Knapsack problem/Continuous: Difference between revisions
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greaves 2.40 45.00 |
greaves 2.40 45.00 |
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welt 3.50 63.38 |
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</pre> |
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=={{header|Phix}}== |
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<lang Phix>constant meats = { |
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--Item Weight (kg) Price (Value) |
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{"beef", 3.8, 36}, |
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{"pork", 5.4, 43}, |
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{"ham", 3.6, 90}, |
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{"greaves", 2.4, 45}, |
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{"flitch", 4.0, 30}, |
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{"brawn", 2.5, 56}, |
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{"welt", 3.7, 67}, |
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{"salami", 3.0, 95}, |
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{"sausage", 5.9, 98}} |
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function by_weighted_value(integer i, j) |
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atom {?,weighti,pricei} = meats[i], |
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{?,weightj,pricej} = meats[j] |
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return compare(pricej/weightj,pricei/weighti) |
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end function |
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sequence tags = custom_sort(routine_id("by_weighted_value"),tagset(length(meats))) |
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atom w = 15, worth = 0 |
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for i=1 to length(tags) do |
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object {desc,wi,price} = meats[tags[i]] |
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atom c = min(wi,w) |
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printf(1,"%3.1fkg%s of %s\n",{c,iff(c=wi?" (all)":""),desc}) |
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worth += (c/wi)*price |
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w -= c |
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if w=0 then exit end if |
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end for |
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printf(1,"Total value: %f\n",{worth})</lang> |
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{{out}} |
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<pre> |
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3.0kg (all) of salami |
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3.6kg (all) of ham |
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2.5kg (all) of brawn |
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2.4kg (all) of greaves |
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3.5kg of welt |
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Total value: 349.378378 |
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</pre> |
</pre> |
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welt 3.50 63.38 |
welt 3.50 63.38 |
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15.00 349.38</pre> |
15.00 349.38</pre> |
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=={{header|PL/I}}== |
=={{header|PL/I}}== |
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<lang pli>*process source xref attributes; |
<lang pli>*process source xref attributes; |
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Revision as of 09:57, 21 March 2017
You are encouraged to solve this task according to the task description, using any language you may know.
A thief burgles a butcher's shop, where he can select from some items.
The thief 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 shop:
| 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 | ? |
- Task
Show which items the thief carries in his knapsack so that their total weight does not exceed 15 kg, and their total value is maximized.
- Related tasks
- See also
- Wikipedia article: continuous knapsack.
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>
GNU APL
<lang APL> ⍝ Data Items←'beef' 'pork' 'ham' 'greaves' 'flitch' 'brawn' 'welt' 'salami' 'sausage' Weights←3.8 5.4 3.6 2.4 4 2.5 3.7 3 5.9 Prices←36 43 90 45 30 56 67 95 98
⍝ Solution Order←⍒Worth←Prices÷Weights ⍝ 'Worth' is each item value for 1 kg. diff←{¯1↓(⍵,0)-0,⍵} ⍝ 'diff' between each item and the prev item (the inverse of '+\'). Filter←×Selected←diff 15⌊+\Weights[Order] ⍝ 'Selected' weights totaling 15kg, others 0. Table←⊃{⍺,⍪⍵}/Items Weights Selected[⍋Order] Take←Filter[⍋Order]/[1]Table TotalCost←+/Prices×Selected[⍋Order]÷Weights </lang>
BBC BASIC
<lang bbcbasic> INSTALL @lib$+"SORTSALIB"
Sort% = FN_sortSAinit(1, 0) : REM Descending
nItems% = 9
maxWeight = 15.0
DIM items{(nItems%-1) name$, weight, price, worth}
FOR item% = 0 TO nItems%-1
READ items{(item%)}.name$, items{(item%)}.weight, items{(item%)}.price
items{(item%)}.worth = items{(item%)}.price / items{(item%)}.weight
NEXT
DATA "beef", 3.8, 36, "pork", 5.4, 43, "ham", 3.6, 90
DATA "greaves", 2.4, 45, "flitch", 4.0, 30, "brawn", 2.5, 56
DATA "welt", 3.7, 67, "salami", 3.0, 95, "sausage", 5.9, 98
C% = nItems% : D% = 0
CALL Sort%, items{()}, items{(0)}.worth
TotalWeight = 0
TotalPrice = 0
FOR i% = 0 TO nItems%-1
IF TotalWeight + items{(i%)}.weight < maxWeight THEN
TotalWeight += items{(i%)}.weight
TotalPrice += items{(i%)}.price
PRINT "Take all the " items{(i%)}.name$
ELSE
weight = maxWeight - TotalWeight
price = weight * items{(i%)}.worth
TotalWeight += weight
TotalPrice += price
PRINT "Take "; weight " kg of " items{(i%)}.name$
EXIT FOR
ENDIF
NEXT
PRINT '"Total weight = " ; TotalWeight " kg"
PRINT "Total price = " ; TotalPrice
END</lang>
Output:
Take all the salami Take all the ham Take all the brawn Take all the greaves Take 3.5 kg of welt Total weight = 15 kg Total price = 349.378379
Befunge
The table of weights and prices are stored as strings to make them easier to edit. Two characters for the weight (with the decimal point dropped), two characters for the price, and then the name of the item. The total numbers of items (9) is specified by the first value on the stack.
<lang befunge>9:02p>:5+::::::0\g68*-55+*\1\g68*-+\0\pv>2gg!*::!2v >\`!v|:-1p\3\0p\2\+-*86g\3\*+55-*86g\2<<1v*g21\*g2< nib@_>0022p6>12p:212gg48*:**012gg/\-:0`3^+>,,55+%6v
- v0pg2231$$_^#`+5g20:+1g21$_+#!:#<0#<<p22<\v84,+*8<
- >22gg+::55*6*`\55*6*-*022gg\-:55+/68*+"."^>*"fo "v
^6*55:,+55$$_,#!1#`+#*:#82#42#:g<g22:4,,,,,,," kg"< 3836beef 5443pork 3690ham 2445greaves 4030flitch 2556brawn 3767welt 3095salami 5998sausage</lang>
- Output:
3.0 kg of salami 3.6 kg of ham 2.5 kg of brawn 2.4 kg of greaves 3.5 kg of welt
Bracmat
<lang bracmat>( ( fixed {function to convert a rational number to fixed point notation.
The second argument is the number of decimals. }
= value decimals powerOf10
. !arg:(?value.?decimals)
& 10^!decimals:?powerOf10
& str
$ ( div$(!value.1)
"."
mod
$ (div$(!value+1/2*!powerOf10^-1.!powerOf10^-1).!powerOf10)
)
)
& (beef.38/10.36)
(pork.54/10.43)
(ham.36/10.90)
(greaves.24/10.45)
(flitch.40/10.30)
(brawn.25/10.56)
(welt.37/10.67)
(salami.30/10.95)
(sausage.59/10.98)
: ?items
& 0:?sorteditems & whl
' ( !items:(?name.?mass.?price) ?items & (!mass*!price^-1.!mass.!name)+!sorteditems:?sorteditems )
& 0:?totalMass & :?stolenItems & whl
' ( !sorteditems:(?massPerPriceunit.?mass.?name)+?sorteditems
& (!mass.!massPerPriceunit.!name) !stolenItems
: ?stolenItems
& !mass+!totalMass:?totalMass:~>15
)
& !stolenItems:(?mass.?massPerPriceunit.?name) ?stolenItems & 15+!mass+-1*!totalMass:?mass & (!mass.!massPerPriceunit.!name) !stolenItems:?stolenItems & 0:?totalPrice & ( !stolenItems
: ?
( (?mass.?massPerPriceunit.?name)
& out$(fixed$(!mass.1) "kg of" !name)
& !mass*!massPerPriceunit^-1+!totalPrice:?totalPrice
& ~
)
?
| out$(fixed$(!totalPrice.2))
)
);</lang>
Output:
3.5 kg of welt 2.4 kg of greaves 2.5 kg of brawn 3.6 kg of ham 3.0 kg of salami 349.38
C
<lang c>#include <stdio.h>
- include <stdlib.h>
struct item { double w, v; const 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
Clojure
<lang Clojure>
- Solve Continuous Knapsack Problem
- Nicolas Modrzyk
- January 2015
(def maxW 15.0) (def 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]})
(defn rob [items maxW]
(let[
val-item
(fn[key]
(- (/ (second (items key)) (first (items key )))))
compare-items
(fn[key1 key2]
(compare (val-item key1) (val-item key2)))
sorted (into (sorted-map-by compare-items) items)]
(loop [current (first sorted)
array (rest sorted)
value 0
weight 0]
(let[new-weight (first (val current))
new-value (second (val current))]
(if (> (- maxW weight new-weight) 0)
(do
(println "Take all " (key current))
(recur
(first array)
(rest array)
(+ value new-value)
(+ weight new-weight)))
(let [t (- maxW weight)] ; else
(println
"Take " t " of "
(key current) "\n"
"Total Value is:"
(+ value (* t (/ new-value new-weight))))))))))
(rob items maxW) </lang>
Output
Take all :salami Take all :ham Take all :brawn Take all :greaves Take 3.5 of :welt Total Value is: 349.3783783783784
Alternate Version
<lang Clojure> (def items
[{:name "beef" :weight 3.8 :price 36}
{:name "pork" :weight 5.4 :price 43}
{:name "ham" :weight 3.6 :price 90}
{:name "graves" :weight 2.4 :price 45}
{:name "flitch" :weight 4.0 :price 30}
{:name "brawn" :weight 2.5 :price 56}
{:name "welt" :weight 3.7 :price 67}
{:name "salami" :weight 3.0 :price 95}
{:name "sausage" :weight 5.9 :price 98}])
(defn per-kg [item] (/ (:price item) (:weight item)))
(defn rob [items capacity]
(let [best-items (reverse (sort-by per-kg items))]
(loop [items best-items cap capacity total 0]
(let [item (first items)]
(if (< (:weight item) cap)
(do (println (str "Take all " (:name item)))
(recur (rest items) (- cap (:weight item)) (+ total (:price item))))
(println (format "Take %.1f kg of %s\nTotal: %.2f monies"
cap (:name item) (+ total (* cap (per-kg item))))))))))
(rob items 15) </lang>
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>
Alternate Version
<lang cpp> // C++11 version
- include <iostream>
- include <vector>
- include <algorithm>
- include <string>
using namespace std;
struct item_type {
double weight, value; string name;
};
vector< item_type > 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 main() {
sort
(
begin( items ), end( items ),
[] (const item_type& a, const item_type& b)
{
return a.value / a.weight > b.value / b.weight;
}
);
double space = 15;
for ( const auto& item : items )
{
if ( space >= item.weight )
cout << "Take all " << item.name << endl;
else
{
cout << "Take " << space << "kg of " << item.name << endl;
break;
}
space -= item.weight;
}
} </lang>
Common Lisp
<lang lisp>(defstruct item
(name nil :type string) (weight nil :type real) (price nil :type real))
(defun price-per-weight (item)
(/ (item-price item) (item-weight item)))
(defun knapsack (items total-weight)
(loop with sorted = (sort items #'> :key #'price-per-weight)
while (plusp total-weight)
for item in sorted
for amount = (min (item-weight item) total-weight)
collect (list (item-name item) amount)
do (decf total-weight amount)))
(defun main ()
(let ((items (list (make-item :name "beef" :weight 3.8 :price 36)
(make-item :name "pork" :weight 5.4 :price 43)
(make-item :name "ham" :weight 3.6 :price 90)
(make-item :name "greaves" :weight 2.4 :price 45)
(make-item :name "flitch" :weight 4.0 :price 30)
(make-item :name "brawn" :weight 2.5 :price 56)
(make-item :name "welt" :weight 3.7 :price 67)
(make-item :name "salami" :weight 3.0 :price 95)
(make-item :name "sausage" :weight 5.9 :price 98))))
(loop for (name amount) in (knapsack items 15)
do (format t "~8A: ~,2F kg~%" name amount))))</lang>
- Output:
salami : 3.00 kg ham : 3.60 kg brawn : 2.50 kg greaves : 2.40 kg welt : 3.50 kg
D
<lang d>import std.stdio, std.algorithm, std.string;
struct Item {
string name; real amount, value;
@property real valuePerKG() @safe const pure nothrow @nogc {
return value / amount;
}
string toString() const pure /*nothrow*/ @safe {
return format("%10s %7.2f %7.2f %7.2f",
name, amount, value, valuePerKG);
}
}
real sumBy(string field)(in Item[] items) @safe pure nothrow @nogc {
return reduce!("a + b." ~ field)(0.0L, items);
}
void main() /*@safe*/ {
const items = [Item("beef", 3.8, 36.0),
Item("pork", 5.4, 43.0),
Item("ham", 3.6, 90.0),
Item("greaves", 2.4, 45.0),
Item("flitch", 4.0, 30.0),
Item("brawn", 2.5, 56.0),
Item("welt", 3.7, 67.0),
Item("salami", 3.0, 95.0),
Item("sausage", 5.9, 98.0)]
.schwartzSort!(it => -it.valuePerKG)
.release;
immutable(Item)[] chosen;
real space = 15.0;
foreach (const 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);
Item("TOTAL", chosen.sumBy!"amount", chosen.sumBy!"value").writeln;
}</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>void main() {
import std.stdio, std.algorithm;
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)]
.schwartzSort!q{ -a.price / a.weight };
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
return writefln("Take %.1fkg %s", left, it.item);
}</lang>
- Output:
Take all the salami Take all the ham Take all the brawn Take all the greaves Take 3.5kg welt
EchoLisp
<lang scheme> (lib 'struct) (lib 'sql) ;; for table
(define T (make-table (struct meal (name poids price))))
(define meals
'((🐂-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)))
(list->table meals T)
- sort table according to best price/poids ratio
(define (price/poids a b ) (- (// (* (meal-price b) (meal-poids a)) (meal-price a) (meal-poids b)) 1)) (table-sort T price/poids)
(define-syntax-rule (name i) (table-xref T i 0)) (define-syntax-rule (poids i) (table-xref T i 1))
- shop
- add items in basket, in order, until W exhausted
(define (shop W ) (for/list ((i (table-count T))) #:break (<= W 0) (begin0 (cons (name i) (if (<= (poids i) W) 'all W)) (set! W (- W (poids i))))))
- output
(shop 15)
→ ((🐃--salami . all) (🍗-ham . all) (brawn . all) (🐪-greaves . all) (welt . 3.5))
</lang>
Eiffel
<lang Eiffel> class CONTINUOUS_KNAPSACK
create make
feature
make local tup: TUPLE [name: STRING; weight: REAL_64; price: REAL_64] do create tup create items.make_filled (tup, 1, 9) create sorted.make sorted.extend (-36.0 / 3.8) sorted.extend (-43.0 / 5.4) sorted.extend (-90.0 / 3.6) sorted.extend (-45.0 / 2.4) sorted.extend (-30.0 / 4.0) sorted.extend (-56.0 / 2.5) sorted.extend (-67.0 / 3.7) sorted.extend (-95.0 / 3.0) sorted.extend (-98.0 / 5.9) tup := ["beef", 3.8, 36.0] items [sorted.index_of (- tup.price / tup.weight, 1)] := tup tup := ["pork", 5.4, 43.0] items [sorted.index_of (- tup.price / tup.weight, 1)] := tup tup := ["ham", 3.6, 90.0] items [sorted.index_of (- tup.price / tup.weight, 1)] := tup tup := ["greaves", 2.4, 45.0] items [sorted.index_of (- tup.price / tup.weight, 1)] := tup tup := ["flitch", 4.0, 30.0] items [sorted.index_of (- tup.price / tup.weight, 1)] := tup tup := ["brawn", 2.5, 56.0] items [sorted.index_of (- tup.price / tup.weight, 1)] := tup tup := ["welt", 3.7, 67.0] items [sorted.index_of (- tup.price / tup.weight, 1)] := tup tup := ["salami", 3.0, 95.0] items [sorted.index_of (- tup.price / tup.weight, 1)] := tup tup := ["sausage", 5.9, 98.0] items [sorted.index_of (- tup.price / tup.weight, 1)] := tup find_solution end
find_solution -- Solution for the continuous Knapsack Problem. local maxW, value: REAL_64 do maxW := 15 across items as c loop if maxW - c.item.weight > 0 then io.put_string ("Take all: " + c.item.name + ".%N") value := value + c.item.price maxW := maxW - c.item.weight elseif maxW /= 0 then io.put_string ("Take " + maxW.truncated_to_real.out + " kg off " + c.item.name + ".%N") io.put_string ("The total value is " + (value + (c.item.price / c.item.weight) * maxW).truncated_to_real.out + ".") maxW := 0 end end end
items: ARRAY [TUPLE [name: STRING; weight: REAL_64; price: REAL_64]]
sorted: SORTED_TWO_WAY_LIST [REAL_64]
end </lang>
- Output:
Take all: salami. Take all: ham. Take all: brawn. Take all: greaves. Take 3.5kg off welt. The total value is 349.378
Elixir
<lang elixir>defmodule KnapsackProblem do
def select( max_weight, items ) do
Enum.sort_by( items, fn {_name, weight, price} -> - price / weight end )
|> Enum.reduce( {max_weight, []}, &select_until/2 )
|> elem(1)
|> Enum.reverse
end
def task( items, max_weight ) do
IO.puts "The robber takes the following to maximize the value"
Enum.each( select( max_weight, items ), fn {name, weight} ->
:io.fwrite("~.2f of ~s~n", [weight, name])
end )
end
defp select_until( {name, weight, _price}, {remains, acc} ) when remains > 0 do
selected_weight = select_until_weight( weight, remains )
{remains - selected_weight, [{name, selected_weight} | acc]}
end
defp select_until( _item, acc ), do: acc
defp select_until_weight( weight, remains ) when weight < remains, do: weight
defp select_until_weight( _weight, remains ), do: remains
end
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} ]
KnapsackProblem.task( items, 15 )</lang>
- Output:
The robber takes the following to maximize the value 3.00 of salami 3.60 of ham 2.50 of brawn 2.40 of greaves 3.50 of welt
Alternate Version
<lang elixir>defmodule KnapsackProblem do
def continuous(items, max_weight) do
Enum.sort_by(items, fn {_item, {weight, price}} -> -price / weight end)
|> Enum.reduce_while({max_weight,0}, fn {item, {weight, price}}, {rest, value} ->
if rest > weight do
IO.puts "Take all #{item}"
{:cont, {rest - weight, value + price}}
else
:io.format "Take ~.3fkg of ~s~n~n", [rest, item]
:io.format "Total value of swag is ~.2f~n", [value + rest*price/weight]
{:halt, :ok}
end
end)
|> case do
{weight, value} ->
:io.format "Total: weight ~.3fkg, value ~p~n", [max_weight-weight, value]
x -> x
end
end
end
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} ]
KnapsackProblem.continuous( items, 15 )</lang>
- Output:
Take all salami Take all ham Take all brawn Take all greaves Take 3.500kg of welt Total value of swag is 349.38
Erlang
Note use of lists:foldr/2, since sort is ascending. <lang Erlang> -module( knapsack_problem_continuous ).
-export( [price_per_weight/1, select/2, task/0] ).
price_per_weight( Items ) -> [{Name, Weight, Price / Weight} || {Name, Weight, Price} <-Items].
select( Max_weight, Items ) -> {_Remains, Selected_items} = lists:foldr( fun select_until/2, {Max_weight, []}, lists:keysort(3, Items) ), Selected_items.
task() -> Items = items(), io:fwrite( "The robber takes the following to maximize the value~n" ), [io:fwrite("~.2f of ~p~n", [Weight, Name]) || {Name, Weight} <- select( 15, price_per_weight(Items) )].
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} ].
select_until( {Name, Weight, _Price}, {Remains, Acc} ) when Remains > 0 -> Selected_weight = select_until_weight( Weight, Remains ), {Remains - Selected_weight, [{Name, Selected_weight} | Acc]}; select_until( _Item, Acc ) -> Acc.
select_until_weight( Weight, Remains ) when Weight < Remains -> Weight; select_until_weight( _Weight, Remains ) -> Remains. </lang>
- Output:
11> knapsack_problem_continuous:task(). The robber takes the following to maximize the value 3.50 of "welt" 2.40 of "greaves" 2.50 of "brawn" 3.60 of "ham" 3.00 of "salami"
F#
<lang fsharp> //Fill a knapsack optimally - Nigel Galloway: February 1st., 2015 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)]
|> List.sortBy(fun(_,weight,value) -> float(-value)/weight)
let knap items maxW=
let rec take(n,g,a) =
match g with
| i::e -> let name, weight, value = i
let total = n + weight
if total <= maxW then
printfn "Take all %s" name
take(total, e, a+float(value))
else
printfn "Take %0.2f kg of %s\nTotal value of swag is %0.2f" (maxW - n) name (a + (float(value)/weight)*(maxW - n))
| [] -> printfn "Everything taken! Total value of swag is £%0.2f; Total weight of bag is %0.2fkg" a n
take(0.0, items, 0.0)
</lang>
- Output:
> knap items 15.0;; Take all salami Take all ham Take all brawn Take all greaves Take 3.50kg of welt Total value of swag is £349.38
Should your burglar be greedy, he may bring a bigger bag.
> knap items 100.0;; Take all salami Take all ham Take all brawn Take all greaves Take all welt Take all sausage Take all beef Take all pork Take all flitch Everything taken! Total value of swag is £560.00; Total weight of bag is 34.30kg
Forth
<lang forth>include lib/selcsort.4th \ use a tiny sorting algorithm
150 value left \ capacity in 1/10th kilo
create items \ list of items
," beef" 38 , 3600 , \ description, weight, price (cents) ," pork" 54 , 4300 , \ weight in 1/10 kilo ," ham" 36 , 9000 , ," greaves" 24 , 4500 , ," flitch" 40 , 3000 , ," brawn" 25 , 5600 , ," welt" 37 , 6700 , ," salami" 30 , 9500 , ," sausage" 59 , 9800 , here items - 3 / constant #items \ total number of items
- redo items swap 3 cells * + ; \ calculate address of record
- items array (items) \ array for sorting
( a -- n)
- price/weight dup 2 cells + @c swap cell+ @c / ;
- weight@ @ cell+ @c ; ( a -- n)
- .item @ @c count type cr ; ( a --)
\ how to sort: on price/weight
- noname >r price/weight r> price/weight > ; is precedes
- knapsack ( --)
(items) dup #items dup 0 ?do i items (items) i th ! loop sort
begin \ use the sorted array
dup weight@ left <= \ still room in the knapsack?
while
." Take all of the " dup .item \ take all of the item
left over weight@ - to left cell+ \ adjust knapsack, increment item
repeat left 100 * dup \ so how much is left?
\ if room, take as much as possible
if ." Take " . ." grams of the " .item else drop drop then
knapsack</lang>
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 descending 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>
See Knapsack_problem/Continuous/J for some comments on intermediate results...
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)
jq
<lang jq># continuous_knapsack(W) expects the input to be
- an array of objects {"name": _, "weight": _, "value": _}
- where "value" is the value of the given weight of the object.
def continuous_knapsack(W):
map( .price = (if .weight > 0 then (.value/.weight) else 0 end) )
| sort_by( .price )
| reverse
| reduce .[] as $item
# state: [array, capacity]
([[], W];
.[1] as $c
| if $c <= 0 then .
else ( [$item.weight, $c] | min) as $min
| [.[0] + [ $item | (.weight = $min) | .value = (.price * $min)],
($c - $min) ]
end)
| .[1] as $remainder
| .[0]
| (.[] | {name, weight}),
"Total value: \( map(.value) | add)",
"Total weight: \(W - $remainder)" ;</lang>
The task: <lang jq>def items: [
{ "name": "beef", "weight": 3.8, "value": 36},
{ "name": "pork", "weight": 5.4, "value": 43},
{ "name": "ham", "weight": 3.6, "value": 90},
{ "name": "greaves", "weight": 2.4, "value": 45},
{ "name": "flitch", "weight": 4.0, "value": 30},
{ "name": "brawn", "weight": 2.5, "value": 56},
{ "name": "welt", "weight": 3.7, "value": 67},
{ "name": "salami", "weight": 3.0, "value": 95},
{ "name": "sausage", "weight": 5.9, "value": 98} ];
items | continuous_knapsack(15)</lang>
- Output:
<lang sh>$ jq -r -c -n -f knapsack_continuous.jq {"name":"salami","weight":3} {"name":"ham","weight":3.6} {"name":"brawn","weight":2.5} {"name":"greaves","weight":2.4} {"name":"welt","weight":3.5000000000000004} Total value: 349.3783783783784 Total weight: 15</lang>
Julia
This solution is built around the immutable type KPCSupply, which holds an item's data including its unit value (uvalue). When the store's inventory is kept in this way, the solution to the continuous knapsack problem (provided by solve), is straightforward. The thief should pack as much of the highest value items as are available until full capacity is reached, topping off with as much of the last item as the knapsack will hold. (If the store contains less than the thief's knapsack will hold, he'll take the store's entire inventory.)
An outer constructor method is used to create instances of KPCSupply when only the item, weight and value are supplied. The isless method is provided for KPCSupply objects so that items are transparently sorted by their unit value. KPCSupply supports any real type for weight, value and uvalue (though this simple implementation does not support mixed types or promotion). This solution uses Rational numbers to avoid rounding errors until the results are printed.
Type and Functions <lang Julia> immutable KPCSupply{S<:String, T<:Real}
item::S weight::T value::T uvalue::T
end Base.isless(a::KPCSupply, b::KPCSupply) = a.uvalue<b.uvalue
function KPCSupply{S<:String, T<:Real}(item::S, weight::T, value::T)
KPCSupply(item, weight, value, value/weight)
end
function solve{S<:String, T<:Real}(store::Array{KPCSupply{S,T},1},
capacity::T)
ksack = KPCSupply{S,T}[]
kweight = zero(T)
for s in sort(store, rev=true)
if kweight + s.weight <= capacity
kweight += s.weight
push!(ksack, s)
else
w = capacity-kweight
v = w*s.uvalue
push!(ksack, KPCSupply(s.item, w, v, s.uvalue))
break
end
end
return ksack
end </lang>
Main <lang Julia> store = [KPCSupply("beef", 38//10, 36//1),
KPCSupply("pork", 54//10, 43//1),
KPCSupply("ham", 36//10, 90//1),
KPCSupply("greaves", 24//10, 45//1),
KPCSupply("flitch", 4//1, 30//1),
KPCSupply("brawn", 25//10, 56//1),
KPCSupply("welt", 37//10, 67//1),
KPCSupply("salami", 3//1, 95//1),
KPCSupply("sausage", 59//10, 98//1)]
sack = solve(store, 15//1)
println("The store contains:") println(" Item Weight Unit Price") for s in store
println(@sprintf("%12s %4.1f %6.2f",
s.item, float(s.weight), float(s.uvalue)))
end
println() println("The thief should take:") println(" Item Weight Value") for s in sack
println(@sprintf("%12s %4.1f %6.2f",
s.item, float(s.weight), float(s.value)))
end w = mapreduce(x->x.weight, +, sack) v = mapreduce(x->x.value, +, sack) println(@sprintf("%12s %4.1f %6.2f",
"Total", float(w), float(v)))
</lang>
- Output:
The store contains:
Item Weight Unit Price
beef 3.8 9.47
pork 5.4 7.96
ham 3.6 25.00
greaves 2.4 18.75
flitch 4.0 7.50
brawn 2.5 22.40
welt 3.7 18.11
salami 3.0 31.67
sausage 5.9 16.61
The thief should take:
Item Weight Value
salami 3.0 95.00
ham 3.6 90.00
brawn 2.5 56.00
greaves 2.4 45.00
welt 3.5 63.38
Total 15.0 349.38
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>
Oforth
<lang oforth>[
[ "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 ]
] const: Items
- rob
| item value |
0.0 ->value
15.0 #[ dup second swap third / ] Items sortBy forEach: item [
dup 0.0 == ifTrue: [ return ]
dup item second >= ifTrue: [
"Taking" . item first . " :" . item second dup .cr -
item third value + ->value continue
]
"And part of" . item first . " :" . dup .cr
item third * item second / value + "Total value :" . .cr break
] ;</lang>
- Output:
>rob Taking salami : 3 Taking ham : 3.6 Taking brawn : 2.5 Taking greaves : 2.4 And part of welt : 3.5 Total value : 349.378378378378 ok
ooRexx
version 1
<lang oorexx>/*--------------------------------------------------------------------
- 20.09.2014 Walter Pachl translated from REXX version 2
- utilizing ooRexx features like objects, array(s) and sort
- -------------------------------------------------------------------*/
maxweight = 15.0
items=.array~new
items~append(.item~new('beef', 3.8, 36.0))
items~append(.item~new('pork', 5.4, 43.0))
items~append(.item~new('ham', 3.6, 90.0))
items~append(.item~new('greaves', 2.4, 45.0))
items~append(.item~new('flitch', 4.0, 30.0))
items~append(.item~new('brawn', 2.5, 56.0))
items~append(.item~new('welt', 3.7, 67.0))
items~append(.item~new('salami', 3.0, 95.0))
items~append(.item~new('sausage', 5.9, 98.0))
/* show the input */ Say '# vpu name weight value' i=0 Do x over items i+=1 Say i format(x~vpu,2,3) left(x~name,7) format(x~weight,2,3) format(x~value,3,3) End
/* sort the items by descending value per unit of weight */ items~sortWith(.DescendingComparator~new)
total_weight=0
total_value =0
Say ' '
Say 'Item Weight Value'
i=0
Do x over items
i+=1
Parse Var item.i name '*' weight '*' value
if total_weight+x~weight<maxweight then Do
total_weight = total_weight + x~weight
total_value = total_value + x~value
Say left(x~name,7) format(x~weight,3,3) format(x~value,3,3)
End
Else Do
weight=maxweight-total_weight
value=weight*x~vpu
total_value = total_value + value
total_weight = maxweight
Say left(x~name,7) format(weight,3,3) format(value,3,3)
Leave
End
End
Say copies('-',23)
Say 'total ' format(total_weight,4,3) format(total_value,3,3)
Exit
- class item
::attribute vpu ::attribute name ::attribute weight ::attribute value
- method init
Expose vpu Use Arg name, weight, value self~name=name self~weight=weight self~value=value self~vpu=value/weight
- CLASS 'DescendingComparator' MIXINCLASS Comparator
- METHOD compare
use strict arg a, b Select
When (a~vpu)<(b~vpu) Then res='1' Otherwise res='-1' End
Return res</lang>
- Output:
# vpu name weight value 1 9.474 beef 3.800 36.000 2 7.963 pork 5.400 43.000 3 25.000 ham 3.600 90.000 4 18.750 greaves 2.400 45.000 5 7.500 flitch 4.000 30.000 6 22.400 brawn 2.500 56.000 7 18.108 welt 3.700 67.000 8 31.667 salami 3.000 95.000 9 16.610 sausage 5.900 98.000 Item Weight Value salami 3.000 95.000 ham 3.600 90.000 brawn 2.500 56.000 greaves 2.400 45.000 welt 3.500 63.378 ----------------------- total 15.000 349.378
version 2
<lang oorexx>/*--------------------------------------------------------------------
- 20.09.2014 Walter Pachl translated from REXX version 2
- utilizing ooRexx features like objects, array(s) and sort
- 21.09.2014 simplified (courtesy Rony Flatscher)
- (sort uses now the method "compareTo" defined for item)
- -------------------------------------------------------------------*/
maxweight = 15.0
items=.array~new
items~append(.item~new('beef', 3.8, 36.0))
items~append(.item~new('pork', 5.4, 43.0))
items~append(.item~new('ham', 3.6, 90.0))
items~append(.item~new('greaves', 2.4, 45.0))
items~append(.item~new('flitch', 4.0, 30.0))
items~append(.item~new('brawn', 2.5, 56.0))
items~append(.item~new('welt', 3.7, 67.0))
items~append(.item~new('salami', 3.0, 95.0))
items~append(.item~new('sausage', 5.9, 98.0))
/* show the input */ Say '# vpu name weight value' i=0 Do x over items i+=1 Say i format(x~vpu,2,3) left(x~name,7) format(x~weight,2,3) format(x~value,3,3) End
/* sort the items by descending value per unit of weight */ items~sort /* using the method compareTo used for item */
total_weight=0
total_value =0
Say ' '
Say 'Item Weight Value'
i=0
Do x over items
i+=1
Parse Var item.i name '*' weight '*' value
if total_weight+x~weight<maxweight then Do
total_weight = total_weight + x~weight
total_value = total_value + x~value
Say left(x~name,7) format(x~weight,3,3) format(x~value,3,3)
End
Else Do
weight=maxweight-total_weight
value=weight*x~vpu
total_value = total_value + value
total_weight = maxweight
Say left(x~name,7) format(weight,3,3) format(value,3,3)
Leave
End
End
Say copies('-',23)
Say 'total ' format(total_weight,4,3) format(total_value,3,3)
Exit
- class item
::attribute vpu ::attribute name ::attribute weight ::attribute value
- method init
Expose vpu Use Arg name, weight, value self~name=name self~weight=weight self~value=value self~vpu=value/weight
- method compareTo -- default sort order
Expose vpu use Arg other return -sign(vpu - other~vpu)</lang>
Output is the same as for version 1.
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, Rat $w, Int $p) {
self.bless(:name($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 gist () { 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
Phix
<lang Phix>constant meats = { --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}}
function by_weighted_value(integer i, j)
atom {?,weighti,pricei} = meats[i],
{?,weightj,pricej} = meats[j]
return compare(pricej/weightj,pricei/weighti)
end function
sequence tags = custom_sort(routine_id("by_weighted_value"),tagset(length(meats)))
atom w = 15, worth = 0 for i=1 to length(tags) do
object {desc,wi,price} = meats[tags[i]]
atom c = min(wi,w)
printf(1,"%3.1fkg%s of %s\n",{c,iff(c=wi?" (all)":""),desc})
worth += (c/wi)*price
w -= c
if w=0 then exit end if
end for printf(1,"Total value: %f\n",{worth})</lang>
- Output:
3.0kg (all) of salami 3.6kg (all) of ham 2.5kg (all) of brawn 2.4kg (all) of greaves 3.5kg of welt Total value: 349.378378
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
PL/I
<lang pli>*process source xref attributes;
KNAPSACK_CONTINUOUS: Proc Options(main); /*-------------------------------------------------------------------- * 19.09.2014 Walter Pachl translated from FORTRAN *-------------------------------------------------------------------*/ Dcl (divide,float,hbound,repeat) Builtin; Dcl SYSPRINT Print;
Dcl maxweight Dec Fixed(15,3); maxweight = 15.0; Dcl (total_weight,total_value) Dec Fixed(15,3) Init(0); Dcl vpu Dec Float(15); Dcl (i,j) Bin Fixed(31);
Dcl 1 item(9),
2 name Char(7),
2 weight Dec Fixed(15,3),
2 value Dec Fixed(15,3);
Dcl temp Like item;
Call init_item(1,'beef', 3.8, 36.0); Call init_item(2,'pork', 5.4, 43.0); Call init_item(3,'ham', 3.6, 90.0); Call init_item(4,'greaves', 2.4, 45.0); Call init_item(5,'flitch', 4.0, 30.0); Call init_item(6,'brawn', 2.5, 56.0); Call init_item(7,'welt', 3.7, 67.0); Call init_item(8,'salami', 3.0, 95.0); Call init_item(9,'sausage', 5.9, 98.0);
/* sort item in descending order of their value per unit weight */
do i = 2 To hbound(item);
j = i - 1;
temp = item(i);
do while(j>=1&item(j).value/item(j).weight<temp.value/temp.weight);
item(j+1) = item(j);
j = j - 1;
end;
item(j+1) = temp;
end;
Do i=1 To hbound(item);
Put Edit(i,item(i))(Skip,f(2),x(2),a(7),2(f(8,3)));
End;
i = 0;
Put Skip;
Put Edit('Item Weight Value')(Skip,a);
do i=1 By 1 while(i < hbound(item) & total_weight < maxweight);
if total_weight+item(i).weight < maxweight then Do;
total_weight = total_weight + item(i).weight;
total_value = total_value + item(i).value;
Put Edit(item(i))(Skip,a(7),2(f(8,3)));
End;
Else Do;
vpu=divide(item(i).value,item(i).weight,15,8);
item(i).weight=maxweight-total_weight;
item(i).value=float(item(i).weight)*vpu;
total_value = total_value + item(i).value;
total_weight = total_weight + item(i).weight;
Put Edit(item(i).name, item(i).weight, item(i).value)
(Skip,a(7),2(f(8,3)));
Leave Loop;
end;
end;
Put Edit(repeat('-',22))(Skip,a);
Put Edit('total',total_weight, total_value)(Skip,a(6),f(9,3),f(8,3));
init_item: Proc(i,name,weight,value); Dcl i Bin Fixed(31); Dcl name Char(*); Dcl (weight,value) Dec Fixed(15,3); item(i).name = name; item(i).weight = weight; item(i).value = value; End; End;</lang>
- Output:
1 salami 3.000 95.000 2 ham 3.600 90.000 3 brawn 2.500 56.000 4 greaves 2.400 45.000 5 welt 3.700 67.000 6 sausage 5.900 98.000 7 beef 3.800 36.000 8 pork 5.400 43.000 9 flitch 4.000 30.000 Item Weight Value salami 3.000 95.000 ham 3.600 90.000 brawn 2.500 56.000 greaves 2.400 45.000 welt 3.500 63.378 ----------------------- total 15.000 349.378
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
R
Translated into r-script by Shana White (vandersm@mail.uc.edu) from pseudocode found in 'Algorithms: Sequential Parallel and Distributed', by Kenneth A. Berman and Jerome L. Paul <lang r> knapsack<- function(Value, Weight, Objects, Capacity){
Fraction = rep(0, length(Value))
Cost = Value/Weight
#print(Cost)
W = Weight[order(Cost, decreasing = TRUE)]
Obs = Objects[order(Cost, decreasing = TRUE)]
Val = Value[order(Cost, decreasing = TRUE)]
#print(W)
RemainCap = Capacity
i = 1
n = length(Cost)
if (W[1] <= Capacity){
Fits <- TRUE
}
else{
Fits <- FALSE
}
while (Fits && i <= n ){
Fraction[i] <- 1
RemainCap <- RemainCap - W[i]
i <- i+1
#print(RemainCap)
if (W[i] <= RemainCap){
Fits <- TRUE
}
else{
Fits <- FALSE
}
}
#print(RemainCap)
if (i <= n){
Fraction[i] <- RemainCap/W[i]
}
names(Fraction) = Obs
Quantity_to_take = W*Fraction
Total_Value = sum(Val*Fraction)
print("Fraction of available quantity to take:")
print(round(Fraction, 3))
print("KG of each to take:")
print(Quantity_to_take)
print("Total value of tasty meats:")
print(Total_Value)
}</lang>
Sample Input
o = c("beef", "pork", "ham", "greaves", "flitch", "brawn", "welt", "salami", "sausage")
w = c(3.8,5.4,3.6,2.4,4.0,2.5,3.7,3.0,5.9)
v = c(36,43,90,45,30,56,67,95,98)
knapsack(v, w, o, 15)
Sample Output
[1] "Fraction of available quantity to take:"
salami ham brawn greaves welt sausage beef pork flitch
1.000 1.000 1.000 1.000 0.946 0.000 0.000 0.000 0.000
[1] "KG of each to take:"
salami ham brawn greaves welt sausage beef pork flitch
3.0 3.6 2.5 2.4 3.5 0.0 0.0 0.0 0.0
[1] "Total value of tasty meats:"
[1] 349.3784
Racket
<lang racket>#lang racket (define shop-inventory
'((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)))
(define (continuous-knapsack shop sack sack-capacity sack-total-value)
;; solved by loading up on the highest value item...
(define (value/kg item) (/ (third item) (second item)))
(if (zero? sack-capacity)
(values (reverse sack) sack-total-value)
(let* ((best-value-item (argmax value/kg shop))
(bvi-full-weight (second best-value-item))
(amount-can-take (min sack-capacity bvi-full-weight))
(bvi-full-value (third best-value-item))
(bvi-taken-value (* bvi-full-value (/ amount-can-take bvi-full-weight))))
(continuous-knapsack (remove best-value-item shop)
(cons (list (first best-value-item)
(if (= amount-can-take bvi-full-weight)
'all-of amount-can-take) bvi-taken-value)
sack)
(- sack-capacity amount-can-take)
(+ sack-total-value bvi-taken-value)))))
(define (report-knapsack sack total-value)
(for-each (lambda (item)
(if (eq? 'all-of (second item))
(printf "Take all of the ~a (for ~a)~%"
(first item) (third item))
(printf "Take ~a of the ~a (for ~a)~%"
(real->decimal-string (second item))
(first item)
(real->decimal-string (third item)))))
sack)
(printf "For a grand total of: ~a" (real->decimal-string total-value)))
(call-with-values (lambda () (continuous-knapsack shop-inventory null 15 0))
report-knapsack)</lang>
- Output:
Take all of the salami (for 95.0) Take all of the ham (for 90.0) Take all of the brawn (for 56.0) Take all of the greaves (for 45.0) Take 3.50 of the welt (for 63.38) For a grand total of: 349.38
REXX
version 1
Originally used the Fortran program as a prototype.
Some amount of code was added to format the output better.
<lang rexx>/*REXX pgm solves the continuous burglar's knapsack problem; items with weight and value*/
@.= /*═══════ 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 '
parse arg maxW d . /*get possible arguments from the C.L. */ if maxW== | maxW=="," then maxW=15 /*the burglar's knapsack maximum weight*/ if d== | d=="," then d= 3 /*number of decimal digits in FORMAT. */ wL=d+length('weight'); nL=d+length("total weight"); vL=d+length('value') totW=0; totV=0
do #=1 while @.#\==; parse var @.# n.# w.# v.# .
end /*#*/ /* [↑] assign item to separate lists. */
- =#-1 /*#: is the number of items in @ list.*/
call show 'unsorted item list' /*display the header and the @ list.*/ call sortD /*invoke sort (which sorts descending).*/ call hdr "burglar's knapsack contents"
do j=1 for # while totW<maxW; f=1 /*process the items. */
if totW+w.j>=maxW then f=(maxW-totW)/w.j /*calculate fraction. */
totW=totW+w.j*f; totV=totV+v.j*f /*add it ───► totals. */
call syf left(word('{all}',1+(f\==1)),5) n.j, w.j*f, v.j*f
end /*j*/ /* [↑] display item. */
call sep; say ' /* [↓] $ suppresses trailing zeroes.*/ call sy left('total weight', nL, "─"), $(format(totW,,d)) call sy left('total value', nL, "─"), , $(format(totV,,d)) exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ sortD: do s=2 to #; a=n.s; !=w.s; u=v.s /* [↓] this is a descending sort. */
do k=s-1 by -1 to 1 while v.k/w.k1 then x=left(strip(strip(x,'T',0),,.),length(x)); return x</lang>
output using the default inputs of: 15 3
────────────────unsorted item list────────────────
item weight value
═══════════════ ═════════ ════════
flitch 4 30
beef 3.8 36
pork 5.4 43
greaves 2.4 45
brawn 2.5 56
welt 3.7 67
ham 3.6 90
salami 3 95
sausage 5.9 98
───────────burglar's knapsack contents────────────
item weight value
═══════════════ ═════════ ════════
{all} salami 3 95
{all} ham 3.6 90
{all} brawn 2.5 56
{all} greaves 2.4 45
welt 3.5 63.378
═══════════════ ═════════ ════════
total weight─── 15
total value─── 349.378
version 2
<lang rexx> /*--------------------------------------------------------------------
* 19.09.2014 Walter Pachl translated from FORTRAN * While this program works with all REXX interpreters, * see section ooRexx for a version that utilizes the ooRexx features *-------------------------------------------------------------------*/ maxweight = 15.0 input.0=0 Call init_input 'beef', 3.8, 36.0 Call init_input 'pork', 5.4, 43.0 Call init_input 'ham', 3.6, 90.0 Call init_input 'greaves', 2.4, 45.0 Call init_input 'flitch', 4.0, 30.0 Call init_input 'brawn', 2.5, 56.0 Call init_input 'welt', 3.7, 67.0 Call init_input 'salami', 3.0, 95.0 Call init_input 'sausage', 5.9, 98.0
/* sort the items by descending value per unit of weight */
Do i = 1 to input.0
Parse Var input.i name '*' weight '*' value
vpu=value/weight;
If i=1 Then Do
item.0=1
item.1=input.1
vpu.1=vpu
End
Else Do
Do ii=1 To item.0
If vpu.ii<vpu Then
Leave
End
Do jj=item.0 To ii By -1
jj1=jj+1
item.jj1=item.jj
vpu.jj1=vpu.jj
End
item.ii=input.i
vpu.ii=vpu
item.0=item.0+1
End
End
Say '# vpu name weight value'
Do i=1 To item.0
Parse Var item.i name '*' weight '*' value
Say i format(vpu.i,2,3) left(name,7) format(weight,2,3) format(value,3,3)
End
total_weight=0
total_value =0
Say ' '
Say 'Item Weight Value'
Do i=1 To item.0
Parse Var item.i name '*' weight '*' value
if total_weight+weight < maxweight then Do
total_weight = total_weight + weight
total_value = total_value + value
Say left(name,7) format(weight,3,3) format(value,3,3)
End
Else Do
weight=maxweight-total_weight
value=weight*vpu.i
total_value = total_value + value
total_weight = maxweight
Say left(name,7) format(weight,3,3) format(value,3,3)
Leave
End
End
Say copies('-',23)
Say 'total ' format(total_weight,4,3) format(total_value,3,3)
Exit
init_input: Procedure Expose input.
Parse Arg name,weight,value i=input.0+1 input.i=name'*'weight'*'value input.0=i Return</lang>
- Output:
# vpu name weight value 1 31.667 salami 3.000 95.000 2 25.000 ham 3.600 90.000 3 22.400 brawn 2.500 56.000 4 18.750 greaves 2.400 45.000 5 18.108 welt 3.700 67.000 6 16.610 sausage 5.900 98.000 7 9.474 beef 3.800 36.000 8 7.963 pork 5.400 43.000 9 7.500 flitch 4.000 30.000 Item Weight Value salami 3.000 95.000 ham 3.600 90.000 brawn 2.500 56.000 greaves 2.400 45.000 welt 3.500 63.378 ----------------------- total 15.000 349.378
Ruby
<lang ruby>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] ].sort_by{|item, weight, price| -price / weight}
maxW, value = 15.0, 0 items.each do |item, weight, price|
if (maxW -= weight) > 0
puts "Take all #{item}"
value += price
else
puts "Take %gkg of %s" % [t=weight+maxW, item], "",
"Total value of swag is %g" % (value+(price/weight)*t)
break
end
end</lang>
- Output:
Take all salami Take all ham Take all brawn Take all greaves Take 3.5kg of welt Total value of swag is 349.378
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
SAS
Use LP solver in SAS/OR: <lang sas>/* create SAS data set */ data mydata;
input item $ weight value; datalines;
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
/* 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};
read data mydata into ITEMS=[item] weight value;
/* declare variables, objective, and constraints */
var WeightSelected {i in ITEMS} >= 0 <= weight[i];
max TotalValue = sum {i in ITEMS} (value[i]/weight[i]) * WeightSelected[i];
con WeightCon:
sum {i in ITEMS} WeightSelected[i] <= 15;
/* call linear programming (LP) solver */ solve;
/* print optimal solution */
print TotalValue;
print {i in ITEMS: WeightSelected[i].sol > 1e-3} WeightSelected;
quit;</lang>
Output:
TotalValue 349.38 [1] WeightSelected brawn 2.5 greaves 2.4 ham 3.6 salami 3.0 welt 3.5
Sidef
<lang ruby>var 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],
].sort {|a,b| b[2]/b[1] <=> a[2]/a[1] }
var (limit, value) = (15, 0); print "Item Fraction Weight Value\n";
items.each { |item|
var ratio = (item[1] > limit ? limit/item[1] : 1);
value += item[2]*ratio;
limit -= item[1];
if (ratio == 1) {
printf("%-8s %4s %7.2f %6.2f\n", item[0], 'all', item[1], item[2]);
}
else {
printf("%-8s %-4.2f %7.2f %6.2f\n", item[0], ratio, item[1]*ratio, item[2]*ratio);
break;
}
}
say "#{'-'*28}\ntotal value: #{'%.14g' % value }"</lang>
- Output:
Item Fraction Weight Value salami all 3.00 95.00 ham all 3.60 90.00 brawn all 2.50 56.00 greaves all 2.40 45.00 welt 0.95 3.50 63.38 ---------------------------- total value: 349.37837837838
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>
XPL0
<lang XPL0>int Name, Price, I, BestItem; real Weight, Best, ItemWt, TotalWt; def Items = 9; real PricePerWt(Items); int Taken(Items); include c:\cxpl\codes;
[Name:= ["beef","pork","ham","greaves","flitch","brawn","welt","salami","sausage"]; Weight:= [ 3.8, 5.4, 3.6, 2.4, 4.0, 2.5, 3.7, 3.0, 5.9]; Price:= [ 36, 43, 90, 45, 30, 56, 67, 95, 98];
for I:= 0 to Items-1 do
[PricePerWt(I):= float(Price(I)) / Weight(I);
Taken(I):= false;
];
Format(2,1); TotalWt:= 0.0; repeat Best:= 0.0;
for I:= 0 to Items-1 do
if not Taken(I) and PricePerWt(I) > Best then
[Best:= PricePerWt(I); BestItem:= I];
Taken(BestItem):= true; \take item
ItemWt:= Weight(BestItem); \get its weight
TotalWt:= TotalWt + ItemWt; \add to total weight
if TotalWt > 15.0 then \if total is too much, reduce
ItemWt:= ItemWt - (TotalWt-15.0); \item weight by amount it's over
RlOut(0, ItemWt); Text(0, " kg of "); \show weight and item
Text(0, Name(BestItem)); CrLf(0);
until TotalWt >= 15.0; \all we can steal ]</lang>
Output:
3.0 kg of salami 3.6 kg of ham 2.5 kg of brawn 2.4 kg of greaves 3.5 kg of welt
zkl
<lang zkl>items:=List( T(3.8, 36.0, "beef"), T(5.4, 43.0, "pork"), // weight, value, name T(3.6, 90.0, "ham"), T(2.4, 45.0, "greaves"), T(4.0, 30.0, "flitch"),T(2.5, 56.0, "brawn"), T(3.7, 67.0, "welt"), T(3.0, 95.0, "salami"), T(5.9, 98.0, "sausage"), ); fcn item_cmp(a,b){ a[1]/a[0] > b[1]/b[0] }
items.sort(item_cmp); space := 15.0; foreach it in (items){ w,_,nm:=it;
if (space >= w){ println("take all ",nm); space-=w }
else{ println("take %gkg of %gkg of %s".fmt(space,w,nm)); break }
}</lang>
- Output:
take all salami take all ham take all brawn take all greaves take 3.5kg of 3.7kg of welt
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