Knapsack problem/Bounded

From Rosetta Code
Task
Knapsack problem/Bounded
You are encouraged to solve this task according to the task description, using any language you may know.

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 some items during the trip.   Food, clothing, etc.   He has a good knapsack for carrying the things, but he knows that he can carry only 4 kg weight in his knapsack, because they will make the trip from morning to evening.

He creates a list of what he wants to bring for the trip, but the total weight of all items is too much.   He adds a value to each item.   The value represents how important the thing for the tourist.

The list contains which items are the wanted things for the trip, what is the weight and value of an item, and how many units does he have from each items.


This is the list:

Table of potential knapsack items
item weight (dag) (each) value (each) piece(s)
map 9 150 1
compass 13 35 1
water 153 200 2
sandwich 50 60 2
glucose 15 60 2
tin 68 45 3
banana 27 60 3
apple 39 40 3
cheese 23 30 1
beer 52 10 3
suntan cream 11 70 1
camera 32 30 1
T-shirt 24 15 2
trousers 48 10 2
umbrella 73 40 1
waterproof trousers 42 70 1
waterproof overclothes 43 75 1
note-case 22 80 1
sunglasses 7 20 1
towel 18 12 2
socks 4 50 1
book 30 10 2
knapsack ≤400 dag ? ?


The tourist can choose to take any combination of items from the list, and some number of each item is available   (see the column   piece(s)   in the list above).

He may not cut the items, so he can only take whole units of any item.


Task

Show which items does the tourist carry in his knapsack so that their total weight does not exceed 4 kg, and their total value is maximized.


Related tasks



11l[edit]

Translation of: Python
V items = [
   ‘sandwich’     = (50, 60, 2),
   ‘map’          = (9, 150, 1),
   ‘compass’      = (13, 35, 1),
   ‘water’        = (153, 200, 3),
   ‘glucose’      = (15, 60, 2),
   ‘tin’          = (68, 45, 3),
   ‘banana’       = (27, 60, 3),
   ‘apple’        = (39, 40, 3),
   ‘cheese’       = (23, 30, 1),
   ‘beer’         = (52, 10, 3),
   ‘suntan cream’ = (11, 70, 1),
   ‘camera’       = (32, 30, 1),
   ‘t-shirt’      = (24, 15, 2),
   ‘trousers’     = (48, 10, 2),
   ‘umbrella’     = (73, 40, 1),
   ‘w-trousers’   = (42, 70, 1),
   ‘w-overcoat’   = (43, 75, 1),
   ‘note-case’    = (22, 80, 1),
   ‘sunglasses’   = (7, 20, 1),
   ‘towel’        = (18, 12, 2),
   ‘socks’        = (4, 50, 1),
   ‘book’         = (30, 10, 2)
]

V item_keys = items.keys()

[(Int, Int) = (Int, [(Int, String)])] cache

F choose_item(weight, idx)
   [(Int, String)] best_list
   I idx < 0
      R (0, best_list)
   V k = (weight, idx)
   V? c = :cache.find(k)
   I c != N
      R c
   V name = :item_keys[idx]
   V (w, v, qty) = :items[name]
   V best_v = 0

   L(i) 0..qty
      V wlim = weight - i * w
      I wlim < 0
         L.break
      V (val, taken) = choose_item(wlim, idx - 1)
      I val + i * v > best_v
         best_v = val + i * v
         best_list = copy(taken)
         best_list.append((i, name))
   :cache[k] = (best_v, best_list)
   R (best_v, best_list)

V (v, lst) = choose_item(400, items.len - 1)
V w = 0
L(cnt, name) lst
   I cnt > 0
      print(cnt‘ ’name)
      w += items[name][0] * cnt

print(‘Total weight: ’w‘ Value: ’v)
Output:
3 banana
1 cheese
1 compass
2 glucose
1 map
1 note-case
1 socks
1 sunglasses
1 suntan cream
1 w-overcoat
1 water
Total weight: 396 Value: 1010

AutoHotkey[edit]

iterative dynamic programming solution

Item = map,compass,water,sandwich,glucose,tin,banana,apple,cheese,beer,suntan cream
      ,camera,tshirt,trousers,umbrella,waterproof trousers,waterproof overclothes,notecase
      ,sunglasses,towel,socks,book
Weight= 9,13,153,50,15,68,27,39,23,52,11,32,24,48,73,42,43,22,7,18,4,30
Value = 150,35,200,60,60,45,60,40,30,10,70,30,15,10,40,70,75,80,20,12,50,10
Bound = 1,1,2,2,2,3,3,3,1,3,1,1,2,2,1,1,1,1,1,2,1,2

StringSplit I, Item,  `, ; Put input in arrays
StringSplit W, Weight,`,
StringSplit V, Value, `,
StringSplit B, Bound, `,

W := 400, N := I0, I0 := V0 := W0 := 0 ; W = total weight allowed, maximize total value
Loop %W%
   m0_%A_Index% := 0

Loop %N% { ; build achievable value matrix m [ N rows, W columns ]
   j := -1+i := A_Index,  m%j%_0 := 0 ; m[i,P] = max value with items 1..i, weight <=P
   Loop %W% {                         ; m[i,P] = max_k {m[i-1,P-k*Wi]}
      p := A_Index, k := 0, y := m%j%_%p%
      While ++k <= B%i% && (r := p - k*W%i%) >= 0
         y := y < (c:=m%j%_%r%+k*V%i%) ? c : y
      m%i%_%p% := y
   }
}

i := 1+j := N, p := W, s := 0
While --i, --j { ; read out solution from value matrix m
   If (m%i%_%p% = m%j%_%p%)
      Continue
   r := p, m := m%i%_%p%, k := 1
   While 0 <= (r-=W%i%)  &&  m%j%_%r% != (m-=V%i%)
      k++ ; find multiplier
   t := k " " I%i% "`n" t, s += k*W%i%, p -= k*W%i%
}

MsgBox % "Value = " m%N%_%W% "`nWeight = " s "`n`n" t

Bracmat[edit]

(knapsack=
  ( things
  =   (map.9.150.1)
      (compass.13.35.1)
      (water.153.200.2)
      (sandwich.50.60.2)
      (glucose.15.60.2)
      (tin.68.45.3)
      (banana.27.60.3)
      (apple.39.40.3)
      (cheese.23.30.1)
      (beer.52.10.3)
      (suntan cream.11.70.1)
      (camera.32.30.1)
      (T-shirt.24.15.2)
      (trousers.48.10.2)
      (umbrella.73.40.1)
      (waterproof trousers.42.70.1)
      (waterproof overclothes.43.75.1)
      (note-case.22.80.1)
      (sunglasses.7.20.1)
      (towel.18.12.2)
      (socks.4.50.1)
      (book.30.10.2)
  )
& 0:?maxvalue
& :?sack
& ( add
  =     cumwght
        cumvalue
        cumsack
        name
        wght
        val
        pcs
        tings
        n
        ncumwght
        ncumvalue
    .     !arg
        : ( ?cumwght
          . ?cumvalue
          . ?cumsack
          . (?name.?wght.?val.?pcs) ?tings
          )
      & -1:?n
      &   whl
        ' ( 1+!n:~>!pcs:?n
          & !cumwght+!n*!wght:~>400:?ncumwght
          & !cumvalue+!n*!val:?ncumvalue
          & (   !tings:
              & (   !ncumvalue:>!maxvalue:?maxvalue
                  &     !cumsack
                        ( !n:0&
                        | (!n.!name)
                        )
                    : ?sack
                | 
                )
            |   add
              $ ( !ncumwght
                . !ncumvalue
                .   !cumsack
                    (!n:0&|(!n.!name))
                . !tings
                )
            )
          )
  )
& add$(0.0..!things)
& out$(!maxvalue.!sack)
);

!knapsack;
Output:
  1010
.   (1.map)
    (1.compass)
    (1.water)
    (2.glucose)
    (3.banana)
    (1.cheese)
    (1.suntan cream)
    (1.waterproof overclothes)
    (1.note-case)
    (1.sunglasses)
    (1.socks)

C[edit]

#include <stdio.h>
#include <stdlib.h>

typedef struct {
    char *name;
    int weight;
    int value;
    int count;
} item_t;

item_t items[] = {
    {"map",                      9,   150,   1},
    {"compass",                 13,    35,   1},
    {"water",                  153,   200,   2},
    {"sandwich",                50,    60,   2},
    {"glucose",                 15,    60,   2},
    {"tin",                     68,    45,   3},
    {"banana",                  27,    60,   3},
    {"apple",                   39,    40,   3},
    {"cheese",                  23,    30,   1},
    {"beer",                    52,    10,   3},
    {"suntan cream",            11,    70,   1},
    {"camera",                  32,    30,   1},
    {"T-shirt",                 24,    15,   2},
    {"trousers",                48,    10,   2},
    {"umbrella",                73,    40,   1},
    {"waterproof trousers",     42,    70,   1},
    {"waterproof overclothes",  43,    75,   1},
    {"note-case",               22,    80,   1},
    {"sunglasses",               7,    20,   1},
    {"towel",                   18,    12,   2},
    {"socks",                    4,    50,   1},
    {"book",                    30,    10,   2},
};

int n = sizeof (items) / sizeof (item_t);

int *knapsack (int w) {
    int i, j, k, v, *mm, **m, *s;
    mm = calloc((n + 1) * (w + 1), sizeof (int));
    m = malloc((n + 1) * sizeof (int *));
    m[0] = mm;
    for (i = 1; i <= n; i++) {
        m[i] = &mm[i * (w + 1)];
        for (j = 0; j <= w; j++) {
            m[i][j] = m[i - 1][j];
            for (k = 1; k <= items[i - 1].count; k++) {
                if (k * items[i - 1].weight > j) {
                    break;
                }
                v = m[i - 1][j - k * items[i - 1].weight] + k * items[i - 1].value;
                if (v > m[i][j]) {
                    m[i][j] = v;
                }
            }
        }
    }
    s = calloc(n, sizeof (int));
    for (i = n, j = w; i > 0; i--) {
        int v = m[i][j];
        for (k = 0; v != m[i - 1][j] + k * items[i - 1].value; k++) {
            s[i - 1]++;
            j -= items[i - 1].weight;
        }
    }
    free(mm);
    free(m);
    return s;
}

int main () {
    int i, tc = 0, tw = 0, tv = 0, *s;
    s = knapsack(400);
    for (i = 0; i < n; i++) {
        if (s[i]) {
            printf("%-22s %5d %5d %5d\n", items[i].name, s[i], s[i] * items[i].weight, s[i] * items[i].value);
            tc += s[i];
            tw += s[i] * items[i].weight;
            tv += s[i] * items[i].value;
        }
    }
    printf("%-22s %5d %5d %5d\n", "count, weight, value:", tc, tw, tv);
    return 0;
}
Output:
map                        1     9   150
compass                    1    13    35
water                      1   153   200
glucose                    2    30   120
banana                     3    81   180
cheese                     1    23    30
suntan cream               1    11    70
waterproof overclothes     1    43    75
note-case                  1    22    80
sunglasses                 1     7    20
socks                      1     4    50
count, weight, value:     14   396  1010

C#[edit]

Similar to Knapsack/0-1.

using System;  // 4790@3.6
class program
{
    static void Main()
    {
        knapSack(40);
        var sw = System.Diagnostics.Stopwatch.StartNew();
        Console.Write(knapSack(400) + "\n" + sw.Elapsed);  // 51 µs
        Console.Read();
    }

    static string knapSack(uint w1)
    {
        init(); change();
        uint n = (uint)w.Length; var K = new uint[n + 1, w1 + 1];
        for (uint vi, wi, w0, x, i = 0; i < n; i++)
            for (vi = v[i], wi = w[i], w0 = 1; w0 <= w1; w0++)
            {
                x = K[i, w0];
                if (wi <= w0) x = max(vi + K[i, w0 - wi], x);
                K[i + 1, w0] = x;
            }
        string str = "";
        for (uint v1 = K[n, w1]; v1 > 0; n--)
            if (v1 != K[n - 1, w1])
            {
                v1 -= v[n - 1]; w1 -= w[n - 1]; str += items[n - 1] + "\n";
            }
        return str;
    }

    static uint max(uint a, uint b) { return a > b ? a : b; }

    static byte[] w, v; static string[] items;

    static byte[] p = { 1, 1, 2, 2, 2, 3, 3, 3, 1, 3, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 2 };

    static void init()
    {
        w = new byte[] { 9, 13, 153, 50, 15, 68, 27, 39, 23, 52, 11,
                          32, 24, 48, 73, 42, 43, 22, 7, 18, 4, 30 };

        v = new byte[] { 150, 35, 200, 60, 60, 45, 60, 40, 30, 10, 70,
                          30, 15, 10, 40, 70, 75, 80, 20, 12, 50, 10 };

        items = new string[] {"map","compass","water","sandwich","glucose","tin",
                              "banana","apple","cheese","beer","suntan cream",
                              "camera","T-shirt","trousers","umbrella",
                              "waterproof trousers","waterproof overclothes",
                              "note-case","sunglasses","towel","socks","book"};
    }

    static void change()
    {
        int n = w.Length, s = 0, i, j, k; byte xi;
        for (i = 0; i < n; i++) s += p[i];
        {
            byte[] x = new byte[s];
            for (k = i = 0; i < n; i++)
                for (xi = w[i], j = p[i]; j > 0; j--) x[k++] = xi;
            w = x;
        }
        {
            byte[] x = new byte[s];
            for (k = i = 0; i < n; i++)
                for (xi = v[i], j = p[i]; j > 0; j--) x[k++] = xi;
            v = x;
        }
        string[] pItems = new string[s]; string itemI;
        for (k = i = 0; i < n; i++)
            for (itemI = items[i], j = p[i]; j > 0; j--) pItems[k++] = itemI;
        items = pItems;
    }
}

C++[edit]

C++ DP solution. Initially taken from C but than fixed and refactored.
Remark: The above comment implies there is a bug in the C code, but refers to a much older and very different version Pete Lomax (talk) 19:28, 20 March 2017 (UTC)

#include <iostream>
#include <vector>
#include <algorithm>
#include <stdexcept>
#include <memory>
#include <sys/time.h>

using std::cout;
using std::endl;

class StopTimer
{
public:
    StopTimer(): begin_(getUsec()) {}
    unsigned long long getTime() const { return getUsec() - begin_; }
private:
    static unsigned long long getUsec()
    {//...you might want to use something else under Windows
        timeval tv;
        const int res = ::gettimeofday(&tv, 0);
        if(res)
            return 0;
        return tv.tv_usec + 1000000 * tv.tv_sec;
    }
    unsigned long long begin_;
};

struct KnapsackTask
{
    struct Item
    {
        std::string name;
        unsigned w, v, qty;
        Item(): w(), v(), qty() {}
        Item(const std::string& iname, unsigned iw, unsigned iv, unsigned iqty):
            name(iname), w(iw), v(iv), qty(iqty)
        {}
    };
    typedef std::vector<Item> Items;
    struct Solution
    {
        unsigned v, w;
        unsigned long long iterations, usec;
        std::vector<unsigned> n;
        Solution(): v(), w(), iterations(), usec() {}
    };
    //...
    KnapsackTask(): maxWeight_(), totalWeight_() {}
    void add(const Item& item)
    {
        const unsigned totalItemWeight = item.w * item.qty;
        if(const bool invalidItem = !totalItemWeight)
            throw std::logic_error("Invalid item: " + item.name);
        totalWeight_ += totalItemWeight;
        items_.push_back(item);
    }
    const Items& getItems() const { return items_; }
    void setMaxWeight(unsigned maxWeight) { maxWeight_ = maxWeight; }
    unsigned getMaxWeight() const { return std::min(totalWeight_, maxWeight_); }

private:
    unsigned maxWeight_, totalWeight_;
    Items items_;
};

class BoundedKnapsackRecursiveSolver
{
public:
    typedef KnapsackTask Task;
    typedef Task::Item Item;
    typedef Task::Items Items;
    typedef Task::Solution Solution;

    void solve(const Task& task)
    {
        Impl(task, solution_).solve();
    }
    const Solution& getSolution() const { return solution_; }
private:
    class Impl
    {
        struct Candidate
        {
            unsigned v, n;
            bool visited;
            Candidate(): v(), n(), visited(false) {}
        };
        typedef std::vector<Candidate> Cache;
    public:
        Impl(const Task& task, Solution& solution):
            items_(task.getItems()),
            maxWeight_(task.getMaxWeight()),
            maxColumnIndex_(task.getItems().size() - 1),
            solution_(solution),
            cache_(task.getMaxWeight() * task.getItems().size()),
            iterations_(0)
        {}
        void solve()
        {
            if(const bool nothingToSolve = !maxWeight_ || items_.empty())
                return;
            StopTimer timer;
            Candidate candidate;
            solve(candidate, maxWeight_, items_.size() - 1);
            convertToSolution(candidate);
            solution_.usec = timer.getTime();
        }
    private:
        void solve(Candidate& current, unsigned reminderWeight, const unsigned itemIndex)
        {
            ++iterations_;

            const Item& item(items_[itemIndex]);

            if(const bool firstColumn = !itemIndex)
            {
                const unsigned maxQty = std::min(item.qty, reminderWeight/item.w);
                current.v = item.v * maxQty;
                current.n = maxQty;
                current.visited = true;
            }
            else
            {
                const unsigned nextItemIndex = itemIndex - 1;
                {
                    Candidate& nextItem = cachedItem(reminderWeight, nextItemIndex);
                    if(!nextItem.visited)
                        solve(nextItem, reminderWeight, nextItemIndex);
                    current.visited = true;
                    current.v = nextItem.v;
                    current.n = 0;
                }
                if(reminderWeight >= item.w)
                {
                    for (unsigned numberOfItems = 1; numberOfItems <= item.qty; ++numberOfItems)
                    {
                        reminderWeight -= item.w;
                        Candidate& nextItem = cachedItem(reminderWeight, nextItemIndex);
                        if(!nextItem.visited)
                            solve(nextItem, reminderWeight, nextItemIndex);

                        const unsigned checkValue = nextItem.v + numberOfItems * item.v;
                        if ( checkValue > current.v)
                        {
                            current.v = checkValue;
                            current.n = numberOfItems;
                        }
                        if(!(reminderWeight >= item.w))
                            break;
                    }
                }
            }
        }
        void convertToSolution(const Candidate& candidate)
        {
            solution_.iterations = iterations_;
            solution_.v = candidate.v;
            solution_.n.resize(items_.size());

            const Candidate* iter = &candidate;
            unsigned weight = maxWeight_, itemIndex = items_.size() - 1;
            while(true)
            {
                const unsigned currentWeight = iter->n * items_[itemIndex].w;
                solution_.n[itemIndex] = iter->n;
                weight -= currentWeight;
                if(!itemIndex--)
                    break;
                iter = &cachedItem(weight, itemIndex);
            }
            solution_.w = maxWeight_ - weight;
        }
        Candidate& cachedItem(unsigned weight, unsigned itemIndex)
        {
            return cache_[weight * maxColumnIndex_ + itemIndex];
        }
        const Items& items_;
        const unsigned maxWeight_;
        const unsigned maxColumnIndex_;
        Solution& solution_;
        Cache cache_;
        unsigned long long iterations_;
    };
    Solution solution_;
};

void populateDataset(KnapsackTask& task)
{
    typedef KnapsackTask::Item Item;
    task.setMaxWeight( 400 );
    task.add(Item("map",9,150,1));
    task.add(Item("compass",13,35,1));
    task.add(Item("water",153,200,2));
    task.add(Item("sandwich",50,60,2));
    task.add(Item("glucose",15,60,2));
    task.add(Item("tin",68,45,3));
    task.add(Item("banana",27,60,3));
    task.add(Item("apple",39,40,3));
    task.add(Item("cheese",23,30,1));
    task.add(Item("beer",52,10,3));
    task.add(Item("suntancream",11,70,1));
    task.add(Item("camera",32,30,1));
    task.add(Item("T-shirt",24,15,2));
    task.add(Item("trousers",48,10,2));
    task.add(Item("umbrella",73,40,1));
    task.add(Item("w-trousers",42,70,1));
    task.add(Item("w-overclothes",43,75,1));
    task.add(Item("note-case",22,80,1));
    task.add(Item("sunglasses",7,20,1));
    task.add(Item("towel",18,12,2));
    task.add(Item("socks",4,50,1));
    task.add(Item("book",30,10,2));
}

int main()
{
    KnapsackTask task;
    populateDataset(task);

    BoundedKnapsackRecursiveSolver solver;
    solver.solve(task);
    const KnapsackTask::Solution& solution = solver.getSolution();

    cout << "Iterations to solve: " << solution.iterations << endl;
    cout << "Time to solve: " << solution.usec << " usec" << endl;
    cout << "Solution:" << endl;
    for (unsigned i = 0; i < solution.n.size(); ++i)
    {
        if (const bool itemIsNotInKnapsack = !solution.n[i])
            continue;
        cout << "  " << solution.n[i] << ' ' << task.getItems()[i].name << " ( item weight = " << task.getItems()[i].w << " )" << endl;
    }

    cout << "Weight: " << solution.w << " Value: " << solution.v << endl;
    return 0;
}

Clojure[edit]

We convert the problem to a Knapsack-0/1 problem by replacing (n-max item) vith n-max identical occurences of 1 item. Adapted Knapsack-0/1 problem solution from [1]

(ns knapsack
  (:gen-class))

(def groupeditems [
                   ["map", 9, 150, 1]
                   ["compass", 13, 35, 1]
                   ["water", 153, 200, 3]
                   ["sandwich", 50, 60, 2]
                   ["glucose", 15, 60, 2]
                   ["tin", 68, 45, 3]
                   ["banana", 27, 60, 3]
                   ["apple", 39, 40, 3]
                   ["cheese", 23, 30, 1]
                   ["beer", 52, 10, 3]
                   ["suntan cream", 11, 70, 1]
                   ["camera", 32, 30, 1]
                   ["t-shirt", 24, 15, 2]
                   ["trousers", 48, 10, 2]
                   ["umbrella", 73, 40, 1]
                   ["waterproof trousers", 42, 70, 1]
                   ["waterproof overclothes", 43, 75, 1]
                   ["note-case", 22, 80, 1]
                   ["sunglasses", 7, 20, 1]
                   ["towel", 18, 12, 2]
                   ["socks", 4, 50, 1]
                   ["book", 30, 10, 2]
                  ])

(defstruct item :name :weight :value)

(def items (->> (for [[item wt val n] groupeditems]
                   (repeat n [item wt val]))
                 (mapcat identity)
                 (map #(apply struct item %))
                 (vec)))

(declare mm) ;forward decl for memoization function
(defn m [i w]
  (cond
    (< i 0) [0 []]
    (= w 0) [0 []]
    :else
    (let [{wi :weight vi :value} (get items i)]
      (if (> wi w)
        (mm (dec i) w)
        (let [[vn sn :as no]  (mm (dec i) w)
              [vy sy :as yes] (mm (dec i) (- w wi))]
          (if (> (+ vy vi) vn)
            [(+ vy vi) (conj sy i)]
            no))))))

(def mm (memoize m))

(let [[value indexes] (mm (-> items count dec) 400)
      names (map (comp :name items) indexes)]
  (println "Items to pack:")
  (doseq [[k v] (frequencies names)]
    (println (format "%d %s" v k)))
  (println "Total value:" value)
  (println "Total weight:" (reduce + (map (comp :weight items) indexes))))
Output:
Items to pack:
1 suntan cream
1 map
3 banana
1 water
1 compass
2 glucose
1 note-case
1 waterproof overclothes
1 cheese
1 sunglasses
1 socks
Total value: 1010
Total weight: 396

Common Lisp[edit]

;;; memoize
(defmacro mm-set (p v) `(if ,p ,p (setf ,p ,v)))

(defun knapsack (max-weight items)
  (let ((cache (make-array (list (1+ max-weight) (1+ (length items)))
			   :initial-element nil)))

    (labels ((knapsack1 (spc items)
	(if (not items) (return-from knapsack1 (list 0 0 '())))
	 (mm-set (aref cache spc (length items))
	   (let* ((i (first items))
		  (w (second i))
		  (v (third i))
		  (x (knapsack1 spc (cdr items))))
	     (loop for cnt from 1 to (fourth i) do
		   (let ((w (* cnt w)) (v (* cnt v)))
		     (if (>= spc w)
		       (let ((y (knapsack1 (- spc w) (cdr items))))
			 (if (> (+ (first y) v) (first x))
			   (setf x (list (+ (first  y) v)
					 (+ (second y) w)
					 (cons (list (first i) cnt) (third y)))))))))
	     x))))
 
      (knapsack1 max-weight items))))
 
(print
  (knapsack 400
	    '((map 9 150 1) (compass 13 35 1) (water 153 200 2) (sandwich 50 60 2)
              (glucose 15 60 2) (tin 68 45 3) (banana 27 60 3) (apple 39 40 3)
	      (cheese 23 30 1) (beer 52 10 3) (cream 11 70 1) (camera 32 30 1)
	      (T-shirt 24 15 2) (trousers 48 10 2) (umbrella 73 40 1)
	      (trousers 42 70 1) (overclothes 43 75 1) (notecase 22 80 1)
	      (glasses 7 20 1) (towel 18 12 2) (socks 4 50 1) (book 30 10 2))))

Crystal[edit]

Branch and Bound solution[edit]

record Item, name : String, weight : Int32, value : Int32, count : Int32

record Selection, mask : Array(Int32), cur_index : Int32, total_value : Int32

class Knapsack
  @threshold_value = 0
  @threshold_choice : Selection?
  getter checked_nodes = 0

  def knapsack_step(taken, items, remaining_weight)
    if taken.total_value > @threshold_value
      @threshold_value = taken.total_value
      @threshold_choice = taken
    end
    candidate_index = items.index(taken.cur_index) { |item| item.weight <= remaining_weight }
    return nil unless candidate_index
    @checked_nodes += 1
    candidate = items[candidate_index]
    # candidate is a best of available items, so if we fill remaining value with
    # and still don't reach the threshold, the branch is wrong
    return nil if taken.total_value + 1.0 * candidate.value / candidate.weight * remaining_weight < @threshold_value
    # now recursively check all variants (from taking maximum count to taking nothing)
    max_count = {candidate.count, remaining_weight // candidate.weight}.min
    (0..max_count).reverse_each do |n|
      mask = taken.mask.clone
      mask[candidate_index] = n
      knapsack_step Selection.new(mask, candidate_index + 1, taken.total_value + n*candidate.value), items, remaining_weight - n*candidate.weight
    end
  end

  def select(items, max_weight)
    @checked_variants = 0
    # sort by descending relative value
    list = items.sort_by { |item| -1.0 * item.value / item.weight }
    nothing = Selection.new(Array(Int32).new(items.size, 0), 0, 0)
    @threshold_value = 0
    @threshold_choice = nothing
    knapsack_step(nothing, list, max_weight)
    selected = @threshold_choice.not_nil!
    result = Hash(Item, Int32).new(0)
    selected.mask.each_with_index { |v, i| result[list[i]] = v if v > 0 }
    result
  end
end

possible = [
  Item.new("map", 9, 150, 1),
  Item.new("compass", 13, 35, 1),
  Item.new("water", 153, 200, 2),
  Item.new("sandwich", 50, 60, 2),
  Item.new("glucose", 15, 60, 2),
  Item.new("tin", 68, 45, 3),
  Item.new("banana", 27, 60, 3),
  Item.new("apple", 39, 40, 3),
  Item.new("cheese", 23, 30, 1),
  Item.new("beer", 52, 10, 3),
  Item.new("suntan cream", 11, 70, 1),
  Item.new("camera", 32, 30, 1),
  Item.new("T-shirt", 24, 15, 2),
  Item.new("trousers", 48, 10, 2),
  Item.new("umbrella", 73, 40, 1),
  Item.new("waterproof trousers", 42, 70, 1),
  Item.new("waterproof overclothes", 43, 75, 1),
  Item.new("note-case", 22, 80, 1),
  Item.new("sunglasses", 7, 20, 1),
  Item.new("towel", 18, 12, 2),
  Item.new("socks", 4, 50, 1),
  Item.new("book", 30, 10, 2),
]

solver = Knapsack.new
used = solver.select(possible, 400)
puts "optimal choice: #{used.map { |item, count| count == 1 ? item.name : "#{count}x #{item.name}" }.join(", ")}"
puts "total weight #{used.sum { |item, count| item.weight*count }}"
puts "total value #{used.sum { |item, count| item.value*count }}"
puts "checked nodes: #{solver.checked_nodes}"
Output:
optimal choice: map, socks, suntan cream, 2x glucose, note-case, sunglasses, compass, 3x banana, waterproof overclothes, water, cheese
total weight 396
total value 1010
checked nodes: 1185

Dynamic programming solution[edit]

Translation of: Ruby
# Item struct to represent each item in the problem
record Item, name : String, weight : Int32, value : Int32, count : Int32

def choose_item(items, weight, id, cache)
  return 0, ([] of Int32) if id < 0

  k = {weight, id}
  cache = cache || {} of Tuple(Int32, Int32) => Tuple(Int32, Array(Int32))
  return cache[k] if cache[k]?
  value = items[id].value
  best_v = 0
  best_list = [] of Int32
  (items[id].count + 1).times do |i|
    wlim = weight - i * items[id].weight
    break if wlim < 0
    val, taken = choose_item(items, wlim, id - 1, cache)
    if val + i * value > best_v
      best_v = val + i * value
      best_list = taken + [i]
    end
  end
  cache[k] = {best_v, best_list}
  return {best_v, best_list}
end

items = [
  Item.new("map", 9, 150, 1),
  Item.new("compass", 13, 35, 1),
  Item.new("water", 153, 200, 2),
  Item.new("sandwich", 50, 60, 2),
  Item.new("glucose", 15, 60, 2),
  Item.new("tin", 68, 45, 3),
  Item.new("banana", 27, 60, 3),
  Item.new("apple", 39, 40, 3),
  Item.new("cheese", 23, 30, 1),
  Item.new("beer", 52, 10, 3),
  Item.new("suntan cream", 11, 70, 1),
  Item.new("camera", 32, 30, 1),
  Item.new("T-shirt", 24, 15, 2),
  Item.new("trousers", 48, 10, 2),
  Item.new("umbrella", 73, 40, 1),
  Item.new("waterproof trousers", 42, 70, 1),
  Item.new("waterproof overclothes", 43, 75, 1),
  Item.new("note-case", 22, 80, 1),
  Item.new("sunglasses", 7, 20, 1),
  Item.new("towel", 18, 12, 2),
  Item.new("socks", 4, 50, 1),
  Item.new("book", 30, 10, 2),
]

val, list = choose_item(items, 400, items.size - 1, nil)
w = 0
list.each_with_index do |cnt, i|
  if cnt > 0
    print "#{cnt} #{items[i].name}\n"
    w += items[i].weight * cnt
  end
end

p "Total weight: #{w}, Value: #{val}"

D[edit]

Translation of: Python

Solution with memoization.

import std.stdio, std.typecons, std.functional;

immutable struct Item {
    string name;
    int weight, value, quantity;
}

immutable Item[] items = [
    {"map",           9, 150, 1}, {"compass",      13,  35, 1},
    {"water",       153, 200, 3}, {"sandwich",     50,  60, 2},
    {"glucose",      15,  60, 2}, {"tin",          68,  45, 3},
    {"banana",       27,  60, 3}, {"apple",        39,  40, 3},
    {"cheese",       23,  30, 1}, {"beer",         52,  10, 3},
    {"suntan cream", 11,  70, 1}, {"camera",       32,  30, 1},
    {"t-shirt",      24,  15, 2}, {"trousers",     48,  10, 2},
    {"umbrella",     73,  40, 1}, {"w-trousers",   42,  70, 1},
    {"w-overcoat",   43,  75, 1}, {"note-case",    22,  80, 1},
    {"sunglasses",    7,  20, 1}, {"towel",        18,  12, 2},
    {"socks",         4,  50, 1}, {"book",         30,  10, 2}];

Tuple!(int, const(int)[]) chooseItem(in int iWeight, in int idx) nothrow @safe {
    alias memoChooseItem = memoize!chooseItem;
    if (idx < 0)
        return typeof(return)();

    int bestV;
    const(int)[] bestList;
    with (items[idx])
        foreach (immutable i; 0 .. quantity + 1) {
            immutable wlim = iWeight - i * weight;
            if (wlim < 0)
                break;

            //const (val, taken) = memoChooseItem(wlim, idx - 1);
            const val_taken = memoChooseItem(wlim, idx - 1);
            if (val_taken[0] + i * value > bestV) {
                bestV = val_taken[0] + i * value;
                bestList = val_taken[1] ~ i;
            }
        }

    return tuple(bestV, bestList);
}

void main() {
    // const (v, lst) = chooseItem(400, items.length - 1);
    const v_lst = chooseItem(400, items.length - 1);

    int w;
    foreach (immutable i, const cnt; v_lst[1])
        if (cnt > 0) {
            writeln(cnt, " ", items[i].name);
            w += items[i].weight * cnt;
        }

    writeln("Total weight: ", w, " Value: ", v_lst[0]);
}
Output:
1 map
1 compass
1 water
2 glucose
3 banana
1 cheese
1 suntan cream
1 w-overcoat
1 note-case
1 sunglasses
1 socks
Total weight: 396 Value: 1010

EchoLisp[edit]

We convert the problem to a Knapsack-0/1 problem by replacing (n-max item) vith n-max identical occurences of 1 item.

(lib 'struct)
(lib 'sql)
(lib 'hash)

(define H (make-hash))
(define T (make-table (struct goodies (name poids valeur qty))))
(define IDX (make-vector 0))


;; convert to 0/1 PB
;; transorm vector [1 2 3 4 (n-max 3) 5 (n-max 2) 6 .. ]
;; into vector of repeated indices : [1 2 3 4 4 4 5 5 6 ... ]

(define (make-01 T)
	 (for ((record T) (i (in-naturals)))
		 (for ((j (in-range 0 (goodies-qty record))))
			 (vector-push IDX i)))
	 IDX)
	
(define-syntax-rule (name i) (table-xref T (vector-ref IDX i) 0))
(define-syntax-rule (poids i) (table-xref T (vector-ref IDX i) 1))
(define-syntax-rule (valeur i) (table-xref T (vector-ref IDX i) 2))

;;
;; code identical to 0/1 problem	
;;

;;  make an unique hash-key from (i rest)
(define (t-idx i r)  (string-append i "|" r))
;; retrieve best core for item i, remaining r availbble weight
(define (t-get i r)  (or (hash-ref H (t-idx i r)) 0))

;; compute best score (i), assuming best (i-1 rest) is known
(define (score i restant)
	 (if (< i 0) 0
	 (hash-ref! H (t-idx i restant)
		 (if ( >= restant (poids i)) 
			 (max 
			 (score (1- i) restant) 
			 (+ (score (1- i) (- restant (poids i))) (valeur i)))
		    (score (1- i) restant)) ;; else not enough
	)))
	
;; compute best scores, starting from last item
(define (task W)
        (define restant W)
        (make-01 T) 
        (define N (1- (vector-length IDX)))
		 (writeln 'total-value (score N W))
		 (group
		 (for/list  ((i (in-range N -1 -1)))
		 #:continue (= (t-get i restant) (t-get (1- i) restant))
		(set! restant (- restant (poids i)))
		(name i))))
Output:
(define goodies
    '((map  9 150 1) (compass 13 35 1)(water 153 200 3)
 (sandwich 50 60 2)(🍰-glucose 15 60 2)(tin  68 45 3)
 (🍌-banana  27 60 3)(🍎-apple 39 40 3)(cheese 23 30 1)
 (beer 52 10 3)(🌞-suntan-cream  11  70 1)(camera 32 30 1)
 (t-shirt 24 15 2)(trousers 48 10 2)(umbrella 73 40 1)
 (☔️-trousers  42 70 1)(☔️--overcoat 43 75 1)(note-case 22 80 1)
 (🌞-sunglasses 7 20 1)(towel 18 12 2)(socks 4 50 1)
 (book 30 10 2)))

(list->table goodies T)

(task 400)
total-value     1010    
    ((socks) (🌞-sunglasses) (note-case) (☔️--overcoat) (🌞-suntan-cream) (cheese)
 (🍌-banana 🍌-banana 🍌-banana) (🍰-glucose 🍰-glucose) (water) (compass) (map))

(length (hash-keys H))
     10827 ;; # of entries in cache

FreeBASIC[edit]

#define PesoMax  400
#define items  22
#define Tabu  Chr(9)

Type Knapsack
    articulo As String*22
    peso As Integer
    valor As Integer
    pieza As Integer
End Type
Dim item(1 To 22) As Knapsack => { _
("map                   ",   9, 150, 0), ("compass               ",  13,  35, 0), _
("water                 ", 153, 200, 0), ("sandwich              ",  50, 160, 0), _
("glucose               ",  15,  60, 0), ("tin                   ",  68,  45, 0), _
("banana                ",  27,  60, 0), ("apple                 ",  39,  40, 0), _
("cheese                ",  23,  30, 0), ("beer                  ",  52,  10, 0), _
("suntan cream          ",  11,  70, 0), ("camera                ",  32,  30, 0), _
("T-shirt               ",  24,  15, 0), ("trousers              ",  48,  10, 0), _
("umbrella              ",  73,  40, 0), ("waterproof trousers   ",  42,  70, 0), _
("waterproof overclothes",  43,  75, 0), ("note-case             ",  22,  80, 0), _
("sunglasses            ",   7,  20, 0), ("towel                 ",  18,  12, 0), _
("socks                 ",   4,  50, 0), ("book                  ",  30,  10, 0)}

Dim As Integer i, tot = 0, TValor = 0
For i =1 To Ubound(item)
    tot += item(i).peso
Next i

Dim As Integer TPeso = tot-PesoMax
Dim As String pr = ""
Dim As Integer c = 0, v, w, k

Do
    v = 1e9 : w = 0 : k = 0
    
    For i = 1 To items
        If item(i).pieza = 0 Then
            w = 1000 * item(i).valor / item(i).peso
            If w < v Then v = w : k = i
        End If
    Next i
    
    If k Then
        TPeso -= item(k).peso
        item(k).pieza = 1
        If TPeso <= 0 Then Exit Do
    End If
    c += 1
Loop Until c>= items

Print "Knapsack contents: "
For i = 1 To items
    If item(i).pieza = 0 Then 
        Print item(i).articulo & Tabu & item(i).peso & Tabu & item(i).valor
        TValor += item(i).valor
    End If
Next i

Print !"\nTotal value: "; TValor
Print "Total weight: "; PesoMax + TPeso
Sleep
Output:
Knapsack contents: 
map                     9       150
compass                 13      35
water                   153     200
sandwich                50      160
glucose                 15      60
banana                  27      60
suntan cream            11      70
waterproof trousers     42      70
waterproof overclothes  43      75
note-case               22      80
sunglasses              7       20
socks                   4       50

Total value: 1030
Total weight: 396

Go[edit]

Solution with caching.

package main

import "fmt"

type Item struct {
	name               string
	weight, value, qty int
}

var items = []Item{
	{"map",			9,	150,	1},
	{"compass",		13,	35,	1},
	{"water",		153,	200,	2},
	{"sandwich",		50,	60,	2},
	{"glucose",		15,	60,	2},
	{"tin",			68,	45,	3},
	{"banana",		27,	60,	3},
	{"apple",		39,	40,	3},
	{"cheese",		23,	30,	1},
	{"beer",		52,	10,	3},
	{"suntancream",		11,	70,	1},
	{"camera",		32,	30,	1},
	{"T-shirt",		24,	15,	2},
	{"trousers",		48,	10,	2},
	{"umbrella",		73,	40,	1},
	{"w-trousers",		42,	70,	1},
	{"w-overclothes",	43,	75,	1},
	{"note-case",		22,	80,	1},
	{"sunglasses",		7,      20,	1},
	{"towel",		18,	12,	2},
	{"socks",		4,      50,	1},
	{"book",		30,	10,	2},
}

type Chooser struct {
	Items []Item
	cache map[key]solution
}

type key struct {
	w, p int
}

type solution struct {
	v, w int
	qty  []int
}

func (c Chooser) Choose(limit int) (w, v int, qty []int) {
	c.cache = make(map[key]solution)
	s := c.rchoose(limit, len(c.Items)-1)
	c.cache = nil // allow cache to be garbage collected
	return s.v, s.w, s.qty
}

func (c Chooser) rchoose(limit, pos int) solution {
	if pos < 0 || limit <= 0 {
		return solution{0, 0, nil}
	}

	key := key{limit, pos}
	if s, ok := c.cache[key]; ok {
		return s
	}

	best_i, best := 0, solution{0, 0, nil}
	for i := 0; i*items[pos].weight <= limit && i <= items[pos].qty; i++ {
		sol := c.rchoose(limit-i*items[pos].weight, pos-1)
		sol.v += i * items[pos].value
		if sol.v > best.v {
			best_i, best = i, sol
		}
	}

	if best_i > 0 {
		// best.qty is used in another cache entry,
		// we need to duplicate it before modifying it to
		// store as our cache entry.
		old := best.qty
		best.qty = make([]int, len(items))
		copy(best.qty, old)
		best.qty[pos] = best_i
		best.w += best_i * items[pos].weight
	}
	c.cache[key] = best
	return best
}

func main() {
	v, w, s := Chooser{Items: items}.Choose(400)

	fmt.Println("Taking:")
	for i, t := range s {
		if t > 0 {
			fmt.Printf("  %d of %d %s\n", t, items[i].qty, items[i].name)
		}
	}
	fmt.Printf("Value: %d; weight: %d\n", v, w)
}

(A simple test and benchmark used while making changes to make sure performance wasn't sacrificed is available at /Go_test.)

Output:
Taking:
  1 of 1 map
  1 of 1 compass
  1 of 2 water
  2 of 2 glucose
  3 of 3 banana
  1 of 1 cheese
  1 of 1 suntancream
  1 of 1 w-overclothes
  1 of 1 note-case
  1 of 1 sunglasses
  1 of 1 socks
Value: 1010; weight: 396

Groovy[edit]

Solution: dynamic programming

def totalWeight = { list -> list.collect{ it.item.weight * it.count }.sum() }
def totalValue = { list -> list.collect{ it.item.value * it.count }.sum() }

def knapsackBounded = { possibleItems ->
    def n = possibleItems.size()
    def m = (0..n).collect{ i -> (0..400).collect{ w -> []} }
    (1..400).each { w ->
        (1..n).each { i ->
            def item = possibleItems[i-1]
            def wi = item.weight, pi = item.pieces
            def bi = [w.intdiv(wi),pi].min()
            m[i][w] = (0..bi).collect{ count ->
                m[i-1][w - wi * count] + [[item:item, count:count]]
            }.max(totalValue).findAll{ it.count }
        }
    }
    m[n][400]
}

Test:

def items = [ 
        [name:"map",                    weight:  9, value:150, pieces:1],
        [name:"compass",                weight: 13, value: 35, pieces:1],
        [name:"water",                  weight:153, value:200, pieces:2],
        [name:"sandwich",               weight: 50, value: 60, pieces:2],
        [name:"glucose",                weight: 15, value: 60, pieces:2],
        [name:"tin",                    weight: 68, value: 45, pieces:3],
        [name:"banana",                 weight: 27, value: 60, pieces:3],
        [name:"apple",                  weight: 39, value: 40, pieces:3],
        [name:"cheese",                 weight: 23, value: 30, pieces:1],
        [name:"beer",                   weight: 52, value: 10, pieces:3],
        [name:"suntan cream",           weight: 11, value: 70, pieces:1],
        [name:"camera",                 weight: 32, value: 30, pieces:1],
        [name:"t-shirt",                weight: 24, value: 15, pieces:2],
        [name:"trousers",               weight: 48, value: 10, pieces:2],
        [name:"umbrella",               weight: 73, value: 40, pieces:1],
        [name:"waterproof trousers",    weight: 42, value: 70, pieces:1],
        [name:"waterproof overclothes", weight: 43, value: 75, pieces:1],
        [name:"note-case",              weight: 22, value: 80, pieces:1],
        [name:"sunglasses",             weight:  7, value: 20, pieces:1],
        [name:"towel",                  weight: 18, value: 12, pieces:2],
        [name:"socks",                  weight:  4, value: 50, pieces:1],
        [name:"book",                   weight: 30, value: 10, pieces:2],
]
 
def start = System.currentTimeMillis()
def packingList = knapsackBounded(items)
def elapsed = System.currentTimeMillis() - start

println "Elapsed Time: ${elapsed/1000.0} s"
println "Total Weight: ${totalWeight(packingList)}"
println " Total Value: ${totalValue(packingList)}"
packingList.each {
    printf ('  item: %-22s  weight:%4d  value:%4d  count:%2d\n',
            it.item.name, it.item.weight, it.item.value, it.count)
}
Output:
Elapsed Time: 0.603 s
Total Weight: 396
 Total Value: 1010
  item: map                     weight:   9  value: 150  count: 1
  item: compass                 weight:  13  value:  35  count: 1
  item: water                   weight: 153  value: 200  count: 1
  item: glucose                 weight:  15  value:  60  count: 2
  item: banana                  weight:  27  value:  60  count: 3
  item: cheese                  weight:  23  value:  30  count: 1
  item: suntan cream            weight:  11  value:  70  count: 1
  item: waterproof overclothes  weight:  43  value:  75  count: 1
  item: note-case               weight:  22  value:  80  count: 1
  item: sunglasses              weight:   7  value:  20  count: 1
  item: socks                   weight:   4  value:  50  count: 1

Haskell[edit]

Directly lifted from 1-0 problem:

inv = 	[("map",9,150,1), ("compass",13,35,1), ("water",153,200,2), ("sandwich",50,60,2),
	("glucose",15,60,2), ("tin",68,45,3), ("banana",27,60,3), ("apple",39,40,3),
	("cheese",23,30,1), ("beer",52,10,3), ("cream",11,70,1), ("camera",32,30,1),
	-- what to do if we end up taking one trouser?
	("tshirt",24,15,2), ("trousers",48,10,2), ("umbrella",73,40,1), ("wtrousers",42,70,1),
	("woverclothes",43,75,1), ("notecase",22,80,1), ("sunglasses",7,20,1), ("towel",18,12,2),
	("socks",4,50,1), ("book",30,10,2)]

knapsack = foldr addItem (repeat (0,[])) where
	addItem (name,w,v,c) old = foldr inc old [1..c] where
		inc i list = left ++ zipWith max right new where
			(left, right) = splitAt (w * i) list
			new = map (\(val,itms)->(val + v * i, (name,i):itms)) old

main = print $ (knapsack inv) !! 400
Output:
(1010,[("socks",1),("sunglasses",1),("notecase",1),("woverclothes",1),("cream",1),("cheese",1),("banana",3),("glucose",2),("water",1),("compass",1),("map",1)])

The above uses merging lists for cache. It's faster, and maybe easier to understand when some constant-time lookup structure is used for cache (same output):

import Data.Array

-- snipped the item list; use the one from above
knapsack items cap = (solu items) ! cap where
	solu = foldr f (listArray (0,cap) (repeat (0,[])))
	f (name,w,v,cnt) ss = listArray (0,cap) $ map optimal [0..] where
		optimal ww = maximum $ (ss!ww):[prepend (v*i,(name,i)) (ss!(ww - i*w))
					| i <- [1..cnt], i*w < ww]
		prepend (x,n) (y,s) = (x+y,n:s)

main = do print $ knapsack inv 400

J[edit]

Brute force solution:

'names numbers'=:|:".;._2]0 :0
  'map';                      9       150        1
  'compass';                 13        35        1
  'water';                  153       200        2
  'sandwich';                50        60        2
  'glucose';                 15        60        2
  'tin';                     68        45        3
  'banana';                  27        60        3
  'apple';                   39        40        3
  'cheese';                  23        30        1
  'beer';                    52        10        3
  'suntan cream';            11        70        1
  'camera';                  32        30        1
  'T-shirt';                 24        15        2
  'trousers';                48        10        2
  'umbrella';                73        40        1
  'waterproof trousers';     42        70        1
  'waterproof overclothes';  43        75        1
  'note-case';               22        80        1
  'sunglasses';               7        20        1
  'towel';                   18        12        2
  'socks';                    4        50        1
  'book';                    30        10        2
)

'weights values pieces'=:|:numbers
decode=: (pieces+1)&#:

pickBest=:4 :0
  NB. given a list of options, return the best option(s)
  n=. decode y
  weight=. n+/ .*weights
  value=.  (x >: weight) * n+/ .*values
  (value = >./value)#y
)

bestCombo=:3 :0
   limit=. */pieces+1
   i=. 0
   step=. 1e6
   best=. ''
   while.i<limit do.
      best=. 400 pickBest best,(#~ limit&>)i+i.step
      i=. i+step
   end.
   best
)

   bestCombo''
978832641

Interpretation of answer:

   decode 978832641
1 1 1 0 2 0 3 0 1 0 1 0 0 0 0 0 1 1 1 0 1 0
    (0<decode 978832641) # (":,.decode 978832641),.' ',.names
1 map                   
1 compass               
1 water                 
2 glucose               
3 banana                
1 cheese                
1 suntan cream          
1 waterproof overclothes
1 note-case             
1 sunglasses            
1 socks                 
   weights +/ .* decode 978832641
396
   values +/ .* decode 978832641
1010

Dynamic programming solution (faster):

dyn=:3 :0
  m=. 0$~1+400,+/pieces NB. maximum value cache
  b=. m                 NB. best choice cache
  opts=.+/\0,pieces     NB. distinct item counts before each piece
  P=. */\1+0,pieces   NB. distinct possibilities before each piece
  for_w.1+i.400 do.
    for_j.i.#pieces do.
      n=. i.1+j{pieces         NB. possible counts for this piece
      W=. n*j{weights          NB. how much they weigh
      s=. w>:W                 NB. permissible options
      v=. s*n*j{values         NB. consequent values
      base=. j{opts            NB. base index for these options
      I=. <"1 w,.n+base        NB. consequent indices
      i0=. <w,base             NB. status quo index
      iN=. <"1 (w-s*W),.base   NB. predecessor indices
      M=. >./\(m{~i0)>.v+m{~iN NB. consequent maximum values
      C=. (n*j{P)+b{~iN        NB. unique encoding for each option
      B=. >./\(b{~i0)>. C * 2 ~:/\ 0,M NB. best options, so far
      m=. M I} m       NB. update with newly computed maxima
      b=. B I} b       NB. same for best choice
    end.
  end.
  |.(1+|.pieces)#:{:{:b
)

   dyn''
1 1 1 0 2 0 3 0 1 0 1 0 0 0 0 0 1 1 1 0 1 0

Note: the brute force approach would return multiple "best answers" if more than one combination of choices would satisfy the "best" constraint. The dynamic approach arbitrarily picks one of those choices. That said, with this particular choice of item weights and values, this is an irrelevant distinction.

Java[edit]

General dynamic solution after wikipedia. The solution extends the method of Knapsack problem/0-1#Java .

package hu.pj.alg.test;

import hu.pj.alg.BoundedKnapsack;
import hu.pj.obj.Item;
import java.util.*;
import java.text.*;

public class BoundedKnapsackForTourists {
    public BoundedKnapsackForTourists() {
        BoundedKnapsack bok = new BoundedKnapsack(400); // 400 dkg = 400 dag = 4 kg

        // making the list of items that you want to bring
        bok.add("map", 9, 150, 1);
        bok.add("compass", 13, 35, 1);
        bok.add("water", 153, 200, 3);
        bok.add("sandwich", 50, 60, 2);
        bok.add("glucose", 15, 60, 2);
        bok.add("tin", 68, 45, 3);
        bok.add("banana", 27, 60, 3);
        bok.add("apple", 39, 40, 3);
        bok.add("cheese", 23, 30, 1);
        bok.add("beer", 52, 10, 3);
        bok.add("suntan cream", 11, 70, 1);
        bok.add("camera", 32, 30, 1);
        bok.add("t-shirt", 24, 15, 2);
        bok.add("trousers", 48, 10, 2);
        bok.add("umbrella", 73, 40, 1);
        bok.add("waterproof trousers", 42, 70, 1);
        bok.add("waterproof overclothes", 43, 75, 1);
        bok.add("note-case", 22, 80, 1);
        bok.add("sunglasses", 7, 20, 1);
        bok.add("towel", 18, 12, 2);
        bok.add("socks", 4, 50, 1);
        bok.add("book", 30, 10, 2);

        // calculate the solution:
        List<Item> itemList = bok.calcSolution();

        // write out the solution in the standard output
        if (bok.isCalculated()) {
            NumberFormat nf  = NumberFormat.getInstance();

            System.out.println(
                "Maximal weight           = " +
                nf.format(bok.getMaxWeight() / 100.0) + " kg"
            );
            System.out.println(
                "Total weight of solution = " +
                nf.format(bok.getSolutionWeight() / 100.0) + " kg"
            );
            System.out.println(
                "Total value              = " +
                bok.getProfit()
            );
            System.out.println();
            System.out.println(
                "You can carry te following materials " +
                "in the knapsack:"
            );
            for (Item item : itemList) {
                if (item.getInKnapsack() > 0) {
                    System.out.format(
                        "%1$-10s %2$-23s %3$-3s %4$-5s %5$-15s \n",
                        item.getInKnapsack() + " unit(s) ",
                        item.getName(),
                        item.getInKnapsack() * item.getWeight(), "dag  ",
                        "(value = " + item.getInKnapsack() * item.getValue() + ")"
                    );
                }
            }
        } else {
            System.out.println(
                "The problem is not solved. " +
                "Maybe you gave wrong data."
            );
        }

    }

    public static void main(String[] args) {
        new BoundedKnapsackForTourists();
    }
} // class
package hu.pj.alg;

import hu.pj.obj.Item;
import java.util.*;

public class BoundedKnapsack extends ZeroOneKnapsack {
    public BoundedKnapsack() {}

    public BoundedKnapsack(int _maxWeight) {
        setMaxWeight(_maxWeight);
    }

    public BoundedKnapsack(List<Item> _itemList) {
        setItemList(_itemList);
    }

    public BoundedKnapsack(List<Item> _itemList, int _maxWeight) {
        setItemList(_itemList);
        setMaxWeight(_maxWeight);
    }

    @Override
    public List<Item> calcSolution() {
        int n = itemList.size();

        // add items to the list, if bounding > 1
        for (int i = 0; i < n; i++) {
            Item item = itemList.get(i);
            if (item.getBounding() > 1) {
                for (int j = 1; j < item.getBounding(); j++) {
                    add(item.getName(), item.getWeight(), item.getValue());
                }
            }
        }
        
        super.calcSolution();

        // delete the added items, and increase the original items
        while (itemList.size() > n) {
            Item lastItem = itemList.get(itemList.size() - 1);
            if (lastItem.getInKnapsack() == 1) {
                for (int i = 0; i < n; i++) {
                    Item iH = itemList.get(i);
                    if (lastItem.getName().equals(iH.getName())) {
                        iH.setInKnapsack(1 + iH.getInKnapsack());
                        break;
                    }
                }
            }
            itemList.remove(itemList.size() - 1);
        }

        return itemList;
    }

    // add an item to the item list
    public void add(String name, int weight, int value, int bounding) {
        if (name.equals(""))
            name = "" + (itemList.size() + 1);
        itemList.add(new Item(name, weight, value, bounding));
        setInitialStateForCalculation();
    }
} // class
package hu.pj.alg;

import hu.pj.obj.Item;
import java.util.*;

public class ZeroOneKnapsack {
    protected List<Item> itemList  = new ArrayList<Item>();
    protected int maxWeight        = 0;
    protected int solutionWeight   = 0;
    protected int profit           = 0;
    protected boolean calculated   = false;

    public ZeroOneKnapsack() {}

    public ZeroOneKnapsack(int _maxWeight) {
        setMaxWeight(_maxWeight);
    }

    public ZeroOneKnapsack(List<Item> _itemList) {
        setItemList(_itemList);
    }

    public ZeroOneKnapsack(List<Item> _itemList, int _maxWeight) {
        setItemList(_itemList);
        setMaxWeight(_maxWeight);
    }

    // calculte the solution of 0-1 knapsack problem with dynamic method:
    public List<Item> calcSolution() {
        int n = itemList.size();

        setInitialStateForCalculation();
        if (n > 0  &&  maxWeight > 0) {
            List< List<Integer> > c = new ArrayList< List<Integer> >();
            List<Integer> curr = new ArrayList<Integer>();

            c.add(curr);
            for (int j = 0; j <= maxWeight; j++)
                curr.add(0);
            for (int i = 1; i <= n; i++) {
                List<Integer> prev = curr;
                c.add(curr = new ArrayList<Integer>());
                for (int j = 0; j <= maxWeight; j++) {
                    if (j > 0) {
                        int wH = itemList.get(i-1).getWeight();
                        curr.add(
                            (wH > j)
                            ?
                            prev.get(j)
                            :
                            Math.max(
                                prev.get(j),
                                itemList.get(i-1).getValue() + prev.get(j-wH)
                            )
                        );
                    } else {
                        curr.add(0);
                    }
                } // for (j...)
            } // for (i...)
            profit = curr.get(maxWeight);

            for (int i = n, j = maxWeight; i > 0  &&  j >= 0; i--) {
                int tempI   = c.get(i).get(j);
                int tempI_1 = c.get(i-1).get(j);
                if (
                    (i == 0  &&  tempI > 0)
                    ||
                    (i > 0  &&  tempI != tempI_1)
                )
                {
                    Item iH = itemList.get(i-1);
                    int  wH = iH.getWeight();
                    iH.setInKnapsack(1);
                    j -= wH;
                    solutionWeight += wH;
                }
            } // for()
            calculated = true;
        } // if()
        return itemList;
    }

    // add an item to the item list
    public void add(String name, int weight, int value) {
        if (name.equals(""))
            name = "" + (itemList.size() + 1);
        itemList.add(new Item(name, weight, value));
        setInitialStateForCalculation();
    }

    // add an item to the item list
    public void add(int weight, int value) {
        add("", weight, value); // the name will be "itemList.size() + 1"!
    }

    // remove an item from the item list
    public void remove(String name) {
        for (Iterator<Item> it = itemList.iterator(); it.hasNext(); ) {
            if (name.equals(it.next().getName())) {
                it.remove();
            }
        }
        setInitialStateForCalculation();
    }

    // remove all items from the item list
    public void removeAllItems() {
        itemList.clear();
        setInitialStateForCalculation();
    }

    public int getProfit() {
        if (!calculated)
            calcSolution();
        return profit;
    }

    public int getSolutionWeight() {return solutionWeight;}
    public boolean isCalculated() {return calculated;}
    public int getMaxWeight() {return maxWeight;}

    public void setMaxWeight(int _maxWeight) {
        maxWeight = Math.max(_maxWeight, 0);
    }

    public void setItemList(List<Item> _itemList) {
        if (_itemList != null) {
            itemList = _itemList;
            for (Item item : _itemList) {
                item.checkMembers();
            }
        }
    }

    // set the member with name "inKnapsack" by all items:
    private void setInKnapsackByAll(int inKnapsack) {
        for (Item item : itemList)
            if (inKnapsack > 0)
                item.setInKnapsack(1);
            else
                item.setInKnapsack(0);
    }

    // set the data members of class in the state of starting the calculation:
    protected void setInitialStateForCalculation() {
        setInKnapsackByAll(0);
        calculated     = false;
        profit         = 0;
        solutionWeight = 0;
    }
} // class
package hu.pj.obj;

public class Item {
    protected String name    = "";
    protected int weight     = 0;
    protected int value      = 0;
    protected int bounding   = 1; // the maximal limit of item's pieces
    protected int inKnapsack = 0; // the pieces of item in solution

    public Item() {}

    public Item(Item item) {
        setName(item.name);
        setWeight(item.weight);
        setValue(item.value);
        setBounding(item.bounding);
    }

    public Item(int _weight, int _value) {
        setWeight(_weight);
        setValue(_value);
    }

    public Item(int _weight, int _value, int _bounding) {
        setWeight(_weight);
        setValue(_value);
        setBounding(_bounding);
    }

    public Item(String _name, int _weight, int _value) {
        setName(_name);
        setWeight(_weight);
        setValue(_value);
    }

    public Item(String _name, int _weight, int _value, int _bounding) {
        setName(_name);
        setWeight(_weight);
        setValue(_value);
        setBounding(_bounding);
    }

    public void setName(String _name) {name = _name;}
    public void setWeight(int _weight) {weight = Math.max(_weight, 0);}
    public void setValue(int _value) {value = Math.max(_value, 0);}

    public void setInKnapsack(int _inKnapsack) {
        inKnapsack = Math.min(getBounding(), Math.max(_inKnapsack, 0));
    }

    public void setBounding(int _bounding) {
        bounding = Math.max(_bounding, 0);
        if (bounding == 0)
            inKnapsack = 0;
    }

    public void checkMembers() {
        setWeight(weight);
        setValue(value);
        setBounding(bounding);
        setInKnapsack(inKnapsack);
    }

    public String getName() {return name;}
    public int getWeight() {return weight;}
    public int getValue() {return value;}
    public int getInKnapsack() {return inKnapsack;}
    public int getBounding() {return bounding;}
} // class
Output:
Maximal weight           = 4 kg
Total weight of solution = 3,96 kg
Total value              = 1010

You can carry te following materials in the knapsack:
1 unit(s)  map                     9   dag   (value = 150)   
1 unit(s)  compass                 13  dag   (value = 35)    
1 unit(s)  water                   153 dag   (value = 200)   
2 unit(s)  glucose                 30  dag   (value = 120)   
3 unit(s)  banana                  81  dag   (value = 180)   
1 unit(s)  cheese                  23  dag   (value = 30)    
1 unit(s)  suntan cream            11  dag   (value = 70)    
1 unit(s)  waterproof overclothes  43  dag   (value = 75)    
1 unit(s)  note-case               22  dag   (value = 80)    
1 unit(s)  sunglasses              7   dag   (value = 20)    
1 unit(s)  socks                   4   dag   (value = 50)   

JavaScript[edit]

Based on the (dynamic) J implementation. Expressed as an htm page:

<html><head><title></title></head><body></body></html>

<script type="text/javascript">
var data= [
  {name: 'map',                    weight:  9, value:150, pieces:1},
  {name: 'compass',                weight: 13, value: 35, pieces:1},
  {name: 'water',                  weight:153, value:200, pieces:2},
  {name: 'sandwich',               weight: 50, value: 60, pieces:2},
  {name: 'glucose',                weight: 15, value: 60, pieces:2},
  {name: 'tin',                    weight: 68, value: 45, pieces:3},
  {name: 'banana',                 weight: 27, value: 60, pieces:3},
  {name: 'apple',                  weight: 39, value: 40, pieces:3},
  {name: 'cheese',                 weight: 23, value: 30, pieces:1},
  {name: 'beer',                   weight: 52, value: 10, pieces:3},
  {name: 'suntan, cream',          weight: 11, value: 70, pieces:1},
  {name: 'camera',                 weight: 32, value: 30, pieces:1},
  {name: 'T-shirt',                weight: 24, value: 15, pieces:2},
  {name: 'trousers',               weight: 48, value: 10, pieces:2},
  {name: 'umbrella',               weight: 73, value: 40, pieces:1},
  {name: 'waterproof, trousers',   weight: 42, value: 70, pieces:1},
  {name: 'waterproof, overclothes',weight: 43, value: 75, pieces:1},
  {name: 'note-case',              weight: 22, value: 80, pieces:1},
  {name: 'sunglasses',             weight:  7, value: 20, pieces:1},
  {name: 'towel',                  weight: 18, value: 12, pieces:2},
  {name: 'socks',                  weight:  4, value: 50, pieces:1},
  {name: 'book',                   weight: 30, value: 10, pieces:2}
];

function findBestPack() {
	var m= [[0]]; // maximum pack value found so far
	var b= [[0]]; // best combination found so far
	var opts= [0]; // item index for 0 of item 0 
	var P= [1]; // item encoding for 0 of item 0
	var choose= 0;
	for (var j= 0; j<data.length; j++) {
		opts[j+1]= opts[j]+data[j].pieces; // item index for 0 of item j+1
		P[j+1]= P[j]*(1+data[j].pieces); // item encoding for 0 of item j+1
	}
	for (var j= 0; j<opts[data.length]; j++) {
		m[0][j+1]= b[0][j+1]= 0; // best values and combos for empty pack: nothing
	}
	for (var w=1; w<=400; w++) {
		m[w]= [0];
		b[w]= [0];
		for (var j=0; j<data.length; j++) {
			var N= data[j].pieces; // how many of these can we have?
			var base= opts[j]; // what is the item index for 0 of these?
			for (var n= 1; n<=N; n++) {
				var W= n*data[j].weight; // how much do these items weigh?
				var s= w>=W ?1 :0; // can we carry this many?
				var v= s*n*data[j].value; // how much are they worth?
				var I= base+n; // what is the item number for this many?
				var wN= w-s*W; // how much other stuff can we be carrying?
				var C= n*P[j] + b[wN][base]; // encoded combination
				m[w][I]= Math.max(m[w][I-1], v+m[wN][base]); // best value
				choose= b[w][I]= m[w][I]>m[w][I-1] ?C :b[w][I-1];
			}
		}
	}
	var best= [];
	for (var j= data.length-1; j>=0; j--) {
		best[j]= Math.floor(choose/P[j]);
		choose-= best[j]*P[j];
	}
	var out='<table><tr><td><b>Count</b></td><td><b>Item</b></td><th>unit weight</th><th>unit value</th>';
	var wgt= 0;
	var val= 0;
	for (var i= 0; i<best.length; i++) {
		if (0==best[i]) continue;
		out+='</tr><tr><td>'+best[i]+'</td><td>'+data[i].name+'</td><td>'+data[i].weight+'</td><td>'+data[i].value+'</td>'
		wgt+= best[i]*data[i].weight;
		val+= best[i]*data[i].value;
	}
	out+= '</tr></table><br/>Total weight: '+wgt;
	out+= '<br/>Total value: '+val;
	document.body.innerHTML= out;
}
findBestPack();
</script>

This will generate (translating html to mediawiki markup):

Count Item unit weight unit value
1 map 9 150
1 compass 13 35
1 water 153 200
2 glucose 15 60
3 banana 27 60
1 cheese 23 30
1 suntan, cream 11 70
1 waterproof, overclothes 43 75
1 note-case 22 80
1 sunglasses 7 20
1 socks 4 50

Total weight: 396
Total value: 1010

jq[edit]

Adapted from Wren

Works with: jq

Works with gojq, the Go implementation of jq

#  Item {name, weight, value, count}

def Item($name; $weight; $value; $count): {$name, $weight, $value, $count};

def items: [
    Item("map"; 9; 150; 1),
    Item("compass"; 13; 35; 1),
    Item("water"; 153; 200; 2),
    Item("sandwich"; 50; 60; 2),
    Item("glucose"; 15; 60; 2),
    Item("tin"; 68; 45; 3),
    Item("banana"; 27; 60; 3),
    Item("apple"; 39; 40; 3),
    Item("cheese"; 23; 30; 1),
    Item("beer"; 52; 10; 3),
    Item("suntan cream"; 11; 70; 1),
    Item("camera"; 32; 30; 1),
    Item("T-shirt"; 24; 15; 2),
    Item("trousers"; 48; 10; 2),
    Item("umbrella"; 73; 40; 1),
    Item("waterproof trousers"; 42; 70; 1),
    Item("waterproof overclothes"; 43; 75; 1),
    Item("note-case"; 22; 80; 1),
    Item("sunglasses"; 7; 20; 1),
    Item("towel"; 18; 12; 2),
    Item("socks"; 4; 50; 1),
    Item("book"; 30; 10; 2)
];
 
def knapsack($w):
  def list($init): [range(0; .) | $init];
  (items|length) as $n
  | {m: ($n+1 | list( $w + 1 | list(0) )) }
  | reduce range(1; $n+1) as $i (.;
      reduce range (0; $w + 1) as $j (.;
            .m[$i][$j] = .m[$i-1][$j]
	    | label $out
            | foreach (range(1; 1 + ((items[$i - 1].count))), null) as $k (.stop = false;
	          if $k == null then .
		  elif ($k * items[$i - 1].weight > $j) then .stop = true
                  else (.m[$i - 1][$j - ($k * items[$i - 1].weight)] + $k * items[$i - 1].value) as $v
                  | if $v > .m[$i][$j] then .m[$i][$j] = $v else . end
		  end;
		if .stop or ($k == null)  then ., break $out else empty end)
            ) )
  | .s = ($n|list(0))
  | .j = $w
  | reduce range($n; 0; -1) as $i (.;
        .m[$i][.j] as $v
        | .k = 0
        | until ($v == .m[$i - 1][.j] + .k * items[$i - 1].value;
            .s[$i - 1] +=  1
            | .j = .j - items[$i - 1].weight
            | .k += 1 ) )
    | .s;

def lpad($len): tostring | ($len - length) as $l | (" " * $l)[:$l] + .;

def task(maxWeight):
  def f(a;b;c;d): "\(a|lpad(22)) \(b|lpad(6)) \(c|lpad(6)) \(d|lpad(6))";
  def f: . as [$a,$b,$c,$d] | f($a;$b;$c;$d);

  (items|length) as $n
  | knapsack(maxWeight) as $s
  | f("Item Chosen";          "Weight";"Value";"Number"),
    "---------------------- ------  ----- ------",
   ({ itemCount:0,
      sumWeight:0,
      sumValue :0,
      sumNumber:0 }
    | reduce range(0; $n) as $i (.;
        if ($s[$i] != 0) 
        then .itemCount += 1
        | .name   = items[$i].name
        | .number = $s[$i]
        | .weight = items[$i].weight * .number
        | .value  = items[$i].value  * .number
        | .sumNumber  += .number
        | .sumWeight  += .weight
        | .sumValue   += .value
	| .emit += [[.name, .weight, .value, .number]]
	else .
	end )
    | (.emit[] | f),
    "---------------------- ------  ----- ------",
     f("Items chosen \(.itemCount|lpad(4))"; .sumWeight; .sumValue; .sumNumber) );

task(400)
Output:
           Item Chosen Weight  Value Number
---------------------- ------  ----- ------
                   map      9    150      1
               compass     13     35      1
                 water    153    200      1
               glucose     30    120      2
                banana     81    180      3
                cheese     23     30      1
          suntan cream     11     70      1
waterproof overclothes     43     75      1
             note-case     22     80      1
            sunglasses      7     20      1
                 socks      4     50      1
---------------------- ------  ----- ------
     Items chosen   11    396   1010     14


Julia[edit]

Works with: Julia version 0.6

The solution uses Julia's MathProgBase. Most of the work is done with this package's mixintprog function.

Type and Function:

using MathProgBase, Cbc

struct KPDSupply{T<:Integer}
    item::String
    weight::T
    value::T
    quant::T
end
Base.show(io::IO, kdps::KPDSupply) = print(io, kdps.quant, " ", kdps.item, " ($(kdps.weight) kg, $(kdps.value) €)")

function solve(gear::Vector{KPDSupply{T}}, capacity::Integer) where T<:Integer
    w = getfield.(gear, :weight)
    v = getfield.(gear, :value)
    q = getfield.(gear, :quant)
    sol = mixintprog(-v, w', '<', capacity, :Int, 0, q, CbcSolver())
    sol.status == :Optimal || error("this problem could not be solved")

    if all(q .== 1) # simpler case
        return gear[sol.sol == 1.0]
    else
        pack = similar(gear, 0)
        s = round.(Int, sol.sol)
        for (i, g) in enumerate(gear)
            iszero(s[i]) && continue
            push!(pack, KPDSupply(g.item, g.weight, g.value, s[i]))
        end
        return pack
    end
end

Main:

gear = [KPDSupply("map", 9, 150, 1),
        KPDSupply("compass", 13, 35, 1),
        KPDSupply("water", 153, 200, 2),
        KPDSupply("sandwich", 50, 60, 2),
        KPDSupply("glucose", 15, 60, 2),
        KPDSupply("tin", 68, 45, 3),
        KPDSupply("banana", 27, 60, 3),
        KPDSupply("apple", 39, 40, 3),
        KPDSupply("cheese", 23, 30, 1),
        KPDSupply("beer", 52, 10, 3),
        KPDSupply("suntan cream", 11, 70, 1),
        KPDSupply("camera", 32, 30, 1),
        KPDSupply("T-shirt", 24, 15, 2),
        KPDSupply("trousers", 48, 10, 2),
        KPDSupply("umbrella", 73, 40, 1),
        KPDSupply("waterproof trousers", 42, 70, 1),
        KPDSupply("waterproof overclothes", 43, 75, 1),
        KPDSupply("note-case", 22, 80, 1),
        KPDSupply("sunglasses", 7, 20, 1),
        KPDSupply("towel", 18, 12, 2),
        KPDSupply("socks", 4, 50, 1),
        KPDSupply("book", 30, 10, 2)]

pack = solve(gear, 400)
println("The hiker should pack: \n - ", join(pack, "\n - "))
println("\nPacked weight: ", sum(getfield.(pack, :weight)), " kg")
println("Packed value: ", sum(getfield.(pack, :value)), " €")
Output:
The hiker should pack:
 - 1 map (9 kg, 150 €)
 - 1 compass (13 kg, 35 €)
 - 1 water (153 kg, 200 €)
 - 2 glucose (15 kg, 60 €)
 - 3 banana (27 kg, 60 €)
 - 1 cheese (23 kg, 30 €)
 - 1 suntan cream (11 kg, 70 €)
 - 1 waterproof overclothes (43 kg, 75 €)
 - 1 note-case (22 kg, 80 €)
 - 1 sunglasses (7 kg, 20 €)
 - 1 socks (4 kg, 50 €)

Packed weight: 327 kg
Packed value: 830 €

Kotlin[edit]

Translation of: C
// version 1.1.2

data class Item(val name: String, val weight: Int, val value: Int, val count: Int)

val items = listOf(
    Item("map", 9, 150, 1),
    Item("compass", 13, 35, 1),
    Item("water", 153, 200, 2),
    Item("sandwich", 50, 60, 2),
    Item("glucose", 15, 60, 2),
    Item("tin", 68, 45, 3),
    Item("banana", 27, 60, 3),
    Item("apple", 39, 40, 3),
    Item("cheese", 23, 30, 1),
    Item("beer", 52, 10, 3),
    Item("suntan cream", 11, 70, 1),
    Item("camera", 32, 30, 1),
    Item("T-shirt", 24, 15, 2),
    Item("trousers", 48, 10, 2),
    Item("umbrella", 73, 40, 1),
    Item("waterproof trousers", 42, 70, 1),
    Item("waterproof overclothes", 43, 75, 1),
    Item("note-case", 22, 80, 1),
    Item("sunglasses", 7, 20, 1),
    Item("towel", 18, 12, 2),
    Item("socks", 4, 50, 1),
    Item("book", 30, 10, 2)
)

val n = items.size

const val MAX_WEIGHT = 400

fun knapsack(w: Int): IntArray {
    val m  = Array(n + 1) { IntArray(w + 1) }
    for (i in 1..n) {
        for (j in 0..w) {
            m[i][j] = m[i - 1][j]
            for (k in 1..items[i - 1].count) {
                if (k * items[i - 1].weight > j) break
                val v = m[i - 1][j - k * items[i - 1].weight] + k * items[i - 1].value
                if (v > m[i][j]) m[i][j] = v
            }
        }
    }
    val s = IntArray(n)
    var j = w
    for (i in n downTo 1) {
        val v = m[i][j]
        var k = 0
        while (v != m[i - 1][j] + k * items[i - 1].value) {
            s[i - 1]++
            j -= items[i - 1].weight
            k++
        }
    }
    return s
}

fun main(args: Array<String>) {
   val s = knapsack(MAX_WEIGHT)
   println("Item Chosen             Weight Value  Number")
   println("---------------------   ------ -----  ------")
   var itemCount = 0
   var sumWeight = 0
   var sumValue  = 0
   var sumNumber = 0
   for (i in 0 until n) {
       if (s[i] == 0) continue
       itemCount++
       val name   = items[i].name
       val number = s[i]
       val weight = items[i].weight * number
       val value  = items[i].value  * number
       sumNumber += number
       sumWeight += weight
       sumValue  += value
       println("${name.padEnd(22)}    ${"%3d".format(weight)}   ${"%4d".format(value)}    ${"%2d".format(number)}")
   }
   println("---------------------   ------ -----  ------")
   println("Items chosen $itemCount           ${"%3d".format(sumWeight)}   ${"%4d".format(sumValue)}    ${"%2d".format(sumNumber)}")
}
Output:
Item Chosen             Weight Value  Number
---------------------   ------ -----  ------
map                         9    150     1
compass                    13     35     1
water                     153    200     1
glucose                    30    120     2
banana                     81    180     3
cheese                     23     30     1
suntan cream               11     70     1
waterproof overclothes     43     75     1
note-case                  22     80     1
sunglasses                  7     20     1
socks                       4     50     1
---------------------   ------ -----  ------
Items chosen 11           396   1010    14

Mathematica/Wolfram Language[edit]

Transpose@{#[[;; , 1]], 
    LinearProgramming[-#[[;; , 3]], -{#[[;; , 2]]}, -{400},
     {0, #[[4]]} & /@ #, Integers]} &@{{"map", 9, 150, 1},
  {"compass", 13, 35, 1},
  {"water", 153, 200, 2},
  {"sandwich", 50, 60, 2},
  {"glucose", 15, 60, 2},
  {"tin", 68, 45, 3},
  {"banana", 27, 60, 3},
  {"apple", 39, 40, 3},
  {"cheese", 23, 30, 1},
  {"beer", 52, 10, 3},
  {"suntan cream", 11, 70, 1},
  {"camera", 32, 30, 1},
  {"T-shirt", 24, 15, 2},
  {"trousers", 48, 10, 2},
  {"umbrella", 73, 40, 1},
  {"waterproof trousers", 42, 70, 1},
  {"waterproof overclothes", 43, 75, 1},
  {"note-case", 22, 80, 1},
  {"sunglasses", 7, 20, 1},
  {"towel", 18, 12, 2},
  {"socks", 4, 50, 1},
  {"book", 30, 10, 2}}
Output:
{{"map", 1}, {"compass", 1}, {"water", 1}, {"sandwich", 
  0}, {"glucose", 2}, {"tin", 0}, {"banana", 3}, {"apple", 
  0}, {"cheese", 1}, {"beer", 0}, {"suntan cream", 1}, {"camera", 
  0}, {"T-shirt", 0}, {"trousers", 0}, {"umbrella", 
  0}, {"waterproof trousers", 0}, {"waterproof overclothes", 
  1}, {"note-case", 1}, {"sunglasses", 1}, {"towel", 0}, {"socks", 
  1}, {"book", 0}}

Mathprog[edit]

/*Knapsack
 
  This model finds the integer optimal packing of a knapsack
 
  Nigel_Galloway
  January 9th., 2012
*/
set Items;
param weight{t in Items};
param value{t in Items};
param quantity{t in Items};

var take{t in Items}, integer, >=0, <=quantity[t];

knap_weight : sum{t in Items} take[t] * weight[t] <= 400;

maximize knap_value: sum{t in Items} take[t] * value[t];

data;

param : Items          : weight   value   quantity :=
         map		  9	   150        1
         compass          13	   35	      1
         water		  153	   200        2
         sandwich	  50	   60	      2
         glucose	  15	   60	      2
         tin		  68	   45	      3
         banana		  27	   60	      3
         apple		  39	   40	      3
         cheese		  23	   30	      1
         beer		  52	   10	      3
         suntancream	  11	   70	      1
         camera		  32	   30	      1
         T-shirt	  24	   15	      2
         trousers	  48	   10	      2
         umbrella	  73	   40	      1
         w-trousers	  42	   70	      1
         w-overclothes	  43	   75	      1
         note-case	  22	   80	      1
         sunglasses	  7        20	      1
         towel		  18	   12	      2
         socks		  4        50	      1
         book		  30	   10	      2
;

end;

The solution produced using glpk is here: Knapsack problem/Bounded/Mathprog

lpsolve may also be used. The result may be found here: File:Knap_objective.png

The constraints may be found here: File:Knap_constraint.png

MiniZinc[edit]

%Solve Knapsack Problem Bounded. Nigel Galloway, Octoer 12th., 2020.
enum Items={map,compass,water,sandwich,glucose,tin,banana,apple,cheese,beer,suntan_cream,camera,t_shirt,trousers,umbrella,waterproof_trousers,waterproof_overclothes,note_case,sunglasses,towel,socks,book};
array[Items] of int: weight  =[9,13,153,50,15,68,27,39,23,52,11,32,24,48,73,42,43,22,7,18,4,30];
array[Items] of int: value   =[150,35,200,60,60,45,60,40,30,10,70,30,15,10,40,70,75,80,20,12,50,10];
array[Items] of int: quantity=[1,1,2,2,2,3,3,3,1,3,1,1,2,2,1,1,1,1,1,2,1,2];
array[Items] of var 0..max(quantity): take; constraint forall(n in Items)(take[n]<=quantity[n]);
int: maxWeight=400;
var int: wTaken=sum(n in Items)(weight[n]*take[n]);
var int: wValue=sum(n in Items)(value[n]*take[n]);
constraint wTaken <= maxWeight;
solve maximize wValue;
output[concat([let {string: g=show(take[n])} in "Take "++(if g==show(quantity[n]) then "all" else g endif)++" of \(n)\n" | n in Items where show(take[n])!="0"])++"\nTotal Weight=\(wTaken) Total Value="++show_float(4,2,wValue)]
Output:
 Take all of map
 Take all of compass
 Take 1 of water
 Take all of glucose
 Take all of banana
 Take all of cheese
 Take all of suntan_cream
 Take all of waterproof_overclothes
 Take all of note_case
 Take all of sunglasses
 Take all of socks

 Total Weight=396 Total Value=1010.00

Nim[edit]

We expand the list of items and apply the 0-1 algorithm.

# Knapsack. Recursive solution.

import strformat
import tables

# Description of an item.
type Item = tuple[name: string; weight, value, pieces: int]

# List of available items.
const Items: seq[Item] = @[("map", 9, 150, 1),
                           ("compass", 13, 35, 1),
                           ("water", 153, 200, 2),
                           ("sandwich", 50, 60, 2),
                           ("glucose", 15, 60, 2),
                           ("tin", 68, 45, 3),
                           ("banana", 27, 60, 3),
                           ("apple", 39, 40, 3),
                           ("cheese", 23, 30, 1),
                           ("beer", 52, 10, 3),
                           ("suntan cream", 11, 70, 1),
                           ("camera", 32, 30, 1),
                           ("T-shirt", 24, 15, 2),
                           ("trousers", 48, 10, 2),
                           ("umbrella", 73, 40, 1),
                           ("waterproof trousers", 42, 70, 1),
                           ("waterproof overclothes", 43, 75, 1),
                           ("note-case", 22, 80, 1),
                           ("sunglasses", 7, 20, 1),
                           ("towel", 18, 12, 2),
                           ("socks", 4, 50, 1),
                           ("book", 30, 10, 2)
                          ]

type

  # Item numbers (used rather than items themselves).
  Number = range[0..Items.high]

  # Description of an expanded item.
  ExpandedItem = tuple[num: Number; weight, value: int]


# Expanded items management.

proc expandedItems(items: seq[Item]): seq[ExpandedItem] =
  ## Expand the list of items.
  for idx, item in Items:
    for _ in 1..item.pieces:
      result.add((idx.Number, item.weight, item.value))

const ItemList = expandedItems(Items)

type

  # Index in the expanded list.
  ExpandedIndex = 0..ItemList.high

  # Chosen items and their total value.
  Choice = tuple[indexes: set[ExpandedIndex]; weight, value: int]

# Cache used to speed up the search.
var cache: Table[tuple[index, weight: int], Choice]


#---------------------------------------------------------------------------------------------------

proc select(idx, weightLimit: int): Choice =
  ## Find the best choice starting from item at index "idx".

  if idx < 0 or weightLimit == 0:
    return

  if (idx, weightLimit) in cache:
    return cache[(idx, weightLimit)]

  let weight = ItemList[idx].weight
  if weight > weightLimit:
    return select(idx - 1, weightLimit)

  # Try by leaving this item and selecting among remaining items.
  result = select(idx - 1, weightLimit)

  # Try by taking this item and completing with some remaining items.
  var result1 = select(idx - 1, weightLimit - weight)
  inc result1.value, ItemList[idx].value

  # Select the best choice (giving the greater value).
  if result1.value > result.value:
    result = (result1.indexes + {idx.ExpandedIndex}, result1.weight + weight, result1.value)

  cache[(idx, weightLimit)] = result

#---------------------------------------------------------------------------------------------------

let (indexes, weight, value) = select(ItemList.high, 400)

# Count the number of pieces for each item.
var pieces = newSeq[int](Items.len)
for idx in indexes:
  inc pieces[ItemList[idx].num]

echo "List of items:"
for num in 0..Items.high:
  if pieces[num] > 0:
    echo fmt"– {pieces[num]} of {Items[num].pieces} {Items[num].name}"
echo ""
echo "Total weight: ", weight
echo "Total value: ", value
Output:
List of items:
– 1 of 1 map
– 1 of 1 compass
– 1 of 2 water
– 2 of 2 glucose
– 3 of 3 banana
– 1 of 1 cheese
– 1 of 1 suntan cream
– 1 of 1 waterproof overclothes
– 1 of 1 note-case
– 1 of 1 sunglasses
– 1 of 1 socks

Total weight: 396
Total value: 1010

OOCalc[edit]

OpenOffice.org Calc has (several) linear solvers. To solve this task, first copy in the table from the task description, then add the extra columns:

  • Number: (How many chosen)
  • weight of n
  • value of n

Add a TOTALS row to sum the weight/value of n.

The sheet should then look like this:

table pre solving

Open the "Tools->Solver..." menu item and fill in the following items:

  • Target Cell: $H$27
  • Optimise result to: maximum
  • By Changing cells: $F$5:$F$26
  • Limiting conditions:
  • $F$5:$F$26 <= $E$5:$E$26
  • $G$27 <= 400
solver menu
  • Options... (opens a separate popup window, then continue)
  • Solver engine: OpenOffice.org Linear Solver
  • Settings:
  • Assume variables as integer: True
  • Assume variables as non-negative: True
  • Epsilon level: 0
  • Limit branch-and-bound depth: True
  • Solving time limit (seconds): 100
solver popup options menu

OK the solver options window leaving the Solver window open, then select solve to produce in seconds:

Table solved

OxygenBasic[edit]

type KnapSackItem string name,sys dag,value,tag

KnapSackItem it[100]

sys dmax=400
sys items=22

it=>

"map",	                  9, 150, 0,
"compass",               13,  35, 0,
"water",                153, 200, 0,
"sandwich",              50, 160, 0,
"glucose",               15,  60, 0,
"tin",                   68,  45, 0,
"banana",                27,  60, 0,
"apple",                 39,  40, 0,
"cheese",                23,  30, 0,
"beer",                  52,  10, 0,
"suntan cream",          11,  70, 0,
"camera",                32,  30, 0,
"T-shirt",               24,  15, 0,
"trousers",              48,  10, 0,
"umbrella",              73,  40, 0,
"waterproof trousers",   42,  70, 0,
"waterproof overclothes",43,  75, 0,
"note-case",             22,  80, 0,
"sunglasses",             7,  20, 0,
"towel",                 18,  12, 0,
"socks",                  4,  50, 0,
"book",                  30,  10, 0

tot=0
for i=1 to items
  tot+=it(i).dag
next
xs=tot-dmax

'REMOVE LOWEST PRIORITY ITEMS TILL XS<=0

cr=chr(13)+chr(10)
tab=chr(9)
pr="remove: " cr
c=0

do
  v=1e9
  w=0
  k=0
  '
  'FIND NEXT LEAST VALUE ITEM
  '
  for i=1 to items
    if it[i].tag=0
      'w=it[i].value               'TEST PRIORITY ONLY
      w=1000*it[i].value/it[i].dag 'TEST PRIORIT/WEIGHT VALUE
      if w<v then v=w : k=i
    end if
  next
  '
  'LOG AND REMOVE FROM LIST
  '
  if k
    xs-=it[k].dag 'deduct from excess weight
    it[k].tag=1
    pr+=it(k).name tab it(k).dag tab it(k).value cr
    if xs<=0 then exit do 'Weight within dmax
  end if
  c++
  if c>=items then exit do
end do
'
pr+=cr "Knapsack contents: " cr
'
for i=1 to items
  if it(i).tag=0
    pr+=it(i).name tab it(i).dag tab it(i).value cr
  end if
next

'TRY FITTING IN LOWER PRIORITY ITEMS

av=-xs

for i=1 to items
  if it[i].tag
    if av-it[i].dag > 0 then
      pr+="Can include: " it(i).name tab it(i).dag tab it(i).value cr
      av-=it[i].dag
    end if
  end if
next
pr+=cr "Weight: " dmax+xs
'putfile "s.txt",pr
print pr
'Knapsack contents: 
'map             	9	150
'compass         	13	35
'water           	153	200
'sandwich        	50	160
'glucose         	15	60
'banana          	27	60
'suntan cream     	11	70
'waterproof trousers	42	70
'waterproof overclothes	43	75
'note-case        	22	80
'sunglasses       	7	20
'socks           	4	50
'
'Weight: 396

Oz[edit]

Using constraint programming.

declare
  %% maps items to tuples of
  %% Weight(hectogram), Value and available Pieces
  Problem = knapsack('map':9#150#1
                     'compass':13#35#1
                     'water':153#200#2
                     'sandwich':50#60#2
                     'glucose':15#60#2
                     'tin':68#45#3
                     'banana':27#60#3 
                     'apple':39#40#3
                     'cheese':23#30#1
                     'beer':52#10#3
                     'suntan cream':11#70#1
                     'camera':32#30#1
                     't-shirt':24#15#2
                     'trousers':48#10#2
                     'umbrella':73#40#1
                     'waterproof trousers':42#70#1
                     'waterproof overclothes':43#75#1 
                     'note-case':22#80#1
                     'sunglasses':7#20#1
                     'towel':18#12#2
                     'socks':4#50#1
                     'book':30#10#2
                    )

  %% item -> Weight
  Weights = {Record.map Problem fun {$ X} X.1 end}
  %% item -> Value
  Values =  {Record.map Problem fun {$ X} X.2 end}

  proc {Knapsack Solution}
     %% a solution maps items to finite domain variables
     %% whose maximum values depend on the item type
     Solution = {Record.map Problem fun {$ _#_#Max} {FD.int 0#Max} end}
     %% no more than 400 hectograms
     {FD.sumC Weights Solution '=<:' 400} 
     %% search through valid solutions
     {FD.distribute naive Solution}
  end
 
  proc {PropagateLargerValue Old New}
     %% propagate that new solutions must yield a higher value
     %% than previously found solutions (essential for performance)
     {FD.sumC Values New '>:' {Value Old}} 
  end

  fun {Value Candidate}
     {Record.foldL {Record.zip Candidate Values Number.'*'} Number.'+' 0}
  end
  
  fun {Weight Candidate}
     {Record.foldL {Record.zip Candidate Weights Number.'*'} Number.'+' 0}
  end

  [Best] = {SearchBest Knapsack PropagateLargerValue}
in
  {System.showInfo "Items: "}
  {Record.forAllInd Best
   proc {$ I V}
      if V > 0 then
	 {System.showInfo I#": "#V}
      end
   end
  }
  {System.printInfo "\n"}
  {System.showInfo "total value: "#{Value Best}}
  {System.showInfo "total weight: "#{Weight Best}}
Output:
Items: 
banana: 3
cheese: 1
compass: 1
glucose: 2
map: 1
note-case: 1
socks: 1
sunglasses: 1
suntan cream: 1
water: 1
waterproof overclothes: 1

total value: 1010
total weight: 396

Takes about 3 seconds on a slow netbook.

Pascal[edit]

Works with: FPC

Dynamic programming solution (tested with FPC 3.2.2).

program KnapsackBounded;
{$mode objfpc}{$j-}
uses
  SysUtils, Math;

type
  TItem = record
    Name: string;
    Weight, Value, Count: Integer;
  end;

const
  NUM_ITEMS = 22;
  ITEMS: array[0..NUM_ITEMS-1] of TItem = (
    (Name: 'map';                    Weight:   9; Value: 150; Count: 1),
    (Name: 'compass';                Weight:  13; Value:  35; Count: 1),
    (Name: 'water';                  Weight: 153; Value: 200; Count: 2),
    (Name: 'sandwich';               Weight:  50; Value:  60; Count: 2),
    (Name: 'glucose';                Weight:  15; Value:  60; Count: 2),
    (Name: 'tin';                    Weight:  68; Value:  45; Count: 3),
    (Name: 'banana';                 Weight:  27; Value:  60; Count: 3),
    (Name: 'apple';                  Weight:  39; Value:  40; Count: 3),
    (Name: 'cheese';                 Weight:  23; Value:  30; Count: 1),
    (Name: 'beer';                   Weight:  52; Value:  10; Count: 3),
    (Name: 'suntan cream';           Weight:  11; Value:  70; Count: 1),
    (Name: 'camera';                 Weight:  32; Value:  30; Count: 1),
    (Name: 'T-shirt';                Weight:  24; Value:  15; Count: 2),
    (Name: 'trousers';               Weight:  48; Value:  10; Count: 2),
    (Name: 'umbrella';               Weight:  73; Value:  40; Count: 1),
    (Name: 'waterproof trousers';    Weight:  42; Value:  70; Count: 1),
    (Name: 'waterproof overclothes'; Weight:  43; Value:  75; Count: 1),
    (Name: 'note-case';              Weight:  22; Value:  80; Count: 1),
    (Name: 'sunglasses';             Weight:   7; Value:  20; Count: 1),
    (Name: 'towel';                  Weight:  18; Value:  12; Count: 2),
    (Name: 'socks';                  Weight:   4; Value:  50; Count: 1),
    (Name: 'book';                   Weight:  30; Value:  10; Count: 2)
  );
  MAX_WEIGHT = 400;

var
  D: array of array of Integer; //DP matrix
  I, W, V, C, MaxWeight: Integer;
begin
  SetLength(D, NUM_ITEMS + 1, MAX_WEIGHT + 1);
  for I := 0 to High(ITEMS) do
    for W := 0 to MAX_WEIGHT do begin
      D[I+1, W] := D[I, W];
      for C := 1 to ITEMS[I].Count do begin
        if ITEMS[I].Weight * C > W then break;
        V := D[I, W - ITEMS[I].Weight * C] + ITEMS[I].Value * C;
        if V > D[I+1, W] then
          D[I+1, W] := V;
      end;
    end;

  W := MAX_WEIGHT;
  MaxWeight := 0;
  WriteLn('bagged:');
  for I := High(ITEMS) downto 0 do begin
    V := D[I+1, W];
    C := 0;
    while V <> D[I, W] + ITEMS[I].Value * C do begin
      Dec(W, ITEMS[I].Weight);
      Inc(C);
    end;
    Inc(MaxWeight, C * ITEMS[I].Weight);
    if C <> 0 then
       WriteLn('  ', C, ' ', ITEMS[I].Name);
  end;
  WriteLn('value  = ', D[NUM_ITEMS, MAX_WEIGHT]);
  WriteLn('weight = ', MaxWeight);
end.
Output:
bagged:
  1 socks
  1 sunglasses
  1 note-case
  1 waterproof overclothes
  1 suntan cream
  1 cheese
  3 banana
  2 glucose
  1 water
  1 compass
  1 map
value  = 1010
weight = 396

Perl[edit]

Recursive solution with caching.

#!/usr/bin/perl

use strict;

my $raw = <<'TABLE';
map     	9       150     1
compass 	13      35      1
water   	153     200     2
sandwich        50      60      2
glucose 	15      60      2
tin     	68      45      3
banana  	27      60      3
apple  		39      40      3
cheese  	23      30      1
beer    	52      10      1
suntancream     11      70      1
camera  	32      30      1
T-shirt 	24      15      2
trousers        48      10      2
umbrella        73      40      1
w_trousers     	42      70      1
w_overcoat  	43      75      1
note-case       22      80      1
sunglasses      7       20      1
towel   	18      12      2
socks   	4       50      1
book    	30      10      2
TABLE
 
my @items;
for (split "\n", $raw) {
        my @x = split /\s+/;
	push @items, {
		name	=> $x[0],
		weight	=> $x[1],
		value	=> $x[2],
		quant	=> $x[3],
	}
}

my $max_weight = 400;

my %cache;
sub pick {
	my ($weight, $pos) = @_;
	if ($pos < 0 or $weight <= 0) {
		return 0, 0, []
	}

	@{ $cache{$weight, $pos} //= [do{	# odd construct: for caching
		my $item = $items[$pos];
		my ($bv, $bi, $bw, $bp) = (0, 0, 0, []);

		for my $i (0 .. $item->{quant}) {
			last if $i * $item->{weight} > $weight;
			my ($v, $w, $p) = pick($weight - $i * $item->{weight}, $pos - 1);
			next if ($v += $i * $item->{value}) <= $bv;

			($bv, $bi, $bw, $bp) = ($v, $i, $w, $p);
		}

		my @picked = ( @$bp, $bi );
		$bv, $bw + $bi * $item->{weight}, \@picked
	}]}
}

my ($v, $w, $p) = pick($max_weight, $#items);
for (0 .. $#$p) {
	if ($p->[$_] > 0) {
		print "$p->[$_] of $items[$_]{name}\n";
	}
}
print "Value: $v; Weight: $w\n";
Output:
1 of map
1 of compass
1 of water
2 of glucose
3 of banana
1 of cheese
1 of suntancream
1 of w_overcoat
1 of note-case
1 of sunglasses
1 of socks
Value: 1010; Weight: 396

Phix[edit]

Very dumb and very slow brute force version[edit]

Of no practical use, except for comparison against improvements.

with javascript_semantics
atom t0 = time()
 
constant goodies = {
-- item                     weight value pieces
{"map",                     9,      150,    1},
{"compass",                 13,     35,     1},
{"sandwich",                50,     60,     2},
{"glucose",                 15,     60,     2},
{"tin",                     68,     45,     3},
{"banana",                  27,     60,     3},
{"apple",                   39,     40,     3},
{"cheese",                  23,     30,     1},
{"beer",                    52,     10,     3},
{"suntan cream",            11,     70,     1},
{"water",                   153,    200,    2},
{"camera",                  32,     30,     1},
{"T-shirt",                 24,     15,     2},
{"trousers",                48,     10,     2},
{"umbrella",                73,     40,     1},
{"waterproof trousers",     42,     70,     1},
{"waterproof overclothes",  43,     75,     1},
{"note-case",               22,     80,     1},
{"sunglasses",              7,      20,     1},
{"towel",                   18,     12,     2},
{"socks",                   4,      50,     1},
{"book",                    30,     10,     2}}
 
function knapsack(integer max_weight, integer at)
    integer best_points = 0, points
    sequence best_choices = {}, choices
    atom act_weight = 0, sub_weight
    if at>=1 then
        integer {?,witem,pitem,imax} = goodies[at]
        for i=0 to imax do
            integer wlim = max_weight-i*witem
            if wlim<0 then exit end if
            {points,sub_weight,choices} = knapsack(wlim, at-1)
            points += i*pitem
            if points>best_points then
                best_points = points
                best_choices = deep_copy(choices)&i
                act_weight = sub_weight+i*witem
            end if
        end for
    end if
    return {best_points, act_weight, best_choices}
end function
 
sequence res = knapsack(400, length(goodies))   -- {points,act_weight,choices}
 
atom weight = 0, witem
atom points = 0, pitem
string idesc
for i=1 to length(goodies) do
    integer c = res[3][i]
    if c then
        {idesc,witem,pitem} = goodies[i]
        printf(1,"%d %s\n",{c,idesc})
        weight += c*witem
        points += c*pitem
    end if
end for
if points!=res[1] then ?9/0 end if  -- sanity check
printf(1,"Value %d, weight %g [%3.2fs]\n",{points,weight,time()-t0})
Output:
1 map
1 compass
2 glucose
3 banana
1 cheese
1 suntan cream
1 water
1 waterproof overclothes
1 note-case
1 sunglasses
1 socks
Value 1010, weight 396 [17.53s]

Dynamic Programming version[edit]

Much faster but limited to integer weights

Translation of: C
with javascript_semantics
sequence items = {
                  {"map",                    9,   150,   1},
                  {"compass",               13,    35,   1},
                  {"water",                153,   200,   2},
                  {"sandwich",              50,    60,   2},
                  {"glucose",               15,    60,   2},
                  {"tin",                   68,    45,   3},
                  {"banana",                27,    60,   3},
                  {"apple",                 39,    40,   3},
                  {"cheese",                23,    30,   1},
                  {"beer",                  52,    10,   3},
                  {"suntan cream",          11,    70,   1},
                  {"camera",                32,    30,   1},
                  {"T-shirt",               24,    15,   2},
                  {"trousers",              48,    10,   2},
                  {"umbrella",              73,    40,   1},
                  {"waterproof trousers",   42,    70,   1},
                  {"waterproof overclothes",43,    75,   1},
                  {"note-case",             22,    80,   1},
                  {"sunglasses",             7,    20,   1},
                  {"towel",                 18,    12,   2},
                  {"socks",                  4,    50,   1},
                  {"book",                  30,    10,   2},
                 };

sequence {names,weights,points,counts} = columnize(items)

constant n = length(items)

function knapsack(int w)
    integer v
    -- m is the achievable points matrix:
    -- Note that Phix uses 1-based indexes, so m[1][1] 
    --  actually holds points for 0 items of weight 0,
    --  and m[n+1][w+1] is for n items at weight w.
    sequence m = repeat(repeat(0,w+1),n+1)
    for i=1 to n do
        for j=1 to w+1 do       -- (0 to w really)
            m[i+1][j] = m[i][j]
            for k=1 to counts[i] do
                if k*weights[i]>j-1 then
                    exit
                end if
                v = m[i][j-k*weights[i]]+k*points[i]
                if v>m[i+1][j] then
                    m[i+1][j] = v
                end if
            end for
        end for
    end for
    sequence s = repeat(0,n)
    integer j = w+1     -- (w -> 0 really)
    for i=n to 1 by -1 do
        v = m[i+1][j]
        integer k = 0
        while v!=m[i][j]+k*points[i] do
            s[i] += 1
            j -= weights[i]
            k += 1
        end while
    end for
    return s
end function

integer tc = 0, tw = 0, tv = 0
sequence s = knapsack(400)
for i=1 to n do
    integer si = s[i]
    if si then
        printf(1,"%-22s %5d %5d %5d\n", {names[i], si, si*weights[i], si*points[i]})
        tc += si
        tw += si*weights[i]
        tv += si*points[i]
    end if
end for
printf(1,"%-22s %5d %5d %5d\n", {"count, weight, points:", tc, tw, tv})
Output:
map                        1     9   150
compass                    1    13    35
water                      1   153   200
glucose                    2    30   120
banana                     3    81   180
cheese                     1    23    30
suntan cream               1    11    70
waterproof overclothes     1    43    75
note-case                  1    22    80
sunglasses                 1     7    20
socks                      1     4    50
count, weight, points:    14   396  1010

Range cache version[edit]

The main problem with the dynamic programming solution is that it is only practical for integer weights. You could multiply by 1000 and truncate to get an approximation to the nearest 0.001kg, but the memory use would obviously increase dramatically. A naive cache could also suffer similarly, if it retained duplicate solutions for w=15.783, 15.784, 15.785, and everything in between. Using a (bespoke) range cache solves this problem, although the simplistic version given could probably be improved with a binary search or similar. It also significantly reduces the workload; for instance if you find a solution for 170 that actually weighs 150 then any subsequent query in 150..170 requires zero work, unlike the naive cache and dp solutions.

-- demo\rosetta\knapsackB.exw
with javascript_semantics
atom t0 = time()

enum HI,PTS,ACTW,SOLN
sequence range_cache = {}

integer cache_entries = 0

procedure add_range(integer at, atom weight, atom actual_weight, atom points, sequence soln)
    assert(actual_weight<=weight)
    for i=length(range_cache)+1 to at do -- (while too small do)
        if i=at then
            range_cache = append(range_cache,{{weight,points,actual_weight,soln}})
            cache_entries += 1
            return
        end if
        range_cache = append(range_cache,{})
    end for
    integer lastHI = -1
    for i=1 to length(range_cache[at]) do
        sequence rcati = range_cache[at][i]
        if weight=rcati[ACTW] then
            assert(rcati[PTS..SOLN]=={points,actual_weight,soln})
            return
        elsif weight<rcati[ACTW] then
            -- (we cannot extend an existing range down...)
            assert(soln!=rcati[SOLN])
            -- insert a new range
            range_cache[at][i..i-1] = {{weight,points,actual_weight,soln}}
            cache_entries += 1
            return
        elsif soln=rcati[SOLN] then
            assert(rcati[PTS..SOLN]=={points,actual_weight,soln})
            if weight>rcati[HI] then        -- extend existing range up
                rcati = {}
                range_cache[at][i][HI] = weight
            end if
            return
        elsif weight<=rcati[HI] then
            crash("duplicate solution??")   -- (or discard as below)
--          return                          -- (discard)
        end if
        assert(rcati[ACTW]>lastHI)
        lastHI = rcati[HI]
    end for
    range_cache[at] = append(range_cache[at],{weight,points,actual_weight,soln})
    cache_entries += 1
end procedure

function in_range(integer at, atom weight)
    if at<=length(range_cache) then
        for i=1 to length(range_cache[at]) do
            sequence rcati = range_cache[at][i]
            if weight<=rcati[HI] then
                if weight>=rcati[ACTW] then
                    return rcati[PTS..SOLN] -- {pts,act_weight,soln}
                end if
                exit
            end if
        end for
    end if
    return {}   -- (no suitable cache entry found)
end function

sequence goodies = ({
-- item                     weight value pieces
{"map",                     9,      150,    1},
{"compass",                 13,     35,     1},
{"sandwich",                50,     60,     2},
{"glucose",                 15,     60,     2},
{"tin",                     68,     45,     3},
{"banana",                  27,     60,     3},
{"apple",                   39,     40,     3},
{"cheese",                  23,     30,     1},
{"beer",                    52,     10,     3},
{"suntan cream",            11,     70,     1},
{"water",                   153,    200,    2},
{"camera",                  32,     30,     1},
{"T-shirt",                 24,     15,     2},
{"trousers",                48,     10,     2},
{"umbrella",                73,     40,     1},
{"waterproof trousers",     42,     70,     1},
{"waterproof overclothes",  43,     75,     1},
{"note-case",               22,     80,     1},
{"sunglasses",              7,      20,     1},
{"towel",                   18,     12,     2},
{"socks",                   4,      50,     1},
{"book",                    30,     10,     2}})

integer cache_hits = 0
integer cache_misses = 0

function knapsack(integer max_weight, integer at)
    integer best_points = 0, points
    sequence best_choices = {}, choices
    atom act_weight = 0, sub_weight
    if at>=1 then
        sequence soln = in_range(at,max_weight)
        if length(soln) then
            cache_hits += 1
            return soln
        end if
        cache_misses += 1
        integer {?,witem,pitem,imax} = goodies[at]
        best_choices = repeat(0,at)
        for i=0 to imax do
            integer wlim = max_weight-i*witem
            if wlim<0 then exit end if
            {points,sub_weight,choices} = knapsack(wlim, at-1)
            points += i*pitem
            if points>best_points then
                best_points = points
                best_choices = deep_copy(choices)&i
                act_weight = sub_weight+i*witem
            end if
        end for
        add_range(at,max_weight,act_weight,best_points,best_choices)
    end if
    return {best_points, act_weight, best_choices}
end function
 
sequence res = knapsack(400, length(goodies))   -- {points,act_weight,choices}
 
atom weight = 0, witem
atom points = 0, pitem
string idesc
sequence choices = res[3]
for i=1 to length(goodies) do
    integer c = choices[i]
    if c then
        {idesc,witem,pitem} = goodies[i]
        printf(1,"%d %s\n",{c,idesc})
        weight += c*witem
        points += c*pitem
    end if
end for
assert({points,weight}==res[1..2]) -- sanity check
printf(1,"Value %d, weight %g [%3.2fs]:\n",{points,weight,time()-t0})

printf(1,"cache_entries:%d, hits:%d, misses:%d\n",{cache_entries,cache_hits,cache_misses})
Output:
1 map
1 compass
2 glucose
3 banana
1 cheese
1 suntan cream
1 water
1 waterproof overclothes
1 note-case
1 sunglasses
1 socks
Value 1010, weight 396 [0.00s]
cache_entries:409, hits:1226, misses:882

The distributed version contains additional comments and extra code for comparing this against a naive cache and no cache (CACHE_RANGE shown above).
CACHE_SIMPLE: (as above but ending)

Value 1010, weight 396 [0.08s]
cache_entries:5549, hits:7707, misses:5549

Even on this simple integer-only case, range cache reduces cache size better than 10-fold and effort 6-fold.
We also know the dp solution matrix has 9223 entries, admittedly each being much smaller than a cache entry.
And finally CACHE_NONE (the dumb version): (as above but ending)

Value 1010, weight 396 [18.14s]
cache_entries:0, hits:0, misses:33615741

Picat[edit]

import mip,util.

go =>
   data(AllItems,MaxWeight),
   [Items,Weights,Values,Pieces] = transpose(AllItems),
   
   knapsack_bounded(Weights,Values,Pieces,MaxWeight, X,TotalWeight,TotalValue),
   
   print_solution(Items,Weights,Values, X,TotalWeight,TotalValue),
   nl.

% Print the solution
print_solution(Items,Weights,Values, X,TotalWeight,TotalValue) ?=> 
   println("\nThese are the items to pick:"),
   println("  Item                         Weight Value"),

   foreach(I in 1..Items.len)
       if X[I] > 0 then
           printf("* %d %-29w %3d %3d\n", X[I],Items[I],Weights[I],Values[I])
       end
   end,
   nl,
   printf("Total weight: %d\n", TotalWeight),
   printf("Total value: %d\n", TotalValue),
   nl.

% Solve knapsack problem
knapsack_bounded(Weights,Values,Pieces,MaxWeight, X,TotalWeight,TotalValue) =>
   NumItems = length(Weights),

   % Variables
   MaxPieces = max(Pieces),
   X = new_list(NumItems),
   X :: 0..MaxPieces,

   % Constraints
   SumValues = sum(Values),
   TotalValue :: 0..SumValues,
   TotalWeight :: 0..MaxWeight,

   scalar_product(Weights,X,TotalWeight),
   scalar_product(Values,X,TotalValue),

   TotalWeight #<= MaxWeight,

   % Check number of pieces
   foreach({XVal,Piece} in zip(X,Pieces))
      XVal #=< Piece
   end,

   % Search
   Vars = X ++ [TotalWeight, TotalValue],
   solve($[max(TotalValue),down],Vars).

% data
data(Items,MaxWeight) => 
   Items = 
      % Item          Weight   Value  Pieces
      [["map",            9,     150,    1],
       ["compass",       13,      35,    1],
       ["water",        153,     200,    2],
       ["sandwich",      50,      60,    2],
       ["glucose",       15,      60,    2],
       ["tin",           68,      45,    3],
       ["banana",        27,      60,    3],
       ["apple",         39,      40,    3],
       ["cheese",        23,      30,    1],
       ["beer",          52,      10,    3],
       ["suntancream",   11,      70,    1],
       ["camera",        32,      30,    1],
       ["T-shirt",       24,      15,    2],
       ["trousers",      48,      10,    2],
       ["umbrella",      73,      40,    1],
       ["waterproof trousers",    42,   70,    1],
       ["waterproof overclothes", 43,   75,    1],
       ["note-case",     22,      80,    1],
       ["sunglasses",     7,      20,    1],
       ["towel",         18,      12,    2],
       ["socks",          4,      50,    1],
       ["book",          30,      10,    2]],
    MaxWeight = 400.
Output:
These are the items to pick:
  Item                         Weight Value
* 1 map                             9 150
* 1 compass                        13  35
* 1 water                         153 200
* 2 glucose                        15  60
* 3 banana                         27  60
* 1 cheese                         23  30
* 1 suntancream                    11  70
* 1 waterproof overclothes         43  75
* 1 note-case                      22  80
* 1 sunglasses                      7  20
* 1 socks                           4  50

Total weight: 396
Total value: 1010

PicoLisp[edit]

(de *Items
   ("map" 9 150 1)                     ("compass" 13 35 1)
   ("water" 153 200 3)                 ("sandwich" 50 60 2)
   ("glucose" 15 60 2)                 ("tin" 68 45 3)
   ("banana" 27 60 3)                  ("apple" 39 40 3)
   ("cheese" 23 30 1)                  ("beer" 52 10 3)
   ("suntan cream" 11 70 1)            ("camera" 32 30 1)
   ("t-shirt" 24 15 2)                 ("trousers" 48 10 2)
   ("umbrella" 73 40 1)                ("waterproof trousers" 42 70 1)
   ("waterproof overclothes" 43 75 1)  ("note-case" 22 80 1)
   ("sunglasses" 7 20 1)               ("towel" 18 12 2)
   ("socks" 4 50 1)                    ("book" 30 10 2) )

# Dynamic programming solution
(de knapsack (Lst W)
   (when Lst
      (cache '*KnapCache (cons W Lst)
         (let X (knapsack (cdr Lst) W)
            (if (ge0 (- W (cadar Lst)))
               (let Y (cons (car Lst) (knapsack (cdr Lst) @))
                  (if (> (sum caddr X) (sum caddr Y)) X Y) )
               X ) ) ) ) )

(let K
   (knapsack
      (mapcan                                   # Expand multiple items
         '((X) (need (cadddr X) NIL X))
         *Items )
      400 )
   (for I K
      (apply tab I (3 -24 6 6) NIL) )
   (tab (27 6 6) NIL (sum cadr K) (sum caddr K)) )
Output:
   map                          9   150
   compass                     13    35
   water                      153   200
   glucose                     15    60
   glucose                     15    60
   banana                      27    60
   banana                      27    60
   banana                      27    60
   cheese                      23    30
   suntan cream                11    70
   waterproof overclothes      43    75
   note-case                   22    80
   sunglasses                   7    20
   socks                        4    50
                              396  1010

Prolog[edit]

Works with: SWI Prolog

Library clpfd[edit]

Library: clpfd

Library clpfd is written by Markus Triska. Takes about 3 seconds to compute the best solution.

:- use_module(library(clpfd)).

% tuples (name, weights, value, nb pieces).
knapsack :-
	L = [(   map, 	        9, 	150, 	1),
	     (   compass, 	13, 	35, 	1),
	     (   water, 	153, 	200, 	2),
	     (   sandwich, 	50, 	60, 	2),
	     (   glucose, 	15, 	60, 	2),
	     (   tin, 	68, 	45, 	3),
	     (   banana, 	27, 	60, 	3),
	     (   apple, 	39, 	40, 	3),
	     (   cheese, 	23, 	30, 	1),
	     (   beer, 	52, 	10, 	3),
	     (   'suntan cream', 	11, 	70, 	1),
	     (   camera, 	32, 	30, 	1),
	     (   'T-shirt', 	24, 	15, 	2),
	     (   trousers, 	48, 	10, 	2),
	     (   umbrella, 	73, 	40, 	1),
	     (   'waterproof trousers', 	42, 	70, 	1),
	     (   'waterproof overclothes', 	43, 	75, 	1),
	     (   'note-case', 	22, 	80, 	1),
	     (   sunglasses, 	7, 	20, 	1),
	     (   towel, 	18, 	12, 	2),
	     (   socks, 	4, 	50, 	1),
	     (   book, 	30, 	10, 	2)],

	% Takes is the list of the numbers of each items
	% these numbers are between 0 and the 4th value of the tuples of the items
        maplist(collect, L, Ws, Vs, Takes),
        scalar_product(Ws, Takes, #=<, 400),
        scalar_product(Vs, Takes, #=, VM),

	% to have statistics on the resolution of the problem.
	time(labeling([max(VM), down], Takes)),
        scalar_product(Ws, Takes, #=, WM),

	%% displayinf of the results.
	compute_lenword(L, 0, Len),
	sformat(A1, '~~w~~t~~~w|', [Len]),
	sformat(A2, '~~t~~w~~~w|', [4]),
	sformat(A3, '~~t~~w~~~w|', [5]),
	print_results(A1,A2,A3, L, Takes, WM, VM).

collect((_, W, V, N), W, V, Take) :-
	Take in 0..N.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
compute_lenword([], N, N).
compute_lenword([(Name, _, _, _)|T], N, NF):-
	atom_length(Name, L),
	(   L > N -> N1 = L; N1 = N),
	compute_lenword(T, N1, NF).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
print_results(A1,A2,A3, [], [], WM, WR) :-
	sformat(W0, '~w ', [' ']),
	sformat(W1, A1, [' ']),
	sformat(W2, A2, [WM]),
	sformat(W3, A3, [WR]),
	format('~w~w~w~w~n', [W0,W1,W2,W3]).


print_results(A1,A2,A3, [_H|T], [0|TR], WM, VM) :-
	!,
	print_results(A1,A2,A3, T, TR, WM, VM).

print_results(A1, A2, A3, [(Name, W, V, _)|T], [N|TR], WM, VM) :-
	sformat(W0, '~w ', [N]),
	sformat(W1, A1, [Name]),
	sformat(W2, A2, [W]),
	sformat(W3, A3, [V]),
	format('~w~w~w~w~n', [W0,W1,W2,W3]),
	print_results(A1, A2, A3, T, TR, WM, VM).
Output:
 ?- knapsack.
% 21,154,932 inferences, 3.187 CPU in 3.186 seconds (100% CPU, 6638515 Lips)
1 map                      9  150
1 compass                 13   35
1 water                  153  200
2 glucose                 15   60
3 banana                  27   60
1 cheese                  23   30
1 suntan cream            11   70
1 waterproof overclothes  43   75
1 note-case               22   80
1 sunglasses               7   20
1 socks                    4   50
                         396 1010
true

Library simplex[edit]

Library simplex is written by Markus Triska. Takes about 10 seconds to compute the best solution.

:- use_module(library(simplex)).
% tuples (name, weights, value, pieces).
knapsack :-
	L = [(map, 	9, 	150, 	1),
	     (   compass, 	13, 	35, 	1),
	     (   water, 	153, 	200, 	2),
	     (   sandwich, 	50, 	60, 	2),
	     (   glucose, 	15, 	60, 	2),
	     (   tin, 	68, 	45, 	3),
	     (   banana, 	27, 	60, 	3),
	     (   apple, 	39, 	40, 	3),
	     (   cheese, 	23, 	30, 	1),
	     (   beer, 	52, 	10, 	3),
	     (   'suntan cream', 	11, 	70, 	1),
	     (   camera, 	32, 	30, 	1),
	     (   'T-shirt', 	24, 	15, 	2),
	     (   trousers, 	48, 	10, 	2),
	     (   umbrella, 	73, 	40, 	1),
	     (   'waterproof trousers', 	42, 	70, 	1),
	     (   'waterproof overclothes', 	43, 	75, 	1),
	     (   'note-case', 	22, 	80, 	1),
	     (   sunglasses, 	7, 	20, 	1),
	     (   towel, 	18, 	12, 	2),
	     (   socks, 	4, 	50, 	1),
	     (   book, 	30, 	10, 	2)],

	 gen_state(S0),
	 length(L, N),
	 numlist(1, N, LN),
	 time(( create_constraint_N(LN, L, S0, S1),
		maplist(create_constraint_WV, LN, L, LW, LV),
		constraint(LW =< 400, S1, S2),
		maximize(LV, S2, S3)
	      )),
	compute_lenword(L, 0, Len),
	sformat(A1, '~~w~~t~~~w|', [Len]),
	sformat(A2, '~~t~~w~~~w|', [4]),
	sformat(A3, '~~t~~w~~~w|', [5]),
	print_results(S3, A1,A2,A3, L, LN, 0, 0).


create_constraint_N([], [], S, S).

create_constraint_N([HN|TN], [(_, _, _, Nb) | TL], S1, SF) :-
	constraint(integral(x(HN)), S1, S2),
	constraint([x(HN)] =< Nb, S2, S3),
	constraint([x(HN)] >= 0, S3, S4),
	create_constraint_N(TN, TL, S4, SF).

create_constraint_WV(N, (_, W, V, _), W * x(N), V * x(N)).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
compute_lenword([], N, N).
compute_lenword([(Name, _, _, _)|T], N, NF):-
	atom_length(Name, L),
	(   L > N -> N1 = L; N1 = N),
	compute_lenword(T, N1, NF).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
print_results(_S, A1, A2, A3, [], [], WM, VM) :-
	sformat(W0, '~w ', [' ']),
	sformat(W1, A1, [' ']),
	sformat(W2, A2, [WM]),
	sformat(W3, A3, [VM]),
	format('~w~w~w~w~n', [W0, W1,W2,W3]).


print_results(S, A1, A2, A3, [(Name, W, V,_)|T], [N|TN], W1, V1) :-
	variable_value(S, x(N), X),
	(   X = 0 -> W1 = W2, V1 = V2
	;   sformat(S0, '~w ', [X]),
	    sformat(S1, A1, [Name]),
	    sformat(S2, A2, [W]),
	    sformat(S3, A3, [V]),
	    format('~w~w~w~w~n', [S0, S1,S2,S3]),
	    W2 is W1 + X * W,
	    V2 is V1 + X * V),
	print_results(S, A1, A2, A3, T, TN, W2, V2).

Python[edit]

Not as dumb a search over all possible combinations under the maximum allowed weight:

from itertools import groupby
from collections import namedtuple

def anyvalidcomb(items, maxwt, val=0, wt=0):
    ' All combinations below the maxwt '
    if not items:
        yield [], val, wt
    else:
        this, *items = items            # car, cdr
        for n in range(this.number + 1):
            w = wt  + n * this.weight
            if w > maxwt:
                break
            v = val + n * this.value
            this_comb = [this] * n
            for comb, value, weight in anyvalidcomb(items, maxwt, v, w):
                yield this_comb + comb, value, weight

maxwt = 400
COMB, VAL, WT = range(3)
Item  = namedtuple('Items', 'name weight value number')
items = [ Item(*x) for x in
          (
            ("map", 9, 150, 1),
            ("compass", 13, 35, 1),
            ("water", 153, 200, 3),
            ("sandwich", 50, 60, 2),
            ("glucose", 15, 60, 2),
            ("tin", 68, 45, 3),
            ("banana", 27, 60, 3),
            ("apple", 39, 40, 3),
            ("cheese", 23, 30, 1),
            ("beer", 52, 10, 3),
            ("suntan cream", 11, 70, 1),
            ("camera", 32, 30, 1),
            ("t-shirt", 24, 15, 2),
            ("trousers", 48, 10, 2),
            ("umbrella", 73, 40, 1),
            ("waterproof trousers", 42, 70, 1),
            ("waterproof overclothes", 43, 75, 1),
            ("note-case", 22, 80, 1),
            ("sunglasses", 7, 20, 1),
            ("towel", 18, 12, 2),
            ("socks", 4, 50, 1),
            ("book", 30, 10, 2),
           ) ]  

bagged = max( anyvalidcomb(items, maxwt), key=lambda c: (c[VAL], -c[WT])) # max val or min wt if values equal
print("Bagged the following %i items" % len(bagged[COMB]))
print('\n\t'.join('%i off: %s' % (len(list(grp)), item.name) for item, grp in groupby(sorted(bagged[COMB]))))
print("for a total value of %i and a total weight of %i" % bagged[1:])
Output:
Bagged the following 14 items
  3 off: banana
  1 off: cheese
  1 off: compass
  2 off: glucose
  1 off: map
  1 off: note-case
  1 off: socks
  1 off: sunglasses
  1 off: suntan cream
  1 off: water
  1 off: waterproof overclothes
for a total value of 1010 and a total weight of 396

Dynamic programming solution[edit]

This is much faster. It expands the multiple possible instances of an item into individual instances then applies the zero-one dynamic knapsack solution:

from itertools import groupby

try:
    xrange
except:
    xrange = range

maxwt = 400

groupeditems = (
            ("map", 9, 150, 1),
            ("compass", 13, 35, 1),
            ("water", 153, 200, 3),
            ("sandwich", 50, 60, 2),
            ("glucose", 15, 60, 2),
            ("tin", 68, 45, 3),
            ("banana", 27, 60, 3),
            ("apple", 39, 40, 3),
            ("cheese", 23, 30, 1),
            ("beer", 52, 10, 3),
            ("suntan cream", 11, 70, 1),
            ("camera", 32, 30, 1),
            ("t-shirt", 24, 15, 2),
            ("trousers", 48, 10, 2),
            ("umbrella", 73, 40, 1),
            ("waterproof trousers", 42, 70, 1),
            ("waterproof overclothes", 43, 75, 1),
            ("note-case", 22, 80, 1),
            ("sunglasses", 7, 20, 1),
            ("towel", 18, 12, 2),
            ("socks", 4, 50, 1),
            ("book", 30, 10, 2),
           )
items = sum( ([(item, wt, val)]*n for item, wt, val,n in groupeditems), [])

def knapsack01_dp(items, limit):
    table = [[0 for w in range(limit + 1)] for j in xrange(len(items) + 1)]
 
    for j in xrange(1, len(items) + 1):
        item, wt, val = items[j-1]
        for w in xrange(1, limit + 1):
            if wt > w:
                table[j][w] = table[j-1][w]
            else:
                table[j][w] = max(table[j-1][w],
                                  table[j-1][w-wt] + val)
 
    result = []
    w = limit
    for j in range(len(items), 0, -1):
        was_added = table[j][w] != table[j-1][w]
 
        if was_added:
            item, wt, val = items[j-1]
            result.append(items[j-1])
            w -= wt
 
    return result
 
 
bagged = knapsack01_dp(items, maxwt)
print("Bagged the following %i items\n  " % len(bagged) +
      '\n  '.join('%i off: %s' % (len(list(grp)), item[0])
                  for item,grp in groupby(sorted(bagged))))
print("for a total value of %i and a total weight of %i" % (
    sum(item[2] for item in bagged), sum(item[1] for item in bagged)))

Non-zero-one solution[edit]

items = {
	"sandwich":	(50,	60,	2),
	"map": 		(9,	150,	1),
	"compass": 	(13,	35,	1),
	"water": 	(153,	200,	3),
	"glucose": 	(15,	60,	2),
	"tin": 		(68,	45,	3),
	"banana": 	(27,	60,	3),
	"apple": 	(39,	40,	3),
	"cheese": 	(23,	30,	1),
	"beer": 	(52,	10,	3),
	"suntan cream": (11,	70,	1),
	"camera": 	(32,	30,	1),
	"t-shirt": 	(24,	15,	2),
	"trousers": 	(48,	10,	2),
	"umbrella": 	(73,	40,	1),
	"w-trousers": 	(42,	70,	1),
	"w-overcoat": 	(43,	75,	1),
	"note-case": 	(22,	80,	1),
	"sunglasses": 	(7,	20,	1),
	"towel": 	(18,	12,	2),
	"socks": 	(4,	50,	1),
	"book": 	(30,	10,	2),
}

item_keys = list(items.keys())

#cache: could just use memoize module, but explicit caching is clearer
def choose_item(weight, idx, cache):
    if idx < 0: return 0, []
 
    k = (weight, idx)
    if k in cache: return cache[k]
 
    name, w, v, qty = item_keys[idx], *items[item_keys[idx]]
    best_v, best_list = 0, []
 
    for i in range(0, qty + 1):
        wlim = weight - i * w
        if wlim < 0: break
 
        val, taken = choose_item(wlim, idx - 1, cache)
        if val + i * v > best_v:
            best_v = val + i * v
            best_list = taken[:]
            best_list.append((i, name))
 
    cache[k] = [best_v, best_list]
    return best_v, best_list
 
 
v, lst = choose_item(400, len(items) - 1, {})
w = 0
for cnt, name in lst:
    if cnt > 0:
        print(cnt, name)        
        w = w + items[name][0] * cnt
 
print("Total weight:", w, "Value:", v)

R[edit]

Reading in task via web scraping.

library(tidyverse)
library(rvest)

task_html= read_html("http://rosettacode.org/wiki/Knapsack_problem/Bounded")
task_table= html_nodes(html, "table")[[1]] %>%
  html_table(table, header= T, trim= T) %>%
  set_names(c("items", "weight", "value", "pieces")) %>%
  filter(items != "knapsack") %>%
  mutate(weight= as.numeric(weight),
         value= as.numeric(value),
         pieces= as.numeric(pieces))


Solution of the task using genetic algorithm.

library(rgenoud)

fitness= function(x= rep(1, nrow(task_table))){
  total_value= sum(task_table$value * x)
  total_weight= sum(task_table$weight * x)
  ifelse(total_weight <= 400, total_value, 400-total_weight)
}

allowed= matrix(c(rep(0, nrow(task_table)), task_table$pieces), ncol = 2)
set.seed(42)
evolution= genoud(fn= fitness, 
                  nvars= nrow(allowed), 
                  max= TRUE,
                  pop.size= 10000,
                  data.type.int= TRUE, 
                  Domains= allowed)

cat("Value: ", evolution$value, "\n")
cat("Weight:", sum(task_table$weight * evolution$par), "dag", "\n")
data.frame(item= task_table$items, pieces= as.integer(solution)) %>%
  filter(solution> 0)
Output:
Value:  1010 
Weight: 396 dag 
                     item pieces
1                     map      1
2                 compass      1
3                sandwich      1
4                 glucose      2
5                  banana      3
6                   apple      2
7                  cheese      1
8            suntan cream      1
9  waterproof overclothes      1
10              note-case      1
11             sunglasses      1
12                  towel      1
13                  socks      1

Racket[edit]

The algorithm is nearly a direct translation of Haskell's array-based implementation. However, the data is taken from the webpage itself.

#lang racket
(require net/url html xml xml/path)

(struct item (name mass value count) #:transparent)

;this section is to convert the web page on the problem into the data for the problem
;i don't got time to manually type tables, nevermind that this took longer
(define (group-n n l)
  (let group-n ([l l] [acc '()])
    (if (null? l)
        (reverse acc)
        (let-values ([(takes drops) (split-at l n)])
          (group-n drops (cons takes acc))))))
(define (atom? x) (not (or (pair? x) (null? x))))
;modified from little schemer...finds nested list where regular member would've returned non-#f
(define (member* x t)
  (cond [(null? t) #f]
        [(atom? (car t)) (or (and (equal? (car t) x) t)
                             (member* x (cdr t)))]
        [else (or (member* x (car t))
                  (member* x (cdr t)))]))
(define (addr->xexpr f) (compose xml->xexpr f read-html-as-xml get-pure-port string->url))
(define (read-page) ((addr->xexpr cadr) "http://rosettacode.org/wiki/Knapsack_problem/Bounded"))
(define (get-xml-table xe) (member* 'table xe))
(define (xml-table->item-list xe-table)
  ;all html table datas
  (let* ([strs (se-path*/list '(td) xe-table)]
         ;4 columns per row
         [rows (group-n 4 strs)]
         ;the last two rows belong to the knapsack entry which we don't want
         [rows (take rows (- (length rows) 2))])
    (for/list ([r rows])
      (let ([r (map string-trim r)])
        ;convert the weight, value, and pieces columns to numbers and structify it
        (apply item (car r) (map string->number (cdr r)))))))
;top-level function to take a string representing a URL and gives the table I didn't want to type
(define addr->item-list (compose xml-table->item-list get-xml-table read-page))

;stores best solution
(struct solution (value choices) #:transparent)

;finds best solution in a list of solutions
(define (best sol-list)
  (let best ([l (cdr sol-list)] [bst (car sol-list)])
    (match l
      ['() bst]
      [(cons s l) (if (> (solution-value s) (solution-value bst))
                      (best l s)
                      (best l bst))])))

;stores the choices leading to best solution...item name and # of that item taken
(struct choice (name count) #:transparent)

;algorithm is derived from Haskell's array-based example
;returns vector of solutions for every natural number capacity up to the input max
(define (solution-vector capacity)
  ;find best value and items that gave it, trying all items
  ;first we set up a vector for the best solution found for every capacity
  ;the best solution found at the beginning has 0 value with no items
  (for/fold ([solutions (make-vector (add1 capacity) (solution 0 '()))])
            ([i (addr->item-list)])
    ;the new solutions aren't accumulated until after processing all the way through a particular
    ;capacity, lest capacity c allow capacity c+1 to reuse an item that had been exhausted, i.e.
    ;we have to move on to the next item before we save ANY solutions found for that item, so they
    ;must be saved "in parallel" by overwriting the entire vector at once
    (for/vector ([target-cap (range 0 (add1 capacity))])
      (match-let ([(item name mass value count) i])
        ;find best solution for item out of every number that can be taken of that item
        ;we'll call an "item and count of item" a choice
        (best (for/list ([n (range 0 (add1 count))])
                ;ensure mass of this choice is not greater than target capacity
                (let ([mass-choice (* n mass)])
                  (if (> mass-choice target-cap)
                      (solution 0 '())
                      ;subtract from target capacity the amount taken up by this choice
                      ;use it to get the best solution for THAT capacity
                      (let ([remaining-cap-solution (vector-ref solutions (- target-cap mass-choice))])
                        ;the new solution found adds the value of this choice to the above value and
                        ;adds the choice itself to the list of choices
                        (solution (+ (* n value) (solution-value remaining-cap-solution))
                                  (cons (choice name n) (solution-choices remaining-cap-solution))))))))))))

;top level function to convert solution vector into single best answer
(define (knapsack capacity)
  ;the best solution is not necessarily the one that uses max capacity,
  ;e.g. if max is 5 and you have items of mass 4 and 5, and the 4 is worth more
  (match-let ([(solution value choices) (best (vector->list (solution-vector capacity)))])
    (solution value (filter (compose positive? choice-count) choices))))
Output:
(solution
 1010
 (list
  (choice "socks" 1)
  (choice "sunglasses" 1)
  (choice "note-case" 1)
  (choice "waterproof overclothes" 1)
  (choice "suntan cream" 1)
  (choice "cheese" 1)
  (choice "banana" 3)
  (choice "glucose" 2)
  (choice "water" 1)
  (choice "compass" 1)
  (choice "map" 1)))

Raku[edit]

(formerly Perl 6)

Original[edit]

Recursive algorithm, with cache. Idiomatic code style, using multi-subs and a class.

Works with: Rakudo version 2017.01
my class KnapsackItem { has $.name; has $.weight; has $.unit; }

multi sub pokem ([],           $,  $v = 0) { $v }
multi sub pokem ([$,  *@],     0,  $v = 0) { $v }
multi sub pokem ([$i, *@rest], $w, $v = 0) {
  my $key = "{+@rest} $w $v";
  (state %cache){$key} or do {
    my @skip = pokem @rest, $w, $v;
    if $w >= $i.weight { # next one fits
      my @put = pokem @rest, $w - $i.weight, $v + $i.unit;
      return (%cache{$key} = |@put, $i.name).list if @put[0] > @skip[0];
    }
    return (%cache{$key} = |@skip).list;
  }
}

my $MAX_WEIGHT = 400;
my @table = flat map -> $name,  $weight,  $unit,     $count {
     KnapsackItem.new( :$name, :$weight, :$unit ) xx $count;
},
        'map',                         9,      150,    1,
        'compass',                     13,     35,     1,
        'water',                       153,    200,    2,
        'sandwich',                    50,     60,     2,
        'glucose',                     15,     60,     2,
        'tin',                         68,     45,     3,
        'banana',                      27,     60,     3,
        'apple',                       39,     40,     3,
        'cheese',                      23,     30,     1,
        'beer',                        52,     10,     3,
        'suntan cream',                11,     70,     1,
        'camera',                      32,     30,     1,
        'T-shirt',                     24,     15,     2,
        'trousers',                    48,     10,     2,
        'umbrella',                    73,     40,     1,
        'waterproof trousers',         42,     70,     1,
        'waterproof overclothes',      43,     75,     1,
        'note-case',                   22,     80,     1,
        'sunglasses',                  7,      20,     1,
        'towel',                       18,     12,     2,
        'socks',                       4,      50,     1,
        'book',                        30,     10,     2
        ;

my ($value, @result) = pokem @table, $MAX_WEIGHT;

(my %hash){$_}++ for @result;

say "Value = $value";
say "Tourist put in the bag:";
say "  # ITEM";
for %hash.sort -> $item {
  say "  {$item.value} {$item.key}";
}
Output:
Value = 1010
Tourist put in the bag:
  # ITEM
  3 banana
  1 cheese
  1 compass
  2 glucose
  1 map
  1 note-case
  1 socks
  1 sunglasses
  1 suntan cream
  1 water
  1 waterproof overclothes

Faster alternative[edit]

Also recursive, with cache, but substantially faster. Code more generic (ported from Perl solution).

my $raw = qq:to/TABLE/;
map             9       150     1
compass         13      35      1
water           153     200     2
sandwich        50      60      2
glucose         15      60      2
tin             68      45      3
banana          27      60      3
apple           39      40      3
cheese          23      30      1
beer            52      10      1
suntancream     11      70      1
camera          32      30      1
T-shirt         24      15      2
trousers        48      10      2
umbrella        73      40      1
w_trousers      42      70      1
w_overcoat      43      75      1
note-case       22      80      1
sunglasses       7      20      1
towel           18      12      2
socks            4      50      1
book            30      10      2
TABLE

my @items;
for split(["\n", /\s+/], $raw, :skip-empty) -> $n,$w,$v,$q {
    @items.push: %{ name => $n, weight => $w, value => $v, quant => $q}
}

my $max_weight = 400;

sub pick ($weight, $pos) {
    state %cache;
    return 0, 0 if $pos < 0 or $weight <= 0;

    my $key = $weight ~ $pos;
    %cache{$key} or do {
        my %item = @items[$pos];
        my ($bv, $bi, $bw, @bp) = (0, 0, 0);

        for 0 .. %item{'quant'} -> $i {
            last if $i * %item{'weight'} > $weight;
            my ($v, $w, @p) = pick($weight - $i * %item{'weight'}, $pos - 1);
            next if ($v += $i * %item{'value'}) <= $bv;

            ($bv, $bi, $bw, @bp) = ($v, $i, $w, |@p);
        }
        %cache{$key} = $bv, $bw + $bi * %item{'weight'}, |@bp, $bi;
    }
}

my ($v, $w, @p) = pick($max_weight, @items.end);
{ say "{@p[$_]} of @items[$_]{'name'}" if @p[$_] } for 0 .. @p.end;
say "Value: $v Weight: $w";
Output:
1 of map
1 of compass
1 of water
2 of glucose
3 of banana
1 of cheese
1 of suntancream
1 of w_overcoat
1 of note-case
1 of sunglasses
1 of socks
Value: 1010 Weight: 396

REXX[edit]

Originally, the combination generator/checker subroutine   findBest   was recursive and made the program solution generic.

However, a recursive solution also made the solution much more slower, so the combination generator/checker was
"unrolled" and converted into discrete combination checks (based on the number of items).   The unused combinatorial
checks were discarded and only the pertinent code was retained.

/*REXX program solves a  knapsack problem  (22 items + repeats, with weight restriction.*/
call @gen                                        /*generate items and initializations.  */
call @sort                                       /*sort items by decreasing their weight*/
call build                                       /*build a list of choices (objects).   */
call showOBJ                                     /*display the list of choices (objects)*/
call findBest                                    /*examine and find the possible choices*/
call showBest                                    /*display best choice  (weight, value).*/
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
@gen: @.=;        @.1  = 'map                       9  150'
                  @.2  = 'compass                  13   35'
                  @.3  = 'water                   153  200   2'
                  @.4  = 'sandwich                 50   60   2'
                  @.5  = 'glucose                  15   60   2'
                  @.6  = 'tin                      68   45   3'
                  @.7  = 'banana                   27   60   3'
                  @.8  = 'apple                    39   40   3'
                  @.9  = 'cheese                   23   30'
                  @.10 = 'beer                     52   10   3'
                  @.11 = 'suntan_cream             11   70'
                  @.12 = 'camera                   32   30'
                  @.13 = 'T-shirt                  24   15   2'
                  @.14 = 'trousers                 48   10   2'
                  @.15 = 'umbrella                 73   40'
                  @.16 = 'waterproof_trousers      42   70'
                  @.17 = 'waterproof_overclothes   43   75'
                  @.18 = 'note-case                22   80'
                  @.19 = 'sunglasses                7   20'
                  @.20 = 'towel                    18   12   2'
                  @.21 = 'socks                     4   50'
                  @.22 = 'book                     30   10   2'
    highQ = 0                                    /*maximum quantity specified (if any). */
     maxL = length('knapsack items')             /* "     "    width for the table names*/
     maxW = length('weight')                     /* "     "      "    "    "   weights. */
     maxV = length('value')                      /* "     "      "    "    "   values.  */
     maxQ = length('pieces')                     /* "     "      "    "    "   quantity.*/
     maxWeight=400                               /*the maximum weight for the knapsack. */
     items= 0;   i.=;    w.=0;    v.=0;   q.=0   /*initialize some stuff and things.    */
        Tw= 0;   Tv=0;   Tq=0;    m=maxWeight    /*     "     more   "    "     "       */
     say;  say 'maximum weight allowed for a knapsack: '   commas(maxWeight);     say
     return
/*──────────────────────────────────────────────────────────────────────────────────────*/
@sort:        do j=1  while @.j\==''             /*process each choice and sort the item*/
              @.j=space(@.j);   _wt=word(@.j, 2) /*choose first item (arbitrary).       */
                  do k=j+1  while @.k\==''       /*find a possible heavier item.        */
                  ?wt=word(@.k, 2)
                  if ?wt>_wt  then do;  _=@.k;  @.k=@.j;  @.j=_;  _wt=?wt;  end   /*swap*/
                  end   /*k*/
              end       /*j*/                    /* [↑]  minimizes the # of combinations*/
      obj=j-1                                    /*adjust for the   DO   loop index.    */
      return
/*──────────────────────────────────────────────────────────────────────────────────────*/
build:        do j=1  for obj                    /*build a list of choices (objects).   */
              parse var  @.j  item  w  v  q  .   /*parse the original choice for table. */
              if w>maxWeight  then iterate       /*Is the weight > maximum?  Then ignore*/
              Tw=Tw+w;  Tv=Tv+v;   Tq=Tq+1       /*add the totals up  (for alignment).  */
              maxL=max(maxL, length(item))       /*find the maximum width for an item.  */
              if q==''  then q=1
              highQ=max(highQ, q)
              items=items+1                      /*bump the item counter.               */
              i.items=item;  w.items=w;  v.items=v;  q.items=q
                  do k=2  to q  ;  items=items+1 /*bump the item counter  (each piece). */
                  i.items=item;  w.items=w;  v.items=v;  q.items=q
                                Tw=Tw+w;    Tv=Tv+v;    Tq=Tq+1
                  end   /*k*/
              end       /*j*/
      maxW = max(maxW, length( commas(Tw) ) )    /*find the maximum width for weight.   */
      maxV = max(maxV, length( commas(Tv) ) )    /*  "   "     "      "    "  value.    */
      maxQ = max(maxQ, length( commas(Tq) ) )    /*  "   "     "      "    "  quantity. */
      maxL = maxL + maxL %4 + 4                  /*extend the width of name for table.  */
      return                                     /* [↑]    %  is REXX integer division. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
commas: procedure;  parse arg _;   n=_'.9';   #=123456789;    b=verify(n, #, "M");   x=','
        e=verify(n, #'0', , verify(n, #"0.", 'M') ) - 4     /* [↓]  add commas to number*/
           do j=e  to b  by -3;    _=insert(x, _, j);  end  /*j*/;                return _
/*──────────────────────────────────────────────────────────────────────────────────────*/
hdr:  parse arg _item_, _;         if highq\==1  then _=center('pieces', maxq)
      call show center(_item_,  maxL), center('weight', maxW), center('value',  maxV), ,
                center(_     ,  maxQ);                          call hdr2;        return
/*──────────────────────────────────────────────────────────────────────────────────────*/
hdr2: _=maxQ;   x='═';          if highq==1  then _=0
      call show copies(x, maxL), copies(x, maxW), copies(x, maxV), copies(x, _);  return
/*──────────────────────────────────────────────────────────────────────────────────────*/
j?:   parse arg _,?;  $=value('Z'_);    do k=1  for _;  ?=? value('J'k);  end;    return
/*──────────────────────────────────────────────────────────────────────────────────────*/
show: parse arg _item, _weight, _value, _quant
      say translate(left(_item, maxL,'─'), ,'_')  right(commas(_weight), maxW),
          right(commas(_value ), maxV)            right(commas(_quant ), maxQ);   return
/*──────────────────────────────────────────────────────────────────────────────────────*/
showOBJ: call hdr 'item';     do j=1  for obj             /*show the formatted choices. */
                              parse var  @.j  item weight value q .
                              if highq==1  then  q=
                                           else  if q==''  then q=1
                              call show  item, weight, value, q
                              end   /*j*/
         say;    say  'number of   unique   named items: '    obj
                 say  'number of items (including reps): '  items;    say;        return
/*──────────────────────────────────────────────────────────────────────────────────────*/
showBest:                     do words(?);  ?=strip(space(?), "L", 0);  end   /*words(?)*/
         bestC=?;   bestW=0;   bestV=$;   highQ=0;   totP=words(bestC);   say
         call hdr 'best choice'
                              do j=1  to totP         /*J  is modified within  DO  loop.*/
                              _=word(bestC, j);     _w=w._;     _v=v._;     q=1
                              if _==0  then iterate
                                  do k=j+1  to totP;  __=word(bestC, k)  /*get a choice.*/
                                  if i._\==i.__  then leave              /*not equal ?  */
                                  j=j+1;       w._=w._+_w;       v._=v._+_v;         q=q+1
                                  end   /*k*/
                              call show  i._,  w._,  v._,  q;            bestW=bestw+w._
                              end       /*j*/
         call hdr2;    say
         call show 'best weight'   ,     bestW        /*show a nicely formatted winnerW.*/
         call show 'best value'    , ,   bestV        /*  "  "    "       "     winnerV.*/
         call show 'knapsack items', , , totP         /*  "  "    "       "     pieces. */
         return
/*─────────────────────────────────────────────────────────────────────────────────────────────────────────*/
findBest:      h=items;      $=0
 do j1 =0  for h+1;                                       w1=    w.j1 ; z1=    v.j1 ;if  z1>$ then call j?  1
 do j2 =j1 +(j1 >0) to h;if w.j2 +w1 >m then iterate  j1; w2=w1 +w.j2 ; z2=z1 +v.j2 ;if  z2>$ then call j?  2
 do j3 =j2 +(j2 >0) to h;if w.j3 +w2 >m then iterate  j2; w3=w2 +w.j3 ; z3=z2 +v.j3 ;if  z3>$ then call j?  3
 do j4 =j3 +(j3 >0) to h;if w.j4 +w3 >m then iterate  j3; w4=w3 +w.j4 ; z4=z3 +v.j4 ;if  z4>$ then call j?  4
 do j5 =j4 +(j4 >0) to h;if w.j5 +w4 >m then iterate  j4; w5=w4 +w.j5 ; z5=z4 +v.j5 ;if  z5>$ then call j?  5
 do j6 =j5 +(j5 >0) to h;if w.j6 +w5 >m then iterate  j5; w6=w5 +w.j6 ; z6=z5 +v.j6 ;if  z6>$ then call j?  6
 do j7 =j6 +(j6 >0) to h;if w.j7 +w6 >m then iterate  j6; w7=w6 +w.j7 ; z7=z6 +v.j7 ;if  z7>$ then call j?  7
 do j8 =j7 +(j7 >0) to h;if w.j8 +w7 >m then iterate  j7; w8=w7 +w.j8 ; z8=z7 +v.j8 ;if  z8>$ then call j?  8
 do j9 =j8 +(j8 >0) to h;if w.j9 +w8 >m then iterate  j8; w9=w8 +w.j9 ; z9=z8 +v.j9 ;if  z9>$ then call j?  9
 do j10=j9 +(j9 >0) to h;if w.j10+w9 >m then iterate  j9;w10=w9 +w.j10;z10=z9 +v.j10;if z10>$ then call j? 10
 do j11=j10+(j10>0) to h;if w.j11+w10>m then iterate j10;w11=w10+w.j11;z11=z10+v.j11;if z11>$ then call j? 11
 do j12=j11+(j11>0) to h;if w.j12+w11>m then iterate j11;w12=w11+w.j12;z12=z11+v.j12;if z12>$ then call j? 12
 do j13=j12+(j12>0) to h;if w.j13+w12>m then iterate j12;w13=w12+w.j13;z13=z12+v.j13;if z13>$ then call j? 13
 do j14=j13+(j13>0) to h;if w.j14+w13>m then iterate j13;w14=w13+w.j14;z14=z13+v.j14;if z14>$ then call j? 14
 do j15=j14+(j14>0) to h;if w.j15+w14>m then iterate j14;w15=w14+w.j15;z15=z14+v.j15;if z15>$ then call j? 15
 do j16=j15+(j15>0) to h;if w.j16+w15>m then iterate j15;w16=w15+w.j16;z16=z15+v.j16;if z16>$ then call j? 16
 do j17=j16+(j16>0) to h;if w.j17+w16>m then iterate j16;w17=w16+w.j17;z17=z16+v.j17;if z17>$ then call j? 17
 do j18=j17+(j17>0) to h;if w.j18+w17>m then iterate j17;w18=w17+w.j18;z18=z17+v.j18;if z18>$ then call j? 18
 do j19=j18+(j18>0) to h;if w.j19+w18>m then iterate j18;w19=w18+w.j19;z19=z18+v.j19;if z19>$ then call j? 19
 do j20=j19+(j19>0) to h;if w.j20+w19>m then iterate j19;w20=w19+w.j20;z20=z19+v.j20;if z20>$ then call j? 20
 do j21=j20+(j20>0) to h;if w.j21+w20>m then iterate j20;w21=w20+w.j21;z21=z20+v.j21;if z21>$ then call j? 21
 do j22=j21+(j21>0) to h;if w.j22+w21>m then iterate j21;w22=w21+w.j22;z22=z21+v.j22;if z22>$ then call j? 22
 do j23=j22+(j22>0) to h;if w.j23+w22>m then iterate j22;w23=w22+w.j23;z23=z22+v.j23;if z23>$ then call j? 23
 do j24=j23+(j23>0) to h;if w.j24+w23>m then iterate j23;w24=w23+w.j24;z24=z23+v.j24;if z24>$ then call j? 24
 do j25=j24+(j24>0) to h;if w.j25+w24>m then iterate j24;w25=w24+w.j25;z25=z24+v.j25;if z25>$ then call j? 25
 do j26=j25+(j25>0) to h;if w.j26+w25>m then iterate j25;w26=w25+w.j26;z26=z25+v.j26;if z26>$ then call j? 26
 do j27=j26+(j26>0) to h;if w.j27+w26>m then iterate j26;w27=w26+w.j27;z27=z26+v.j27;if z27>$ then call j? 27
 do j28=j27+(j27>0) to h;if w.j28+w27>m then iterate j27;w28=w27+w.j28;z28=z27+v.j28;if z28>$ then call j? 28
 do j29=j28+(j28>0) to h;if w.j29+w28>m then iterate j28;w29=w28+w.j29;z29=z28+v.j29;if z29>$ then call j? 29
 do j30=j29+(j29>0) to h;if w.j30+w29>m then iterate j29;w30=w29+w.j30;z30=z29+v.j30;if z30>$ then call j? 30
 do j31=j30+(j30>0) to h;if w.j31+w30>m then iterate j30;w31=w30+w.j31;z31=z30+v.j31;if z31>$ then call j? 31
 do j32=j31+(j31>0) to h;if w.j32+w31>m then iterate j31;w32=w31+w.j32;z32=z31+v.j32;if z32>$ then call j? 32
 do j33=j32+(j32>0) to h;if w.j33+w32>m then iterate j32;w33=w32+w.j33;z33=z32+v.j33;if z33>$ then call j? 33
 do j34=j33+(j33>0) to h;if w.j34+w33>m then iterate j33;w34=w33+w.j34;z34=z33+v.j34;if z34>$ then call j? 34
 do j35=j34+(j34>0) to h;if w.j35+w34>m then iterate j34;w35=w34+w.j35;z35=z34+v.j35;if z35>$ then call j? 35
 do j36=j35+(j35>0) to h;if w.j36+w35>m then iterate j35;w36=w35+w.j36;z36=z35+v.j36;if z36>$ then call j? 36
 do j37=j36+(j36>0) to h;if w.j37+w36>m then iterate j36;w37=w36+w.j37;z37=z36+v.j37;if z37>$ then call j? 37
 end;end;end;end;end;end;end;end;end;end;end;end;end;end;end;end;end;end;end;end;end;end;end;end;end;end;end
     end;end;end;end;end;end;end;end;end;end;        return      /* [↑]  there is one END for each DO loop.*/

output

maximum weight allowed for a knapsack:  400


             item               weight value pieces
═══════════════════════════════ ══════ ═════ ══════
water──────────────────────────    153   200      2
umbrella───────────────────────     73    40      1
tin────────────────────────────     68    45      3
beer───────────────────────────     52    10      3
sandwich───────────────────────     50    60      2
trousers───────────────────────     48    10      2
waterproof overclothes─────────     43    75      1
waterproof trousers────────────     42    70      1
apple──────────────────────────     39    40      3
camera─────────────────────────     32    30      1
book───────────────────────────     30    10      2
banana─────────────────────────     27    60      3
T-shirt────────────────────────     24    15      2
cheese─────────────────────────     23    30      1
note-case──────────────────────     22    80      1
towel──────────────────────────     18    12      2
glucose────────────────────────     15    60      2
compass────────────────────────     13    35      1
suntan cream───────────────────     11    70      1
map────────────────────────────      9   150      1
sunglasses─────────────────────      7    20      1
socks──────────────────────────      4    50      1

number of items (with repetitions):  37



          best choice           weight value pieces
═══════════════════════════════ ══════ ═════ ══════
water──────────────────────────    153   200      1
waterproof overclothes─────────     43    75      1
banana─────────────────────────     81   180      3
cheese─────────────────────────     23    30      1
note-case──────────────────────     22    80      1
glucose────────────────────────     30   120      2
compass────────────────────────     13    35      1
suntan cream───────────────────     11    70      1
map────────────────────────────      9   150      1
sunglasses─────────────────────      7    20      1
socks──────────────────────────      4    50      1
═══════════════════════════════ ══════ ═════ ══════

best weight────────────────────    396
best value─────────────────────        1,010
knapsack items─────────────────                  14

Ruby[edit]

Translation of: Python
# Item struct to represent each item in the problem
Struct.new('Item', :name, :weight, :value, :count)

$items = [
  Struct::Item.new('map', 9, 150, 1),
  Struct::Item.new('compass', 13, 35, 1),
  Struct::Item.new('water', 153, 200, 3),
  Struct::Item.new('sandwich', 50, 60, 2),
  Struct::Item.new('glucose', 15, 60, 2),
  Struct::Item.new('tin', 68, 45, 3),
  Struct::Item.new('banana', 27, 60, 3),
  Struct::Item.new('apple', 39, 40, 3),
  Struct::Item.new('cheese', 23, 30, 1),
  Struct::Item.new('beer', 52, 10, 3),
  Struct::Item.new('suntan cream', 11, 70, 1),
  Struct::Item.new('camera', 32, 30, 1),
  Struct::Item.new('t-shirt', 24, 15, 2),
  Struct::Item.new('trousers', 48, 10, 2),
  Struct::Item.new('umbrella', 73, 40, 1),
  Struct::Item.new('w-trousers', 42, 70, 1),
  Struct::Item.new('w-overcoat', 43, 75, 1),
  Struct::Item.new('note-case', 22, 80, 1),
  Struct::Item.new('sunglasses', 7, 20, 1),
  Struct::Item.new('towel', 18, 12, 2),
  Struct::Item.new('socks', 4, 50, 1),
  Struct::Item.new('book', 30, 10, 2)
]

def choose_item(weight, id, cache)
  return 0, [] if id < 0

  k = [weight, id]
  return cache[k] unless cache[k].nil?
  value = $items[id].value
  best_v = 0
  best_list = []
  ($items[id].count+1).times do |i|
    wlim = weight - i * $items[id].weight
    break if wlim < 0
    val, taken = choose_item(wlim, id - 1, cache)
    if val + i * value > best_v
      best_v = val + i * value
      best_list = taken + [i]
    end
  end
  cache[k] = [best_v, best_list]
  return [best_v, best_list]
end

val, list = choose_item(400, $items.length - 1, {})
w = 0
list.each_with_index do |cnt, i|
  if cnt > 0
    print "#{cnt} #{$items[i].name}\n"
    w += $items[i][1] * cnt
  end
end

p "Total weight: #{w}, Value: #{val}"

SAS[edit]

Use MILP solver in SAS/OR:

/* create SAS data set */
data mydata;
   input item $1-23 weight value pieces;
   datalines;
map                      9  150  1  
compass                 13   35  1  
water                  153  200  2  
sandwich                50   60  2  
glucose                 15   60  2  
tin                     68   45  3  
banana                  27   60  3  
apple                   39   40  3  
cheese                  23   30  1  
beer                    52   10  3  
suntan cream            11   70  1  
camera                  32   30  1  
T-shirt                 24   15  2  
trousers                48   10  2  
umbrella                73   40  1  
waterproof trousers     42   70  1  
waterproof overclothes  43   75  1  
note-case               22   80  1  
sunglasses               7   20  1  
towel                   18   12  2  
socks                    4   50  1  
book                    30   10  2  
;

/* call OPTMODEL procedure in SAS/OR */
proc optmodel;
   /* declare sets and parameters, and read input data */
   set <str> ITEMS;
   num weight {ITEMS};
   num value {ITEMS};
   num pieces {ITEMS};
   read data mydata into ITEMS=[item] weight value pieces;

   /* declare variables, objective, and constraints */
   var NumSelected {i in ITEMS} >= 0 <= pieces[i] integer;
   max TotalValue = sum {i in ITEMS} value[i] * NumSelected[i];
   con WeightCon:
      sum {i in ITEMS} weight[i] * NumSelected[i] <= 400;

   /* call mixed integer linear programming (MILP) solver */
   solve;

   /* print optimal solution */
   print TotalValue;
   print {i in ITEMS: NumSelected[i].sol > 0.5} NumSelected;
quit;

Output:

TotalValue 
1010 

[1] NumSelected 
banana 3 
cheese 1 
compass 1 
glucose 2 
map 1 
note-case 1 
socks 1 
sunglasses 1 
suntan cream 1 
water 1 
waterproof overclothes 1 

Sidef[edit]

Translation of: Perl
var raw = <<'TABLE'
map           9     150      1
compass      13      35      1
water       153     200      2
sandwich     50      60      2
glucose      15      60      2
tin          68      45      3
banana       27      60      3
apple        39      40      3
cheese       23      30      1
beer         52      10      1
suntancream  11      70      1
camera       32      30      1
T-shirt      24      15      2
trousers     48      10      2
umbrella     73      40      1
w_trousers   42      70      1
w_overcoat   43      75      1
note-case    22      80      1
sunglasses    7      20      1
towel        18      12      2
socks         4      50      1
book         30      10      2
TABLE

struct KnapsackItem {
    String name,
    Number weight,
    Number value,
    Number quant,
}

var items = []
raw.each_line{ |row|
    var fields = row.words;
    items << KnapsackItem(
          name: fields[0],
        weight: fields[1].to_n,
         value: fields[2].to_n,
         quant: fields[3].to_n,
    )
}

func pick(weight, pos) is cached {

    if (pos.is_neg || weight.is_neg || weight.is_zero) {
        return (0, 0, [])
    }

    var (bv=0, bi=0, bw=0, bp=[])
    var item = items[pos];

    for i in range(0, item.quant) {
        break if (i*item.weight > weight)
        var (v, w, p) = pick(weight - i*item.weight, pos.dec)
        next if ((v += i*item.value) <= bv)
        (bv, bi, bw, bp) = (v, i, w, p)
    }

    (bv, bw + bi*item.weight, [bp..., bi])
}

var (v, w, p) = pick(400, items.end)
p.range.each { |i|
    say "#{p[i]} of #{items[i].name}" if p[i].is_pos
}
say "Value: #{v}; Weight: #{w}"
Output:
1 of map
1 of compass
1 of water
2 of glucose
3 of banana
1 of cheese
1 of suntancream
1 of w_overcoat
1 of note-case
1 of sunglasses
1 of socks
Value: 1010; Weight: 396

Swift[edit]

Translation of: Python
public struct KnapsackItem: Hashable {
  public var name: String
  public var weight: Int
  public var value: Int

  public init(name: String, weight: Int, value: Int) {
    self.name = name
    self.weight = weight
    self.value = value
  }
}

public func knapsack(items: [KnapsackItem], limit: Int) -> [KnapsackItem] {
  var table = Array(repeating: Array(repeating: 0, count: limit + 1), count: items.count + 1)

  for j in 1...items.count {
    let item = items[j-1]

    for w in 1...limit {
      if item.weight > w {
        table[j][w] = table[j-1][w]
      } else {
        table[j][w] = max(table[j-1][w], table[j-1][w-item.weight] + item.value)
      }
    }
  }

  var result = [KnapsackItem]()
  var w = limit

  for j in stride(from: items.count, to: 0, by: -1) where table[j][w] != table[j-1][w] {
    let item = items[j-1]

    result.append(item)

    w -= item.weight
  }

  return result
}

typealias GroupedItem = (name: String, weight: Int, val: Int, n: Int)

let groupedItems: [GroupedItem] = [
  ("map", 9, 150, 1),
  ("compass", 13, 35, 1),
  ("water", 153, 200, 3),
  ("sandwich", 50, 60, 2),
  ("glucose", 15, 60, 2),
  ("tin", 68, 45, 3),
  ("banana", 27, 60, 3),
  ("apple", 39, 40, 3),
  ("cheese", 23, 30, 1),
  ("beer", 52, 10, 3),
  ("suntan cream", 11, 70, 1),
  ("camera", 32, 30, 1),
  ("t-shirt", 24, 15, 2),
  ("trousers", 48, 10, 2),
  ("umbrella", 73, 40, 1),
  ("waterproof trousers", 42, 70, 1),
  ("waterproof overclothes", 43, 75, 1),
  ("note-case", 22, 80, 1),
  ("sunglasses", 7, 20, 1),
  ("towel", 18, 12, 2),
  ("socks", 4, 50, 1),
  ("book", 30, 10, 2)
]

let items = groupedItems.flatMap({item in
  (0..<item.n).map({_ in KnapsackItem(name: item.name, weight: item.weight, value: item.val) })
})

let bagged = knapsack(items: items, limit: 400)
let (totalVal, totalWeight) = bagged.reduce((0, 0), {cur, item in (cur.0 + item.value, cur.1 + item.weight) })

print("Bagged the following \(bagged.count) items:")

for item in bagged {
  print("\t\(item.name)")
}

print("For a total value of \(totalVal) and weight of \(totalWeight)")
Output:
Bagged the following 14 items:
	socks
	sunglasses
	note-case
	waterproof overclothes
	suntan cream
	cheese
	banana
	banana
	banana
	glucose
	glucose
	water
	compass
	map
For a total value of 1010 and weight of 396

Tcl[edit]

Classic dumb search algorithm:

# The list of items to consider, as list of lists
set items {
    {map			9	150	1}
    {compass			13	35	1}
    {water			153	200	2}
    {sandwich			50	60	2}
    {glucose			15	60	2}
    {tin			68	45	3}
    {banana			27	60	3}
    {apple			39	40	3}
    {cheese			23	30	1}
    {beer			52	10	3}
    {{suntan cream}		11	70	1}
    {camera			32	30	1}
    {t-shirt			24	15	2}
    {trousers			48	10	2}
    {umbrella			73	40	1}
    {{waterproof trousers}	42	70	1}
    {{waterproof overclothes}	43	75	1}
    {note-case			22	80	1}
    {sunglasses			7	20	1}
    {towel			18	12	2}
    {socks			4	50	1}
    {book			30	10	2}
}

# Simple extraction functions
proc countednames {chosen} {
    set names {}
    foreach item $chosen {
	lappend names [lindex $item 3] [lindex $item 0]
    }
    return $names
}
proc weight {chosen} {
    set weight 0
    foreach item $chosen {
	incr weight [expr {[lindex $item 3] * [lindex $item 1]}]
    }
    return $weight
}
proc value {chosen} {
    set value 0
    foreach item $chosen {
	incr value [expr {[lindex $item 3] * [lindex $item 2]}]
    }
    return $value
}

# Recursive function for searching over all possible choices of items
proc knapsackSearch {items {chosen {}}} {
    # If we've gone over the weight limit, stop now
    if {[weight $chosen] > 400} {
	return
    }
    # If we've considered all of the items (i.e., leaf in search tree)
    # then see if we've got a new best choice.
    if {[llength $items] == 0} {
	global best max
	set v [value $chosen]
	if {$v > $max} {
	    set max $v
	    set best $chosen
	}
	return
    }
    # Branch, so recurse for chosing the current item or not
    set this [lindex $items 0]
    set rest [lrange $items 1 end]
    knapsackSearch $rest $chosen
    for {set i 1} {$i<=[lindex $this 3]} {incr i} {
	knapsackSearch $rest \
	    [concat $chosen [list [lreplace $this end end $i]]]
    }
}

# Initialize a few global variables
set best {}
set max 0
# Do the brute-force search
knapsackSearch $items
# Pretty-print the results
puts "Best filling has weight of [expr {[weight $best]/100.0}]kg and score [value $best]"
puts "Best items:"
foreach {count item} [countednames $best] {
    puts "\t$count * $item"
}
Output:
Best filling has weight of 3.96kg and score 1010
Best items:
	1 * map
	1 * compass
	1 * water
	2 * glucose
	3 * banana
	1 * cheese
	1 * suntan cream
	1 * waterproof overclothes
	1 * note-case
	1 * sunglasses
	1 * socks

Ursala[edit]

Instead of an ad-hoc solution, we can convert this task to a mixed integer programming problem instance and solve it with the lpsolve library, which is callable in Ursala.

#import std
#import flo
#import lin

items = # name: (weight,value,limit)

<
   'map': (9,150,1),
   'compass': (13,35,1),
   'water': (153,200,3),
   'sandwich': (50,60,2),
   'glucose': (15,60,2),
   'tin': (68,45,3),
   'banana': (27,60,3),
   'apple': (39,40,3),
   'cheese': (23,30,1),
   'beer': (52,10,3),
   'suntan cream': (11,70,1),
   'camera': (32,30,1),
   't-shirt': (24,15,2),
   'trousers': (48,10,2),
   'umbrella': (73,40,1),
   'waterproof trousers': (42,70,1),
   'waterproof overclothes': (43,75,1),
   'note-case': (22,80,1),
   'sunglasses': (7,20,1),
   'towel': (18,12,2),
   'socks': (4,50,1),
   'book': (30,10,2)>

system = # convert the item list to mixed integer programming problem specification

linear_system$[
   integers: ~&nS,
   upper_bounds: * ^|/~& float@rr,
   lower_bounds: @nS ~&\*0.+ :/'(slack)',
   costs: * ^|/~& negative+ float@rl,
   equations: ~&iNC\400.+ :/(1.,'(slack)')+ * ^|rlX/~& float@l]

format = @t --^*p\pad` @nS @mS printf/*'%0.0f '

#show+

main =  format solution system items

The linear_system$[...] function takes the item list as an argument and returns a data structure with the following fields, which is passed to the solution function, which calls the lpsolve routines.

  • integers declares that all variables in the list take integer values
  • upper_bounds associates an upper bound for each variable directly as given
  • lower_bounds are zero for all variables in the table, and for an additional slack variable, which is not required to be an integer and won't appear in the solution
  • costs are also taken from the table and negated because their value is to be maximized rather than minimized as in the standard formulation
  • equations specifies the single constraint that the total weights of the selected items in the selected quantities plus the slack equal 400

The format function converts the output list to a readable form.

Output:
3 banana                
1 cheese                
1 compass               
2 glucose               
1 map                   
1 note-case             
1 socks                 
1 sunglasses            
1 suntan cream          
1 water                 
1 waterproof overclothes

Wren[edit]

Translation of: Kotlin
Library: Wren-fmt
import "/fmt" for Fmt

class Item {
    construct new(name, weight, value, count) {
        _name   = name
        _weight = weight
        _value  = value
        _count  = count
    }

    name   { _name   }
    weight { _weight }
    value  { _value  }
    count  { _count  }
}

var items = [
    Item.new("map", 9, 150, 1),
    Item.new("compass", 13, 35, 1),
    Item.new("water", 153, 200, 2),
    Item.new("sandwich", 50, 60, 2),
    Item.new("glucose", 15, 60, 2),
    Item.new("tin", 68, 45, 3),
    Item.new("banana", 27, 60, 3),
    Item.new("apple", 39, 40, 3),
    Item.new("cheese", 23, 30, 1),
    Item.new("beer", 52, 10, 3),
    Item.new("suntan cream", 11, 70, 1),
    Item.new("camera", 32, 30, 1),
    Item.new("T-shirt", 24, 15, 2),
    Item.new("trousers", 48, 10, 2),
    Item.new("umbrella", 73, 40, 1),
    Item.new("waterproof trousers", 42, 70, 1),
    Item.new("waterproof overclothes", 43, 75, 1),
    Item.new("note-case", 22, 80, 1),
    Item.new("sunglasses", 7, 20, 1),
    Item.new("towel", 18, 12, 2),
    Item.new("socks", 4, 50, 1),
    Item.new("book", 30, 10, 2)
]

var n = items.count
var maxWeight = 400

var knapsack = Fn.new { |w|
    var m = List.filled(n + 1, null)
    for (i in 0..n) m[i] = List.filled(w + 1, 0)
    for (i in 1..n) {
        for (j in 0..w) {
            m[i][j] = m[i-1][j]
            for (k in 1..items[i - 1].count) {
                if (k * items[i - 1].weight > j) break
                var v = m[i - 1][j - k * items[i - 1].weight] + k * items[i - 1].value
                if (v > m[i][j]) m[i][j] = v
            }
        }
    }
    var s = List.filled(n, 0)
    var j = w
    for (i in n..1) {
        var v = m[i][j]
        var k = 0
        while (v != m[i - 1][j] + k * items[i - 1].value) {
            s[i - 1] = s[i - 1] + 1
            j = j - items[i - 1].weight
            k = k + 1
        }
    }
    return s
}

var s = knapsack.call(maxWeight)
System.print("Item Chosen             Weight Value  Number")
System.print("---------------------   ------ -----  ------")
var itemCount = 0
var sumWeight = 0
var sumValue  = 0
var sumNumber = 0
for (i in 0... n) {
       if (s[i] != 0) {
           itemCount  = itemCount + 1
           var name   = items[i].name
           var number = s[i]
           var weight = items[i].weight * number
           var value  = items[i].value  * number
           sumNumber  = sumNumber + number
           sumWeight  = sumWeight + weight
           sumValue   = sumValue + value
           Fmt.print("$-22s    $3d   $4d    $2d", name, weight, value, number)
       }
}
System.print("---------------------   ------ -----  ------")
Fmt.print("Items chosen $d           $3d   $4d    $2d", itemCount, sumWeight, sumValue, sumNumber)
Output:
Item Chosen             Weight Value  Number
---------------------   ------ -----  ------
map                         9    150     1
compass                    13     35     1
water                     153    200     1
glucose                    30    120     2
banana                     81    180     3
cheese                     23     30     1
suntan cream               11     70     1
waterproof overclothes     43     75     1
note-case                  22     80     1
sunglasses                  7     20     1
socks                       4     50     1
---------------------   ------ -----  ------
Items chosen 11           396   1010    14

zkl[edit]

Translation of: Haskell
fcn addItem(old,[(nm,w,v,c)]){  // old:list:(cost of:,(name,#,...))
   [1..c].reduce(fcn(list,i,nm,w,v,old){
      wi,left,right:=w*i,list[0,wi],list[wi,*];
      new:=old.apply('wrap([(val,itms)]){ T(val + v*i,itms.append(nm,i)) });
      left.extend(right.zipWith(  // inc
	    fcn([(v1,_)]a,[(v2,_)]b){ v1>v2 and a or b  },new));
   },old,nm,w,v,old);
}//--> new list
items:=T( // item: (name,weight,value,#)
   T("apple",      39, 40,3),T("banana",   27,60,3),
   T("beer",       52, 10,3),T("book",     30,10,2),T("camera",      32, 30,1),
   T("cheese",     23, 30,1),T("compass",  13,35,1),T("glucose",     15, 60,2),
   T("map",         9,150,1),T("note-case",22,80,1),T("sandwich",    50, 60,2),
   T("socks",       4, 50,1),T("sunglasses",7,20,1),T("suntan cream",11, 70,1),
   T("t-shirt",    24, 15,2),T("tin",      68,45,3),T("towel",       18, 12,2),
   T("trousers",   48, 10,2),T("umbrella", 73,40,1),T("water",      153,200,2),
   T("overclothes",43, 75,1),T("waterproof trousers",42,70,1) );
weight:='wrap(knapsack){ // knapsack is (cost, (nm,#,nm,#...))
   knapsack[1].pump(List,Void.Read, // read nm,#, first read is implicit
      'wrap(nm,n){ items[items.filter1n(fcn(it,nm){ it[0]==nm },nm)][1]*n })
      .sum(0)
};
const MAX_WEIGHT=400;
knapsack:=items.reduce(addItem,(MAX_WEIGHT).pump(List,T(0,T).copy))[-1];
knapsack.toString(*).println(weight(knapsack));
Output:
L(1010,L("banana",3,"cheese",1,"compass",1,"glucose",2,"map",1,"note-case",1,"socks",1,"sunglasses",1,"suntan cream",1,"water",1,"overclothes",1))396