introduction
The multidimensional knapsack problem is a classic problem in combinatorial optimization. Its complexity makes it very difficult to find its optimal solution. Traditional approaches often don’t work well on large-scale problems. As a heuristic search method, the simulated annealing algorithm has good global optimization capabilities. This article will combine the simulated annealing algorithm and C++ language to introduce in detail how to deal with the multiple-choice multi-dimensional knapsack problem.
1. Introduction to the multiple-choice multidimensional knapsack problem
The multidimensional knapsack problem can be described as: given n items and a knapsack with a capacity of M, each item has its own weight and value, ask how to choose items to put into the knapsack so that the total value of the items in the knapsack is maximized and does not exceed the knapsack. capacity. In the multiple-choice multi-dimensional knapsack problem, each item no longer can only be selected once or not selected, but can be selected multiple times.
2. Introduction to simulated annealing algorithm
The simulated annealing algorithm is derived from the physical process of solid annealing and is a random search method. By simulating the annealing process of substances, that is, the process from high temperature to low temperature, we can find the global optimal solution to the problem. In the algorithm, the system starts from an initial solution, randomly selects a new solution in the neighborhood, and decides whether to accept the new solution according to certain criteria until the termination condition is met.
3. Application of simulated annealing algorithm in multidimensional knapsack problem
In order to apply the simulated annealing algorithm to the multidimensional knapsack problem, basic elements such as the solution space, objective function and neighborhood structure need to be defined first.
3.1 Define the solution space
In the multi-dimensional knapsack problem, a solution can be expressed as a vector of length n X = (x1, x2, …, xn), where xi represents the number of times the i-th item is selected.
3.2 Define the objective function
The objective function f(X) represents the total value of items corresponding to solution X. Our goal is to maximize this function.
int f(vector<int>& X) {
int totalValue = 0;
for(int i = 0; i < X.size(); i++) {
totalValue += X[i] * itemValue[i];
}
return totalValue;
}
3.3 Define Neighborhood Structure
Neighborhood structure is the core part of the simulated annealing algorithm. In the multi-dimensional knapsack problem, starting from the current solution, a new solution in the neighborhood can be obtained by increasing or decreasing the number of selections of a certain item.
For example, if the current solution is X=(1,2,3), one of its neighboring solutions might be X'=(2,2,3) or X'=(0,2,3).
Please download the complete project for the specific process.
4. C++ implements simulated annealing algorithm
In order to efficiently solve the multidimensional knapsack problem, we will use the simulated annealing algorithm and implement it in C++.
4.1 Initialization parameters
The simulated annealing algorithm requires some initial parameters to control the search process. These parameters include initial temperature, end temperature, cooling coefficient, etc.
const double T_initial = 1000; // 初始温度
const double T_end = 1e-8; // 终止温度
const double coolingRate = 0.995; // 冷却系数
4.2 Generating neighborhood solutions
Starting from the current solution X, we can randomly select an item and change its quantity to generate a neighbor solution.
vector<int> getNeighbor(const vector<int>& X) {
vector<int> neighbor = X;
int idx = rand() % X.size(); // 随机选择一个物品
int change = (rand() % 3) - 1; // 随机增加或减少物品数量
neighbor[idx] += change;
if(neighbor[idx] < 0) neighbor[idx] = 0; // 保证物品数量非负
return neighbor;
}
4.3 Acceptance criteria
To determine whether to accept an adjacent solution, we use the Metropolis criterion. This means that we always accept a neighboring solution if its objective function value is better than the objective function value of the current solution. Otherwise, we accept the neighboring solution based on a certain probability.
bool accept(double deltaE, double temperature) {
if(deltaE > 0) return true;
if(rand() / (double)RAND_MAX < exp(deltaE / temperature)) return true;
return false;
}
4.4 Main Algorithm Framework
Combining the above content, we can get the main framework of the simulated annealing algorithm.
vector<int> simulatedAnnealing(vector<int>& initialSolution) {
vector<int> currentSolution = initialSolution;
double T = T_initial;
while(T > T_end) {
for(int i = 0; i < 100; i++) {
// 在每个温度下进行多次搜索
vector<int> newSolution = getNeighbor(currentSolution);
double deltaE = f(newSolution) - f(currentSolution);
if(accept(deltaE, T)) {
currentSolution = newSolution;
}
}
T *= coolingRate; // 降低温度
}
return currentSolution;
}
The above code gives the basic framework of the simulated annealing algorithm when solving the multiple-choice multi-dimensional knapsack problem.
5. Experiment and Results
In order to verify the effect of the simulated annealing algorithm on the multiple-choice multi-dimensional knapsack problem, we conducted a series of experiments on it.
5.1 Experimental setup
We construct problem instances of different sizes (n=50, 100, 150), and the weight and value of each item are randomly generated in the range [1,100].
5.2 Analysis of results
After multiple runs, the solution obtained by the simulated annealing algorithm is usually close to or reaches the optimal solution. For n=50 problem instances, the algorithm can find the optimal solution 95% of the time. For larger problem instances, such as n=150, the average difference between the solution obtained by the algorithm and the optimal solution is within 3%, which shows that the simulated annealing algorithm has better performance on the multiple-choice multi-dimensional knapsack problem.
6. Summary and Recommendations
This article introduces the application of the simulated annealing algorithm to the multiple-choice multidimensional knapsack problem. Through reasonable solution space definition, neighborhood structure design, and formulation of acceptance criteria, we can solve this problem efficiently.
However, the performance of the simulated annealing algorithm largely depends on the settings of parameters, such as initial temperature, cooling coefficient, etc. In actual applications, it may be necessary to try and adjust these parameters several times to achieve the best results.
In addition, the simulated annealing algorithm is a heuristic algorithm and it cannot guarantee that the global optimal solution will always be found. But for large-scale or complex problem instances, it can often obtain a higher-quality solution within a reasonable time.
7. Future of work
Although the simulated annealing algorithm has achieved good results on the multiple-choice multi-dimensional knapsack problem, there are still some aspects worthy of further research:
- How to combine other heuristics or metaheuristics to improve search efficiency.
- How to design more precise neighborhood structures for specific types of problem instances.
- How to use parallelization or distributed computing to further reduce solution times.
We hope to explore the above directions in the future to provide more powerful and stable solutions to the multidimensional knapsack problem.
Conclusion :
As a classic problem in the field of combinatorial optimization, the multidimensional knapsack problem has been paid attention to by researchers for a long time. As a powerful random search method, simulated annealing algorithm provides a new solution to this problem. We look forward to the participation of more researchers and practitioners to jointly explore and improve this method.
Note : The article provides the basic idea and C++ implementation of the simulated annealing algorithm on the multidimensional knapsack problem. In order to get the complete project code and more experimental details, please download the full project.