i am building a max heap with class item which has (name ,price ,category) one time by comparing the name and one time by comparing the price but there is a problem with the name comparing

here is the class item

class Item
{
    int Price;

public:
    Item(string name, string category, int price);
    string itemName;
    string Category;

    string get_category();
// operators to compare between prices
    bool operator<(Item &another);
    bool operator>(Item &another);
    bool operator<=(Item &another);
    bool operator>=(Item &another);
// operators to compare between names
    bool operator>(string &another);
    bool operator<(string &another);
    bool operator<=(string &another_name);
    bool operator>=(string &another_name);
    void print();
    bool operator==(Item &another);
};

and i have two functions, one to make the max heap by comparing prices which works fine

void Heap::Max_heapify(int size, int i)
{
    int left = 2*i+1;
    int right = 2*i+2;
    int Max = i;

    if (heap[Max] < heap[left] and left < size)
    {
        Max = left;
    }
    if (heap[Max] < heap[right] and right < size)
    {
        Max = right;
    }

    if (Max != i)
    {
        swap(heap[i], heap[Max]);
        Max_heapify(size, Max);
    }
void Heap::Build_Max()
{
    for (int i = (heap.size() / 2 - 1); i >= 0; i--)
    {
        Max_heapify(heap.size(), i);
    }
}

and another one to compare by names

void Heap::Max_heapify_By_Name(int size, int i)
{

    int left = 2*i+1;
    int right = 2*i+2;
    int Max = i;
    if ((heap[Max] < heap[left].itemName) and left <= size)
    {
        Max = left;
    }
    if ((heap[Max] < heap[right].itemName) and right <= size)
    {
        Max = right;
    }
    if (Max != i)
    {
        swap(heap[i], heap[Max]);
        Max_heapify_By_Name(size, Max);
    }
}
void Heap::Build_Max_By_Name()
{
    int n = heap.size();
    for (int i = floor(n / 2) - 1; i >= 0; i--)
    {
        cout << i << endl;
        Max_heapify_By_Name(n, i);
    }
}

here is the class heap

class Heap
{
    vector<Item> heap;

public:
    Heap(vector<Item>);
    void insert_item(Item item);
    void Delete_item(int index);
    int Get_parent(int num);
    int Get_Left_Chlid(int num);
    int Get_Right_Child(int num);
    void Insert_Item(Item item);
    void Heap_sort();    
    void Heap_sort_Name();
    void Max_heapify(int size ,int i );
    void Min_heapify(int i);
    void Build_Max();
    void Build_Min();
    void Max_heapify_By_Name(int size,int i);
    void Min_heapify_By_Name(int i);
    void Build_Max_By_Name();
    void Build_Min_By_Name();
    void print_heap();
};

the problem is that when i try to call the build_Max_by_name i just don’t get any output
here is the main.cpp

int main()
{
    Item i1("item1", "food", 123);
    Item i2("item5", "food", 120);
    Item i3("item4", "food", 156);
    Item i4("item6", "food", 1234);
    Item i5("item3", "food", 122);
    Item i6("item2", "food", 111);
    Item i7("item7", "food", 161);
    vector<Item> arr;
    //
    arr.push_back(i1);
    arr.push_back(i2);
    arr.push_back(i3);
    arr.push_back(i4);
    arr.push_back(i5);
    arr.push_back(i6);
    arr.push_back(i7);
    //
    Heap a(arr);
    //
    a.Build_Max_By_Name();
    a.print_heap();
    }

i thought that the problem was the overloaded operator but when i checked it was working fine
and here is its implementation

bool Item::operator<(Item &another)
{ 
// to compare prices
    return (this->Price < another.Price);
}
bool Item::operator<(string &another_name)
{
    return (this->itemName < another_name);
}

the max heap with price compare is working fine but the name one just doesn’t give any output i’d appreciate the help.