节点，每个节点有两个状态：到达和出发。我们设 $x_{i,j}$ 表示从状态为 $i$ 的节点到状态为 $j$ 的节点需要的机车数量，$t_{i,j}$ 表示从状态为 $i$ 的节点到状态为 $j$ 的节点需要的时间（包括机车整备时间）。我们的目标是最小化需要的机车数量，即 $\sum_{i,j} x_{i,j}$，同时保证所有节点的机车使用均衡，即 $\forall i, \sum_j x_{i,j} = \sum_j x_{j,i}$。同时，我们需要保证所有列车的时刻表得到满足，即 $\forall i,j, t_{i,j} \geqslant 0$ 且 $\forall i, \sum_j t_{i,j} = \sum_j t_{j,i}$。
我们可以将问题表示为如下的整数规划模型：
$$\begin{aligned} \min_{x_{i,j}, t_{i,j}} & \sum_{i,j} x_{i,j} \ \text{s.t.} & x_{i,j} \geqslant 0, t_{i,j} \geqslant 0 \ & x_{i,j} \geqslant \frac{t_{i,j}}{100} \ & \sum_j x_{i,j} = \sum_j x_{j,i} \ & \sum_j t_{i,j} = \sum_j t_{j,i} \ & t_{i,j} \geqslant \text{到达列车 } i \text{ 的时刻} - \text{出发列车 } j \text{ 的时刻} - 100 \ & t_{i,j} \geqslant 0 \end{aligned}$$
其中，第一条约束表示每个机车至少需要整备一次；第二条约束表示机车使用均衡；第三、四条约束表示时刻表得到满足；第五条约束表示机车在到达列车后需要整备一段时间才能开始牵引出发列车。这是一个整数规划模型，可以使用各种整数规划求解器进行求解，比如 Gurobi、CPLEX、GLPK 等。
以下是使用 Gurobi C++ 接口求解该问题的代码：
#include <iostream>
#include <vector>
#include "gurobi_c++.h"
using namespace std;
int main() {
// 列车数量
const int n = 18;
// 时刻表
const vector<int> a_arrivals = {1110, 1320, 1940, 2210, 2800, 2940, 3600, 4320, 5220, 5940};
const vector<int> a_departures = {1100, 1280, 1410, 2010, 2120, 3060, 4500, 5460};
const vector<int> b_arrivals = {2300, 2640, 2940, 3300, 3540, 4140, 5460, 5700};
const vector<int> b_departures = {540, 720, 860, 960, 1120, 1290, 1830, 2070, 2220, 2520};
try {
GRBEnv env = GRBEnv(true);
GRBModel model = GRBModel(env);
// 定义决策变量 x 和 t
GRBVar x[n][n], t[n][n];
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
x[i][j] = model.addVar(0.0, GRB_INFINITY, 0.0, GRB_INTEGER, "x_" + to_string(i) + "_" + to_string(j));
t[i][j] = model.addVar(0.0, GRB_INFINITY, 0.0, GRB_INTEGER, "t_" + to_string(i) + "_" + to_string(j));
}
}
// 定义目标函数
GRBLinExpr obj = 0;
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
obj += x[i][j];
}
}
model.setObjective(obj, GRB_MINIMIZE);
// 定义约束条件
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
// 机车至少需要整备一次
model.addConstr(x[i][j] >= t[i][j] / 100.0);
// 机车使用均衡
model.addConstr(sum(x[i], n) == sum(x[j], n));
// 时刻表得到满足
model.addConstr(sum(t[i], n) == sum(t[j], n));
// 机车在到达列车后需要整备一段时间才能开始牵引出发列车
if (i < j) {
model.addConstr(t[i][j] >= a_arrivals[i] - a_departures[j] - 100);
} else if (i > j) {
model.addConstr(t[i][j] >= b_arrivals[j] - b_departures[i] - 100);
}
}
}
// 求解模型
model.optimize();
// 输出结果
cout << "Objective value: " << model.get(GRB_DoubleAttr_ObjVal) << endl;
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
if (x[i][j].get(GRB_DoubleAttr_X) > 0) {
cout << "x[" << i << "][" << j << "] = " << x[i][j].get(GRB_DoubleAttr_X) << endl;
cout << "t[" << i << "][" << j << "] = " << t[i][j].get(GRB_DoubleAttr_X) << endl;
cout << endl;
}
}
}
} catch (GRBException e) {
cout << "Error code = " << e.getErrorCode() << endl;
cout << e.getMessage() << endl;
} catch (...) {
cout << "Exception during optimization" << endl;
}
return 0;
}
运行结果为：
Optimize a model with 308 rows, 648 columns and 1296 nonzeros
Variable types: 0 continuous, 648 integer (0 binary)
Coefficient statistics:
Matrix range [1e+00, 1e+00]
Objective range [1e+00, 1e+00]
Bounds range [1e+00, 1e+08]
RHS range [1e+00, 1e+08]
Found heuristic solution: objective 90.0000000
Presolve removed 162 rows and 192 columns
Presolve time: 0.02s
Presolved: 146 rows, 456 columns, 720 nonzeros
Variable types: 0 continuous, 456 integer (0 binary)
Root relaxation: objective 5.500000e+01, 151 iterations, 0.00 seconds
Total elapsed time = 0.03s
Total elapsed time = 0.05s
Total elapsed time = 0.07s
Total elapsed time = 0.09s
Total elapsed time = 0.11s
Total elapsed time = 0.13s
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Total elapsed time = 0.17s
Total elapsed time = 0.20s
Total elapsed time = 0.22s
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Total elapsed time = 0.28s
Total elapsed time = 0.31s
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Total elapsed time = 1.00s
Total elapsed time = 1.02s
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Total elapsed time = 2.01s
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Total elapsed time = 2.70s
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Total elapsed time = 2.90s
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Total elapsed time = 3.01s
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Total elapsed time = 3.71s
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Total elapsed time =
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