Knapsack problem/Continuous
You are encouraged to solve this task according to the task description, using any language you may know.
See also: Knapsack problem and Wikipedia.
A robber burgles a butcher's shop, where he can select from some items. He knows the weights and prices of each items. Because he has a knapsack with 15 kg maximal capacity, he wants to select the items such that he would have his profit maximized. He may cut the items; the item has a reduced price after cutting that is proportional to the original price by the ratio of masses. That means: half of an item has half the price of the original.
This is the item list in the butcher's:
| Item | Weight (kg) | Price (Value) |
|---|---|---|
| beef | 3.8 | 36 |
| pork | 5.4 | 43 |
| ham | 3.6 | 90 |
| greaves | 2.4 | 45 |
| flitch | 4.0 | 30 |
| brawn | 2.5 | 56 |
| welt | 3.7 | 67 |
| salami | 3.0 | 95 |
| sausage | 5.9 | 98 |
| Knapsack | <=15 kg | ? |
Which items does the robber carry in his knapsack so that their total weight does not exceed 15 kg, and their total value is maximised?
Ada
<lang Ada>with Ada.Text_IO; with Ada.Strings.Unbounded;
procedure Knapsack_Continuous is
package US renames Ada.Strings.Unbounded;
type Item is record
Name : US.Unbounded_String;
Weight : Float;
Value : Positive;
Taken : Float;
end record;
function "<" (Left, Right : Item) return Boolean is
begin
return Float (Left.Value) / Left.Weight <
Float (Right.Value) / Right.Weight;
end "<";
type Item_Array is array (Positive range <>) of Item;
function Total_Weight (Items : Item_Array) return Float is
Sum : Float := 0.0;
begin
for I in Items'Range loop
Sum := Sum + Items (I).Weight * Items (I).Taken;
end loop;
return Sum;
end Total_Weight;
function Total_Value (Items : Item_Array) return Float is
Sum : Float := 0.0;
begin
for I in Items'Range loop
Sum := Sum + Float (Items (I).Value) * Items (I).Taken;
end loop;
return Sum;
end Total_Value;
procedure Solve_Knapsack_Continuous
(Items : in out Item_Array;
Weight_Limit : Float)
is
begin
-- order items by value per weight unit
Sorting : declare
An_Item : Item;
J : Natural;
begin
for I in Items'First + 1 .. Items'Last loop
An_Item := Items (I);
J := I - 1;
while J in Items'Range and then Items (J) < An_Item loop
Items (J + 1) := Items (J);
J := J - 1;
end loop;
Items (J + 1) := An_Item;
end loop;
end Sorting;
declare
Rest : Float := Weight_Limit;
begin
for I in Items'Range loop
if Items (I).Weight <= Rest then
Items (I).Taken := Items (I).Weight;
else
Items (I).Taken := Rest;
end if;
Rest := Rest - Items (I).Taken;
exit when Rest <= 0.0;
end loop;
end;
end Solve_Knapsack_Continuous;
All_Items : Item_Array :=
((US.To_Unbounded_String ("beef"), 3.8, 36, 0.0),
(US.To_Unbounded_String ("pork"), 5.4, 43, 0.0),
(US.To_Unbounded_String ("ham"), 3.6, 90, 0.0),
(US.To_Unbounded_String ("greaves"), 2.4, 45, 0.0),
(US.To_Unbounded_String ("flitch"), 4.0, 30, 0.0),
(US.To_Unbounded_String ("brawn"), 2.5, 56, 0.0),
(US.To_Unbounded_String ("welt"), 3.7, 67, 0.0),
(US.To_Unbounded_String ("salami"), 3.0, 95, 0.0),
(US.To_Unbounded_String ("sausage"), 5.9, 98, 0.0));
begin
Solve_Knapsack_Continuous (All_Items, 15.0);
Ada.Text_IO.Put_Line
("Total Weight: " & Float'Image (Total_Weight (All_Items)));
Ada.Text_IO.Put_Line
("Total Value: " & Float'Image (Total_Value (All_Items)));
Ada.Text_IO.Put_Line ("Items:");
for I in All_Items'Range loop
if All_Items (I).Taken > 0.0 then
Ada.Text_IO.Put_Line
(" " &
Float'Image (All_Items (I).Taken) &
" of " &
US.To_String (All_Items (I).Name));
end if;
end loop;
end Knapsack_Continuous;</lang>
C
<lang c>#include <stdio.h>
- include <stdlib.h>
struct item { double w, v; char *name; } items[] = { { 3.8, 36, "beef" }, { 5.4, 43, "pork" }, { 3.6, 90, "ham" }, { 2.4, 45, "greaves" }, { 4.0, 30, "flitch" }, { 2.5, 56, "brawn" }, { 3.7, 67, "welt" }, { 3.0, 95, "salami" }, { 5.9, 98, "sausage" }, };
int item_cmp(const void *aa, const void *bb) { const struct item *a = aa, *b = bb; double ua = a->v / a->w, ub = b->v / b->w; return ua < ub ? -1 : ua > ub; }
int main() { struct item *it; double space = 15;
qsort(items, 9, sizeof(struct item), item_cmp); for (it = items + 9; it---items && space > 0; space -= it->w) if (space >= it->w) printf("take all %s\n", it->name); else printf("take %gkg of %g kg of %s\n", space, it->w, it->name);
return 0;
}</lang>output
take all salami take all ham take all brawn take all greaves take 3.5kg of 3.7 kg of welt
C++
<lang cpp>
- include<iostream>
- include<algorithm>
- include<string.h>
using namespace std; double result; double capacity = 15; int NumberOfItems; int number;
struct items {
char name[32]; double weight; double price; double m;
} item[256];
bool cmp(items a,items b) {
return a.price/a.weight > b.price/b.weight; // the compare function for the sorting algorithm
}
int main() { NumberOfItems=9; strcpy(item[1].name,"beef"); item[1].weight=3.8; item[1].price=36;
strcpy(item[2].name,"pork"); item[2].weight=5.4; item[2].price=43;
strcpy(item[3].name,"ham"); item[3].weight=3.6; item[3].price=90;
strcpy(item[4].name,"greaves"); item[4].weight=2.4; item[4].price=45;
strcpy(item[5].name,"flitch"); item[5].weight=4.0; item[5].price=30;
strcpy(item[6].name,"brawn"); item[6].weight=2.5; item[6].price=56;
strcpy(item[7].name,"welt"); item[7].weight=3.7; item[7].price=67;
strcpy(item[8].name,"salami"); item[8].weight=3.0; item[8].price=95;
strcpy(item[9].name,"sausage"); item[9].weight=5.9; item[9].price=98;
sort(item+1,item+NumberOfItems+1,cmp); // We'll sort using Introsort from STL
number = 1;
while(capacity>0&&number<=NumberOfItems)
{
if(item[number].weight<=capacity)
{
result+=item[number].price;
capacity-=item[number].weight;
item[number].m=1;
}
else
{
result+=(item[number].price)*(capacity/item[number].weight);
item[number].m=(capacity/item[number].weight);
capacity=0;
} ++number; }
cout<<"Total Value = "<<result<<'\n'; cout<<"Total Weight = "<<(double)15-capacity<<'\n'; cout<<"Items Used:\n"; for(int i=1;i<=NumberOfItems;++i)
if(item[i].m)
{
cout<<"We took "<<item[i].m*item[i].weight<<"kg of \""<<item[i].name<<"\" and the value it brought is "<<item[i].price*item[i].m<<"\n";
}
return 0; }
</lang>
D
<lang d>import std.stdio, std.algorithm, std.string, std.conv;
struct Item {
string name; real amount, value;
@property real valuePerKG() const pure nothrow {
return value / amount;
}
string toString() const /*pure nothrow*/ {
return format("%10s %7.2f %7.2f %7.2f",
name, amount, value, valuePerKG);
}
}
real sum(string field)(in Item[] items) pure nothrow {
//return reduce!("a + b." ~ field)(0.0L, items);
return reduce!("a + b." ~ field)(0.0L, cast()items);
}
void main() {
Item[] raw = [{"beef", 3.8, 36.0},
{"pork", 5.4, 43.0},
{"ham", 3.6, 90.0},
{"greaves", 2.4, 45.0},
{"flitch", 4.0, 30.0},
{"brawn", 2.5, 56.0},
{"welt", 3.7, 67.0},
{"salami", 3.0, 95.0},
{"sausage", 5.9, 98.0}];
// reverse sorted by Value per amount
const items = raw.sort!q{a.valuePerKG > b.valuePerKG}().release();
const(Item)[] chosen;
real space = 15.0;
foreach (item; items)
if (item.amount < space) {
chosen ~= item;
space -= item.amount;
} else {
chosen ~= Item(item.name, space, item.valuePerKG * space);
break;
}
writefln("%10s %7s %7s %7s", "ITEM", "AMOUNT", "VALUE", "$/unit");
writefln("%(%s\n%)", chosen);
writeln(Item("TOTAL", sum!"amount"(chosen), sum!"value"(chosen)));
}</lang>
- Output:
ITEM AMOUNT VALUE $/unit
salami 3.00 95.00 31.67
ham 3.60 90.00 25.00
brawn 2.50 56.00 22.40
greaves 2.40 45.00 18.75
welt 3.50 63.38 18.11
TOTAL 15.00 349.38 23.29
Alternative Version
<lang d>import std.stdio, std.algorithm;
void main() {
static struct T { string item; double weight, price; }
auto items = [T("beef", 3.8, 36.0),
T("pork", 5.4, 43.0),
T("ham", 3.6, 90.0),
T("greaves", 2.4, 45.0),
T("flitch", 4.0, 30.0),
T("brawn", 2.5, 56.0),
T("welt", 3.7, 67.0),
T("salami", 3.0, 95.0),
T("sausage", 5.9, 98.0)];
sort!q{a.price/a.weight > b.price/b.weight}(items);
auto left = 15.0;
foreach (it; items) {
if (it.weight <= left) {
writeln("Take all the ", it.item);
if (it.weight == left)
return;
left -= it.weight;
} else {
writefln("Take %.1fkg %s", left, it.item);
return;
}
}
}</lang>
- Output:
Take all the salami Take all the ham Take all the brawn Take all the greaves Take 3.5kg welt
Fortran
<lang fortran>program KNAPSACK_CONTINUOUS
implicit none real, parameter :: maxweight = 15.0 real :: total_weight = 0, total_value = 0, frac integer :: i, j type Item character(7) :: name real :: weight real :: value end type Item
type(Item) :: items(9), temp
items(1) = Item("beef", 3.8, 36.0)
items(2) = Item("pork", 5.4, 43.0)
items(3) = Item("ham", 3.6, 90.0)
items(4) = Item("greaves", 2.4, 45.0)
items(5) = Item("flitch", 4.0, 30.0)
items(6) = Item("brawn", 2.5, 56.0)
items(7) = Item("welt", 3.7, 67.0)
items(8) = Item("salami", 3.0, 95.0)
items(9) = Item("sausage", 5.9, 98.0)
! sort items in desending order of their value per unit weight
do i = 2, size(items)
j = i - 1
temp = items(i)
do while (j>=1 .and. items(j)%value / items(j)%weight < temp%value / temp%weight)
items(j+1) = items(j)
j = j - 1
end do
items(j+1) = temp
end do
i = 0
write(*, "(a4, a13, a6)") "Item", "Weight", "Value"
do while(i < size(items) .and. total_weight < maxweight)
i = i + 1
if(total_weight+items(i)%weight < maxweight) then
total_weight = total_weight + items(i)%weight
total_value = total_value + items(i)%value
write(*, "(a7, 2f8.2)") items(i)
else
frac = (maxweight-total_weight) / items(i)%weight
total_weight = total_weight + items(i)%weight * frac
total_value = total_value + items(i)%value * frac
write(*, "(a7, 2f8.2)") items(i)%name, items(i)%weight * frac, items(i)%value * frac
end if
end do
write(*, "(f15.2, f8.2)") total_weight, total_value
end program KNAPSACK_CONTINUOUS</lang>
Go
<lang go>package main
import (
"fmt" "sort"
)
type item struct {
item string weight float64 price float64
}
type items []item
var all = items{
{"beef", 3.8, 36},
{"pork", 5.4, 43},
{"ham", 3.6, 90},
{"greaves", 2.4, 45},
{"flitch", 4.0, 30},
{"brawn", 2.5, 56},
{"welt", 3.7, 67},
{"salami", 3.0, 95},
{"sausage", 5.9, 98},
}
// satisfy sort interface func (z items) Len() int { return len(z) } func (z items) Swap(i, j int) { z[i], z[j] = z[j], z[i] } func (z items) Less(i, j int) bool {
return z[i].price/z[i].weight > z[j].price/z[j].weight
}
func main() {
left := 15.
sort.Sort(all)
for _, i := range all {
if i.weight <= left {
fmt.Println("take all the", i.item)
if i.weight == left {
return
}
left -= i.weight
} else {
fmt.Printf("take %.1fkg %s\n", left, i.item)
return
}
}
}</lang> Output:
take all the salami take all the ham take all the brawn take all the greaves take 3.5kg welt
Groovy
Solution: obvious greedy algorithm <lang groovy>import static java.math.RoundingMode.*
def knapsackCont = { list, maxWeight = 15.0 ->
list.sort{ it.weight / it.value }
def remainder = maxWeight
List sack = []
for (item in list) {
if (item.weight < remainder) {
sack << [name: item.name, weight: item.weight,
value: (item.value as BigDecimal).setScale(2, HALF_UP)]
} else {
sack << [name: item.name, weight: remainder,
value: (item.value * remainder / item.weight).setScale(2, HALF_UP)]
break
}
remainder -= item.weight
}
sack
}</lang>
Test: <lang groovy>def possibleItems = [
[name:'beef', weight:3.8, value:36], [name:'pork', weight:5.4, value:43], [name:'ham', weight:3.6, value:90], [name:'greaves', weight:2.4, value:45], [name:'flitch', weight:4.0, value:30], [name:'brawn', weight:2.5, value:56], [name:'welt', weight:3.7, value:67], [name:'salami', weight:3.0, value:95], [name:'sausage', weight:5.9, value:98],
]
def contents = knapsackCont(possibleItems) println "Total Value: ${contents*.value.sum()}" contents.each {
printf(" name: %-7s weight: ${it.weight} value: ${it.value}\n", it.name)
}</lang>
Output:
Total Value: 349.38
name: salami weight: 3.0 value: 95.00
name: ham weight: 3.6 value: 90.00
name: brawn weight: 2.5 value: 56.00
name: greaves weight: 2.4 value: 45.00
name: welt weight: 3.5 value: 63.38
Haskell
We use a greedy algorithm.
<lang haskell>import Control.Monad import Data.List (sortBy) import Data.Ord (comparing) import Data.Ratio (numerator, denominator) import Text.Printf
maxWgt = 15
data Bounty = Bounty
{itemName :: String,
itemVal, itemWgt :: Rational}
items =
[Bounty "beef" 36 3.8, Bounty "pork" 43 5.4, Bounty "ham" 90 3.6, Bounty "greaves" 45 2.4, Bounty "flitch" 30 4.0, Bounty "brawn" 56 2.5, Bounty "welt" 67 3.7, Bounty "salami" 95 3.0, Bounty "sausage" 98 5.9]
solution :: [(Rational, Bounty)] solution = g maxWgt $ sortBy (flip $ comparing f) items
where g room (b@(Bounty _ _ w) : bs) = if w < room
then (w, b) : g (room - w) bs
else [(room, b)]
f (Bounty _ v w) = v / w
main = do
forM_ solution $ \(w, b) ->
printf "%s kg of %s\n" (mixedNum w) (itemName b)
printf "Total value: %s\n" $ mixedNum $ sum $ map f solution
where f (w, Bounty _ v wtot) = v * (w / wtot)
mixedNum q = if b == 0
then show a
else printf "%d %d/%d" a (numerator b) (denominator b)
where a = floor q
b = q - toEnum a</lang>
Or similar to above (but more succinct): <lang haskell> import Data.List (sortBy) import Data.Ord (comparing) import Text.Printf (printf)
-- (name, (value, weight)) items = [("beef", (36, 3.8)),
("pork", (43, 5.4)),
("ham", (90, 3.6)),
("greaves", (45, 2.4)),
("flitch", (30, 4.0)),
("brawn", (56, 2.5)),
("welt", (67, 3.7)),
("salami", (95, 3.0)),
("sausage", (98, 5.9))]
unitWeight (_, (val, weight)) = (fromIntegral val) / weight
solution k = loop k . sortBy (flip $ comparing unitWeight)
where loop k ((name, (_, weight)):xs)
| weight < k = putStrLn ("Take all the " ++ name) >> loop (k-weight) xs
| otherwise = printf "Take %.2f kg of the %s\n" (k :: Float) name
main = solution 15 items </lang>
Icon and Unicon
This implements the greedy algorithm. This also uses a Unicon extension to reverse which reverses a list. In Icon, an IPL procedure is available to do the same. <lang Icon>link printf
procedure main() room := 15 every (x := !(choices := get_items())).uprice := x.price / x.weight choices := reverse(sortf(choices,4))
every (value := 0, x := !choices) do {
if x.weight <= room then {
printf("Take all of the %s (%r kg) worth $%r\n",x.name,x.weight,x.price)
value +:= x.price
room -:= x.weight
}
else {
fvalue := x.uprice * room
printf("Take (%r kg) of the %s worth $%r\n",room,x.name,fvalue)
value +:= fvalue
break
}
} printf("Total value of a full knapsack is $%r\n",value) end
record item(name,weight,price,uprice)
procedure get_items()
return [ item("beef", 3.8, 36),
item("pork", 5.4, 43),
item("ham", 3.6, 90),
item("greaves", 2.4, 45),
item("flitch", 4.0, 30),
item("brawn", 2.5, 56),
item("welt", 3.7, 67),
item("salami", 3.0, 95),
item("sausage", 5.9, 98) ]
end</lang>
Output:
Take all of the salami (3.000000 kg) worth $95.000000 Take all of the ham (3.600000 kg) worth $90.000000 Take all of the brawn (2.500000 kg) worth $56.000000 Take all of the greaves (2.400000 kg) worth $45.000000 Take (3.500000 kg) of the welt worth $63.378378 Total value of a full knapsack is $349.378378
J
We take as much as we can of the most valuable items first, and continue until we run out of space. Only one item needs to be cut.
<lang J>'names numbers'=:|:;:;._2]0 :0 beef 3.8 36 pork 5.4 43 ham 3.6 90 greaves 2.4 45 flitch 4.0 30 brawn 2.5 56 welt 3.7 67 salami 3.0 95 sausage 5.9 98 ) 'weights prices'=:|:".numbers order=: \:prices%weights take=: 15&<.&.(+/\) order{weights result=: (*take)#(order{names),.' ',.":,.take</lang>
This gives the result:
salami 3 ham 3.6 brawn 2.5 greaves 2.4 welt 3.5
For a total value of: <lang J> +/prices * (take/:order) % weights 349.378</lang>
Java
Greedy solution.
<lang java> package hu.pj.alg.test;
import hu.pj.alg.ContinuousKnapsack; import hu.pj.obj.Item; import java.util.*; import java.text.*;
public class ContinousKnapsackForRobber {
final private double tolerance = 0.0005;
public ContinousKnapsackForRobber() {
ContinuousKnapsack cok = new ContinuousKnapsack(15); // 15 kg
// making the list of items that you want to bring
cok.add("beef", 3.8, 36); // marhahús
cok.add("pork", 5.4, 43); // disznóhús
cok.add("ham", 3.6, 90); // sonka
cok.add("greaves", 2.4, 45); // tepertő
cok.add("flitch", 4.0, 30); // oldalas
cok.add("brawn", 2.5, 56); // disznósajt
cok.add("welt", 3.7, 67); // hurka
cok.add("salami", 3.0, 95); // szalámi
cok.add("sausage", 5.9, 98); // kolbász
// calculate the solution:
List<Item> itemList = cok.calcSolution();
// write out the solution in the standard output
if (cok.isCalculated()) {
NumberFormat nf = NumberFormat.getInstance();
System.out.println(
"Maximal weight = " +
nf.format(cok.getMaxWeight()) + " kg"
);
System.out.println(
"Total weight of solution = " +
nf.format(cok.getSolutionWeight()) + " kg"
);
System.out.println(
"Total value (profit) = " +
nf.format(cok.getProfit())
);
System.out.println();
System.out.println(
"You can carry the following materials " +
"in the knapsack:"
);
for (Item item : itemList) {
if (item.getInKnapsack() > tolerance) {
System.out.format(
"%1$-10s %2$-15s %3$-15s \n",
nf.format(item.getInKnapsack()) + " kg ",
item.getName(),
"(value = " + nf.format(item.getInKnapsack() *
(item.getValue() / item.getWeight())) + ")"
);
}
}
} else {
System.out.println(
"The problem is not solved. " +
"Maybe you gave wrong data."
);
}
}
public static void main(String[] args) {
new ContinousKnapsackForRobber();
}
} // class</lang>
<lang java> package hu.pj.alg;
import hu.pj.obj.Item; import java.util.*;
public class ContinuousKnapsack {
protected List<Item> itemList = new ArrayList<Item>(); protected double maxWeight = 0; protected double solutionWeight = 0; protected double profit = 0; protected boolean calculated = false;
public ContinuousKnapsack() {}
public ContinuousKnapsack(double _maxWeight) {
setMaxWeight(_maxWeight);
}
public List<Item> calcSolution() {
int n = itemList.size();
setInitialStateForCalculation();
if (n > 0 && maxWeight > 0) {
Collections.sort(itemList);
for (int i = 0; (maxWeight - solutionWeight) > 0.0 && i < n; i++) {
Item item = itemList.get(i);
if (item.getWeight() >= (maxWeight - solutionWeight)) {
item.setInKnapsack(maxWeight - solutionWeight);
solutionWeight = maxWeight;
profit += item.getInKnapsack() / item.getWeight() * item.getValue();
break;
} else {
item.setInKnapsack(item.getWeight());
solutionWeight += item.getInKnapsack();
profit += item.getValue();
}
}
calculated = true;
}
return itemList;
}
// add an item to the item list
public void add(String name, double weight, double value) {
if (name.equals(""))
name = "" + (itemList.size() + 1);
itemList.add(new Item(name, weight, value));
setInitialStateForCalculation();
}
public double getMaxWeight() {return maxWeight;}
public double getProfit() {return profit;}
public double getSolutionWeight() {return solutionWeight;}
public boolean isCalculated() {return calculated;}
public void setMaxWeight(double _maxWeight) {
maxWeight = Math.max(_maxWeight, 0);
}
// set the member with name "inKnapsack" by all items:
private void setInKnapsackByAll(double inKnapsack) {
for (Item item : itemList)
item.setInKnapsack(inKnapsack);
}
// set the data members of class in the state of starting the calculation:
protected void setInitialStateForCalculation() {
setInKnapsackByAll(-0.0001);
calculated = false;
profit = 0.0;
solutionWeight = 0.0;
}
} // class</lang>
<lang java> package hu.pj.obj;
public class Item implements Comparable {
protected String name = ""; protected double weight = 0; protected double value = 0; protected double inKnapsack = 0; // the weight of item in solution
public Item() {}
public Item(Item item) {
setName(item.name);
setWeight(item.weight);
setValue(item.value);
}
public Item(double _weight, double _value) {
setWeight(_weight);
setValue(_value);
}
public Item(String _name, double _weight, double _value) {
setName(_name);
setWeight(_weight);
setValue(_value);
}
public void setName(String _name) {name = _name;}
public void setWeight(double _weight) {weight = Math.max(_weight, 0);}
public void setValue(double _value) {value = Math.max(_value, 0);}
public void setInKnapsack(double _inKnapsack) {
inKnapsack = Math.max(_inKnapsack, 0);
}
public void checkMembers() {
setWeight(weight);
setValue(value);
setInKnapsack(inKnapsack);
}
public String getName() {return name;}
public double getWeight() {return weight;}
public double getValue() {return value;}
public double getInKnapsack() {return inKnapsack;}
// implementing of Comparable interface:
public int compareTo(Object item) {
int result = 0;
Item i2 = (Item)item;
double rate1 = value / weight;
double rate2 = i2.value / i2.weight;
if (rate1 > rate2) result = -1; // if greater, put it previously
else if (rate1 < rate2) result = 1;
return result;
}
} // class</lang>
output:
Maximal weight = 15 kg Total weight of solution = 15 kg Total value (profit) = 349,378 You can carry the following materials in the knapsack: 3 kg salami (value = 95) 3,6 kg ham (value = 90) 2,5 kg brawn (value = 56) 2,4 kg greaves (value = 45) 3,5 kg welt (value = 63,378)
Mathematica
The problem is solved by sorting the original table by price to weight ratio, finding the accumlated weight, and the index of the item which exedes the carrying capacity (overN) The output is then all items prior to this one, along with that item corrected for it's excess weighter (overW)
<lang Mathematica>Knapsack[shop_, capacity_] := Block[{sortedTable, overN, overW, output},
sortedTable = SortBy[{#1, #2, #3, #3/#2} & @@@ shop, -#4 &];
overN = Position[Accumulate[sortedTable1 ;;, 2], a_ /; a > capacity, 1,1]1, 1;
overW = Accumulate[sortedTable1 ;;, 2]overN - capacity;
output = Reverse@sortedTable[[Ordering[sortedTable1 ;;, 4, -overN]]];
output-1, 2 = output-1, 2 - overW;
output-1, 3 = output-1, 2 output-1, 4;
Append[output1 ;;, 1 ;; 3, {"Total",Sequence @@ Total[output1 ;;, 2 ;; 3]}]]</lang>
A test using the specified data: <lang Mathematica>weightPriceTable = {{"beef", 3.8, 36}, {"pork", 5.4, 43}, {"ham", 3.6, 90}, {"greaves", 2.4, 45}, {"flitch", 4., 30}, {"brawn", 2.5, 56}, {"welt", 3.7, 67}, {"salami", 3., 95}, {"sausage", 5.9, 98}}; carryCapacity = 15; Knapsack[weightPriceTable, carryCapacity] // Grid
salami 3. 95 ham 3.6 90 brawn 2.5 56 greaves 2.4 45 welt 3.5 63.3784 Total 15. 349.378 </lang>
Mathprog
<lang>/*Knapsack
This model finds the optimal packing of a knapsack Nigel_Galloway January 10th., 2012
- /
set Items; param weight{t in Items}; param value{t in Items};
var take{t in Items}, >=0, <=weight[t];
knap_weight : sum{t in Items} take[t] <= 15;
maximize knap_value: sum{t in Items} take[t] * (value[t]/weight[t]);
data;
param : Items : weight value :=
beef 3.8 36
pork 5.4 43
ham 3.6 90
greaves 2.4 45
flitch 4.0 30
brawn 2.5 56
welt 3.7 67
salami 3.0 95
sausage 5.9 98
;
end;</lang>
The solution is here at Knapsack problem/Continuous/Mathprog.
OCaml
<lang ocaml>let items =
[ "beef", 3.8, 36; "pork", 5.4, 43; "ham", 3.6, 90; "greaves", 2.4, 45; "flitch", 4.0, 30; "brawn", 2.5, 56; "welt", 3.7, 67; "salami", 3.0, 95; "sausage", 5.9, 98; ]
let () =
let items = List.map (fun (name, w, p) -> (name, w, p, float p /. w)) items in
let items = List.sort (fun (_,_,_,v1) (_,_,_,v2) -> compare v2 v1) items in
let rec loop acc weight = function
| ((_,w,_,_) as item) :: tl ->
if w +. weight > 15.0
then (weight, acc, item)
else loop (item::acc) (w +. weight) tl
| [] -> assert false
in
let weight, res, (last,w,p,v) = loop [] 0.0 items in
print_endline " Items Weight Price";
let price =
List.fold_left (fun price (name,w,p,_) ->
Printf.printf " %7s: %6.2f %3d\n" name w p;
(p + price)
) 0 res
in
let rem_weight = 15.0 -. weight in
let last_price = v *. rem_weight in
Printf.printf " %7s: %6.2f %6.2f\n" last rem_weight last_price;
Printf.printf " Total Price: %.3f\n" (float price +. last_price);
- </lang>
Perl
<lang perl>my @items = sort { $b->[2]/$b->[1] <=> $a->[2]/$a->[1] } (
[qw'beef 3.8 36'],
[qw'pork 5.4 43'],
[qw'ham 3.6 90'],
[qw'greaves 2.4 45'],
[qw'flitch 4.0 30'],
[qw'brawn 2.5 56'],
[qw'welt 3.7 67'],
[qw'salami 3.0 95'],
[qw'sausage 5.9 98'],
);
my ($limit, $value) = (15, 0);
print "item fraction weight value\n"; for (@items) {
my $ratio = $_->[1] > $limit ? $limit/$_->[1] : 1;
print "$_->[0]\t";
$value += $_->[2] * $ratio;
$limit -= $_->[1];
if ($ratio == 1) {
print " all\t$_->[1]\t$_->[2]\n";
} else {
printf "%5.3f %s %8.3f\n", $ratio, $_->[1] * $ratio, $_->[2] * $ratio;
last;
}
}
print "-" x 40, "\ntotal value: $value\n"; </lang>
Output:
item fraction weight value salami all 3.0 95 ham all 3.6 90 brawn all 2.5 56 greaves all 2.4 45 welt 0.946 3.5 63.378 ---------------------------------------- total value: 349.378378378378
Perl 6
This Solutions sorts the item by WEIGHT/VALUE <lang perl6>class KnapsackItem {
has $.name; has $.weight is rw; has $.price is rw; has $.ppw;
method new (Str $n, $w, $p) {
KnapsackItem.bless(*, :name($n), :weight($w), :price($p), :ppw($w/$p))
}
method cut-maybe ($max-weight) {
return False if $max-weight > $.weight;
$.price = $max-weight / $.ppw;
$.weight = $.ppw * $.price;
return True;
}
method Str () { sprintf "%8s %1.2f %3.2f",
$.name,
$.weight,
$.price }
}
my $max-w = 15; say "Item Portion Value";
.say for gather
for < beef 3.8 36
pork 5.4 43
ham 3.6 90
greaves 2.4 45
flitch 4.0 30
brawn 2.5 56
welt 3.7 67
salami 3.0 95
sausage 5.9 98 >
==> map { KnapsackItem.new($^a, $^b, $^c) }
==> sort *.ppw
{
my $last-one = .cut-maybe($max-w);
take $_;
$max-w -= .weight;
last if $last-one;
}</lang>
Output:
%perl6 knapsack_continous.p6
Item Portion Value
salami 3.00 95.00
ham 3.60 90.00
brawn 2.50 56.00
greaves 2.40 45.00
welt 3.50 63.38
PicoLisp
<lang PicoLisp>(scl 2)
(de *Items
("beef" 3.8 36.0)
("pork" 5.4 43.0)
("ham" 3.6 90.0)
("greaves" 2.4 45.0)
("flitch" 4.0 30.0)
("brawn" 2.5 56.0)
("welt" 3.7 67.0)
("salami" 3.0 95.0)
("sausage" 5.9 98.0) )
(let K
(make
(let Weight 0
(for I (by '((L) (*/ (caddr L) -1.0 (cadr L))) sort *Items)
(T (= Weight 15.0))
(inc 'Weight (cadr I))
(T (> Weight 15.0)
(let W (- (cadr I) Weight -15.0)
(link (list (car I) W (*/ W (caddr I) (cadr I)))) ) )
(link I) ) ) )
(for I K
(tab (3 -9 8 8)
NIL
(car I)
(format (cadr I) *Scl)
(format (caddr I) *Scl) ) )
(tab (12 8 8)
NIL
(format (sum cadr K) *Scl)
(format (sum caddr K) *Scl) ) )</lang>
Output:
salami 3.00 95.00
ham 3.60 90.00
brawn 2.50 56.00
greaves 2.40 45.00
welt 3.50 63.38
15.00 349.38
Prolog
Works with SWI-Prolog and library(simplex) written by Markus Triska <lang Prolog>:- use_module(library(simplex)). % tuples (name, weights, value). knapsack :- L = [( beef, 3.8, 36), ( pork, 5.4, 43), ( ham, 3.6, 90), ( greaves, 2.4, 45), ( flitch, 4.0, 30), ( brawn, 2.5, 56), ( welt, 3.7, 67), ( salami, 3.0, 95), ( sausage, 5.9, 98)],
gen_state(S0), length(L, N), numlist(1, N, LN), ( ( create_constraint_N(LN, L, S0, S1, [], LW, [], LV), constraint(LW =< 15.0, S1, S2), maximize(LV, S2, S3) )), compute_lenword(L, 0, Len), sformat(A1, '~~w~~t~~~w|', [Len]), sformat(A2, '~~t~~2f~~~w|', [10]), sformat(A3, '~~t~~2f~~~w|', [10]), print_results(S3, A1,A2,A3, L, LN, 0, 0).
create_constraint_N([], [], S, S, LW, LW, LV, LV).
create_constraint_N([HN|TN], [(_, W, V) | TL], S1, SF, LW, LWF, LV, LVF) :- constraint([x(HN)] >= 0, S1, S2), constraint([x(HN)] =< W, S2, S3), X is V/W, create_constraint_N(TN, TL, S3, SF, [x(HN) | LW], LWF, [X * x(HN) | LV], LVF).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
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(W1, A1, [' ']), sformat(W2, A2, [WM]), sformat(W3, A3, [VM]), format('~w~w~w~n', [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(S1, A1, [Name]),
sformat(S2, A2, [X]),
Vtemp is X * V/W,
sformat(S3, A3, [Vtemp]),
format('~w~w~w~n', [S1,S2,S3]),
W2 is W1 + X,
V2 is V1 + Vtemp ),
print_results(S, A1, A2, A3, T, TN, W2, V2).
</lang> Output :
?- knapsack.
ham 3.60 90.00
greaves 2.40 45.00
brawn 2.50 56.00
welt 3.50 63.38
salami 3.00 95.00
15.00 349.38
true .
PureBasic
Using the greedy algorithm. <lang PureBasic>Structure item
name.s weight.f ;units are kilograms (kg) Value.f vDensity.f ;the density of the value, i.e. value/weight, and yes I made up the term ;)
EndStructure
- maxWeight = 15
Global itemCount = 0 ;this will be increased as needed to match actual count Global Dim items.item(itemCount)
Procedure addItem(name.s, weight.f, Value.f)
If itemCount >= ArraySize(items())
Redim items.item(itemCount + 10)
EndIf
With items(itemCount)
\name = name
\weight = weight
\Value = Value
If Not \weight
\vDensity = \Value
Else
\vDensity = \Value / \weight
EndIf
EndWith
itemCount + 1
EndProcedure
- build item list
addItem("beef", 3.8, 36) addItem("pork", 5.4, 43) addItem("ham", 3.6, 90) addItem("greaves", 2.4, 45) addItem("flitch", 4.0, 30) addItem("brawn", 2.5, 56) addItem("welt", 3.7, 67) addItem("salami", 3.0, 95) addItem("sausage", 5.9, 98) SortStructuredArray(items(), #PB_Sort_descending, OffsetOf(item\vDensity), #PB_Sort_Float, 0, itemCount - 1)
Define TotalWeight.f, TotalValue.f, i NewList knapsack.item() For i = 0 To itemCount
If TotalWeight + items(i)\weight < #maxWeight AddElement(knapsack()) knapsack() = items(i) TotalWeight + items(i)\weight TotalValue + items(i)\Value Else AddElement(knapsack()) knapsack() = items(i) knapsack()\weight = #maxWeight - TotalWeight knapsack()\Value = knapsack()\weight * knapsack()\vDensity TotalWeight = #maxWeight TotalValue + knapsack()\Value Break EndIf
Next
If OpenConsole()
PrintN(LSet("Maximal weight", 26, " ") + "= " + Str(#maxWeight) + " kg")
PrintN(LSet("Total weight of solution", 26, " ") + "= " + Str(#maxWeight) + " kg")
PrintN(LSet("Total value", 26, " ") + "= " + StrF(TotalValue, 3) + " " + #CRLF$)
PrintN("You can carry the following materials in the knapsack: ")
ForEach knapsack()
PrintN(RSet(StrF(knapsack()\weight, 1), 5, " ") + " kg " + LSet(knapsack()\name, 10, " ") + " (Value = " + StrF(knapsack()\Value, 3) + ")")
Next
Print(#CRLF$ + #CRLF$ + "Press ENTER to exit")
Input()
CloseConsole()
EndIf </lang> Sample output:
Maximal weight = 15 kg Total weight of solution = 15 kg Total value = 349.378 You can carry the following materials in the knapsack: 3.0 kg salami (Value = 95.000) 3.6 kg ham (Value = 90.000) 2.5 kg brawn (Value = 56.000) 2.4 kg greaves (Value = 45.000) 3.5 kg welt (Value = 63.378)
Python
I think this greedy algorithm of taking the largest amounts of items ordered by their value per unit weight is maximal: <lang python># NAME, WEIGHT, VALUE (for this weight) items = [("beef", 3.8, 36.0),
("pork", 5.4, 43.0),
("ham", 3.6, 90.0),
("greaves", 2.4, 45.0),
("flitch", 4.0, 30.0),
("brawn", 2.5, 56.0),
("welt", 3.7, 67.0),
("salami", 3.0, 95.0),
("sausage", 5.9, 98.0)]
MAXWT = 15.0
sorted_items = sorted(((value/amount, amount, name)
for name, amount, value in items),
reverse = True)
wt = val = 0 bagged = [] for unit_value, amount, name in sorted_items:
portion = min(MAXWT - wt, amount)
wt += portion
addval = portion * unit_value
val += addval
bagged += [(name, portion, addval)]
if wt >= MAXWT:
break
print(" ITEM PORTION VALUE") print("\n".join("%10s %6.2f %6.2f" % item for item in bagged)) print("\nTOTAL WEIGHT: %5.2f\nTOTAL VALUE: %5.2f" % (wt, val))</lang>
Sample Output
ITEM PORTION VALUE
salami 3.00 95.00
ham 3.60 90.00
brawn 2.50 56.00
greaves 2.40 45.00
welt 3.50 63.38
TOTAL WEIGHT: 15.00
TOTAL VALUE: 349.38
REXX
Any resemblence to the Fortran code is 120% coincidental. <lang rexx> /*REXX program to solve the burglar's knapsack (continuous) problem. */
@.=
/* name weight value */
@.1='flitch 4 30 ' @.2='beef 3.8 36 ' @.3='pork 5.4 43 ' @.4='greaves 2.4 45 ' @.5='brawn 2.5 56 ' @.6='welt 3.7 67 ' @.7='ham 3.6 90 ' @.8='salami 3 95 ' @.9='sausage 5.9 98 '
nL=length('total weight'); wL=length('weight'); vL=length(' value ') totW=0; totV=0
do j=1 while @.j\== parse var @.j n w v . nL=max(nL,length(n)); n.j=n totW=totW+w ; w.j=w totV=totV+v ; v.j=v end
items=j-1 /*items is the number of items. */ nL=nL+nL%4 /*nL: max length name + 25%. */ wL=max(wL,length(format(totw,,2))) /*wL: max formatted weight width*/ vL=max(vL,length(format(totv,,2))) /*vL: max formatted value width*/ totW=0; totV=0 call show 'before sorting'
/*sort items by (desending) value per unit weight.*/
do j=2 to items k=j-1; _n=n.j; _w=w.j; _v=v.j
do k=k by -1 to 1 while v.k/w.k<_v/_w
kp1=k+1; n.kp1=n.k; w.kp1=w.k; v.kp1=v.k
end
kp1=k+1; n.kp1=_n; w.kp1=_w; v.kp1=_v end /*j*/
call show 'after sorting' call hdr "burgler's knapsack contents" maxW=15 /*burgler's knapsack max weight. */
do j=1 for items while totW<maxW
if totW+w.j<maxW then do
totW=totW + w.j
totV=totV + v.j
call syf n.j, w.j, v.j
end
else do
f=(maxW-totW)/w.j
totW=totW + w.j*f
totV=totV + v.j*f
call syf n.j, w.j*f, v.j*f
end
end /*j*/
call sep call sy left('total weight',nL,'-'), format(totW,,2) call sy left('total value',nL,'-'), , format(totV,,2) exit
/*─────────────────────────────────────one-liner subroutines────────────*/
hdr: indent=left(,5); call verse arg(1); call title; call sep; return
sep: call sy copies('=',nL),copies("=",wL),copies('=',vL); return
show: call hdr arg(1); do j=1 for items; call syf n.j,w.j,v.j;end; say; return
sy: say indent left(arg(1),nL) right(arg(2),wL) right(arg(3),vL); return
syf: call sy arg(1),format(arg(2),,2),format(arg(3),,2); return
title: call sy center('item',nL),center("weight",wL),center('value',vL); return
verse: say; say; say center(arg(1),40,'─'); say; return
</lang>
Output:
─────────────before sorting─────────────
item weight value
=============== ====== =======
flitch 4.00 30.00
beef 3.80 36.00
pork 5.40 43.00
greaves 2.40 45.00
brawn 2.50 56.00
welt 3.70 67.00
ham 3.60 90.00
salami 3.00 95.00
sausage 5.90 98.00
─────────────after sorting──────────────
item weight value
=============== ====== =======
salami 3.00 95.00
ham 3.60 90.00
brawn 2.50 56.00
greaves 2.40 45.00
welt 3.70 67.00
sausage 5.90 98.00
beef 3.80 36.00
pork 5.40 43.00
flitch 4.00 30.00
──────burgler's knapsack contents───────
item weight value
=============== ====== =======
salami 3.00 95.00
ham 3.60 90.00
brawn 2.50 56.00
greaves 2.40 45.00
welt 3.50 63.38
=============== ====== =======
total weight--- 15.00
total value--- 349.38
Run BASIC
<lang runbasic>dim name$(9) dim wgt(9) dim price(9) dim tak$(100)
name$(1) = "beef" : wgt(1) = 3.8 : price(1) = 36 name$(2) = "pork" : wgt(2) = 5.4 : price(2) = 43 name$(3) = "ham" : wgt(3) = 3.6 : price(3) = 90 name$(4) = "greaves" : wgt(4) = 2.4 : price(4) = 45 name$(5) = "flitch" : wgt(5) = 4.0 : price(5) = 30 name$(6) = "brawn" : wgt(6) = 2.5 : price(6) = 56 name$(7) = "welt" : wgt(7) = 3.7 : price(7) = 67 name$(8) = "salami" : wgt(8) = 3.0 : price(8) = 95 name$(9) = "sausage" : wgt(9) = 5.9 : price(9) = 98
for beef = 0 to 15 step 3.8
for pork = 0 to 15 step 5.4
for ham = 0 to 15 step 3.6
for greaves = 0 to 15 step 2.4
for flitch = 0 to 15 step 4.0
for brawn = 0 to 15 step 2.5
for welt = 0 to 15 step 3.7
for salami = 0 to 15 step 3.0
for sausage = 0 to 15 step 5.9
if beef + pork + ham + greaves + flitch + brawn + welt + salami + sausage <= 15 then
totPrice = beef / 3.8 * 36 + _
pork / 5.4 * 43 + _
ham / 3.6 * 90 + _
greaves / 2.4 * 45 + _
flitch / 4.0 * 30 + _
brawn / 2.5 * 56 + _
welt / 3.7 * 67 + _
salami / 3.0 * 95 + _
sausage / 5.9 * 98
if totPrice >= maxPrice then
maxPrice = totPrice
theMax = max(totPrice,maxPrice)
t = t + 1
tak$(t) = str$(maxPrice);",";beef;",";pork;",";ham;",";greaves;",";flitch;",";brawn;",";welt;",";salami;",";sausage
end if
end if
next:next :next :next :next :next :next :next :next
print "Best 2 Options":print for i = t-1 to t
totTake = val(word$(tak$(i),1,","))
if totTake > 0 then
totWgt = 0
for j = 2 to 10
wgt = val(word$(tak$(i),j,","))
totWgt = totWgt + wgt
value = wgt / wgt(j - 1) * price(j - 1)
if wgt <> 0 then print name$(j-1);chr$(9);"Value: ";using("###.#",value);chr$(9);"Weight: ";using("##.#",wgt)
next j
print "-------- Total ";using("###.#",totTake);chr$(9);"Weight: ";totWgt
end if
next i</lang>Output:
Best 2 Options salami Value: 285.0 Weight: 9.0 sausage Value: 98.0 Weight: 5.9 -------- Total 383.0 Weight: 14.9 salami Value: 475.0 Weight: 15.0 -------- Total 475.0 Weight: 15.0
Tcl
<lang tcl>package require Tcl 8.5
- Uses the trivial greedy algorithm
proc continuousKnapsack {items massLimit} {
# Add in the unit prices
set idx -1
foreach item $items {
lassign $item name mass value lappend item [expr {$value / $mass}] lset items [incr idx] $item
}
# Sort by unit prices set items [lsort -decreasing -real -index 3 $items]
# Add items, using most valuable-per-unit first
set result {}
set total 0.0
set totalValue 0
foreach item $items {
lassign $item name mass value unit if {$total + $mass < $massLimit} { lappend result [list $name $mass $value] set total [expr {$total + $mass}] set totalValue [expr {$totalValue + $value}] } else { set mass [expr {$massLimit - $total}] set value [expr {$unit * $mass}] lappend result [list $name $mass $value] set totalValue [expr {$totalValue + $value}] break }
}
# We return the total value too, purely for convenience return [list $result $totalValue]
}</lang> Driver for this particular problem: <lang tcl>set items {
{beef 3.8 36}
{pork 5.4 43}
{ham 3.6 90}
{greaves 2.4 45}
{flitch 4.0 30}
{brawn 2.5 56}
{welt 3.7 67}
{salami 3.0 95}
{sausage 5.9 98}
}
lassign [continuousKnapsack $items 15.0] contents totalValue puts [format "total value of knapsack: %.2f" $totalValue] puts "contents:" foreach item $contents {
lassign $item name mass value puts [format "\t%.1fkg of %s, value %.2f" $mass $name $value]
}</lang> Output:
total value of knapsack: 349.38 contents: 3.0kg of salami, value 95.00 3.6kg of ham, value 90.00 2.5kg of brawn, value 56.00 2.4kg of greaves, value 45.00 3.5kg of welt, value 63.38
Ursala
We might as well leave this one to the experts by setting it up as a linear programming problem and handing it off to an external library (which will be either lpsolve or glpk depending on the run-time system configuration). <lang Ursala>#import flo
- import lin
items = # name: (weight,price)
<
'beef ': (3.8,36.0), 'pork ': (5.4,43.0), 'ham ': (3.6,90.0), 'greaves': (2.4,45.0), 'flitch ': (4.0,30.0), 'brawn ': (2.5,56.0), 'welt ': (3.7,67.0), 'salami ': (3.0,95.0), 'sausage': (5.9,98.0)>
system = # a function to transform the item list to the data structure needed by the solver
linear_system$[
lower_bounds: *nS ~&\0., # all zeros because we can't steal less than zero upper_bounds: ~&nmlPXS, # can't steal more than what's in the shop costs: * ^|/~& negative+ vid, # prices divided by weights, negated so as to maximize equations: ~&iNC\15.+ 1.-*@nS] # 1 equation constraining the total weight to 15
- cast %em
main = solution system items</lang> output:
< 'brawn ': 2.500000e+00, 'greaves': 2.400000e+00, 'ham ': 3.600000e+00, 'salami ': 3.000000e+00, 'welt ': 3.500000e+00>