变邻域搜索算法python解决背包问题

### 变邻域搜索算法简介

变邻域搜索(VNS)是一种元启发式优化技术，旨在通过改变邻居结构来克服局部搜索容易陷入局部最优的问题。VNS利用一系列不同的邻域结构，在每次迭代中随机选择一个邻域进行探索，当遇到更好的解决方案时接受它并继续搜索。

对于0-1背包问题而言，其核心在于决定哪些物品应该被放入容量有限的背包内以使得总价值最大[^1]。

### Python实现变邻域搜索解决0-1背包问题

下面是一个简单的Python程序示例，用于展示如何应用变邻域搜索算法求解0-1背包问题：

import random
from copy import deepcopy


def knapsack_value(solution, weights, values, capacity):
    total_weight = sum([weights[i] * solution[i] for i in range(len(solution))])
    if total_weight > capacity:
        return float('-inf')  # 超过重量则返回负无穷大作为惩罚
    else:
        return sum([values[i] * solution[i] for i in range(len(solution))])


def generate_neighbor(current_solution, method='swap'):
    neighbor = deepcopy(current_solution)
    
    if method == 'flip':
        index_to_flip = random.randint(0, len(neighbor)-1)
        neighbor[index_to_flip] = 1 - neighbor[index_to_flip]

    elif method == 'swap':  
        idxs = list(range(len(neighbor)))
        a, b = random.sample(idxs, 2)
        neighbor[a], neighbor[b] = neighbor[b], neighbor[a]
        
    return neighbor


def vns_for_knapsack(weights, values, capacity, max_iter=1000):
    n_items = len(values)

    current_solution = [random.choice([0, 1]) for _ in range(n_items)]
    best_solution = current_solution[:]
    best_value = knapsack_value(best_solution, weights, values, capacity)

    k_max = min(3, n_items//2)  # 邻域变化的最大次数
    
    iteration = 0
    while iteration < max_iter:
        improved = False
        
        for k in range(k_max):            
            new_solution = generate_neighbor(current_solution, 'flip')
            local_best_val = knapsack_value(new_solution, weights, values, capacity)

            if local_best_val > best_value:
                best_solution = new_solution[:]
                best_value = local_best_val
                
                current_solution = new_solution[:]                
                improved = True
                break
            
        if not improved:
            # 如果没有改进，则尝试更大幅度的变化（增加k）
            pass
        
        iteration += 1

    return best_solution, best_value


if __name__ == "__main__":
    # 测试数据集
    item_weights = [2, 3, 4, 5]
    item_values = [3, 4, 5, 6]
    bag_capacity = 7

    optimal_selection, max_profit = vns_for_knapsack(item_weights, item_values, bag_capacity)
    print(f'Optimal selection: {optimal_selection}')
    print(f'Maximum profit: {max_profit}')

此代码实现了基本形式下的变邻域搜索算法，并将其应用于0-1背包问题上。注意这里只采用了两种最基础的方法——翻转(`flip`)和交换(`swap`) 来生成新的候选方案；实际应用中可能还需要考虑更多复杂的变换方式以及参数调整机制。