A tourist wants to make a good trip at the weekend with his friends. They will go to the mountains to see the wonders of nature, so he needs to pack well for the trip. He has a good knapsack for carrying things, but knows that he can carry a maximum of only 4kg in it and it will have to last the whole day. He creates a list of what he wants to bring for the trip but the total weight of all items is too much. He then decides to add columns to his initial list detailing their weights and a numerical value representing how important the item is for the trip.

Here is the list:

item | weight (dag) | value |
---|---|---|

map | 9 | 150 |

compass | 13 | 35 |

water | 153 | 200 |

sandwich | 50 | 160 |

glucose | 15 | 60 |

tin | 68 | 45 |

banana | 27 | 60 |

apple | 39 | 40 |

cheese | 23 | 30 |

beer | 52 | 10 |

suntan cream | 11 | 70 |

camera | 32 | 30 |

T-shirt | 24 | 15 |

trousers | 48 | 10 |

umbrella | 73 | 40 |

waterproof trousers | 42 | 70 |

waterproof overclothes | 43 | 75 |

note-case | 22 | 80 |

sunglasses | 7 | 20 |

towel | 18 | 12 |

socks | 4 | 50 |

book | 30 | 10 |

knapsack | ≤400 dag | ? |

The tourist can choose to take any combination of items from the list, but only one of each item is available. He may not cut or diminish the items, so he can only take whole units of any item.

**Which items does the tourist carry in his knapsack **so that their total weight does not exceed 400 dag [4 kg], and their total value is maximised?

[dag = decagram = 10 grams]

#include <vector> #include <string> #include <iostream> #include <boost/tuple/tuple.hpp> #include <set> int findBestPack( const std::vector<boost::tuple<std::string , int , int> > & , std::set<int> & , const int ) ; int main( ) { std::vector<boost::tuple<std::string , int , int> > items ; //===========fill the vector with data==================== items.push_back( boost::make_tuple( "" , 0 , 0 ) ) ; items.push_back( boost::make_tuple( "map" , 9 , 150 ) ) ; items.push_back( boost::make_tuple( "compass" , 13 , 35 ) ) ; items.push_back( boost::make_tuple( "water" , 153 , 200 ) ) ; items.push_back( boost::make_tuple( "sandwich", 50 , 160 ) ) ; items.push_back( boost::make_tuple( "glucose" , 15 , 60 ) ) ; items.push_back( boost::make_tuple( "tin", 68 , 45 ) ) ; items.push_back( boost::make_tuple( "banana", 27 , 60 ) ) ; items.push_back( boost::make_tuple( "apple" , 39 , 40 ) ) ; items.push_back( boost::make_tuple( "cheese" , 23 , 30 ) ) ; items.push_back( boost::make_tuple( "beer" , 52 , 10 ) ) ; items.push_back( boost::make_tuple( "suntan creme" , 11 , 70 ) ) ; items.push_back( boost::make_tuple( "camera" , 32 , 30 ) ) ; items.push_back( boost::make_tuple( "T-shirt" , 24 , 15 ) ) ; items.push_back( boost::make_tuple( "trousers" , 48 , 10 ) ) ; items.push_back( boost::make_tuple( "umbrella" , 73 , 40 ) ) ; items.push_back( boost::make_tuple( "waterproof trousers" , 42 , 70 ) ) ; items.push_back( boost::make_tuple( "waterproof overclothes" , 43 , 75 ) ) ; items.push_back( boost::make_tuple( "note-case" , 22 , 80 ) ) ; items.push_back( boost::make_tuple( "sunglasses" , 7 , 20 ) ) ; items.push_back( boost::make_tuple( "towel" , 18 , 12 ) ) ; items.push_back( boost::make_tuple( "socks" , 4 , 50 ) ) ; items.push_back( boost::make_tuple( "book" , 30 , 10 ) ) ; const int maximumWeight = 400 ; std::set<int> bestItems ; //these items will make up the optimal value int bestValue = findBestPack( items , bestItems , maximumWeight ) ; std::cout << "The best value that can be packed in the given knapsack is " << bestValue << " !\n" ; int totalweight = 0 ; std::cout << "The following items should be packed in the knapsack:\n" ; for ( std::set<int>::const_iterator si = bestItems.begin( ) ; si != bestItems.end( ) ; si++ ) { std::cout << (items.begin( ) + *si)->get<0>( ) << "\n" ; totalweight += (items.begin( ) + *si)->get<1>( ) ; } std::cout << "The total weight of all items is " << totalweight << " !\n" ; return 0 ; } int findBestPack( const std::vector<boost::tuple<std::string , int , int> > & items ,std::set<int> & bestItems , const int weightlimit ) { //dynamic programming approach sacrificing storage space for execution //time , creating a table of optimal values for every weight and a //second table of sets with the items collected so far in the knapsack //the best value is in the bottom right corner of the values table, //the set of items in the bottom right corner of the sets' table. const int n = items.size( ) ; int bestValues [ n ][ weightlimit ] ; std::set<int> solutionSets[ n ][ weightlimit ] ; std::set<int> emptyset ; for ( int i = 0 ; i < n ; i++ ) { for ( int j = 0 ; j < weightlimit ; j++ ) { bestValues[ i ][ j ] = 0 ; solutionSets[ i ][ j ] = emptyset ; } } for ( int i = 0 ; i < n ; i++ ) { for ( int weight = 0 ; weight < weightlimit ; weight++ ) { if ( i == 0 ) bestValues[ i ][ weight ] = 0 ; else { int itemweight = (items.begin( ) + i)->get<1>( ) ; if ( weight < itemweight ) { bestValues[ i ][ weight ] = bestValues[ i - 1 ][ weight ] ; solutionSets[ i ][ weight ] = solutionSets[ i - 1 ][ weight ] ; } else { // weight >= itemweight if ( bestValues[ i - 1 ][ weight - itemweight ] + (items.begin( ) + i)->get<2>( ) > bestValues[ i - 1 ][ weight ] ) { bestValues[ i ][ weight ] = bestValues[ i - 1 ][ weight - itemweight ] + (items.begin( ) + i)->get<2>( ) ; solutionSets[ i ][ weight ] = solutionSets[ i - 1 ][ weight - itemweight ] ; solutionSets[ i ][ weight ].insert( i ) ; } else { bestValues[ i ][ weight ] = bestValues[ i - 1 ][ weight ] ; solutionSets[ i ][ weight ] = solutionSets[ i - 1 ][ weight ] ; } } } } } bestItems.swap( solutionSets[ n - 1][ weightlimit - 1 ] ) ; return bestValues[ n - 1 ][ weightlimit - 1 ] ; }

- Output:

The best value that can be packed in the given knapsack is 1030 ! The following items should be packed in the knapsack: map compass water sandwich glucose banana suntan creme waterproof trousers waterproof overclothes note-case sunglasses socks The total weight of all items is 396 !

Content is available under GNU Free Documentation License 1.2