Knapsack problem/Unbounded
You are encouraged to solve this task according to the task description, using any language you may know.
A traveler gets diverted and has to make an unscheduled stop in what turns out to be Shangri La. Opting to leave, he is allowed to take as much as he likes of the following items, so long as it will fit in his knapsack, and he can carry it.
He knows that he can carry no more than 25 'weights' in total; and that the capacity of his knapsack is 0.25 'cubic lengths'.
Looking just above the bar codes on the items he finds their weights and volumes. He digs out his recent copy of a financial paper and gets the value of each item.
| Item | Explanation | Value (each) | weight | Volume (each) |
| panacea (vials of) | Incredible healing properties | 3000 | 0.3 | 0.025 |
| ichor (ampules of) | Vampires blood | 1800 | 0.2 | 0.015 |
| gold (bars) | Shiney shiney | 2500 | 2.0 | 0.002 |
| Knapsack | For the carrying of | - | <=25 | <=0.25 |
He can only take whole units of any item, but there is much more of any item than he could ever carry
- Task
Show how many of each item does he take to maximize the value of items he is carrying away with him.
- Note
- There are four solutions that maximize the value taken. Only one need be given.
- Related tasks
11l
T Bounty
Int value
Float weight, volume
F (value, weight, volume)
(.value, .weight, .volume) = (value, weight, volume)
V panacea = Bounty(3000, 0.3, 0.025)
V ichor = Bounty(1800, 0.2, 0.015)
V gold = Bounty(2500, 2.0, 0.002)
V sack = Bounty( 0, 25.0, 0.25)
V best = Bounty( 0, 0, 0)
V current = Bounty( 0, 0, 0)
V best_amounts = (0, 0, 0)
V max_panacea = Int(min(sack.weight I/ panacea.weight, sack.volume I/ panacea.volume))
V max_ichor = Int(min(sack.weight I/ ichor.weight, sack.volume I/ ichor.volume))
V max_gold = Int(min(sack.weight I/ gold.weight, sack.volume I/ gold.volume))
L(npanacea) 0 .< max_panacea
L(nichor) 0 .< max_ichor
L(ngold) 0 .< max_gold
current.value = npanacea * panacea.value + nichor * ichor.value + ngold * gold.value
current.weight = npanacea * panacea.weight + nichor * ichor.weight + ngold * gold.weight
current.volume = npanacea * panacea.volume + nichor * ichor.volume + ngold * gold.volume
I current.value > best.value & current.weight <= sack.weight & current.volume <= sack.volume
best = current
best_amounts = (npanacea, nichor, ngold)
print(‘Maximum value achievable is ’best.value)
print(‘This is achieved by carrying (one solution) #. panacea, #. ichor and #. gold’.format(best_amounts[0], best_amounts[1], best_amounts[2]))
print(‘The weight to carry is #2.1 and the volume used is #.3’.format(best.weight, best.volume))- Output:
Maximum value achievable is 54500 This is achieved by carrying (one solution) 0 panacea, 15 ichor and 11 gold The weight to carry is 25.0 and the volume used is 0.247
360 Assembly
The program uses two ASSIST macros (XDECO,XPRNT) to keep the code as short as possible.
* Knapsack problem/Unbounded 04/02/2017
KNAPSACK CSECT
USING KNAPSACK,R13 base register
B 72(R15) skip savearea
DC 17F'0' savearea
STM R14,R12,12(R13) prolog
ST R13,4(R15) " <-
ST R15,8(R13) " ->
LR R13,R15 " addressability
MVC S,=F'0' s(1,kva)=0;
LA R11,0 ns=0
LA R1,KW kw
SLA R1,2 *4
L R2,PANACEA-4(R1) panacea(kw)
L R4,SACKW sackw
SRDA R4,32 ~
DR R4,R2 sackw/panacea(kw)
ST R5,XP xp=sackw/panacea(kw)
LA R1,KV kv
SLA R1,2 *4
L R2,PANACEA-4(R1) panacea(kv)
L R4,SACKV sackv
SRDA R4,32 ~
DR R4,R2 r5=sackv/panacea(kv)
C R5,XP if r5<xp
BNL EMINXP
ST R5,XP xp=min(sackw/panacea(kw),sackv/panacea(kv))
EMINXP LA R1,KW kw
SLA R1,2 *4
L R2,ICHOR-4(R1) ichor(kw)
L R4,SACKW sackw
SRDA R4,32 ~
DR R4,R2 sackw/ichor(kw)
ST R5,XI xi=sackw/ichor(kw)
LA R1,KV kv
SLA R1,2 *4
L R2,ICHOR-4(R1) ichor(kv)
L R4,SACKV sackv
SRDA R4,32 ~
DR R4,R2 r5=sackv/ichor(kv)
C R5,XI if r5<xi
BNL EMINXI
ST R5,XI xi=min(sackw/ichor(kw),sackv/ichor(kv))
EMINXI LA R1,KW kw
SLA R1,2 *4
L R2,GOLD-4(R1) gold(kw)
L R4,SACKW sackw
SRDA R4,32 ~
DR R4,R2 sackw/gold(kw)
ST R5,XG xg=sackw/gold(kw)
LA R1,KV kv
SLA R1,2 *4
L R2,GOLD-4(R1) gold(kv)
L R4,SACKV sackv
SRDA R4,32 ~
DR R4,R2 r5=sackv/gold(kv)
C R5,XG if r5<xg
BNL EMINXG
ST R5,XG xg=min(sackw/gold(kw),sackv/gold(kv))
EMINXG SR R10,R10 ip=0
LOOPIP C R10,XP do ip=0 to xp
BH ELOOPIP
SR R9,R9 ii=0
LOOPII C R9,XI do ii=0 to xi
BH ELOOPII
SR R8,R8 ig=0
LOOPIG C R8,XG do ig=0 to xg
BH ELOOPIG
LA R7,KVA m=kva
LOOPM C R7,=A(KV) do m=kva to kv
BH ELOOPM
LR R1,R7 m
SLA R1,2 *4
LR R5,R8 ig
M R4,GOLD-4(R1) *gold(m)
LR R2,R5 r2=ig*gold(m)
LR R5,R9 ii
M R4,ICHOR-4(R1) *ichor(m)
AR R2,R5 r2=ig*gold(m)+ii*ichor(m)
LR R5,R10 ip
M R4,PANACEA-4(R1) *panacea(m)
AR R2,R5 r2=r2+ip*panacea(m)
ST R2,CUR-4(R1) cur(m)=r2
LA R7,1(R7) m=m+1
B LOOPM
ELOOPM LA R1,KVA kva
SLA R1,2 *4
L R2,CUR-4(R1) cur(kva)
C R2,S-4(R1) if cur(kva)>=s(1,kva)
BL ENDIF
LA R1,KW kw
SLA R1,2 *4
L R2,CUR-4(R1) cur(kw)
C R2,SACKW if cur(kw)<=sackw
BH ENDIF
LA R1,KV kv
SLA R1,2 *4
L R2,CUR-4(R1) cur(kv)
C R2,SACKV if cur(kv)<=sackv
BH ENDIF
LR R6,R11 j=ns
LOOPJ C R6,=F'1' do j=ns to 1 by -1
BL ELOOPJ
LR R1,R6 j
MH R1,=H'24' *24
LA R2,S(R1) s(j+1,1)
LA R3,S-24(R1) s(j,1)
MVC 0(24,R2),0(R3) s(j+1,*)=s(j,*)
BCTR R6,0 j=j-1
B LOOPJ
ELOOPJ LA R1,KVA kva
SLA R1,2 *4
L R2,CUR-4(R1) cur(kva)
ST R2,S-4(R1) s(1,kva)=cur(kva)
LA R1,KW kw
SLA R1,2 *4
L R2,CUR-4(R1) cur(kw)
ST R2,S-4(R1) s(1,kw)=cur(kw)
LA R1,KV kv
SLA R1,2 *4
L R2,CUR-4(R1) cur(kv)
ST R2,S-4(R1) s(1,kv)=cur(kv)
LA R1,KP kp
SLA R1,2 *4
ST R10,S-4(R1) s(1,kp)=ip
LA R1,KI ki
SLA R1,2 *4
ST R9,S-4(R1) s(1,ki)=ii
LA R1,KG kg
SLA R1,2 *4
ST R8,S-4(R1) s(1,kg)=ig
L R2,S r2=s(1,1)
C R2,S+24 if s(1,1)>s(2,1)
BNH ELSE
LA R11,1 ns=1
B ENDIF
ELSE LA R11,1(R11) ns+1
ENDIF LA R8,1(R8) ig=ig+1
B LOOPIG
ELOOPIG LA R9,1(R9) ii=ii+1
B LOOPII
ELOOPII LA R10,1(R10) ip=ip+1
B LOOPIP
ELOOPIP XPRNT TITLE,72
LA R6,1 j=1
LA R3,S-4 r3=@item
LOOPJP CR R6,R11 do j=1 to ns
BH ELOOPJP
LA R3,4(R3) ++
L R1,0(R3) s(j,kva)
XDECO R1,PG edit
LA R3,4(R3) ++
L R1,0(R3) s(j,kw)
XDECO R1,PG+12 edit
LA R3,4(R3) ++
L R1,0(R3) s(j,kv)
XDECO R1,PG+24 edit
MVC PG+20(2),PG+21 shift
MVI PG+22,C'.' decimal point
LA R3,4(R3) ++
L R1,0(R3) s(j,kp)
XDECO R1,PG+36 edit
MVC PG+31(2),=C'0.' decimal point
LA R3,4(R3) ++
L R1,0(R3) s(j,ki)
XDECO R1,PG+48 edit
LA R3,4(R3) ++
L R1,0(R3) s(j,kg)
XDECO R1,PG+60 edit
XPRNT PG,L'PG print buffer
LA R6,1(R6) j=j+1
B LOOPJP
ELOOPJP L R13,4(0,R13) epilog
LM R14,R12,12(R13) " restore
XR R15,R15 " rc=0
BR R14 exit
KVA EQU 1
KW EQU 2
KV EQU 3
KP EQU 4
KI EQU 5
KG EQU 6
SACKW DC F'250'
SACKV DC F'250'
PANACEA DC F'3000',F'3',F'25'
ICHOR DC F'1800',F'2',F'15'
GOLD DC F'2500',F'20',F'2'
XP DS F
XI DS F
XG DS F
CUR DS 3F
S DS 60F
TITLE DC CL36' Value Weight Volume'
DC CL36' Panacea Ichor Gold'
PG DS CL72
YREGS
END KNAPSACK- Output:
Value Weight Volume Panacea Ichor Gold
54500 24.7 0.247 9 0 11
54500 24.8 0.247 6 5 11
54500 24.9 0.247 3 10 11
54500 25.0 0.247 0 15 11
Ada
with Ada.Text_IO;
procedure Knapsack_Unbounded is
type Bounty is record
Value : Natural;
Weight : Float;
Volume : Float;
end record;
function Min (A, B : Float) return Float is
begin
if A < B then
return A;
else
return B;
end if;
end Min;
Panacea : Bounty := (3000, 0.3, 0.025);
Ichor : Bounty := (1800, 0.2, 0.015);
Gold : Bounty := (2500, 2.0, 0.002);
Limits : Bounty := ( 0, 25.0, 0.250);
Best : Bounty := ( 0, 0.0, 0.000);
Current : Bounty := ( 0, 0.0, 0.000);
Best_Amounts : array (1 .. 3) of Natural := (0, 0, 0);
Max_Panacea : Natural := Natural (Float'Floor (Min
(Limits.Weight / Panacea.Weight,
Limits.Volume / Panacea.Volume)));
Max_Ichor : Natural := Natural (Float'Floor (Min
(Limits.Weight / Ichor.Weight,
Limits.Volume / Ichor.Volume)));
Max_Gold : Natural := Natural (Float'Floor (Min
(Limits.Weight / Gold.Weight,
Limits.Volume / Gold.Volume)));
begin
for Panacea_Count in 0 .. Max_Panacea loop
for Ichor_Count in 0 .. Max_Ichor loop
for Gold_Count in 0 .. Max_Gold loop
Current.Value := Panacea_Count * Panacea.Value +
Ichor_Count * Ichor.Value +
Gold_Count * Gold.Value;
Current.Weight := Float (Panacea_Count) * Panacea.Weight +
Float (Ichor_Count) * Ichor.Weight +
Float (Gold_Count) * Gold.Weight;
Current.Volume := Float (Panacea_Count) * Panacea.Volume +
Float (Ichor_Count) * Ichor.Volume +
Float (Gold_Count) * Gold.Volume;
if Current.Value > Best.Value and
Current.Weight <= Limits.Weight and
Current.Volume <= Limits.Volume then
Best := Current;
Best_Amounts := (Panacea_Count, Ichor_Count, Gold_Count);
end if;
end loop;
end loop;
end loop;
Ada.Text_IO.Put_Line ("Maximum value:" & Natural'Image (Best.Value));
Ada.Text_IO.Put_Line ("Panacea:" & Natural'Image (Best_Amounts (1)));
Ada.Text_IO.Put_Line ("Ichor: " & Natural'Image (Best_Amounts (2)));
Ada.Text_IO.Put_Line ("Gold: " & Natural'Image (Best_Amounts (3)));
end Knapsack_Unbounded;
ALGOL 68
MODE BOUNTY = STRUCT(STRING name, INT value, weight, volume);
[]BOUNTY items = (
("panacea", 3000, 3, 25),
("ichor", 1800, 2, 15),
("gold", 2500, 20, 2)
);
BOUNTY sack := ("sack", 0, 250, 250);
OP * = ([]INT a,b)INT: ( # dot product operator #
INT sum := 0;
FOR i TO UPB a DO sum +:= a[i]*b[i] OD;
sum
);
OP INIT = (REF[]INT vector)VOID:
FOR index FROM LWB vector TO UPB vector DO
vector[index]:=0
OD;
OP INIT = (REF[,]INT matrix)VOID:
FOR row index FROM LWB matrix TO UPB matrix DO
INIT matrix[row index,]
OD;
PROC total value = ([]INT items count, []BOUNTY items, BOUNTY sack) STRUCT(INT value, weight, volume):(
###
Given the count of each item in the sack return -1 if they can"t be carried or their total value.
(also return the negative of the weight and the volume so taking the max of a series of return
values will minimise the weight if values tie, and minimise the volume if values and weights tie).
###
INT weight = items count * weight OF items;
INT volume = items count * volume OF items;
IF weight > weight OF sack OR volume > volume OF sack THEN
(-1, 0, 0)
ELSE
( items count * value OF items, -weight, -volume)
FI
);
PRIO WRAP = 5; # wrap negative array indices as per python's indexing regime #
OP WRAP = (INT index, upb)INT:
IF index>=0 THEN index ELSE upb + index + 1 FI;
PROC knapsack dp = ([]BOUNTY items, BOUNTY sack)[]INT:(
###
Solves the Knapsack problem, with two sets of weights,
using a dynamic programming approach
###
# (weight+1) x (volume+1) table #
# table[w,v] is the maximum value that can be achieved #
# with a sack of weight w and volume v. #
# They all start out as 0 (empty sack) #
[0:weight OF sack, 0:volume OF sack]INT table; INIT table;
FOR w TO 1 UPB table DO
FOR v TO 2 UPB table DO
### Consider the optimal solution, and consider the "last item" added
to the sack. Removing this item must produce an optimal solution
to the subproblem with the sack"s weight and volume reduced by that
of the item. So we search through all possible "last items": ###
FOR item index TO UPB items DO
BOUNTY item := items[item index];
# Only consider items that would fit: #
IF w >= weight OF item AND v >= volume OF item THEN
# Optimal solution to subproblem + value of item: #
INT candidate := table[w-weight OF item,v-volume OF item] + value OF item;
IF candidate > table[w,v] THEN
table[w,v] := candidate
FI
FI
OD
OD
OD;
[UPB items]INT result; INIT result;
INT w := weight OF sack, v := volume OF sack;
WHILE table[w,v] /= 0 DO
# Find the last item that was added: #
INT needle = table[w,v];
INT item index;
FOR i TO UPB items WHILE
item index := i;
BOUNTY item = items[item index];
INT candidate = table[w-weight OF item WRAP UPB table, v-volume OF item WRAP 2 UPB table] + value OF item;
# WHILE # candidate NE needle DO
SKIP
OD;
# Record it in the result, and remove it: #
result[item index] +:= 1;
w -:= weight OF items[item index];
v -:= volume OF items[item index]
OD;
result
);
[]INT max items = knapsack dp(items, sack);
STRUCT (INT value, weight, volume) max := total value(max items, items, sack);
max := (value OF max, -weight OF max, -volume OF max);
FORMAT d = $zz-d$;
printf(($"The maximum value achievable (by dynamic programming) is "gl$, value OF max));
printf(($" The number of ("n(UPB items-1)(g", ")g") items to achieve this is: ("n(UPB items-1)(f(d)",")f(d)") respectively"l$,
name OF items, max items));
printf(($" The weight to carry is "f(d)", and the volume used is "f(d)l$,
weight OF max, volume OF max))Output:
The maximum value achievable (by dynamic programming) is +54500 The number of (panacea, ichor, gold) items to achieve this is: ( 9, 0, 11) respectively The weight to carry is 247, and the volume used is 247
AutoHotkey
Brute Force.
Item = Panacea,Ichor,Gold
Value = 3000,1800,2500
Weight= 3,2,20 ; *10
Volume= 25,15,2 ; *1000
StringSplit I, Item, `, ; Put input in arrays
StringSplit W, Weight,`,
StringSplit $, Value, `,
StringSplit V, Volume,`,
SetFormat Float, 0.3
W := 250, V := 250, sW:=.1, sV:=.001 ; limits for the total, scale factors
p := -1, Wp := -W1, Vp := -V1 ; initial values
While (Wp+=W1) <= W && (Vp+=V1) <= V {
p++, Wi := Wp-W2, Vi := Vp-V2, i := -1
While (Wi+=W2) <= W && (Vi+=V2) <= V {
i++, Wg := Wi-W3, Vg := Vi-V3, g := -1
While (Wg+=W3) <= W && (Vg+=V3) <= V
If ($ <= Val := p*$1 + i*$2 + ++g*$3)
t := ($=Val ? t "`n " : " ")
. p "`t " i "`t " g "`t " Wg*sW "`t " Vg*sV
, $ := Val
}
}
MsgBox Value = %$%`n`nPanacea`tIchor`tGold`tWeight`tVolume`n%t%
Bracmat
(knapsack=
( things
= (panacea.3000.3/10.25/1000)
(ichor.1800.2/10.15/1000)
(gold.2500.2.2/1000)
)
& 0:?maxvalue
& :?sack
& ( add
= cumwght
cumvol
cumvalue
cumsack
name
wght
val
vol
tings
n
ncumwght
ncumvalue
ncumvol
. !arg
: ( ?cumwght
. ?cumvol
. ?cumvalue
. ?cumsack
. (?name.?val.?wght.?vol) ?tings
)
& -1:?n
& whl
' ( 1+!n:?n
& !cumwght+!n*!wght:~>25:?ncumwght
& !cumvol+!n*!vol:~>250/1000:?ncumvol
& !cumvalue+!n*!val:?ncumvalue
& ( !tings:
& ( !ncumvalue:>!maxvalue:?maxvalue
& !cumsack
( !n:0&
| ( !cumsack:&Take
| Finally
)
" take "
!n
" items of "
!name
".\n"
)
: ?sack
|
)
| add
$ ( !ncumwght
. !ncumvol
. !ncumvalue
. !cumsack
( !n:0&
| "Take " !n " items of " !name ".\n"
)
. !tings
)
)
)
)
& add$(0.0.0..!things)
& out$(str$(!sack "The value in the knapsack is " !maxvalue "."))
&
);
!knapsack;Output:
Take 15 items of ichor. Finally take 11 items of gold. The value in the knapsack is 54500.
C
figures out the best (highest value) set by brute forcing every possible subset.
#include <stdio.h>
#include <stdlib.h>
typedef struct {
char *name;
double value;
double weight;
double volume;
} item_t;
item_t items[] = {
{"panacea", 3000.0, 0.3, 0.025},
{"ichor", 1800.0, 0.2, 0.015},
{"gold", 2500.0, 2.0, 0.002},
};
int n = sizeof (items) / sizeof (item_t);
int *count;
int *best;
double best_value;
void knapsack (int i, double value, double weight, double volume) {
int j, m1, m2, m;
if (i == n) {
if (value > best_value) {
best_value = value;
for (j = 0; j < n; j++) {
best[j] = count[j];
}
}
return;
}
m1 = weight / items[i].weight;
m2 = volume / items[i].volume;
m = m1 < m2 ? m1 : m2;
for (count[i] = m; count[i] >= 0; count[i]--) {
knapsack(
i + 1,
value + count[i] * items[i].value,
weight - count[i] * items[i].weight,
volume - count[i] * items[i].volume
);
}
}
int main () {
count = malloc(n * sizeof (int));
best = malloc(n * sizeof (int));
best_value = 0;
knapsack(0, 0.0, 25.0, 0.25);
int i;
for (i = 0; i < n; i++) {
printf("%d %s\n", best[i], items[i].name);
}
printf("best value: %.0f\n", best_value);
free(count); free(best);
return 0;
}
- Output:
9 panacea0 ichor 11 gold best value: 54500
C#
/* Items Value Weight Volume
a 30 3 25
b 18 2 15
c 25 20 2
<=250 <=250 */
using System;
class Program
{
static void Main()
{
uint[] r = items1();
Console.WriteLine(r[0] + " v " + r[1] + " a " + r[2] + " b"); // 0 15 11
var sw = System.Diagnostics.Stopwatch.StartNew();
for (int i = 1000; i > 0; i--) items1();
Console.Write(sw.Elapsed); Console.Read();
}
static uint[] items0() // 1.2 µs
{
uint v, v0 = 0, a, b, c, a0 = 0, b0 = 0, c0 = 0;
for (a = 0; a <= 10; a++)
for (b = 0; a * 5 + b * 3 <= 50; b++)
for (c = 0; a * 25 + b * 15 + c * 2 <= 250 && a * 3 + b * 2 + c * 20 <= 250; c++)
if (v0 < (v = a * 30 + b * 18 + c * 25))
{
v0 = v; a0 = a; b0 = b; c0 = c;
//Console.WriteLine("{0,5} {1,5} {2,5} {3,5}", v, a, b, c);
}
return new uint[] { a0, b0, c0 };
}
static uint[] items1() // 0,22 µs
{
uint v, v0 = 0, a, b, c, a0 = 0, b0 = 0, c0 = 0, c1 = 0;
for (a = 0; a <= 10; a++)
for (b = 0; a * 5 + b * 3 <= 50; b++)
{
c = (250 - a * 25 - b * 15) / 2;
if ((c1 = (250 - a * 3 - b * 2) / 20) < c) c = c1;
if (v0 < (v = a * 30 + b * 18 + c * 25))
{ v0 = v; a0 = a; b0 = b; c0 = c; }
}
return new uint[] { a0, b0, c0 };
}
}
C_sharp
/*Knapsack
This model finds the integer optimal packing of a knapsack
Nigel_Galloway
January 29th., 2012
*/
using Microsoft.SolverFoundation.Services;
namespace KnapU
{
class Item {
public string Name {get; set;}
public int Value {get; set;}
public double Weight {get; set;}
public double Volume {get; set;}
public Item(string name, int value, double weight, double volume) {
Name = name;
Value = value;
Weight = weight;
Volume = volume;
}
}
class Program
{
static void Main(string[] args)
{
SolverContext context = SolverContext.GetContext();
Model model = context.CreateModel();
Item[] Knapsack = new Item[] {
new Item("Panacea", 3000, 0.3, 0.025),
new Item("Ichor", 1800, 0.2, 0.015),
new Item("Gold", 2500, 2.0, 0.002)
};
Set items = new Set(Domain.Any, "items");
Decision take = new Decision(Domain.IntegerNonnegative, "take", items);
model.AddDecision(take);
Parameter value = new Parameter(Domain.IntegerNonnegative, "value", items);
value.SetBinding(Knapsack, "Value", "Name");
Parameter weight = new Parameter(Domain.RealNonnegative, "weight", items);
weight.SetBinding(Knapsack, "Weight", "Name");
Parameter volume = new Parameter(Domain.RealNonnegative, "volume", items);
volume.SetBinding(Knapsack, "Volume", "Name");
model.AddParameters(value, weight, volume);
model.AddConstraint("knap_weight", Model.Sum(Model.ForEach(items, t => take[t] * weight[t])) <= 25);
model.AddConstraint("knap_vol", Model.Sum(Model.ForEach(items, t => take[t] * volume[t])) <= 0.25);
model.AddGoal("knap_value", GoalKind.Maximize, Model.Sum(Model.ForEach(items, t => take[t] * value[t])));
Solution solution = context.Solve(new SimplexDirective());
Report report = solution.GetReport();
System.Console.Write("{0}", report);
}
}
}
Produces:
===Solver Foundation Service Report=== Date: 28/01/2012 17:18:56 Version: Microsoft Solver Foundation 3.0.1.10599 Express Edition Model Name: DefaultModel Capabilities Applied: MILP Solve Time (ms): 210 Total Time (ms): 376 Solve Completion Status: Optimal Solver Selected: Microsoft.SolverFoundation.Solvers.SimplexSolver Directives: Simplex(TimeLimit = -1, MaximumGoalCount = -1, Arithmetic = Default, Pricing = D efault, IterationLimit = -1, Algorithm = Default, Basis = Default, GetSensitivit y = False) Algorithm: Dual Arithmetic: Double Variables: 3 -> 3 + 3 Rows: 3 -> 3 Nonzeros: 9 Eliminated Slack Variables: 0 Pricing (double): SteepestEdge Basis: Current Pivot Count: 0 Phase 1 Pivots: 0 + 0 Phase 2 Pivots: 0 + 0 Factorings: 3 + 0 Degenerate Pivots: 0 (0.00 %) Branches: 21 ===Solution Details=== Goals: knap_value: 54500 Decisions: take(Panacea): 9 take(Ichor): 0 take(Gold): 11
C++
#include <algorithm>
#include <cmath>
#include <cstdint>
#include <iomanip>
#include <iostream>
#include <string>
#include <vector>
struct Item {
std::string name;
int32_t value;
double weight;
double volume;
};
const std::vector<Item> items = {
Item("panacea", 3000, 0.3, 0.025),
Item("ichor", 1800, 0.2, 0.015),
Item("gold", 2500, 2.0, 0.002)
};
constexpr double MAX_WEIGHT = 25.0;
constexpr double MAX_VOLUME = 0.25;
std::vector<int32_t> count(items.size(), 0);
std::vector<int32_t> best(items.size(), 0);
int32_t best_value = 0;
void knapsack(const uint64_t& i, const int32_t& value, const double& weight, const double& volume) {
if ( i == items.size() ) {
if ( value > best_value ) {
best_value = value;
best = count;
}
return;
}
int32_t measure1 = (int32_t) std::floor( weight / items[i].weight );
int32_t measure2 = (int32_t) std::floor( volume / items[i].volume );
int32_t measure = std::min(measure1, measure2);
count[i] = measure;
while ( count[i] >= 0 ) {
knapsack(
i + 1,
value + count[i] * items[i].value,
weight - count[i] * items[i].weight,
volume - count[i] * items[i].volume
);
count[i]--;
}
}
int main() {
knapsack(0, 0, MAX_WEIGHT, MAX_VOLUME);
std::cout << "Item Chosen Number Value Weight Volume" << std::endl;
std::cout << "----------- ------ ----- ------ ------" << std::endl;
int32_t item_count = 0;
int32_t number_sum = 0;
double weight_sum = 0.0;
double volume_sum = 0.0;
for ( uint64_t i = 0; i < items.size(); ++i ) {
if ( best[i] == 0 ) {
continue;
}
item_count++;
std::string name = items[i].name;
int32_t number = best[i];
int32_t value = items[i].value * number;
double weight = items[i].weight * number;
double volume = items[i].volume * number;
number_sum += number;
weight_sum += weight;
volume_sum += volume;
std::cout << std::setw(11) << name << std::setw(6) << number << std::setw(8) << value << std::fixed
<< std::setw(8) << std::setprecision(2) << weight
<< std::setw(8) << std::setprecision(2) << volume << std::endl;
}
std::cout << "----------- ------ ----- ------ ------" << std::endl;
std::cout << std::setw(5) << item_count << " items" << std::setw(6) << number_sum << std::setw(8) << best_value
<< std::setw(8) << weight_sum << std::setw(8) << volume_sum << std::endl;
}
- Output:
Item Chosen Number Value Weight Volume
----------- ------ ----- ------ ------
panacea 9 27000 2.70 0.23
gold 11 27500 22.00 0.02
----------- ------ ----- ------ ------
2 items 20 54500 24.70 0.25
Clojure
(defstruct item :value :weight :volume)
(defn total [key items quantities]
(reduce + (map * quantities (map key items))))
(defn max-count [item max-weight max-volume]
(let [mcw (/ max-weight (:weight item))
mcv (/ max-volume (:volume item))]
(min mcw mcv)))
We have an item struct to contain the data for both contents and the knapsack. The total function returns the sum of a particular attribute across all items times their quantities. Finally, the max-count function returns the most of that item that could fit given the constraints (used as the upper bound on the combination). Now the real work:
(defn knapsacks []
(let [pan (struct item 3000 0.3 0.025)
ich (struct item 1800 0.2 0.015)
gol (struct item 2500 2.0 0.002)
types [pan ich gol]
max-w 25.0
max-v 0.25
iters #(range (inc (max-count % max-w max-v)))]
(filter (complement nil?)
(pmap
#(let [[p i g] %
w (total :weight types %)
v (total :volume types %)]
(if (and (<= w max-w) (<= v max-v))
(with-meta (struct item (total :value types %) w v) {:p p :i i :g g})))
(for [p (iters pan)
i (iters ich)
g (iters gol)]
[p i g])))))
The knapsacks function returns a lazy sequence of all valid knapsacks, with the particular content quantities as metadata. The work of realizing each knapsack is done in parallel via the pmap function. The following then finds the best by value, and prints the result.
(defn best-by-value [ks]
(reduce #(if (> (:value %1) (:value %2)) %1 %2) ks))
(defn print-knapsack[k]
(let [ {val :value w :weight v :volume} k
{p :p i :i g :g} ^k]
(println "Maximum value:" (float val))
(println "Total weight: " (float w))
(println "Total volume: " (float v))
(println "Containing: " p "Panacea," i "Ichor," g "Gold")))
Calling (print-knapsack (best-by-value (knapsacks))) would result in something like:
Maximum value: 54500 Total weight: 24.7 Total volume: 0.247 Containing: 9 Panacea, 0 Ichor, 11 Gold
Further, we could find all "best" knapsacks rather simply (albeit at the cost of some efficiency):
(defn all-best-by-value [ks]
(let [b (best-by-value ks)]
(filter #(= (:value b) (:value %)) ks)))
(defn print-knapsacks [ks]
(doseq [k ks]
(print-knapsack k)
(println)))
Calling (print-knapsacks (all-best-by-value (knapsacks))) would result in something like:
Maximum value: 54500.0 Total weight: 25.0 Total volume: 0.247 Containing: 0 Panacea, 15 Ichor, 11 Gold Maximum value: 54500.0 Total weight: 24.9 Total volume: 0.247 Containing: 3 Panacea, 10 Ichor, 11 Gold Maximum value: 54500.0 Total weight: 24.8 Total volume: 0.247 Containing: 6 Panacea, 5 Ichor, 11 Gold Maximum value: 54500.0 Total weight: 24.7 Total volume: 0.247 Containing: 9 Panacea, 0 Ichor, 11 Gold
Common Lisp
A dynamic programming O(maxVolume × maxWeight × nItems) solution, where volumes and weights are integral values.
(defun fill-knapsack (items max-volume max-weight)
"Items is a list of lists of the form (name value weight volume) where weight
and value are integers. max-volume and max-weight, also integers, are the
maximum volume and weight of the knapsack. fill-knapsack returns a list of the
form (total-value inventory total-volume total-weight) where total-value is the
total-value of a knapsack packed with inventory (a list whose elements are
elements of items), and total-weight and total-volume are the total weights and
volumes of the inventory."
;; maxes is a table indexed by volume and weight, where maxes[volume,weight]
;; is a list of the form (value inventory used-volume used-weight) where
;; inventory is a list of items of maximum value fitting within volume and
;; weight, value is the maximum value, and used-volume/used-weight are the
;; actual volume/weight of the inventory.
(let* ((VV (1+ max-volume))
(WW (1+ max-weight))
(maxes (make-array (list VV WW))))
;; fill in the base cases where volume or weight is 0
(dotimes (v VV) (setf (aref maxes v 0) (list 0 '() 0 0)))
(dotimes (w WW) (setf (aref maxes 0 w) (list 0 '() 0 0)))
;; populate the rest of the table. The best value for a volume/weight
;; combination is the best way of adding an item to any of the inventories
;; from [volume-1,weight], [volume,weight-1], or [volume-1,weight-1], or the
;; best of these, if no items can be added.
(do ((v 1 (1+ v))) ((= v VV) (aref maxes max-volume max-weight))
(do ((w 1 (1+ w))) ((= w WW))
(let ((options (sort (list (aref maxes v (1- w))
(aref maxes (1- v) w)
(aref maxes (1- v) (1- w)))
'> :key 'first)))
(destructuring-bind (b-value b-items b-volume b-weight) (first options)
(dolist (option options)
(destructuring-bind (o-value o-items o-volume o-weight) option
(dolist (item items)
(destructuring-bind (_ i-value i-volume i-weight) item
(declare (ignore _))
(when (and (<= (+ o-volume i-volume) v)
(<= (+ o-weight i-weight) w)
(> (+ o-value i-value) b-value))
(setf b-value (+ o-value i-value)
b-volume (+ o-volume i-volume)
b-weight (+ o-weight i-weight)
b-items (list* item o-items)))))))
(setf (aref maxes v w)
(list b-value b-items b-volume b-weight))))))))
Use, having multiplied volumes and weights as to be integral:
> (pprint (fill-knapsack '((panacea 3000 3 25)
(ichor 1800 2 15)
(gold 2500 20 2))
250
250))
(54500 ; total-value
((ICHOR 1800 2 15) ; 15 ichor
...
(ICHOR 1800 2 15)
(GOLD 2500 20 2) ; 11 gold
...
(GOLD 2500 20 2))
250 ; total volume
247) ; total weight
D
void main() @safe /*@nogc*/ {
import std.stdio, std.algorithm, std.typecons, std.conv;
static struct Bounty {
int value;
double weight, volume;
}
immutable Bounty panacea = {3000, 0.3, 0.025};
immutable Bounty ichor = {1800, 0.2, 0.015};
immutable Bounty gold = {2500, 2.0, 0.002};
immutable Bounty sack = { 0, 25.0, 0.25 };
immutable maxPanacea = min(sack.weight / panacea.weight,
sack.volume / panacea.volume).to!int;
immutable maxIchor = min(sack.weight / ichor.weight,
sack.volume / ichor.volume).to!int;
immutable maxGold = min(sack.weight / gold.weight,
sack.volume / gold.volume).to!int;
Bounty best = {0, 0, 0};
Tuple!(int, int, int) bestAmounts;
foreach (immutable nPanacea; 0 .. maxPanacea)
foreach (immutable nIchor; 0 .. maxIchor)
foreach (immutable nGold; 0 .. maxGold) {
immutable Bounty current = {
value: nPanacea * panacea.value +
nIchor * ichor.value +
nGold * gold.value,
weight: nPanacea * panacea.weight +
nIchor * ichor.weight +
nGold * gold.weight,
volume: nPanacea * panacea.volume +
nIchor * ichor.volume +
nGold * gold.volume};
if (current.value > best.value &&
current.weight <= sack.weight &&
current.volume <= sack.volume) {
best = Bounty(current.value, current.weight, current.volume);
bestAmounts = tuple(nPanacea, nIchor, nGold);
}
}
writeln("Maximum value achievable is ", best.value);
writefln("This is achieved by carrying (one solution) %d" ~
" panacea, %d ichor and %d gold", bestAmounts[]);
writefln("The weight to carry is %4.1f and the volume used is %5.3f",
best.weight, best.volume);
}
- Output:
Maximum value achievable is 54500 This is achieved by carrying (one solution) 0 panacea, 15 ichor and 11 gold The weight to carry is 25.0 and the volume used is 0.247
Alternative Version
The output is the same.
void main() {
import std.stdio, std.algorithm, std.typecons, std.range, std.conv;
alias Bounty = Tuple!(int,"value", double,"weight", double,"volume");
immutable panacea = Bounty(3000, 0.3, 0.025);
immutable ichor = Bounty(1800, 0.2, 0.015);
immutable gold = Bounty(2500, 2.0, 0.002);
immutable sack = Bounty( 0, 25.0, 0.25);
immutable maxPanacea = min(sack.weight / panacea.weight, sack.volume / panacea.volume).to!int;
immutable maxIchor = min(sack.weight / ichor.weight, sack.volume / ichor.volume).to!int;
immutable maxGold = min(sack.weight / gold.weight, sack.volume / gold.volume).to!int;
immutable best =
cartesianProduct(maxPanacea.iota, maxIchor.iota, maxGold.iota)
.map!(t => tuple(Bounty(t[0] * panacea.value + t[1] * ichor.value + t[2] * gold.value,
t[0] * panacea.weight + t[1] * ichor.weight + t[2] * gold.weight,
t[0] * panacea.volume + t[1] * ichor.volume + t[2] * gold.volume), t))
.filter!(t => t[0].weight <= sack.weight && t[0].volume <= sack.volume)
.reduce!max;
writeln("Maximum value achievable is ", best[0].value);
writefln("This is achieved by carrying (one solution) %d panacea, %d ichor and %d gold", best[1][]);
writefln("The weight to carry is %4.1f and the volume used is %5.3f", best[0][1..$]);
}
E
This is a mostly brute-force general solution (the first author of this example does not know dynamic programming); the only optimization is that when considering the last (third) treasure type, it does not bother filling with anything but the maximum amount.
pragma.enable("accumulator")
/** A data type representing a bunch of stuff (or empty space). */
def makeQuantity(value, weight, volume, counts) {
def quantity {
to __printOn(out) {
for name => n in counts { out.print(`$n $name `) }
out.print(`(val=$value wt=$weight vol=$volume)`)
}
to value () { return value }
to weight() { return weight }
to volume() { return volume }
to counts() { return counts }
to subtract(other) { return quantity + other * -1 }
to add(other) {
return makeQuantity(value + other.value (),
weight + other.weight(),
volume + other.volume(),
accum counts for name => n in other.counts() { _.with(name, n+counts.fetch(name, fn {0})) })
}
to multiply(scalar) {
return makeQuantity(value * scalar,
weight * scalar,
volume * scalar,
accum [].asMap() for name => n in counts { _.with(name, n*scalar) })
}
/** a.fit(b) the greatest integer k such that a - b * k does not have negative weight or volume. */
to fit(item) {
return (weight // item.weight()) \
.min(volume // item.volume())
}
}
return quantity
}
/** Fill the space with the treasures, returning candidate results as spaceAvailable - the items. */
def fill(spaceAvailable, treasures) {
if (treasures.size().isZero()) { # nothing to pick
return [spaceAvailable]
}
# Pick one treasure type
def [unit] + otherTreasures := treasures
var results := []
for count in (0..spaceAvailable.fit(unit)).descending() {
results += fill(spaceAvailable - unit * count, otherTreasures)
if (otherTreasures.size().isZero()) {
break # If there are no further kinds, there is no point in taking less than the most
}
}
return results
}
def chooseBest(emptyKnapsack, treasures) {
var maxValue := 0
var best := []
for result in fill(emptyKnapsack, treasures) {
def taken := emptyKnapsack - result # invert the backwards result fill() returns
if (taken.value() > maxValue) {
best := [taken]
maxValue := taken.value()
} else if (taken.value() <=> maxValue) {
best with= taken
}
}
return best
}
def printBest(emptyKnapsack, treasures) {
for taken in chooseBest(emptyKnapsack, treasures) { println(` $taken`) }
}
def panacea := makeQuantity(3000, 0.3, 0.025, ["panacea" => 1])
def ichor := makeQuantity(1800, 0.2, 0.015, ["ichor" => 1])
def gold := makeQuantity(2500, 2.0, 0.002, ["gold" => 1])
def emptyKnapsack \
:= makeQuantity( 0, 25, 0.250, [].asMap())
printBest(emptyKnapsack, [panacea, ichor, gold])EasyLang
names$[] = [ "panacea" "ichor" "gold" ]
vals[] = [ 3000 1800 2500 ]
wgts[] = [ 0.3 0.2 2.0 ]
vols[] = [ 0.025 0.015 0.002 ]
maxwgt = 25
maxvol = 0.25
#
n = len names$[]
len cnt[] n
global best[] bestval .
#
proc knapsack i val wgt vol . .
if i > n
if val > bestval
bestval = val
best[] = cnt[]
.
return
.
cnt[i] = lower floor (wgt / wgts[i]) floor (vol / vols[i])
while cnt[i] >= 0
knapsack i + 1 val + cnt[i] * vals[i] wgt - cnt[i] * wgts[i] vol - cnt[i] * vols[i]
cnt[i] -= 1
.
.
knapsack 1 0 maxwgt maxvol
for i to n
if best[i] > 0
print best[i] & " " & names$[i]
.
.
- Output:
9 panacea 11 gold
EchoLisp
Use a cache, and multiply by 10^n to get an integer problem.
(require 'struct)
(require 'hash)
(require 'sql)
(define H null) ;; cache
(define T (make-table (struct goodies (name valeur poids volume ))))
(define-syntax-rule (name i) (table-xref T i 0))
(define-syntax-rule (valeur i) (table-xref T i 1))
(define-syntax-rule (poids i) (table-xref T i 2))
(define-syntax-rule (volume i) (table-xref T i 3))
(define goodies
'(("🍁-panacea" 3000 300 25)
("🌵-ichor" 1800 200 15)
("⭐️-gold" 2500 2000 2)))
(list->table goodies T)
;; i = item index, p= remaining weight, v = remaining volume
;; make an unique hash-key from (i p v)
(define (t-key i p v) (string-append i "|" p "|" v))
;; retrieve best core for item i
;; returns ( score . quantity)
(define (t-get i p v)
(if ( < i 0) (cons 0 0)
(hash-ref H (t-key i p v )))) ;; may be #f
;; compute best quantity.score (i), assuming best (i-1 p v) is known
(define (score-qty i p v (q) (score)(smax)(qmax))
(or
(t-get i p v) ;; already known
(begin
(set! q (min (quotient p (poids i)) (quotient v (volume i)))) ;; max possible q
(set! smax -Infinity)
( for ((k (1+ q))) ;; try all legal quantities
(set! score (+
(first (score-qty (1- i) (- p (* k (poids i))) (- v (* k (volume i)))))
(* k (valeur i))))
#:continue (< score smax)
(set! smax score)
(set! qmax k))
(hash-set H (t-key i p v) (cons smax qmax)))))
;; compute best scores, starting from last item
(define (task P V)
(define N (1- (table-count T)))
(define qty 0)
(set! H (make-hash))
(writeln 'total-value (first (score-qty N P V)))
(for/list ((i (in-range N -1 -1)))
(set! qty (rest (t-get i P V)))
#:continue (= qty 0)
(begin0
(cons (name i) (t-get i P V))
(set! P (- P (* (poids i) qty)))
(set! V (- V (* (volume i) qty))))))
;; output
(task 25000 250)
total-value 54500
→ (("⭐️-gold" 54500 . 11) ("🌵-ichor" 27000 . 15))
(length (hash-keys H)) ;; # entries in cache
→ 218
Eiffel
class
KNAPSACK
create
make
feature
make
do
create panacea;
panacea := [3000, 0.3, 0.025]
create ichor;
ichor := [1800, 0.2, 0.015]
create gold;
gold := [2500, 2.0, 0.002]
create sack;
sack := [0, 25.0, 0.25]
find_solution
end
feature {NONE}
panacea: TUPLE [value: INTEGER; weight: REAL_64; volume: REAL_64]
ichor: TUPLE [value: INTEGER; weight: REAL_64; volume: REAL_64]
gold: TUPLE [value: INTEGER; weight: REAL_64; volume: REAL_64]
sack: TUPLE [value: INTEGER; weight: REAL_64; volume: REAL_64]
find_solution
-- Solution for unbounded Knapsack Problem.
local
totalweight, totalvolume: REAL_64
maxpanacea, maxichor, maxvalue, maxgold: INTEGER
n: ARRAY [INTEGER]
r: TUPLE [value: INTEGER; weight: REAL_64; volume: REAL_64]
do
maxpanacea := minimum (sack.weight / panacea.weight, sack.volume / panacea.volume).rounded
maxichor := minimum (sack.weight / ichor.weight, sack.volume / ichor.volume).rounded
maxgold := minimum (sack.weight / gold.weight, sack.volume / gold.volume).rounded
create n.make_filled (0, 1, 3)
create r
across
0 |..| maxpanacea as p
loop
across
0 |..| maxichor as i
loop
across
0 |..| maxgold as g
loop
r.value := g.item * gold.value + i.item * ichor.value + p.item * panacea.value
r.weight := g.item * gold.weight + i.item * ichor.weight + p.item * panacea.weight
r.volume := g.item * gold.volume + i.item * ichor.volume + p.item * panacea.volume
if r.value > maxvalue and r.weight <= sack.weight and r.volume <= sack.volume then
maxvalue := r.value
totalweight := r.weight
totalvolume := r.volume
n [1] := p.item
n [2] := i.item
n [3] := g.item
end
end
end
end
io.put_string ("Maximum value achievable is " + maxValue.out + ".%N")
io.put_string ("This is achieved by carrying " + n [1].out + " panacea, " + n [2].out + " ichor and " + n [3].out + " gold.%N")
io.put_string ("The weight is " + totalweight.out + " and the volume is " + totalvolume.truncated_to_real.out + ".")
end
minimum (a, b: REAL_64): REAL_64
-- Smaller of 'a' and 'b'.
do
Result := a
if a > b then
Result := b
end
end
end
- Output:
Maximum value achievable is 54500. This is achieved by carrying 0 panacea, 15 ichor and 11 gold. The weight is 25 and the volume is 0.247.
Elixir
Brute Force:
defmodule Item do
defstruct volume: 0.0, weight: 0.0, value: 0
def new(volume, weight, value) do
%__MODULE__{volume: volume, weight: weight, value: value}
end
end
defmodule Knapsack do
def solve_unbounded(items, maximum) do
{max_volume, max_weight} = {maximum.volume, maximum.weight}
max_items = Enum.map(items, fn {name,item} ->
{name, trunc(min(max_volume / item.volume, max_weight / item.weight))}
end)
Enum.map(max_items, fn {name,max} -> for i <- 0..max, do: {name,i} end)
|> product
|> total(items)
|> Enum.filter(fn {_kw, {volume,weight,_}} -> volume <= max_volume and
weight <= max_weight end)
|> Enum.group_by(fn {_kw, {_,_,value}} -> value end)
|> Enum.max
|> print
end
defp product([x]), do: x
defp product([a,b]), do: for x <- a, y <- b, do: [x,y]
defp product([h|t]), do: for x <- h, y <- product(t), do: [x | y]
defp total(lists, items) do
Enum.map(lists, fn kwlist ->
total = Enum.reduce(kwlist, {0,0,0}, fn {name,n},{volume,weight,value} ->
{volume + n * items[name].volume,
weight + n * items[name].weight,
value + n * items[name].value}
end)
{kwlist, total}
end)
end
defp print({max_value, data}) do
IO.puts "Maximum value achievable is #{max_value}\tvolume weight value"
Enum.each(data, fn {kw,{volume,weight,value}} ->
:io.format "~s =>\t~6.3f, ~5.1f, ~6w~n", [(inspect kw), volume, weight, value]
end)
end
end
items = %{panacea: Item.new(0.025, 0.3, 3000),
ichor: Item.new(0.015, 0.2, 1800),
gold: Item.new(0.002, 2.0, 2500) }
maximum = Item.new(0.25, 25, 0)
Knapsack.solve_unbounded(items, maximum)
- Output:
Maximum value achievable is 54500 volume weight value [gold: 11, ichor: 0, panacea: 9] => 0.247, 24.7, 54500 [gold: 11, ichor: 5, panacea: 6] => 0.247, 24.8, 54500 [gold: 11, ichor: 10, panacea: 3] => 0.247, 24.9, 54500 [gold: 11, ichor: 15, panacea: 0] => 0.247, 25.0, 54500
Factor
This is a brute force solution. It is general enough to be able to provide solutions for any number of different items.
USING: accessors combinators kernel locals math math.order
math.vectors sequences sequences.product combinators.short-circuit ;
IN: knapsack
CONSTANT: values { 3000 1800 2500 }
CONSTANT: weights { 0.3 0.2 2.0 }
CONSTANT: volumes { 0.025 0.015 0.002 }
CONSTANT: max-weight 25.0
CONSTANT: max-volume 0.25
TUPLE: bounty amounts value weight volume ;
: <bounty> ( items -- bounty )
[ bounty new ] dip {
[ >>amounts ]
[ values v. >>value ]
[ weights v. >>weight ]
[ volumes v. >>volume ]
} cleave ;
: valid-bounty? ( bounty -- ? )
{ [ weight>> max-weight <= ]
[ volume>> max-volume <= ] } 1&& ;
M:: bounty <=> ( a b -- <=> )
a valid-bounty? [
b valid-bounty? [
a b [ value>> ] compare
] [ +gt+ ] if
] [ b valid-bounty? +lt+ +eq+ ? ] if ;
: find-max-amounts ( -- amounts )
weights volumes [
[ max-weight swap / ]
[ max-volume swap / ] bi* min >integer
] 2map ;
: best-bounty ( -- bounty )
find-max-amounts [ 1 + iota ] map <product-sequence>
[ <bounty> ] [ max ] map-reduce ;
Forth
\ : value ; immediate
: weight cell+ ;
: volume 2 cells + ;
: number 3 cells + ;
\ item value weight volume number
create panacea 30 , 3 , 25 , 0 ,
create ichor 18 , 2 , 15 , 0 ,
create gold 25 , 20 , 2 , 0 ,
create sack 0 , 250 , 250 ,
: fits? ( item -- ? )
dup weight @ sack weight @ > if drop false exit then
volume @ sack volume @ > 0= ;
: add ( item -- )
dup @ sack +!
dup weight @ negate sack weight +!
dup volume @ negate sack volume +!
1 swap number +! ;
: take ( item -- )
dup @ negate sack +!
dup weight @ sack weight +!
dup volume @ sack volume +!
-1 swap number +! ;
variable max-value
variable max-pan
variable max-ich
variable max-au
: .solution
cr
max-pan @ . ." Panaceas, "
max-ich @ . ." Ichors, and "
max-au @ . ." Gold for a total value of "
max-value @ 100 * . ;
: check
sack @ max-value @ <= if exit then
sack @ max-value !
panacea number @ max-pan !
ichor number @ max-ich !
gold number @ max-au !
( .solution ) ; \ and change <= to < to see all solutions
: solve-gold
gold fits? if gold add recurse gold take
else check then ;
: solve-ichor
ichor fits? if ichor add recurse ichor take then
solve-gold ;
: solve-panacea
panacea fits? if panacea add recurse panacea take then
solve-ichor ;
solve-panacea .solution
Or like this...
0 VALUE vials
0 VALUE ampules
0 VALUE bars
0 VALUE bag
#250 3 / #250 #25 / MIN 1+ CONSTANT maxvials
#250 2/ #250 #15 / MIN 1+ CONSTANT maxampules
#250 #20 / #250 2/ MIN 1+ CONSTANT maxbars
: RESULTS ( v a b -- k )
3DUP #20 * SWAP 2* + SWAP 3 * + #250 > IF 3DROP -1 EXIT ENDIF
3DUP 2* SWAP #15 * + SWAP #25 * + #250 > IF 3DROP -1 EXIT ENDIF
#2500 * SWAP #1800 * + SWAP #3000 * + ;
: .SOLUTION ( -- )
CR ." The traveller's knapsack contains "
vials DEC. ." vials of panacea, "
ampules DEC. ." ampules of ichor, "
CR bars DEC. ." bars of gold, a total value of "
vials ampules bars RESULTS 0DEC.R ." ." ;
: KNAPSACK ( -- )
-1 TO bag
maxvials 0 ?DO
maxampules 0 ?DO
maxbars 0 ?DO
K J I RESULTS DUP
bag > IF TO bag K TO vials J TO ampules I TO bars
ELSE DROP
ENDIF
LOOP
LOOP
LOOP
.SOLUTION ;
With the result...
FORTH> knapsack The traveller's knapsack contains 0 vials of panacea, 15 ampules of ichor, 11 bars of gold, a total value of 54500. ok
Fortran
A straight forward 'brute force' approach
PROGRAM KNAPSACK
IMPLICIT NONE
REAL :: totalWeight, totalVolume
INTEGER :: maxPanacea, maxIchor, maxGold, maxValue = 0
INTEGER :: i, j, k
INTEGER :: n(3)
TYPE Bounty
INTEGER :: value
REAL :: weight
REAL :: volume
END TYPE Bounty
TYPE(Bounty) :: panacea, ichor, gold, sack, current
panacea = Bounty(3000, 0.3, 0.025)
ichor = Bounty(1800, 0.2, 0.015)
gold = Bounty(2500, 2.0, 0.002)
sack = Bounty(0, 25.0, 0.25)
maxPanacea = MIN(sack%weight / panacea%weight, sack%volume / panacea%volume)
maxIchor = MIN(sack%weight / ichor%weight, sack%volume / ichor%volume)
maxGold = MIN(sack%weight / gold%weight, sack%volume / gold%volume)
DO i = 0, maxPanacea
DO j = 0, maxIchor
Do k = 0, maxGold
current%value = k * gold%value + j * ichor%value + i * panacea%value
current%weight = k * gold%weight + j * ichor%weight + i * panacea%weight
current%volume = k * gold%volume + j * ichor%volume + i * panacea%volume
IF (current%weight > sack%weight .OR. current%volume > sack%volume) CYCLE
IF (current%value > maxValue) THEN
maxValue = current%value
totalWeight = current%weight
totalVolume = current%volume
n(1) = i ; n(2) = j ; n(3) = k
END IF
END DO
END DO
END DO
WRITE(*, "(A,I0)") "Maximum value achievable is ", maxValue
WRITE(*, "(3(A,I0),A)") "This is achieved by carrying ", n(1), " panacea, ", n(2), " ichor and ", n(3), " gold items"
WRITE(*, "(A,F4.1,A,F5.3)") "The weight to carry is ", totalWeight, " and the volume used is ", totalVolume
END PROGRAM KNAPSACK
Sample output
Maximum value achievable is 54500 This is achieved by carrying 0 panacea, 15 ichor and 11 gold items The weight to carry is 25.0 and the volume used is 0.247
FreeBASIC
#define min(a, b) iif((a) < (b), (a), (b))
Dim As Single totalPeso, totalVolumen
Dim As Integer maxPanacea, maxIchor, maxGold, maxValor
Dim As Integer i, j ,k
Dim As Integer n(2)
Type Bounty
articulo As String*7
valor As Integer
peso As Single
volumen As Single
End Type
Dim item(1 To 5) As Bounty => { _
("panacea", 3000, 0.3, 0.025), ("ichor", 1800, 0.2, 0.015), _
("gold", 2500, 2.0, 0.002), ("sack", 0, 25.0, 0.25 )}
maxPanacea = min(item(4).peso/item(1).peso, item(4).volumen/item(1).volumen)
maxIchor = min(item(4).peso/item(2).peso, item(4).volumen/item(2).volumen)
maxGold = min(item(4).peso/item(3).peso, item(4).volumen/item(3).volumen)
For i = 0 To maxPanacea
For j = 0 To maxIchor
For k = 0 To maxGold
item(0).valor = k*item(3).valor + j*item(2).valor + i*item(1).valor
item(0).peso = k*item(3).peso + j*item(2).peso + i*item(1).peso
item(0).volumen = k*item(3).volumen + j*item(2).volumen + i*item(1).volumen
If item(0).peso > item(4).peso Or item(0).volumen > item(4).volumen Then
Continue For
End If
If item(0).valor > maxValor Then
maxValor = item(0).valor
totalPeso = item(0).peso
totalVolumen = item(0).volumen
n(0) = i: n(1) = j: n(2) = k
End If
Next k
Next j
Next i
Print "Maximum valor achievable is "; Str(maxValor)
Print "This is achieved by carrying "; Str(n(0));
Print " panacea, "; Str(n(1)); " ichor and "; Str(n(2)); " gold items."
Print "The peso to carry is "; Str(totalPeso);
Print " and the volume used is "; Str(totalVolumen)
Sleep
- Output:
Maximum valor achievable is 54500 This is achieved by carrying 0 panacea, 15 ichor and 11 gold items The peso to carry is 25 and the volume used is 0.247
Go
Recursive brute-force.
package main
import "fmt"
type Item struct {
Name string
Value int
Weight, Volume float64
}
type Result struct {
Counts []int
Sum int
}
func min(a, b int) int {
if a < b {
return a
}
return b
}
func Knapsack(items []Item, weight, volume float64) (best Result) {
if len(items) == 0 {
return
}
n := len(items) - 1
maxCount := min(int(weight/items[n].Weight), int(volume/items[n].Volume))
for count := 0; count <= maxCount; count++ {
sol := Knapsack(items[:n],
weight-float64(count)*items[n].Weight,
volume-float64(count)*items[n].Volume)
sol.Sum += items[n].Value * count
if sol.Sum > best.Sum {
sol.Counts = append(sol.Counts, count)
best = sol
}
}
return
}
func main() {
items := []Item{
{"Panacea", 3000, 0.3, 0.025},
{"Ichor", 1800, 0.2, 0.015},
{"Gold", 2500, 2.0, 0.002},
}
var sumCount, sumValue int
var sumWeight, sumVolume float64
result := Knapsack(items, 25, 0.25)
for i := range result.Counts {
fmt.Printf("%-8s x%3d -> Weight: %4.1f Volume: %5.3f Value: %6d\n",
items[i].Name, result.Counts[i], items[i].Weight*float64(result.Counts[i]),
items[i].Volume*float64(result.Counts[i]), items[i].Value*result.Counts[i])
sumCount += result.Counts[i]
sumValue += items[i].Value * result.Counts[i]
sumWeight += items[i].Weight * float64(result.Counts[i])
sumVolume += items[i].Volume * float64(result.Counts[i])
}
fmt.Printf("TOTAL (%3d items) Weight: %4.1f Volume: %5.3f Value: %6d\n",
sumCount, sumWeight, sumVolume, sumValue)
}
Output:
Panacea x 9 -> Weight: 2.7 Volume: 0.225 Value: 27000 Ichor x 0 -> Weight: 0.0 Volume: 0.000 Value: 0 Gold x 11 -> Weight: 22.0 Volume: 0.022 Value: 27500 TOTAL ( 20 items) Weight: 24.7 Volume: 0.247 Value: 54500
Groovy
Solution: dynamic programming
def totalWeight = { list -> list.collect{ it.item.weight * it.count }.sum() }
def totalVolume = { list -> list.collect{ it.item.volume * it.count }.sum() }
def totalValue = { list -> list.collect{ it.item.value * it.count }.sum() }
def knapsackUnbounded = { possibleItems, BigDecimal weightMax, BigDecimal volumeMax ->
def n = possibleItems.size()
def wm = weightMax.unscaledValue()
def vm = volumeMax.unscaledValue()
def m = (0..n).collect{ i -> (0..wm).collect{ w -> (0..vm).collect{ v -> [] } } }
(1..wm).each { w ->
(1..vm).each { v ->
(1..n).each { i ->
def item = possibleItems[i-1]
def wi = item.weight.unscaledValue()
def vi = item.volume.unscaledValue()
def bi = [w.intdiv(wi),v.intdiv(vi)].min()
m[i][w][v] = (0..bi).collect{ count ->
m[i-1][w - wi * count][v - vi * count] + [[item:item, count:count]]
}.max(totalValue).findAll{ it.count }
}
}
}
m[n][wm][vm]
}
Test:
Set solutions = []
items.eachPermutation { itemList ->
def start = System.currentTimeMillis()
def packingList = knapsackUnbounded(itemList, 25.0, 0.250)
def elapsed = System.currentTimeMillis() - start
println "\n Item Order: ${itemList.collect{ it.name.split()[0] }}"
println "Elapsed Time: ${elapsed/1000.0} s"
solutions << (packingList as Set)
}
solutions.each { packingList ->
println "\nTotal Weight: ${totalWeight(packingList)}"
println "Total Volume: ${totalVolume(packingList)}"
println " Total Value: ${totalValue(packingList)}"
packingList.each {
printf (' item: %-22s count:%2d weight:%4.1f Volume:%5.3f\n',
it.item.name, it.count, it.item.weight * it.count, it.item.volume * it.count)
}
}
Output:
Item Order: [panacea, ichor, gold] Elapsed Time: 26.883 s Item Order: [panacea, gold, ichor] Elapsed Time: 27.17 s Item Order: [ichor, panacea, gold] Elapsed Time: 25.884 s Item Order: [ichor, gold, panacea] Elapsed Time: 26.126 s Item Order: [gold, panacea, ichor] Elapsed Time: 26.596 s Item Order: [gold, ichor, panacea] Elapsed Time: 26.47 s Total Weight: 25.0 Total Volume: 0.247 Total Value: 54500 item: gold (bars) count:11 weight:22.0 Volume:0.022 item: ichor (ampules of) count:15 weight: 3.0 Volume:0.225 Total Weight: 24.7 Total Volume: 0.247 Total Value: 54500 item: gold (bars) count:11 weight:22.0 Volume:0.022 item: panacea (vials of) count: 9 weight: 2.7 Volume:0.225
While this solver can only be used to detect two of the four possible solutions, the other two may be discovered by noting that 5 ampules of ichor and 3 vials of panacea have the same value and the same volume and only differ by 0.1 in weight. Thus the other two solutions can be derived by substitution as follows:
Total Weight: 24.9 Total Volume: 0.247 Total Value: 54500 item: gold (bars) count:11 weight:22.0 Volume:0.022 item: ichor (ampules of) count:10 weight: 2.0 Volume:0.150 item: panacea (vials of) count: 3 weight: 0.9 Volume:0.075 Total Weight: 24.8 Total Volume: 0.247 Total Value: 54500 item: gold (bars) count:11 weight:22.0 Volume:0.022 item: ichor (ampules of) count: 5 weight: 1.0 Volume:0.075 item: panacea (vials of) count: 6 weight: 1.8 Volume:0.150
Haskell
This is a brute-force solution: it generates a list of every legal combination of items (options) and then finds the option of greatest value.
import Data.List (maximumBy)
import Data.Ord (comparing)
(maxWgt, maxVol) = (25, 0.25)
items =
[Bounty "panacea" 3000 0.3 0.025,
Bounty "ichor" 1800 0.2 0.015,
Bounty "gold" 2500 2.0 0.002]
data Bounty = Bounty
{itemName :: String,
itemVal :: Int,
itemWgt, itemVol :: Double}
names = map itemName items
vals = map itemVal items
wgts = map itemWgt items
vols = map itemVol items
dotProduct :: (Num a, Integral b) => [a] -> [b] -> a
dotProduct factors = sum . zipWith (*) factors . map fromIntegral
options :: [[Int]]
options = filter fits $ mapM f items
where f (Bounty _ _ w v) = [0 .. m]
where m = floor $ min (maxWgt / w) (maxVol / v)
fits opt = dotProduct wgts opt <= maxWgt &&
dotProduct vols opt <= maxVol
showOpt :: [Int] -> String
showOpt opt = concat (zipWith showItem names opt) ++
"total weight: " ++ show (dotProduct wgts opt) ++
"\ntotal volume: " ++ show (dotProduct vols opt) ++
"\ntotal value: " ++ show (dotProduct vals opt) ++ "\n"
where showItem name num = name ++ ": " ++ show num ++ "\n"
main = putStr $ showOpt $ best options
where best = maximumBy $ comparing $ dotProduct vals
Output:
panacea: 9 ichor: 0 gold: 11 total weight: 24.7 total volume: 0.247 total value: 54500
HicEst
CHARACTER list*1000
NN = ALIAS($Panacea, $Ichor, $Gold, wSack, wPanacea, wIchor, wGold, vSack, vPanacea, vIchor, vGold)
NN = (3000, 1800, 2500, 25, 0.3, 0.2, 2.0, 0.25, 0.025, 0.015, 0.002)
maxItems = ALIAS(maxPanacea, maxIchor, maxGold)
maxItems = ( MIN( wSack/wPanacea, vSack/vPanacea), MIN( wSack/wIchor, vSack/vIchor), MIN( wSack/wGold, vSack/vGold) )
maxValue = 0
DO Panaceas = 0, maxPanacea
DO Ichors = 0, maxIchor
DO Golds = 0, maxGold
weight = Panaceas*wPanacea + Ichors*wIchor + Golds*wGold
IF( weight <= wSack ) THEN
volume = Panaceas*vPanacea + Ichors*vIchor + Golds*vGold
IF( volume <= vSack ) THEN
value = Panaceas*$Panacea + Ichors*$Ichor + Golds*$Gold
IF( value > maxValue ) THEN
maxValue = value
! this restarts the list, removing all previous entries:
WRITE(Text=list, Name) value, Panaceas, Ichors, Golds, weight, volume, $CR//$LF
ELSEIF( value == maxValue ) THEN
WRITE(Text=list, Name, APPend) value, Panaceas, Ichors, Golds, weight, volume, $CR//$LF
ENDIF
ENDIF
ENDIF
ENDDO
ENDDO
ENDDOvalue=54500; Panaceas=0; Ichors=15; Golds=11; weight=25; volume=0.247;
value=54500; Panaceas=3; Ichors=10; Golds=11; weight=24.9; volume=0.247;
value=54500; Panaceas=6; Ichors=5; Golds=11; weight=24.8; volume=0.247;
value=54500; Panaceas=9; Ichors=0; Golds=11; weight=24.7; volume=0.247;J
Brute force solution.
mwv=: 25 0.25
prods=: <;. _1 ' panacea: ichor: gold:'
hdrs=: <;. _1 ' weight: volume: value:'
vls=: 3000 1800 2500
ws=: 0.3 0.2 2.0
vs=: 0.025 0.015 0.002
ip=: +/ .*
prtscr=: (1!:2)&2
KS=: 3 : 0
os=. (#:i.@(*/)) mwv >:@<.@<./@:% ws,:vs
bo=.os#~(ws,:vs) mwv&(*./@:>)@ip"_ 1 os
mo=.bo{~{.\: vls ip"1 bo
prtscr &.> prods ([,' ',":@])&.>mo
prtscr &.> hdrs ('total '&,@[,' ',":@])&.> mo ip"1 ws,vs,:vls
LF
)
Example output:
KS'' panacea: 3 ichor: 10 gold: 11 total weight: 24.9 total volume: 0.247 total value: 54500
Java
With recursion for more than 3 items.
package hu.pj.alg;
import hu.pj.obj.Item;
import java.text.*;
public class UnboundedKnapsack {
protected Item [] items = {
new Item("panacea", 3000, 0.3, 0.025),
new Item("ichor" , 1800, 0.2, 0.015),
new Item("gold" , 2500, 2.0, 0.002)
};
protected final int n = items.length; // the number of items
protected Item sack = new Item("sack" , 0, 25.0, 0.250);
protected Item best = new Item("best" , 0, 0.0, 0.000);
protected int [] maxIt = new int [n]; // maximum number of items
protected int [] iIt = new int [n]; // current indexes of items
protected int [] bestAm = new int [n]; // best amounts
public UnboundedKnapsack() {
// initializing:
for (int i = 0; i < n; i++) {
maxIt [i] = Math.min(
(int)(sack.getWeight() / items[i].getWeight()),
(int)(sack.getVolume() / items[i].getVolume())
);
} // for (i)
// calc the solution:
calcWithRecursion(0);
// Print out the solution:
NumberFormat nf = NumberFormat.getInstance();
System.out.println("Maximum value achievable is: " + best.getValue());
System.out.print("This is achieved by carrying (one solution): ");
for (int i = 0; i < n; i++) {
System.out.print(bestAm[i] + " " + items[i].getName() + ", ");
}
System.out.println();
System.out.println("The weight to carry is: " + nf.format(best.getWeight()) +
" and the volume used is: " + nf.format(best.getVolume())
);
}
// calculation the solution with recursion method
// item : the number of item in the "items" array
public void calcWithRecursion(int item) {
for (int i = 0; i <= maxIt[item]; i++) {
iIt[item] = i;
if (item < n-1) {
calcWithRecursion(item+1);
} else {
int currVal = 0; // current value
double currWei = 0.0; // current weight
double currVol = 0.0; // current Volume
for (int j = 0; j < n; j++) {
currVal += iIt[j] * items[j].getValue();
currWei += iIt[j] * items[j].getWeight();
currVol += iIt[j] * items[j].getVolume();
}
if (currVal > best.getValue()
&&
currWei <= sack.getWeight()
&&
currVol <= sack.getVolume()
)
{
best.setValue (currVal);
best.setWeight(currWei);
best.setVolume(currVol);
for (int j = 0; j < n; j++) bestAm[j] = iIt[j];
} // if (...)
} // else
} // for (i)
} // calcWithRecursion()
// the main() function:
public static void main(String[] args) {
new UnboundedKnapsack();
} // main()
} // class
package hu.pj.obj;
public class Item {
protected String name = "";
protected int value = 0;
protected double weight = 0;
protected double volume = 0;
public Item() {
}
public Item(String name, int value, double weight, double volume) {
setName(name);
setValue(value);
setWeight(weight);
setVolume(volume);
}
public int getValue() {
return value;
}
public void setValue(int value) {
this.value = Math.max(value, 0);
}
public double getWeight() {
return weight;
}
public void setWeight(double weight) {
this.weight = Math.max(weight, 0);
}
public double getVolume() {
return volume;
}
public void setVolume(double volume) {
this.volume = Math.max(volume, 0);
}
public String getName() {
return name;
}
public void setName(String name) {
this.name = name;
}
} // class
output:
Maximum value achievable is: 54500 This is achieved by carrying (one solution): 0 panacea, 15 ichor, 11 gold, The weight to carry is: 25 and the volume used is: 0,247
JavaScript
Brute force.
var gold = { 'value': 2500, 'weight': 2.0, 'volume': 0.002 },
panacea = { 'value': 3000, 'weight': 0.3, 'volume': 0.025 },
ichor = { 'value': 1800, 'weight': 0.2, 'volume': 0.015 },
items = [gold, panacea, ichor],
knapsack = {'weight': 25, 'volume': 0.25},
max_val = 0,
solutions = [],
g, p, i, item, val;
for (i = 0; i < items.length; i += 1) {
item = items[i];
item.max = Math.min(
Math.floor(knapsack.weight / item.weight),
Math.floor(knapsack.volume / item.volume)
);
}
for (g = 0; g <= gold.max; g += 1) {
for (p = 0; p <= panacea.max; p += 1) {
for (i = 0; i <= ichor.max; i += 1) {
if (i * ichor.weight + g * gold.weight + p * panacea.weight > knapsack.weight) {
continue;
}
if (i * ichor.volume + g * gold.volume + p * panacea.volume > knapsack.volume) {
continue;
}
val = i * ichor.value + g * gold.value + p * panacea.value;
if (val > max_val) {
solutions = [];
max_val = val;
}
if (val === max_val) {
solutions.push([g, p, i]);
}
}
}
}
document.write("maximum value: " + max_val + '<br>');
for (i = 0; i < solutions.length; i += 1) {
item = solutions[i];
document.write("(gold: " + item[0] + ", panacea: " + item[1] + ", ichor: " + item[2] + ")<br>");
}
output:
<pre>maximum value: 54500
(gold: 11, panacea: 0, ichor: 15)
(gold: 11, panacea: 3, ichor: 10)
(gold: 11, panacea: 6, ichor: 5)
(gold: 11, panacea: 9, ichor: 0)</pre>
jq
Works with gojq, the Go implementation of jq
def Item($name; $value; $weight; $volume):
{$name, $value, $weight, $volume};
def items:[
Item("panacea"; 3000; 0.3; 0.025),
Item("ichor"; 1800; 0.2; 0.015),
Item("gold"; 2500; 2; 0.002)
];
def array($init): [range(0; .) | $init];
# input: {count, best, bestvalue}
def knapsack($i; $value; $weight; $volume):
(items|length) as $n
| if $i == $n
then if $value > .bestValue
then .bestValue = $value
| reduce range(0; $n) as $j (.; .best[$j] = .count[$j])
else .
end
else (($weight / items[$i].weight)|floor) as $m1
| (($volume / items[$i].volume)|floor) as $m2
| .count[$i] = ([$m1, $m2] | min)
| until (.count[$i] < 0;
knapsack(
$i + 1;
$value + .count[$i] * (items[$i].value);
$weight - .count[$i] * (items[$i].weight);
$volume - .count[$i] * (items[$i].volume)
)
| .count[$i] += -1
)
end ;
def lpad($len): tostring | ($len - length) as $l | (" " * $l)[:$l] + .;
def solve($maxWeight; $maxVolume):
def rnd: 100 * . | round / 100;
def rnd($width): if type == "string" then lpad($width) else rnd|lpad($width) end;
def f(a;b;c;d;f):
"\(a|lpad(11)) \(b|rnd(6)) \(c|rnd(6)) \(d|rnd(6)) \(f|rnd(6))" ;
def f: . as [$a,$b,$c,$d,$f] | f($a;$b;$c;$d;$f);
(items|length) as $n
| def init:
{ count: ($n|array(0)),
best : ($n|array(0)),
bestValue: 0,
maxWeight: $maxWeight,
maxVolume: $maxVolume };
f("Item Chosen"; "Number"; "Value"; "Weight"; "Volume"),
"----------- ------ ------ ------ ------",
( init
| knapsack(0; 0; $maxWeight; $maxVolume)
| reduce range(0; $n) as $i (
. + {itemCount:0, sumNumber:0, sumWeight:0, sumVolume:0 };
if (.best[$i]) != 0
then .itemCount += 1
| .name = items[$i].name
| .number = .best[$i]
| .value = items[$i].value * .number
| .weight = items[$i].weight * .number
| .volume = items[$i].volume * .number
| .sumNumber += .number
| .sumWeight += .weight
| .sumVolume += .volume
| .emit += [ f(.name; .number; .value; .weight; .volume) ]
else .
end)
| .emit[],
"----------- ------ ------ ------ ------",
f(.itemCount; .sumNumber; .bestValue; .sumWeight; .sumVolume) );
solve(25; 0.25)- Output:
Item Chosen Number Value Weight Volume
----------- ------ ------ ------ ------
panacea 9 27000 2.7 0.23
gold 11 27500 22 0.02
----------- ------ ------ ------ ------
2 20 54500 24.7 0.25
Julia
using JuMP
using GLPKMathProgInterface
model = Model(solver=GLPKSolverMIP())
@variable(model, vials_of_panacea >= 0, Int)
@variable(model, ampules_of_ichor >= 0, Int)
@variable(model, bars_of_gold >= 0, Int)
@objective(model, Max, 3000*vials_of_panacea + 1800*ampules_of_ichor + 2500*bars_of_gold)
@constraint(model, 0.3*vials_of_panacea + 0.2*ampules_of_ichor + 2.0*bars_of_gold <= 25.0)
@constraint(model, 0.025*vials_of_panacea + 0.015*ampules_of_ichor + 0.002*bars_of_gold <= 0.25)
println("The optimization problem to be solved is:")
println(model)
status = solve(model)
println("Objective value: ", getobjectivevalue(model))
println("vials of panacea = ", getvalue(vials_of_panacea))
println("ampules of ichor = ", getvalue(ampules_of_ichor))
println("bars of gold = ", getvalue(bars_of_gold))
- Output:
The optimization problem to be solved is: Max 3000 vials_of_panacea + 1800 ampules_of_ichor + 2500 bars_of_gold Subject to 0.3 vials_of_panacea + 0.2 ampules_of_ichor + 2 bars_of_gold <= 25 0.025 vials_of_panacea + 0.015 ampules_of_ichor + 0.002 bars_of_gold <= 0.25 vials_of_panacea >= 0, integer ampules_of_ichor >= 0, integer bars_of_gold >= 0, integer Objective value: 54500.0 vials of panacea = 9.0 ampules of ichor = 0.0 bars of gold = 11.0
Kotlin
Recursive brute force approach:
// version 1.1.2
data class Item(val name: String, val value: Double, val weight: Double, val volume: Double)
val items = listOf(
Item("panacea", 3000.0, 0.3, 0.025),
Item("ichor", 1800.0, 0.2, 0.015),
Item("gold", 2500.0, 2.0, 0.002)
)
val n = items.size
val count = IntArray(n)
val best = IntArray(n)
var bestValue = 0.0
const val MAX_WEIGHT = 25.0
const val MAX_VOLUME = 0.25
fun knapsack(i: Int, value: Double, weight: Double, volume: Double) {
if (i == n) {
if (value > bestValue) {
bestValue = value
for (j in 0 until n) best[j] = count[j]
}
return
}
val m1 = Math.floor(weight / items[i].weight).toInt()
val m2 = Math.floor(volume / items[i].volume).toInt()
val m = minOf(m1, m2)
count[i] = m
while (count[i] >= 0) {
knapsack(
i + 1,
value + count[i] * items[i].value,
weight - count[i] * items[i].weight,
volume - count[i] * items[i].volume
)
count[i]--
}
}
fun main(args: Array<String>) {
knapsack(0, 0.0, MAX_WEIGHT, MAX_VOLUME)
println("Item Chosen Number Value Weight Volume")
println("----------- ------ ----- ------ ------")
var itemCount = 0
var sumNumber = 0
var sumWeight = 0.0
var sumVolume = 0.0
for (i in 0 until n) {
if (best[i] == 0) continue
itemCount++
val name = items[i].name
val number = best[i]
val value = items[i].value * number
val weight = items[i].weight * number
val volume = items[i].volume * number
sumNumber += number
sumWeight += weight
sumVolume += volume
print("${name.padEnd(11)} ${"%2d".format(number)} ${"%5.0f".format(value)} ${"%4.1f".format(weight)}")
println(" ${"%4.2f".format(volume)}")
}
println("----------- ------ ----- ------ ------")
print("${itemCount} items ${"%2d".format(sumNumber)} ${"%5.0f".format(bestValue)} ${"%4.1f".format(sumWeight)}")
println(" ${"%4.2f".format(sumVolume)}")
}
- Output:
Item Chosen Number Value Weight Volume ----------- ------ ----- ------ ------ panacea 9 27000 2.7 0.23 gold 11 27500 22.0 0.02 ----------- ------ ----- ------ ------ 2 items 20 54500 24.7 0.25
Lua
items = { ["panaea"] = { ["value"] = 3000, ["weight"] = 0.3, ["volume"] = 0.025 },
["ichor"] = { ["value"] = 1800, ["weight"] = 0.2, ["volume"] = 0.015 },
["gold"] = { ["value"] = 2500, ["weight"] = 2.0, ["volume"] = 0.002 }
}
max_weight = 25
max_volume = 0.25
max_num_items = {}
for i in pairs( items ) do
max_num_items[i] = math.floor( math.min( max_weight / items[i].weight, max_volume / items[i].volume ) )
end
best = { ["value"] = 0.0, ["weight"] = 0.0, ["volume"] = 0.0 }
best_amounts = {}
for i = 1, max_num_items["panaea"] do
for j = 1, max_num_items["ichor"] do
for k = 1, max_num_items["gold"] do
current = { ["value"] = i*items["panaea"]["value"] + j*items["ichor"]["value"] + k*items["gold"]["value"],
["weight"] = i*items["panaea"]["weight"] + j*items["ichor"]["weight"] + k*items["gold"]["weight"],
["volume"] = i*items["panaea"]["volume"] + j*items["ichor"]["volume"] + k*items["gold"]["volume"]
}
if current.value > best.value and current.weight <= max_weight and current.volume <= max_volume then
best = { ["value"] = current.value, ["weight"] = current.weight, ["volume"] = current.volume }
best_amounts = { ["panaea"] = i, ["ichor"] = j, ["gold"] = k }
end
end
end
end
print( "Maximum value:", best.value )
for k, v in pairs( best_amounts ) do
print( k, v )
end
MiniZinc
%Knapsack problem/Unbounded. Nigel Galloway, August 13th., 2021
enum Items ={panacea,ichor,gold};
array[Items] of float: weight =[0.3,0.2,2.0]; constraint sum(n in Items)(take[n]*weight[n])<=25.0;
array[Items] of int: value =[3000,1800,2500];
array[Items] of float: volume =[0.025,0.015,0.002]; constraint sum(n in Items)(take[n]*volume[n])<=0.25;
array[Items] of var 0..floor(25.0/min(weight)): take;
solve maximize sum(n in Items)(value[n]*take[n]);
output(["Take "++show(take[panacea])++" vials of panacea\nTake "++show(take[ichor])++" ampules of ichor\nTake "++ show(take[gold])++" bars of gold\n"])- Output:
Take 0 vials of panacea Take 15 ampules of ichor Take 11 bars of gold ---------- ========== Finished in 159msec
M4
A brute force solution:
divert(-1)
define(`set2d',`define(`$1[$2][$3]',`$4')')
define(`get2d',`defn(`$1[$2][$3]')')
define(`for',
`ifelse($#,0,``$0'',
`ifelse(eval($2<=$3),1,
`pushdef(`$1',$2)$4`'popdef(`$1')$0(`$1',incr($2),$3,`$4')')')')
define(`min',
`define(`ma',eval($1))`'define(`mb',eval($2))`'ifelse(eval(ma<mb),1,ma,mb)')
define(`setv',
`set2d($1,$2,1,$3)`'set2d($1,$2,2,$4)`'set2d($1,$2,3,$5)`'set2d($1,$2,4,$6)')
dnl name,value (each),weight,volume
setv(a,0,`knapsack',0,250,250)
setv(a,1,`panacea',3000,3,25)
setv(a,2,`ichor',1800,2,15)
setv(a,3,`gold',2500,20,2)
define(`mv',0)
for(`x',0,min(get2d(a,0,3)/get2d(a,1,3),get2d(a,0,4)/get2d(a,1,4)),
`for(`y',0,min((get2d(a,0,3)-x*get2d(a,1,3))/get2d(a,2,3),
(get2d(a,0,4)-x*get2d(a,1,4))/get2d(a,2,4)),
`
define(`z',min((get2d(a,0,3)-x*get2d(a,1,3)-y*get2d(a,2,3))/get2d(a,3,3),
(get2d(a,0,4)-x*get2d(a,1,4)-y*get2d(a,2,4))/get2d(a,3,4)))
define(`cv',eval(x*get2d(a,1,2)+y*get2d(a,2,2)+z*get2d(a,3,2)))
ifelse(eval(cv>mv),1,
`define(`mv',cv)`'define(`best',(x,y,z))',
`ifelse(cv,mv,`define(`best',best (x,y,z))')')
')')
divert
mv bestOutput:
54500 (0,15,11) (3,10,11) (6,5,11) (9,0,11)
Mathematica /Wolfram Language
Brute force algorithm:
{pva,pwe,pvo}={3000,3/10,1/40};
{iva,iwe,ivo}={1800,2/10,3/200};
{gva,gwe,gvo}={2500,2,2/1000};
wemax=25;
vomax=1/4;
{pmax,imax,gmax}=Floor/@{Min[vomax/pvo,wemax/pwe],Min[vomax/ivo,wemax/iwe],Min[vomax/gvo,wemax/gwe]};
data=Flatten[Table[{{p,i,g}.{pva,iva,gva},{p,i,g}.{pwe,iwe,gwe},{p,i,g}.{pvo,ivo,gvo},{p,i,g}},{p,0,pmax},{i,0,imax},{g,0,gmax}],2];
data=Select[data,#[[2]]<=25&&#[[3]]<=1/4&];
First[SplitBy[Sort[data,First[#1]>First[#2]&],First]]
gives back an array of the best solution(s), with each element being value, weight, volume, {number of vials, number of ampules, number of bars}:
{{54500,247/10,247/1000,{9,0,11}},{54500,124/5,247/1000,{6,5,11}},{54500,249/10,247/1000,{3,10,11}},{54500,25,247/1000,{0,15,11}}}
if we call the three items by their first letters the best packings are:
p:9 i:0 v:11
p:6 i:5 v:11
p:3 i:10 v:11
p:0 i:15 v:11
The volume for all of those is the same, the 'best' solution would be the one that has the least weight: that would the first solution.
Mathprog
/*Knapsack
This model finds the integer optimal packing of a knapsack
Nigel_Galloway
January 9th., 2012
*/
set Items;
param weight{t in Items};
param value{t in Items};
param volume{t in Items};
var take{t in Items}, integer, >=0;
knap_weight : sum{t in Items} take[t] * weight[t] <= 25;
knap_vol : sum{t in Items} take[t] * volume[t] <= 0.25;
maximize knap_value: sum{t in Items} take[t] * value[t];
data;
param : Items : weight value volume :=
panacea 0.3 3000 0.025
ichor 0.2 1800 0.015
gold 2.0 2500 0.002
;
end;The solution produced is at Knapsack problem/Unbounded/Mathprog.
Modula-3
Note that unlike Fortran and C, Modula-3 does not do any hidden casting, which is why FLOAT and FLOOR are used.
MODULE Knapsack EXPORTS Main;
FROM IO IMPORT Put;
FROM Fmt IMPORT Int, Real;
TYPE Bounty = RECORD
value: INTEGER;
weight, volume: REAL;
END;
VAR totalWeight, totalVolume: REAL;
maxPanacea, maxIchor, maxGold, maxValue: INTEGER := 0;
n: ARRAY [1..3] OF INTEGER;
panacea, ichor, gold, sack, current: Bounty;
BEGIN
panacea := Bounty{3000, 0.3, 0.025};
ichor := Bounty{1800, 0.2, 0.015};
gold := Bounty{2500, 2.0, 0.002};
sack := Bounty{0, 25.0, 0.25};
maxPanacea := FLOOR(MIN(sack.weight / panacea.weight, sack.volume / panacea.volume));
maxIchor := FLOOR(MIN(sack.weight / ichor.weight, sack.volume / ichor.volume));
maxGold := FLOOR(MIN(sack.weight / gold.weight, sack.volume / gold.volume));
FOR i := 0 TO maxPanacea DO
FOR j := 0 TO maxIchor DO
FOR k := 0 TO maxGold DO
current.value := k * gold.value + j * ichor.value + i * panacea.value;
current.weight := FLOAT(k) * gold.weight + FLOAT(j) * ichor.weight + FLOAT(i) * panacea.weight;
current.volume := FLOAT(k) * gold.volume + FLOAT(j) * ichor.volume + FLOAT(i) * panacea.volume;
IF current.weight > sack.weight OR current.volume > sack.volume THEN
EXIT;
END;
IF current.value > maxValue THEN
maxValue := current.value;
totalWeight := current.weight;
totalVolume := current.volume;
n[1] := i; n[2] := j; n[3] := k;
END;
END;
END;
END;
Put("Maximum value achievable is " & Int(maxValue) & "\n");
Put("This is achieved by carrying " & Int(n[1]) & " panacea, " & Int(n[2]) & " ichor and " & Int(n[3]) & " gold items\n");
Put("The weight of this carry is " & Real(totalWeight) & " and the volume used is " & Real(totalVolume) & "\n");
END Knapsack.
Output:
Maximum value achievable is 54500 This is achieved by carrying 0 panacea, 15 ichor and 11 gold items The weight of this carry is 25 and the volume used is 0.247
Nim
This is a brute-force solution translated from Pascal to Nim, which is straightforward.
# Knapsack unbounded. Brute force solution.
import lenientops # Mixed float/int operators.
import strformat
type Bounty = tuple[value: int; weight, volume: float]
const
Panacea: Bounty = (value: 3000, weight: 0.3, volume: 0.025)
Ichor: Bounty = (value: 1800, weight: 0.2, volume: 0.015)
Gold: Bounty = (value: 2500, weight: 2.0, volume: 0.002)
Sack: Bounty = (value: 0, weight: 25.0, volume: 0.25)
MaxPanacea = min(Sack.weight / Panacea.weight, Sack.volume / Panacea.volume).toInt
MaxIchor = min(Sack.weight / Ichor.weight, Sack.volume / Ichor.volume).toInt
MaxGold = min(Sack.weight / Gold.weight, Sack.volume / Gold.volume).toInt
var
totalWeight, totalVolume: float
n: array[1..3, int] # Number of panacea, ichor and gold.
maxValue = 0
for i in 0..MaxPanacea:
for j in 0..MaxIchor:
for k in 0..MaxGold:
var current: Bounty
current.value = k * Gold.value + j * Ichor.value + i * Panacea.value
current.weight = k * Gold.weight + j * Ichor.weight + i * Panacea.weight
current.volume = k * Gold.volume + j * Ichor.volume + i * Panacea.volume
if current.value > maxValue and current.weight <= Sack.weight and current.volume <= Sack.volume:
maxvalue = current.value
totalweight = current.weight
totalvolume = current.volume
n = [i, j, k]
echo fmt"Maximum value achievable is {maxValue}."
echo fmt"This is achieved by carrying {n[1]} panacea, {n[2]} ichor and {n[3]} gold items."
echo fmt"The weight of this carry is {totalWeight:6.3f} and the volume used is {totalVolume:6.4f}."
- Output:
Maximum value achievable is 54500. This is achieved by carrying 0 panacea, 15 ichor and 11 gold items. The weight of this carry is 25.000 and the volume used is 0.2470.
OCaml
This is a brute-force solution: it generates a list of every legal combination of items and then finds the best results:
type bounty = { name:string; value:int; weight:float; volume:float }
let bounty n d w v = { name = n; value = d; weight = w; volume = v }
let items =
[ bounty "panacea" 3000 0.3 0.025;
bounty "ichor" 1800 0.2 0.015;
bounty "gold" 2500 2.0 0.002; ]
let max_wgt = 25.0 and max_vol = 0.25
let itmax =
let f it =
let rec aux n =
if float n *. it.weight >= max_wgt
|| float n *. it.volume >= max_vol
then (n)
else aux (succ n)
in
aux 0
in
List.map f items
let mklist n m =
let rec aux i acc =
if i > m then (List.rev acc)
else aux (succ i) (i::acc)
in
aux n []
let comb_items = List.map (mklist 0) itmax
let combs ll =
let f hd acc =
List.concat
(List.map (fun l -> List.map (fun v -> (v::l)) hd) acc)
in
List.fold_right f ll [[]]
let possibles = combs comb_items
let packs =
let f l =
let g (v, wgt, vol) n it =
(v + n * it.value,
wgt +. float n *. it.weight,
vol +. float n *. it.volume)
in
List.fold_left2 g (0, 0.0, 0.0) l items
in
List.map f possibles
let packs = List.combine packs possibles
let results =
let f (_, wgt, vol) = (wgt <= max_wgt && vol <= max_vol) in
List.filter (fun v -> f(fst v)) packs
let best_results =
let max_value = List.fold_left (fun v1 ((v2,_,_),_) -> max v1 v2) 0 results in
List.filter (fun ((v,_,_),_) -> v = max_value) results
let items_name = List.map (fun it -> it.name) items
let print ((v, wgt, vol), ns) =
Printf.printf "\
Maximum value: %d \n \
Total weight: %g \n \
Total volume: %g \n \
Containing: " v wgt vol;
let f n name = string_of_int n ^ " " ^ name in
let ss = List.map2 f ns items_name in
print_endline(String.concat ", " ss);
print_newline()
let () = List.iter print best_results
outputs:
Maximum value: 54500 Total weight: 24.7 Total volume: 0.247 Containing: 9 panacea, 0 ichor, 11 gold Maximum value: 54500 Total weight: 24.8 Total volume: 0.247 Containing: 6 panacea, 5 ichor, 11 gold Maximum value: 54500 Total weight: 24.9 Total volume: 0.247 Containing: 3 panacea, 10 ichor, 11 gold Maximum value: 54500 Total weight: 25 Total volume: 0.247 Containing: 0 panacea, 15 ichor, 11 gold
OOCalc
OpenOffice.org Calc has (several) linear solvers. To solve this task, first copy in the table from the task description, then add the extra columns:
- Number: (How many chosen, n)
- value of n
- weight of n
- volume of n
Add a TOTALS row to sum the value/weight/volume of n.
The sheet should then look like this:
table pre solving
Open the "Tools->Solver..." menu item and fill in the following items:
solver menu
- Options... (opens a separate popup window, then continue)
solver popup options menu
OK the solver options window leaving the Solver window open, then select solve to produce in seconds:
Table solved
Oz
Using constraint propagation and branch and bound search:
declare
proc {Knapsack Sol}
solution(panacea:P = {FD.decl}
ichor: I = {FD.decl}
gold: G = {FD.decl} ) = Sol
in
{Show 0#Sol}
3 * P + 2 * I + 20 * G =<: 250 {Show 1#Sol}
25 * P + 15 * I + 2 * G =<: 250 {Show 2#Sol}
{FD.distribute naive Sol} {Show d#Sol}
end
fun {Value solution(panacea:P ichor:I gold:G)}
3000 * P + 1800 * I + 2500 * G
end
{System.showInfo "Search:"}
[Best] = {SearchBest Knapsack proc {$ Old New}
{Value Old} <: {Value New}
end}
in
{System.showInfo "\nResult:"}
{Show Best}
{System.showInfo "total value: "#{Value Best}}If you study the output, you see how the weight and volume equations automagically constrain the domain of the three variables. Afterwards SearchBest only has to evaluate 38 different combinations to find an optimal solution:
Search:
0#solution(gold:_{0#134217726} ichor:_{0#134217726} panacea:_{0#134217726})
1#solution(gold:_{0#12} ichor:_{0#125} panacea:_{0#83})
2#solution(gold:_{0#12} ichor:_{0#16} panacea:_{0#10})
d#solution(gold:0 ichor:0 panacea:0)
d#solution(gold:0 ichor:1 panacea:0)
d#solution(gold:0 ichor:2 panacea:0)
d#solution(gold:0 ichor:3 panacea:0)
d#solution(gold:0 ichor:4 panacea:0)
d#solution(gold:0 ichor:5 panacea:0)
d#solution(gold:0 ichor:6 panacea:0)
d#solution(gold:0 ichor:7 panacea:0)
d#solution(gold:0 ichor:8 panacea:0)
d#solution(gold:0 ichor:9 panacea:0)
d#solution(gold:0 ichor:10 panacea:0)
d#solution(gold:0 ichor:11 panacea:0)
d#solution(gold:0 ichor:12 panacea:0)
d#solution(gold:0 ichor:13 panacea:0)
d#solution(gold:0 ichor:14 panacea:0)
d#solution(gold:0 ichor:15 panacea:0)
d#solution(gold:0 ichor:16 panacea:0)
d#solution(gold:1 ichor:15 panacea:0)
d#solution(gold:1 ichor:16 panacea:0)
d#solution(gold:2 ichor:15 panacea:0)
d#solution(gold:2 ichor:16 panacea:0)
d#solution(gold:3 ichor:15 panacea:0)
d#solution(gold:3 ichor:16 panacea:0)
d#solution(gold:4 ichor:15 panacea:0)
d#solution(gold:4 ichor:16 panacea:0)
d#solution(gold:5 ichor:15 panacea:0)
d#solution(gold:5 ichor:16 panacea:0)
d#solution(gold:6 ichor:15 panacea:0)
d#solution(gold:7 ichor:14 panacea:0)
d#solution(gold:7 ichor:15 panacea:0)
d#solution(gold:8 ichor:14 panacea:0)
d#solution(gold:8 ichor:15 panacea:0)
d#solution(gold:9 ichor:14 panacea:0)
d#solution(gold:9 ichor:15 panacea:0)
d#solution(gold:10 ichor:14 panacea:0)
d#solution(gold:10 ichor:15 panacea:0)
d#solution(gold:11 ichor:14 panacea:0)
d#solution(gold:11 ichor:15 panacea:0)
Result:
solution(gold:11 ichor:15 panacea:0)
total value: 54500
Pascal
With ideas from C, Fortran and Modula-3.
Program Knapsack(output);
uses
math;
type
bounty = record
value: longint;
weight, volume: real;
end;
const
panacea: bounty = (value:3000; weight: 0.3; volume: 0.025);
ichor: bounty = (value:1800; weight: 0.2; volume: 0.015);
gold: bounty = (value:2500; weight: 2.0; volume: 0.002);
sack: bounty = (value: 0; weight: 25.0; volume: 0.25);
var
totalweight, totalvolume: real;
maxpanacea, maxichor, maxgold: longint;
maxvalue: longint = 0;
n: array [1..3] of longint;
current: bounty;
i, j, k: longint;
begin
maxpanacea := round(min(sack.weight / panacea.weight, sack.volume / panacea.volume));
maxichor := round(min(sack.weight / ichor.weight, sack.volume / ichor.volume));
maxgold := round(min(sack.weight / gold.weight, sack.volume / gold.volume));
for i := 0 to maxpanacea do
for j := 0 to maxichor do
for k := 0 to maxgold do
begin
current.value := k * gold.value + j * ichor.value + i * panacea.value;
current.weight := k * gold.weight + j * ichor.weight + i * panacea.weight;
current.volume := k * gold.volume + j * ichor.volume + i * panacea.volume;
if (current.value > maxvalue) and
(current.weight <= sack.weight) and
(current.volume <= sack.volume) then
begin
maxvalue := current.value;
totalweight := current.weight;
totalvolume := current.volume;
n[1] := i;
n[2] := j;
n[3] := k;
end;
end;
writeln ('Maximum value achievable is ', maxValue);
writeln ('This is achieved by carrying ', n[1], ' panacea, ', n[2], ' ichor and ', n[3], ' gold items');
writeln ('The weight of this carry is ', totalWeight:6:3, ' and the volume used is ', totalVolume:6:4);
end.
Output:
:> ./Knapsack Maximum value achievable is 54500 This is achieved by carrying 0 panacea, 15 ichor and 11 gold items The weight of this carry is 25.000 and the volume used is 0.2470
Perl
Dynamic programming solution. Before you ask, no, it's actually slower for the given data set. See the alternate data set.
my (@names, @val, @weight, @vol, $max_vol, $max_weight, $vsc, $wsc);
if (1) { # change 1 to 0 for different data set
@names = qw(panacea icor gold);
@val = qw(3000 1800 2500);
@weight = qw(3 2 20 );
@vol = qw(25 15 2 );
$max_weight = 250;
$max_vol = 250;
$vsc = 1000;
$wsc = 10;
} else { # with these numbers cache would have been useful
@names = qw(panacea icor gold banana monkey );
@val = qw(17 11 5 3 34 );
@weight = qw(14 3 2 2 10 );
@vol = qw(3 4 2 1 12 );
$max_weight = 150;
$max_vol = 100;
$vsc = $wsc = 1;
}
my @cache;
my ($hits, $misses) = (0, 0);
sub solu {
my ($i, $w, $v) = @_;
return [0, []] if $i < 0;
if ($cache[$i][$w][$v]) {
$hits ++;
return $cache[$i][$w][$v]
}
$misses ++;
my $x = solu($i - 1, $w, $v);
my ($w1, $v1);
for (my $t = 1; ; $t++) {
last if ($w1 = $w - $t * $weight[$i]) < 0;
last if ($v1 = $v - $t * $vol[$i]) < 0;
my $y = solu($i - 1, $w1, $v1);
if ( (my $tmp = $y->[0] + $val[$i] * $t) > $x->[0] ) {
$x = [ $tmp, [ @{$y->[1]}, [$i, $t] ] ];
}
}
$cache[$i][$w][$v] = $x
}
my $x = solu($#names, $max_weight, $max_vol);
print "Max value $x->[0], with:\n",
" Item\tQty\tWeight Vol Value\n", '-'x 50, "\n";
my ($wtot, $vtot) = (0, 0);
for (@{$x->[1]}) {
my $i = $_->[0];
printf " $names[$i]:\t% 3d % 8d% 8g% 8d\n",
$_->[1],
$weight[$i] * $_->[1] / $wsc,
$vol[$i] * $_->[1] / $vsc,
$val[$i] * $_->[1];
$wtot += $weight[$i] * $_->[1];
$vtot += $vol[$i] * $_->[1];
}
print "-" x 50, "\n";
printf " Total:\t % 8d% 8g% 8d\n",
$wtot/$wsc, $vtot/$vsc, $x->[0];
print "\nCache hit: $hits\tmiss: $misses\n";
Output:
Max value 54500, with:
Item Qty Weight Vol Value
--------------------------------------------------
panacea: 9 2 0.225 27000
gold: 11 22 0.022 27500
--------------------------------------------------
Total: 24 0.247 54500
Cache hit: 0 miss: 218
Cache info is not pertinent to this task, just some info.
Phix
For each goodie, fill yer boots, then (except for the last) recursively try with fewer and fewer.
Increase profit and decrease weight/volume to pick largest profit for least weight and space.
-- demo\rosetta\knapsack.exw with javascript_semantics function knapsack(sequence res, goodies, atom profit, weight, volume, at=1, sequence chosen={}) atom {?,pitem,witem,vitem} = goodies[at] integer n = min(floor(weight/witem),floor(volume/vitem)) chosen &= n profit += n*pitem -- increase profit weight -= n*witem -- decrease weight left volume -= n*vitem -- decrease space left if at=length(goodies) then sequence pwvc = {profit,weight,volume,chosen} if length(res)=0 or profit>res[1][1] then res = {pwvc} elsif profit=res[1][1] then res = append(res,pwvc) end if else while n>=0 do res = knapsack(res,goodies,profit,weight,volume,at+1,deep_copy(chosen)) n -= 1 chosen[$] = n profit -= pitem weight += witem volume += vitem end while end if return res end function constant goodies = {-- item profit weight volume {"ichor", 1800, 0.2, 0.015}, {"panacea", 3000, 0.3, 0.025}, {"shiney shiney", 2500, 2.0, 0.002}}, {descs,profits,wts,vols} = columnize(goodies) --res is {{profit,(weight left),(space left),{counts}}} sequence res = knapsack({},goodies,0,25,0.25) for r=1 to length(res) do integer profit = res[r][1] sequence counts = res[r][4] atom weight = sum(sq_mul(counts,wts)), volume = sum(sq_mul(counts,vols)) string what = join(apply(true,sprintf,{{"%2d %s"},columnize({counts,descs})}),", ") printf(1,"Profit %d: %s [weight:%.1f, volume:%g]\n",{profit,what,weight,volume}) end for
- Output:
Profit 54500: 15 ichor, 0 panacea, 11 shiney shiney [weight:25.0, volume:0.247] Profit 54500: 10 ichor, 3 panacea, 11 shiney shiney [weight:24.9, volume:0.247] Profit 54500: 5 ichor, 6 panacea, 11 shiney shiney [weight:24.8, volume:0.247] Profit 54500: 0 ichor, 9 panacea, 11 shiney shiney [weight:24.7, volume:0.247]
Picat
This is a typical constraint modelling problem. We must use the MIP solver - using the mip module - for this problem since the data contains float values (which neither of the CP/SAT/SMT solvers support).
import mip.
go =>
data(Items,Value,Weight,Volume,MaxWeight,MaxVolume),
knapsack_problem(Value,Weight,Volume,MaxWeight,MaxVolume, X,Z),
println(z=Z),
println(x=X),
N = Items.len,
foreach({Item,Num} in zip(Items,X), Num > 0)
printf("Take %d of %w\n", Num,Item)
end,
print("\nTotal volume: "),
println(sum([X[I]*Volume[I] : I in 1..N])),
print("Total weight: "),
println(sum([X[I]*Weight[I] : I in 1..N])),
print("Total cost: "),
println(sum([X[I]*Value[I] : I in 1..N])),
nl.
knapsack_problem(Value,Weight,Volume,MaxWeight,MaxVolume, X,Z) =>
println([max_weight=MaxWeight,max_volume=MaxVolume,z=Z]),
N = Value.length,
X = new_list(N),
X :: 0..1000,
Z #= sum([X[I]*Value[I] : I in 1..N]),
foreach(I in 1..N)
X[I] #>= 0
end,
limit(Weight, X, MaxWeight),
limit(Volume, X, MaxVolume),
if var(Z) then
println(maximize),
solve($[glpk,max(Z)], X)
else
solve($[glpk], X)
end.
limit(W, Take, WTMax) =>
sum([W[I]*Take[I] : I in 1..W.length]) #<= WTMax.
% data
data(Items,Value,Weight,Volume,MaxWeight,MaxVolume) =>
Items = ["panacea","ichor","gold"],
Value = [3000.0, 1800.0, 2500.0 ],
Weight = [ 0.3, 0.2, 2.0 ],
Volume = [ 0.025, 0.015, 0.002],
MaxWeight = 25.0,
MaxVolume = 0.25.- Output:
z = 54500 x = [9,0,11] Take 9 of panacea Take 11 of gold Total volume: 0.247 Total weight: 24.699999999999999 Total cost: 54500.0
PicoLisp
Brute force solution
(de *Items
("panacea" 3 25 3000)
("ichor" 2 15 1800)
("gold" 20 2 2500) )
(de knapsack (Lst W V)
(when Lst
(let X (knapsack (cdr Lst) W V)
(if (and (ge0 (dec 'W (cadar Lst))) (ge0 (dec 'V (caddar Lst))))
(maxi
'((L) (sum cadddr L))
(list
X
(cons (car Lst) (knapsack (cdr Lst) W V))
(cons (car Lst) (knapsack Lst W V)) ) )
X ) ) ) )
(let K (knapsack *Items 250 250)
(for (L K L)
(let (N 1 X)
(while (= (setq X (pop 'L)) (car L))
(inc 'N) )
(apply tab X (4 2 8 5 5 7) N "x") ) )
(tab (14 5 5 7) NIL (sum cadr K) (sum caddr K) (sum cadddr K)) )Output:
15 x ichor 2 15 1800
11 x gold 20 2 2500
250 247 54500
PowerShell
# Define the items to pack
$Item = @(
[pscustomobject]@{ Name = 'panacea'; Unit = 'vials' ; value = 3000; Weight = 0.3; Volume = 0.025 }
[pscustomobject]@{ Name = 'ichor' ; Unit = 'ampules'; value = 1800; Weight = 0.2; Volume = 0.015 }
[pscustomobject]@{ Name = 'gold' ; Unit = 'bars' ; value = 2500; Weight = 2.0; Volume = 0.002 }
)
# Define our maximums
$MaxWeight = 25
$MaxVolume = 0.25
# Set our default value to beat
$OptimalValue = 0
# Iterate through the possible quantities of item 0, without going over the weight or volume limit
ForEach ( $Qty0 in 0..( [math]::Min( [math]::Truncate( $MaxWeight / $Item[0].Weight ), [math]::Truncate( $MaxVolume / $Item[0].Volume ) ) ) )
{
# Calculate the remaining space
$RemainingWeight = $MaxWeight - $Qty0 * $Item[0].Weight
$RemainingVolume = $MaxVolume - $Qty0 * $Item[0].Volume
# Iterate through the possible quantities of item 1, without going over the weight or volume limit
ForEach ( $Qty1 in 0..( [math]::Min( [math]::Truncate( $RemainingWeight / $Item[1].Weight ), [math]::Truncate( $RemainingVolume / $Item[1].Volume ) ) ) )
{
# Calculate the remaining space
$RemainingWeight2 = $RemainingWeight - $Qty1 * $Item[1].Weight
$RemainingVolume2 = $RemainingVolume - $Qty1 * $Item[1].Volume
# Calculate the maximum quantity of item 2 for the remaining space, without going over the weight or volume limit
$Qty2 = [math]::Min( [math]::Truncate( $RemainingWeight2 / $Item[2].Weight ), [math]::Truncate( $RemainingVolume2 / $Item[2].Volume ) )
# Calculate the total value of the items packed
$TrialValue = $Qty0 * $Item[0].Value +
$Qty1 * $Item[1].Value +
$Qty2 * $Item[2].Value
# Describe the trial solution
$Solution = "$Qty0 $($Item[0].Unit) of $($Item[0].Name), "
$Solution += "$Qty1 $($Item[1].Unit) of $($Item[1].Name), and "
$Solution += "$Qty2 $($Item[2].Unit) of $($Item[2].Name) worth a total of $TrialValue."
# If the trial value is higher than previous most valuable trial...
If ( $TrialValue -gt $OptimalValue )
{
# Set the new number to beat
$OptimalValue = $TrialValue
# Overwrite the previous optimal solution(s) with the trial solution
$Solutions = @( $Solution )
}
# Else if the trial value matches the previous most valuable trial...
ElseIf ( $TrialValue -eq $OptimalValue )
{
# Add the trial solution to the list of optimal solutions
$Solutions += @( $Solution )
}
}
}
# Show the results
$Solutions
- Output:
0 vials of panacea, 15 ampules of ichor, and 11 bars of gold worth a total of 54500. 3 vials of panacea, 10 ampules of ichor, and 11 bars of gold worth a total of 54500. 6 vials of panacea, 5 ampules of ichor, and 11 bars of gold worth a total of 54500. 9 vials of panacea, 0 ampules of ichor, and 11 bars of gold worth a total of 54500.
Prolog
Works with SWI-Prolog and library simplex written by Markus Triska.
:- use_module(library(simplex)).
% tuples (name, Explantion, Value, weights, volume).
knapsack :-
L =[( panacea, 'Incredible healing properties', 3000, 0.3, 0.025),
( ichor, 'Vampires blood', 1800, 0.2, 0.015),
( gold , 'Shiney shiney', 2500, 2.0, 0.002)],
gen_state(S0),
length(L, N),
numlist(1, N, LN),
% to get statistics
time((create_constraint_N(LN, L, S0, S1, [], LVa, [], LW, [], LVo),
constraint(LW =< 25.0, S1, S2),
constraint(LVo =< 0.25, S2, S3),
maximize(LVa, S3, S4)
)),
% we display the results
compute_lenword(L, 0, Len),
sformat(A0, '~~w~~t~~~w|', [3]),
sformat(A1, '~~w~~t~~~w|', [Len]),
sformat(A2, '~~t~~w~~~w|', [10]),
sformat(A3, '~~t~~2f~~~w|', [10]),
sformat(A4, '~~t~~3f~~~w|', [10]),
sformat(A33, '~~t~~w~~~w|', [10]),
sformat(A44, '~~t~~w~~~w|', [10]),
sformat(W0, A0, ['Nb']),
sformat(W1, A1, ['Items']),
sformat(W2, A2, ['Value']),
sformat(W3, A33, ['Weigth']),
sformat(W4, A44, ['Volume']),
format('~w~w~w~w~w~n', [W0, W1,W2,W3,W4]),
print_results(S4, A0, A1, A2, A3, A4, L, LN, 0, 0, 0).
create_constraint_N([], [], S, S, LVa, LVa, LW, LW, LVo, LVo).
create_constraint_N([HN|TN], [(_, _,Va, W, Vo) | TL], S1, SF, LVa, LVaF, LW, LWF, LVo, LVoF) :-
constraint(integral(x(HN)), S1, S2),
constraint([x(HN)] >= 0, S2, S3),
create_constraint_N(TN, TL, S3, SF,
[Va * x(HN) | LVa], LVaF,
[W * x(HN) | LW], LWF,
[Vo * x(HN) | LVo], LVoF).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
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, A0, A1, A2, A3, A4, [], [], VaM, WM, VoM) :-
sformat(W0, A0, [' ']),
sformat(W1, A1, [' ']),
sformat(W2, A2, [VaM]),
sformat(W3, A3, [WM]),
sformat(W4, A4, [VoM]),
format('~w~w~w~w~w~n', [W0, W1,W2,W3,W4]).
print_results(S, A0, A1, A2, A3, A4, [(Name, _, Va, W, Vo)|T], [N|TN], Va1, W1, Vo1) :-
variable_value(S, x(N), X),
( X = 0 -> Va1 = Va2, W1 = W2, Vo1 = Vo2
;
sformat(S0, A0, [X]),
sformat(S1, A1, [Name]),
Vatemp is X * Va,
Wtemp is X * W,
Votemp is X * Vo,
sformat(S2, A2, [Vatemp]),
sformat(S3, A3, [Wtemp]),
sformat(S4, A4, [Votemp]),
format('~w~w~w~w~w~n', [S0,S1,S2,S3,S4]),
Va2 is Va1 + Vatemp,
W2 is W1 + Wtemp,
Vo2 is Vo1 + Votemp ),
print_results(S, A0, A1, A2, A3, A4, T, TN, Va2, W2, Vo2).
Output :
?- knapsack.
% 145,319 inferences, 0.078 CPU in 0.079 seconds (99% CPU, 1860083 Lips)
Nb Items Value Weigth Volume
15 ichor 27000 3.00 0.225
11 gold 27500 22.00 0.022
54500 25.00 0.247
true
PureBasic
Define.f TotalWeight, TotalVolyme
Define.i maxPanacea, maxIchor, maxGold, maxValue
Define.i i, j ,k
Dim n.i(2)
Enumeration
#Panacea
#Ichor
#Gold
#Sack
#Current
EndEnumeration
Structure Bounty
value.i
weight.f
volyme.f
EndStructure
Dim Item.Bounty(4)
CopyMemory(?panacea,@Item(#Panacea),SizeOf(Bounty))
CopyMemory(?ichor, @Item(#Ichor), SizeOf(Bounty))
CopyMemory(?gold, @Item(#gold), SizeOf(Bounty))
CopyMemory(?sack, @Item(#Sack), SizeOf(Bounty))
Procedure.f min(a.f, b.f)
If a<b
ProcedureReturn a
Else
ProcedureReturn b
EndIf
EndProcedure
maxPanacea=min(Item(#Sack)\weight/Item(#Panacea)\weight,Item(#Sack)\volyme/Item(#Panacea)\volyme)
maxIchor =min(Item(#Sack)\weight/Item(#Ichor)\weight, Item(#Sack)\volyme/Item(#Ichor)\volyme)
maxGold =min(Item(#Sack)\weight/Item(#Gold)\weight, Item(#Sack)\volyme/Item(#Gold)\volyme)
For i=0 To maxPanacea
For j=0 To maxIchor
For k=0 To maxGold
Item(#Current)\value=k*Item(#Gold)\value +j*item(#Ichor)\value +i*item(#Panacea)\value
Item(#Current)\weight=k*Item(#Gold)\weight+j*Item(#Ichor)\weight+i*Item(#Panacea)\weight
Item(#Current)\volyme=k*Item(#Gold)\volyme+j*Item(#Ichor)\volyme+i*Item(#Panacea)\volyme
If Item(#Current)\weight>Item(#Sack)\weight Or Item(#Current)\volyme>Item(#Sack)\volyme
Continue
EndIf
If Item(#Current)\value>maxValue
maxValue=Item(#Current)\value
TotalWeight=Item(#Current)\weight
TotalVolyme=Item(#Current)\volyme
n(#Panacea)=i: n(#Ichor)=j: n(#Gold)=k
EndIf
Next k
Next j
Next i
If OpenConsole()
Define txt$
txt$="Maximum value achievable is "+Str(maxValue)+#CRLF$
txt$+"This is achieved by carrying "+Str(n(#Panacea))+" panacea, "
txt$+Str(n(#Ichor))+" ichor and "+Str(n(#Gold))+" gold items."+#CRLF$
txt$+"The weight to carry is "+StrF(totalWeight,2)
txt$+" and the volume used is "+StrF(TotalVolyme,2)
PrintN(txt$)
Print(#CRLF$+"Press Enter to quit"): Input()
EndIf
DataSection
panacea:
Data.i 3000
Data.f 0.3, 0.025
ichor:
Data.i 1800
Data.f 0.2, 0.015
gold:
Data.i 2500
Data.f 2.0, 0.002
sack:
Data.i 0
Data.f 25.0, 0.25
EndDataSection
Outputs
Maximum value achievable is 54500 This is achieved by carrying 0 panacea, 15 ichor and 11 gold items The weight to carry is 25.00 and the volume used is 0.25 Press Enter to quit
Python
R
Brute force method
# Define consts
weights <- c(panacea=0.3, ichor=0.2, gold=2.0)
volumes <- c(panacea=0.025, ichor=0.015, gold=0.002)
values <- c(panacea=3000, ichor=1800, gold=2500)
sack.weight <- 25
sack.volume <- 0.25
max.items <- floor(pmin(sack.weight/weights, sack.volume/volumes))
# Some utility functions
getTotalValue <- function(n) sum(n*values)
getTotalWeight <- function(n) sum(n*weights)
getTotalVolume <- function(n) sum(n*volumes)
willFitInSack <- function(n) getTotalWeight(n) <= sack.weight && getTotalVolume(n) <= sack.volume
# Find all possible combination, then eliminate those that won't fit in the sack
knapsack <- expand.grid(lapply(max.items, function(n) seq.int(0, n)))
ok <- apply(knapsack, 1, willFitInSack)
knapok <- knapsack[ok,]
# Find the solutions with the highest value
vals <- apply(knapok, 1, getTotalValue)
knapok[vals == max(vals),]
panacea ichor gold 2067 9 0 11 2119 6 5 11 2171 3 10 11 2223 0 15 11
Using Dynamic Programming
Data_<-structure(list(item = c("Panacea", "Ichor", "Gold"), value = c(3000,
1800, 2500), weight = c(3, 2, 20), volume = c(25, 15, 2)), .Names = c("item",
"value", "weight", "volume"), row.names = c(NA, 3L), class = "data.frame")
knapsack_volume<-function(Data, W, Volume, full_K)
{
# Data must have the colums with names: item, value, weight and volume.
K<-list() # hightest values
K_item<-list() # itens that reach the hightest value
K<-rep(0,W+1) # The position '0'
K_item<-rep('',W+1) # The position '0'
for(w in 1:W)
{
temp_w<-0
temp_item<-''
temp_value<-0
for(i in 1:dim(Data)[1]) # each row
{
wi<-Data$weight[i] # item i
vi<- Data$value[i]
item<-Data$item[i]
volume_i<-Data$volume[i]
if(wi<=w & volume_i <= Volume)
{
back<- full_K[[Volume-volume_i+1]][w-wi+1]
temp_wi<-vi + back
if(temp_w < temp_wi)
{
temp_value<-temp_wi
temp_w<-temp_wi
temp_item <- item
}
}
}
K[[w+1]]<-temp_value
K_item[[w+1]]<-temp_item
}
return(list(K=K,Item=K_item))
}
Un_knapsack<-function(Data,W,V)
{
K<-list();K_item<-list()
K[[1]]<-rep(0,W+1) #the line 0
K_item[[1]]<-rep('', W+1) #the line 0
for(v in 1:V)
{
best_volum_v<-knapsack_volume(Data, W, v, K)
K[[v+1]]<-best_volum_v$K
K_item[[v+1]]<-best_volum_v$Item
}
return(list(K=data.frame(K),Item=data.frame(K_item,stringsAsFactors=F)))
}
retrieve_info<-function(knapsack, Data)
{
W<-dim(knapsack$K)[1]
itens<-c()
col<-dim(knapsack$K)[2]
selected_item<-knapsack$Item[W,col]
while(selected_item!='')
{
selected_item<-knapsack$Item[W,col]
if(selected_item!='')
{
selected_item_value<-Data[Data$item == selected_item,]
W <- W - selected_item_value$weight
itens<-c(itens,selected_item)
col <- col - selected_item_value$volume
}
}
return(itens)
}
main_knapsack<-function(Data, W, Volume)
{
knapsack_result<-Un_knapsack(Data,W,Volume)
items<-table(retrieve_info(knapsack_result, Data))
K<-knapsack_result$K[W+1, Volume+1]
cat(paste('The Total profit is: ', K, '\n'))
cat(paste('You must carry:', names(items), '(x',items, ') \n'))
}
main_knapsack(Data_, 250, 250)
Output:
The Total profit is: 54500
You must carry: Gold (x 11 )
You must carry: Panacea (x 9 )
Racket
#lang racket
(struct item (name explanation value weight volume) #:prefab)
(define items
(list
(item "panacea (vials of)" "Incredible healing properties" 3000 0.3 0.025)
(item "ichor (ampules of)" "Vampires blood" 1800 0.2 0.015)
(item "gold (bars)" "Shiney shiney" 2500 2.0 0.002)))
(define (fill-sack items volume-left weight-left sack sack-value)
(match items
['() (values (list sack) sack-value)]
[(cons (and (item _ _ item-val weight volume) item) items)
(define max-q-wgt (floor (/ weight-left weight)))
(define max-q-vol (floor (/ volume-left volume)))
(for/fold ([best (list sack)] [best-val sack-value])
([n (exact-round (add1 (min max-q-vol max-q-wgt)))])
(define-values [best* best-val*]
(fill-sack items
(- volume-left (* n volume))
(- weight-left (* n weight))
(cons (cons n item) sack)
(+ sack-value (* n item-val))))
(cond [(> best-val* best-val) (values best* best-val*)]
[(= best-val* best-val) (values (append best best*) best-val*)]
[else (values best best-val)]))]))
(define (display-sack sack total)
(for ([sk sack])
(define qty (car sk))
(define name (item-name (cdr sk)))
(if (zero? qty)
(printf "Leave ~a\n" name)
(printf "Take ~a ~a\n" qty name)))
(printf "GRAND TOTAL: ~a\n\n" total))
(call-with-values (λ() (fill-sack items 0.25 25 '() 0))
(λ(sacks total) (for ([s sacks]) (display-sack s total))))
- Output:
Take 11 gold (bars) Take 15 ichor (ampules of) Leave panacea (vials of) GRAND TOTAL: 54500 Take 11 gold (bars) Take 10 ichor (ampules of) Take 3 panacea (vials of) GRAND TOTAL: 54500 Take 11 gold (bars) Take 5 ichor (ampules of) Take 6 panacea (vials of) GRAND TOTAL: 54500 Take 11 gold (bars) Leave ichor (ampules of) Take 9 panacea (vials of) GRAND TOTAL: 54500
Raku
(formerly Perl 6)
Brute force, looked a lot at the Ruby solution.
class KnapsackItem {
has $.volume;
has $.weight;
has $.value;
has $.name;
method new($volume,$weight,$value,$name) {
self.bless(:$volume, :$weight, :$value, :$name)
}
};
my KnapsackItem $panacea .= new: 0.025, 0.3, 3000, "panacea";
my KnapsackItem $ichor .= new: 0.015, 0.2, 1800, "ichor";
my KnapsackItem $gold .= new: 0.002, 2.0, 2500, "gold";
my KnapsackItem $maximum .= new: 0.25, 25, 0 , "max";
my $max_val = 0;
my @solutions;
my %max_items;
for $panacea, $ichor, $gold -> $item {
%max_items{$item.name} = floor min
$maximum.volume / $item.volume,
$maximum.weight / $item.weight;
}
for 0..%max_items<panacea>
X 0..%max_items<ichor>
X 0..%max_items<gold>
-> ($p, $i, $g)
{
next if $panacea.volume * $p + $ichor.volume * $i + $gold.volume * $g > $maximum.volume;
next if $panacea.weight * $p + $ichor.weight * $i + $gold.weight * $g > $maximum.weight;
given $panacea.value * $p + $ichor.value * $i + $gold.value * $g {
if $_ > $max_val { $max_val = $_; @solutions = (); }
when $max_val { @solutions.push: $[$p,$i,$g] }
}
}
say "maximum value is $max_val\npossible solutions:";
say "panacea\tichor\tgold";
.join("\t").say for @solutions;
Output:
maximum value is 54500 possible solutions: panacea ichor gold 0 15 11 3 10 11 6 5 11 9 0 11
REXX
displays 1st solution
/*REXX program solves the knapsack/unbounded problem: highest value, weight, and volume.*/
/* value weight volume */
maxPanacea= 0 /* ═══════ ══════ ══════ */
maxIchor = 0; panacea.$ = 3000 ; panacea.w = 0.3 ; panacea.v = 0.025
maxGold = 0; ichor.$ = 1800 ; ichor.w = 0.2 ; ichor.v = 0.015
max$ = 0; gold.$ = 2500 ; gold.w = 2 ; gold.v = 0.002
now. = 0; sack.$ = 0 ; sack.w = 25 ; sack.v = 0.25
maxPanacea= min(sack.w / panacea.w, sack.v / panacea.v)
maxIchor = min(sack.w / ichor.w, sack.v / ichor.v)
maxGold = min(sack.w / gold.w, sack.v / gold.v)
do p=0 to maxPanacea
do i=0 to maxIchor
do g=0 to maxGold
now.$= g * gold.$ + i * ichor.$ + p * panacea.$
now.w= g * gold.w + i * ichor.w + p * panacea.w
now.v= g * gold.v + i * ichor.v + p * panacea.v
if now.w > sack.w | now.v > sack.v then iterate
if now.$ > max$ then do; maxP= p; maxI= i; maxG= g
max$= now.$; maxW= now.w; maxV= now.v
end
end /*g (gold) */
end /*i (ichor) */
end /*p (panacea)*/
Ctot = maxP + maxI + maxG; L = length(Ctot) + 1
say ' panacea in sack:' right(maxP, L)
say ' ichors in sack:' right(maxI, L)
say ' gold items in sack:' right(maxG, L)
say '════════════════════' copies("═", L)
say 'carrying a total of:' right(cTot, L)
say left('', 40) "total value: " max$ / 1
say left('', 40) "total weight: " maxW / 1
say left('', 40) "total volume: " maxV / 1
/*stick a fork in it, we're all done. */
- output when using the internal default input:
panacea in sack: 0
ichors in sack: 15
gold items in sack: 11
════════════════════ ═══
carrying a total of: 26
total value: 54500
total weight: 25
total volume: 0.247
displays all solutions
/*REXX program solves the knapsack/unbounded problem: highest value, weight, and volume.*/
maxPanacea= 0
maxIchor = 0; /* value weight volume */
maxGold = 0; /* ═══════ ══════ ══════ */
max$ = 0; panacea.$ = 3000 ; panacea.w = 0.3 ; panacea.v = 0.025
now. = 0; ichor.$ = 1800 ; ichor.w = 0.2 ; ichor.v = 0.015
# = 0; gold.$ = 2500 ; gold.w = 2 ; gold.v = 0.002
L = 0; sack.$ = 0 ; sack.w = 25 ; sack.v = 0.25
maxPanacea= min(sack.w / panacea.w, sack.v / panacea.v)
maxIchor = min(sack.w / ichor.w, sack.v / ichor.v)
maxGold = min(sack.w / gold.w, sack.v / gold.v)
do p=0 to maxPanacea
do i=0 to maxIchor
do g=0 to maxGold
now.$ = g * gold.$ + i * ichor.$ + p * panacea.$
now.w = g * gold.w + i * ichor.w + p * panacea.w
now.v = g * gold.v + i * ichor.v + p * panacea.v
if now.w > sack.w | now.v > sack.v then iterate i
if now.$ > max$ then do; #= 0; max$= now.$; end
if now.$ = max$ then do; #= # + 1; maxP.#= p; maxI.#= i; maxG.#= g
max$.#= now.$; maxW.#= now.w; maxV.#= now.v
L= max(L, length(p + i + g) )
end
end /*g (gold) */
end /*i (ichor) */
end /*p (panacea)*/
L= L + 1
do j=1 for #; say; say copies('▒', 70) "solution" j
say ' panacea in sack:' right(maxP.j, L)
say ' ichors in sack:' right(maxI.j, L)
say ' gold items in sack:' right(maxG.j, L)
say '════════════════════' copies("═", L)
say 'carrying a total of:' right(maxP.j + maxI.j + maxG.j, L)
say left('', 40) "total value: " max$.j / 1
say left('', 40) "total weight: " maxW.j / 1
say left('', 40) "total volume: " maxV.j / 1
end /*j*/
/*stick a fork in it, we're all done. */
- output when using the internal default input:
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒ solution 1
panacea in sack: 0
ichors in sack: 15
gold items in sack: 11
════════════════════ ═══
carrying a total of: 26
total value: 54500
total weight: 25
total volume: 0.247
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒ solution 2
panacea in sack: 3
ichors in sack: 10
gold items in sack: 11
════════════════════ ═══
carrying a total of: 24
total value: 54500
total weight: 24.9
total volume: 0.247
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒ solution 3
panacea in sack: 6
ichors in sack: 5
gold items in sack: 11
════════════════════ ═══
carrying a total of: 22
total value: 54500
total weight: 24.8
total volume: 0.247
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒ solution 4
panacea in sack: 9
ichors in sack: 0
gold items in sack: 11
════════════════════ ═══
carrying a total of: 20
total value: 54500
total weight: 24.7
total volume: 0.247
Ruby
Brute force method,
KnapsackItem = Struct.new(:volume, :weight, :value)
panacea = KnapsackItem.new(0.025, 0.3, 3000)
ichor = KnapsackItem.new(0.015, 0.2, 1800)
gold = KnapsackItem.new(0.002, 2.0, 2500)
maximum = KnapsackItem.new(0.25, 25, 0)
max_items = {}
for item in [panacea, ichor, gold]
max_items[item] = [(maximum.volume/item.volume).to_i, (maximum.weight/item.weight).to_i].min
end
maxval = 0
solutions = []
0.upto(max_items[ichor]) do |i|
0.upto(max_items[panacea]) do |p|
0.upto(max_items[gold]) do |g|
break if i*ichor.weight + p*panacea.weight + g*gold.weight > maximum.weight
break if i*ichor.volume + p*panacea.volume + g*gold.volume > maximum.volume
val = i*ichor.value + p*panacea.value + g*gold.value
if val > maxval
maxval = val
solutions = [[i, p, g]]
elsif val == maxval
solutions << [i, p, g]
end
end
end
end
puts "The maximal solution has value #{maxval}"
solutions.each do |i, p, g|
printf " ichor=%2d, panacea=%2d, gold=%2d -- weight:%.1f, volume=%.3f\n",
i, p, g,
i*ichor.weight + p*panacea.weight + g*gold.weight,
i*ichor.volume + p*panacea.volume + g*gold.volume
end
- Output:
The maximal solution has value 54500 ichor= 0, panacea= 9, gold=11 -- weight:24.7, volume=0.247 ichor= 5, panacea= 6, gold=11 -- weight:24.8, volume=0.247 ichor=10, panacea= 3, gold=11 -- weight:24.9, volume=0.247 ichor=15, panacea= 0, gold=11 -- weight:25.0, volume=0.247
SAS
This is yet another brute force solution.
data one;
wtpanacea=0.3; wtichor=0.2; wtgold=2.0;
volpanacea=0.025; volichor=0.015; volgold=0.002;
valpanacea=3000; valichor=1800; valgold=2500;
maxwt=25; maxvol=0.25;
/* we can prune the possible selections */
maxpanacea = floor(min(maxwt/wtpanacea, maxvol/volpanacea));
maxichor = floor(min(maxwt/wtichor, maxvol/volichor));
maxgold = floor(min(maxwt/wtgold, maxvol/volgold));
do i1 = 0 to maxpanacea;
do i2 = 0 to maxichor;
do i3 = 0 to maxgold;
panacea = i1; ichor=i2; gold=i3; output;
end;
end;
end;
run;
data one; set one;
vals = valpanacea*panacea + valichor*ichor + valgold*gold;
totalweight = wtpanacea*panacea + wtichor*ichor + wtgold*gold;
totalvolume = volpanacea*panacea + volichor*ichor + volgold*gold;
if (totalweight le maxwt) and (totalvolume le maxvol);
run;
proc sort data=one;
by descending vals;
run;
proc print data=one (obs=4);
var panacea ichor gold vals;
run;
Output:
Obs panacea ichor gold vals 1 0 15 11 54500 2 3 10 11 54500 3 6 5 11 54500 4 9 0 11 54500
Use SAS/OR:
/* create SAS data set */
data mydata;
input Item $1-19 Value weight Volume;
datalines;
panacea (vials of) 3000 0.3 0.025
ichor (ampules of) 1800 0.2 0.015
gold (bars) 2500 2.0 0.002
;
/* call OPTMODEL procedure in SAS/OR */
proc optmodel;
/* declare sets and parameters, and read input data */
set <str> ITEMS;
num value {ITEMS};
num weight {ITEMS};
num volume {ITEMS};
read data mydata into ITEMS=[item] value weight volume;
/* declare variables, objective, and constraints */
var NumSelected {ITEMS} >= 0 integer;
max TotalValue = sum {i in ITEMS} value[i] * NumSelected[i];
con WeightCon:
sum {i in ITEMS} weight[i] * NumSelected[i] <= 25;
con VolumeCon:
sum {i in ITEMS} volume[i] * NumSelected[i] <= 0.25;
/* call mixed integer linear programming (MILP) solver */
solve;
/* print optimal solution */
print TotalValue;
print NumSelected;
/* to get all optimal solutions, call CLP solver instead */
solve with CLP / findallsolns;
/* print all optimal solutions */
print TotalValue;
for {s in 1.._NSOL_} print {i in ITEMS} NumSelected[i].sol[s];
quit;
MILP solver output:
TotalValue 54500 [1] NumSelected gold (bars) 11 ichor (ampules of) 0 panacea (vials of) 9
CLP solver output:
TotalValue 54500 [1] gold (bars) 11 ichor (ampules of) 15 panacea (vials of) 0 [1] gold (bars) 11 ichor (ampules of) 10 panacea (vials of) 3 [1] gold (bars) 11 ichor (ampules of) 5 panacea (vials of) 6 [1] gold (bars) 11 ichor (ampules of) 0 panacea (vials of) 9
Scala
Functional approach (Tail recursive)
import scala.annotation.tailrec
object UnboundedKnapsack extends App {
private val (maxWeight, maxVolume) = (BigDecimal(25.0), BigDecimal(0.25))
private val items = Seq(Item("panacea", 3000, 0.3, 0.025), Item("ichor", 1800, 0.2, 0.015), Item("gold", 2500, 2.0, 0.002))
@tailrec
private def packer(notPacked: Seq[Knapsack], packed: Seq[Knapsack]): Seq[Knapsack] = {
def fill(knapsack: Knapsack): Seq[Knapsack] = items.map(i => Knapsack(i +: knapsack.bagged))
def stuffer(Seq: Seq[Knapsack]): Seq[Knapsack] = // Cause brute force
Seq.map(k => Knapsack(k.bagged.sortBy(_.name))).distinct
if (notPacked.isEmpty) packed.sortBy(-_.totValue).take(4)
else packer(stuffer(notPacked.flatMap(fill)).filter(_.isNotFull), notPacked ++ packed)
}
private case class Item(name: String, value: Int, weight: BigDecimal, volume: BigDecimal)
private case class Knapsack(bagged: Seq[Item]) {
def isNotFull: Boolean = totWeight <= maxWeight && totVolume <= maxVolume
override def toString = s"[${show(bagged)} | value: $totValue, weight: $totWeight, volume: $totVolume]"
def totValue: Int = bagged.map(_.value).sum
private def totVolume = bagged.map(_.volume).sum
private def totWeight = bagged.map(_.weight).sum
private def show(is: Seq[Item]) =
(items.map(_.name) zip items.map(i => is.count(_ == i)))
.map { case (i, c) => f"$i:$c%3d" }
.mkString(", ")
}
packer(items.map(i => Knapsack(Seq(i))), Nil).foreach(println)
}
- Output:
[panacea: 0, ichor: 15, gold: 11 | value: 54500, weight: 25.0, volume: 0.247] [panacea: 3, ichor: 10, gold: 11 | value: 54500, weight: 24.9, volume: 0.247] [panacea: 6, ichor: 5, gold: 11 | value: 54500, weight: 24.8, volume: 0.247] [panacea: 9, ichor: 0, gold: 11 | value: 54500, weight: 24.7, volume: 0.247]
- Output:
See it in running in your browser by ScalaFiddle (JavaScript) or by Scastie (JVM).
Seed7
$ include "seed7_05.s7i";
include "float.s7i";
const type: bounty is new struct
var integer: value is 0;
var float: weight is 0.0;
var float: volume is 0.0;
end struct;
const func bounty: bounty (in integer: value, in float: weight, in float: volume) is func
result
var bounty: bountyVal is bounty.value;
begin
bountyVal.value := value;
bountyVal.weight := weight;
bountyVal.volume := volume;
end func;
const proc: main is func
local
const bounty: panacea is bounty(3000, 0.3, 0.025);
const bounty: ichor is bounty(1800, 0.2, 0.015);
const bounty: gold is bounty(2500, 2.0, 0.002);
const bounty: sack is bounty(0, 25.0, 0.25);
const integer: maxPanacea is trunc(min(sack.weight / panacea.weight, sack.volume / panacea.volume));
const integer: maxIchor is trunc(min(sack.weight / ichor.weight, sack.volume / ichor.volume));
const integer: maxGold is trunc(min(sack.weight / gold.weight, sack.volume / gold.volume));
var bounty: current is bounty.value;
var bounty: best is bounty.value;
var array integer: bestAmounts is 3 times 0;
var integer: numPanacea is 0;
var integer: numIchor is 0;
var integer: numGold is 0;
begin
for numPanacea range 0 to maxPanacea do
for numIchor range 0 to maxIchor do
for numGold range 0 to maxGold do
current.value := numGold * gold.value + numIchor * ichor.value + numPanacea * panacea.value;
current.weight := flt(numGold) * gold.weight + flt(numIchor) * ichor.weight + flt(numPanacea) * panacea.weight;
current.volume := flt(numGold) * gold.volume + flt(numIchor) * ichor.volume + flt(numPanacea) * panacea.volume;
if current.value > best.value and current.weight <= sack.weight and current.volume <= sack.volume then
best := current;
bestAmounts := [] (numPanacea, numIchor, numGold);
end if;
end for;
end for;
end for;
writeln("Maximum value achievable is " <& best.value);
writeln("This is achieved by carrying " <& bestAmounts[1] <& " panacea, " <& bestAmounts[2] <& " ichor and " <& bestAmounts[3] <& " gold items");
writeln("The weight of this carry is " <& best.weight <& " and the volume used is " <& best.volume digits 4);
end func;Output:
Maximum value achievable is 54500 This is achieved by carrying 0 panacea, 15 ichor and 11 gold items The weight of this carry is 25.0 and the volume used is 0.2470
Sidef
struct KnapsackItem {
Number volume,
Number weight,
Number value,
String name,
}
var items = [
KnapsackItem(25, 3, 3000, "panacea")
KnapsackItem(15, 2, 1800, "ichor" )
KnapsackItem( 2, 20, 2500, "gold" )
]
var (
max_weight = 250,
max_vol = 250,
vsc = 1000,
wsc = 10
)
func solve(i, w, v) is cached {
return [0, []] if i.is_neg;
var x = solve(i.dec, w, v);
var (w1, v1);
Inf.times { |t|
var item = items[i];
break if ((w1 = (w - t*item.weight)).is_neg)
break if ((v1 = (v - t*item.volume)).is_neg)
var y = solve(i.dec, w1, v1);
if ((var tmp = (y[0] + t*item.value)) > x[0]) {
x = [tmp, [y[1]..., [i, t]]];
}
}
return x
}
var x = solve(items.end, max_weight, max_vol)
print <<"EOT"
Max value #{x[0]}, with:
Item Qty Weight Vol Value
#{"-" * 50}
EOT
var (wtot=0, vtot=0);
x[1].each { |s|
var item = items[s[0]];
" #{item.name}:\t% 3d % 8d% 8g% 8d\n".printf(
s[1],
item.weight * s[1] / wsc,
item.volume * s[1] / vsc,
item.value * s[1]
);
wtot += (item.weight * s[1]);
vtot += (item.volume * s[1]);
}
print <<"EOT"
#{"-" * 50}
Total:\t #{"%8d%8g%8d" % (wtot/wsc, vtot/vsc, x[0])}
EOT
- Output:
Max value 54500, with:
Item Qty Weight Vol Value
--------------------------------------------------
panacea: 9 2 0.225 27000
gold: 11 22 0.022 27500
--------------------------------------------------
Total: 24 0.247 54500
Tcl
The following code uses brute force, but that's tolerable as long as it takes just a split second to find all 4 solutions. The use of arrays makes the code quite legible:
#!/usr/bin/env tclsh
proc main argv {
array set value {panacea 3000 ichor 1800 gold 2500}
array set weight {panacea 0.3 ichor 0.2 gold 2.0 max 25}
array set volume {panacea 0.025 ichor 0.015 gold 0.002 max 0.25}
foreach i {panacea ichor gold} {
set max($i) [expr {min(int($volume(max)/$volume($i)),
int($weight(max)/$weight($i)))}]
}
set maxval 0
for {set i 0} {$i < $max(ichor)} {incr i} {
for {set p 0} {$p < $max(panacea)} {incr p} {
for {set g 0} {$g < $max(gold)} {incr g} {
if {$i*$weight(ichor) + $p*$weight(panacea) + $g*$weight(gold)
> $weight(max)} continue
if {$i*$volume(ichor) + $p*$volume(panacea) + $g*$volume(gold)
> $volume(max)} continue
set val [expr {$i*$value(ichor)+$p*$value(panacea)+$g*$value(gold)}]
if {$val == $maxval} {
lappend best [list i $i p $p g $g]
} elseif {$val > $maxval} {
set maxval $val
set best [list [list i $i p $p g $g]]
}
}
}
}
puts "maxval: $maxval, best: $best"
}
main $argv
$ time tclsh85 /Tcl/knapsack.tcl
maxval: 54500, best: {i 0 p 9 g 11} {i 5 p 6 g 11} {i 10 p 3 g 11} {i 15 p 0 g 11}
real 0m0.188s
user 0m0.015s
sys 0m0.015s
Ursala
The algorithm is to enumerate all packings with up to the maximum of each item, filter them by the volume and weight restrictions, partition the remaining packings by value, and search for the maximum value class.
#import nat
#import flo
vol = iprod/<0.025,0.015,0.002>+ float*
val = iprod/<3000.,1800.,2500.>+ float*
wgt = iprod/<0.3,0.2,2.0>+ float*
packings = ~&lrlrNCCPCS ~&K0=> iota* <11,17,13>
solutions = fleq$^rS&hl |=&l ^(val,~&)* (fleq\25.+ wgt)*~ (fleq\0.25+ vol)*~ packings
#cast %nmL
human_readable = ~&p/*<'panacea','ichor','gold'> solutionsoutput:
< <'panacea': 0,'ichor': 15,'gold': 11>, <'panacea': 3,'ichor': 10,'gold': 11>, <'panacea': 6,'ichor': 5,'gold': 11>, <'panacea': 9,'ichor': 0,'gold': 11>>
Visual Basic
See: Knapsack Problem/Visual Basic
The above Link contains a longer version (which perhaps runs a bit faster), whilst the one below is focussing more on expressing/solving the problem in less lines of code.
Function Min(E1, E2): Min = IIf(E1 < E2, E1, E2): End Function 'small Helper-Function
Sub Main()
Const Value = 0, Weight = 1, Volume = 2, PC = 3, IC = 4, GC = 5
Dim P&, I&, G&, A&, M, Cur(Value To Volume)
Dim S As New Collection: S.Add Array(0) '<- init Solutions-Coll.
Const SackW = 25, SackV = 0.25
Dim Panacea: Panacea = Array(3000, 0.3, 0.025)
Dim Ichor: Ichor = Array(1800, 0.2, 0.015)
Dim Gold: Gold = Array(2500, 2, 0.002)
For P = 0 To Int(Min(SackW / Panacea(Weight), SackV / Panacea(Volume)))
For I = 0 To Int(Min(SackW / Ichor(Weight), SackV / Ichor(Volume)))
For G = 0 To Int(Min(SackW / Gold(Weight), SackV / Gold(Volume)))
For A = Value To Volume: Cur(A) = G * Gold(A) + I * Ichor(A) + P * Panacea(A): Next
If Cur(Value) >= S(1)(Value) And Cur(Weight) <= SackW And Cur(Volume) <= SackV Then _
S.Add Array(Cur(Value), Cur(Weight), Cur(Volume), P, I, G), , 1
Next G, I, P
Debug.Print "Value", "Weight", "Volume", "PanaceaCount", "IchorCount", "GoldCount"
For Each M In S '<- enumerate the Attributes of the Maxima
If M(Value) = S(1)(Value) Then Debug.Print M(Value), M(Weight), M(Volume), M(PC), M(IC), M(GC)
Next
End Sub
Output:
Value Weight Volume PanaceaCount IchorCount GoldCount 54500 24.7 0.247 9 0 11 54500 24.8 0.247 6 5 11 54500 24.9 0.247 3 10 11 54500 25 0.247 0 15 11
Wren
import "./fmt" for Fmt
class Item {
construct new(name, value, weight, volume) {
_name = name
_value = value
_weight = weight
_volume = volume
}
name { _name }
value { _value }
weight { _weight }
volume { _volume }
}
var items = [
Item.new("panacea", 3000, 0.3, 0.025),
Item.new("ichor", 1800, 0.2, 0.015),
Item.new("gold", 2500, 2, 0.002)
]
var n = items.count
var count = List.filled(n, 0)
var best = List.filled(n, 0)
var bestValue = 0
var maxWeight = 25
var maxVolume = 0.25
var knapsack // recursive
knapsack = Fn.new { |i, value, weight, volume|
if (i == n) {
if (value > bestValue) {
bestValue = value
for (j in 0...n) best[j] = count[j]
}
return
}
var m1 = (weight / items[i].weight).floor
var m2 = (volume / items[i].volume).floor
count[i] = m1.min(m2)
while (count[i] >= 0) {
knapsack.call(
i + 1,
value + count[i] * items[i].value,
weight - count[i] * items[i].weight,
volume - count[i] * items[i].volume
)
count[i] = count[i] - 1
}
}
knapsack.call(0, 0, maxWeight, maxVolume)
System.print("Item Chosen Number Value Weight Volume")
System.print("----------- ------ ----- ------ ------")
var itemCount = 0
var sumNumber = 0
var sumWeight = 0
var sumVolume = 0
for (i in 0... n) {
if (best[i] != 0) {
itemCount = itemCount + 1
var name = items[i].name
var number = best[i]
var value = items[i].value * number
var weight = items[i].weight * number
var volume = items[i].volume * number
sumNumber = sumNumber + number
sumWeight = sumWeight + weight
sumVolume = sumVolume + volume
Fmt.write("$-11s $2d $5.0f $4.1f", name, number, value, weight)
Fmt.print(" $4.2f", volume)
}
}
System.print("----------- ------ ----- ------ ------")
Fmt.write("$d items $2d $5.0f $4.1f", itemCount, sumNumber, bestValue, sumWeight)
Fmt.print(" $4.2f", sumVolume)
- Output:
Item Chosen Number Value Weight Volume ----------- ------ ----- ------ ------ panacea 9 27000 2.7 0.23 gold 11 27500 22.0 0.02 ----------- ------ ----- ------ ------ 2 items 20 54500 24.7 0.25
XPL0
func Min(A, B);
real A, B;
return fix(if A < B then A else B);
int Panacea, Ichor, Gold, Panacea_, Ichor_, Gold_, Val, Max;
real Weight, Volume;
[Max:= 0;
for Panacea:= 0 to Min(25.0/0.3, 0.25/0.025) do
[for Ichor:= 0 to Min(25.0/0.2, 0.25/0.015) do
[for Gold:= 0 to Min(25.0/2.0, 0.25/0.002) do
[Val:= Panacea*3000 + Ichor*1800 + Gold*2500;
Weight:= float(Panacea)*0.3 + float(Ichor)*0.2 + float(Gold)*2.0;
Volume:= float(Panacea)*0.025 + float(Ichor)*0.015 + float(Gold)*0.002;
if Val>Max and Weight<=25.0 and Volume<= 0.25 then
[Max:= Val;
Panacea_:= Panacea; Ichor_:= Ichor; Gold_:= Gold;
];
];
];
];
Text(0, "The traveler carries ");
IntOut(0, Panacea_); Text(0, " vials of panacea, ");
IntOut(0, Ichor_); Text(0, " ampules of ichor, and ");
IntOut(0, Gold_); Text(0, " bars of gold"); CrLf(0);
Text(0, "for a maximum value of "); IntOut(0, Max); CrLf(0);
]- Output:
The traveler carries 0 vials of panacea, 15 ampules of ichor, and 11 bars of gold for a maximum value of 54500
zkl
panacea:=T(3000, 0.3, 0.025); // (value,weight,volume)
ichor :=T(1800, 0.2, 0.015);
gold :=T(2500, 2.0, 0.002);
sack :=T( 0, 25.0, 0.250); const VAL=0, W=1, VOL=2;
maxes:=T(panacea,ichor,gold)
.apply('wrap(t){ (sack[W]/t[W]).min(sack[VOL]/t[VOL]).toInt().walker() });
best:=Utils.Helpers.cprod3(maxes.xplode())
.apply('wrap(t){
T(T(panacea[VAL]*t[0] + ichor[VAL]*t[1] + gold[VAL]*t[2],
panacea[W] *t[0] + ichor[W] *t[1] + gold[W] *t[2],
panacea[VOL]*t[0] + ichor[VOL]*t[1] + gold[VOL]*t[2]), t)
})
.filter('wrap(t){ t[0][W]<=sack[W] and t[0][VOL]<=sack[VOL] })
.reduce(fcn(a,b){ a[0][VAL] > b[0][VAL] and a or b });
println("Maximum value achievable is %,d".fmt(best[0][VAL]));
println(("This is achieved by carrying (one solution):"
" %d panacea, %d ichor and %d gold").fmt(best[1].xplode()));
println("The weight to carry is %4.1f and the volume used is %5.3f"
.fmt(best[0][1,*].xplode()));cprod3 is the Cartesian product of three lists or iterators.
- Output:
Maximum value achievable is 54,500 This is achieved by carrying (one solution): 9 panacea, 0 ichor and 11 gold The weight to carry is 24.7 and the volume used is 0.247
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