每个物品可以放多个的背包问题用分支限界法实现matlab程序
时间: 2024-02-11 17:03:44 浏览: 34
以下是用分支限界法实现多重背包问题的 Matlab 程序:
```matlab
function [max_value, selected_items] = knapsack(values, weights, capacities)
n = length(values);
selected_items = zeros(n, 1);
max_value = 0;
% sort items by unit value in descending order
unit_values = values ./ weights;
[~, sorted_indices] = sort(unit_values, 'descend');
% initialize priority queue with empty state
queue = PriorityQueue();
queue.insert(struct('index', 0, 'weight', 0, 'value', 0, 'items', selected_items, 'bound', 0));
while ~queue.isempty()
% remove state with highest priority from queue
state = queue.pop();
if state.bound < max_value
% prune this state, as it cannot lead to a better solution
continue;
end
if state.value > max_value
% update best solution found so far
max_value = state.value;
selected_items = state.items;
end
if state.index < n
% generate child states by adding each item to the current state
j = sorted_indices(state.index+1); % index of next item to consider
for k = 0:min(1, floor((capacities(j)-state.weight)/weights(j)))
% add k items of type j to the current state
new_items = state.items;
new_items(j) = new_items(j) + k;
% compute bound for the new state
new_value = state.value + k * values(j);
new_weight = state.weight + k * weights(j);
remaining_capacities = capacities(j+1:end) - k * weights(j);
if state.index < n-1
% estimate maximum value for remaining items using fractional knapsack
remaining_values = unit_values(sorted_indices(state.index+2:end)) .* weights(sorted_indices(state.index+2:end));
[~, idx] = sort(remaining_values, 'descend');
remaining_values = remaining_values(idx);
remaining_weights = weights(sorted_indices(state.index+2:end));
remaining_weights = remaining_weights(idx);
while ~isempty(remaining_weights)
take = min(remaining_capacities ./ remaining_weights(1), 1);
new_value = new_value + sum(take .* remaining_values);
remaining_capacities = remaining_capacities - take .* remaining_weights;
remaining_weights(take == 1) = [];
remaining_values(take == 1) = [];
end
end
% add new state to the queue
queue.insert(struct('index', state.index+1, 'weight', new_weight, 'value', new_value, 'items', new_items, 'bound', new_value + remaining_values(1)));
end
end
end
end
```
其中,`values` 和 `weights` 分别是物品的价值和重量,`capacities` 是背包的容量数组,表示每个物品的最大数量。输出的 `max_value` 和 `selected_items` 分别是最大价值和所选物品的数量数组。这里使用了优先队列来维护搜索状态,每次取出优先级最高的状态进行拓展。在拓展状态时,我们只考虑添加当前物品的 0 个或 1 个,因为添加更多的物品会超过背包的容量限制。同时,我们使用了一个估价函数来计算剩余物品的最大价值,其中使用了一个类似于分数背包的贪心算法来计算。
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