---
id: 5a23c84252665b21eecc7ed2
title: Knapsack problem/Bounded
challengeType: 5
forumTopicId: 323652
dashedName: knapsack-problembounded
---

# --description--

The bounded knapsack problem is defined as follows:

You are given an array of objects representing items to be put in a knapsack. The objects have 4 attributes: name, pieces (the number of the particular item), weight, and value. The items need to be selected so that the total weight does not exceed the maximum weight and the value is maximized. Keep in mind that each item can appear between 0 and `pieces` times.

# --instructions--

Write a function to solve the knapsack problem. The function is given the array of objects and the maximum weight as parameters. It should return the maximum total value possible.

# --hints--

`findBestPack([{ name:'map', weight:9, value:150, pieces:1 }, { name:'compass', weight:13, value:35, pieces:1 }, { name:'water', weight:153, value:200, pieces:2 }, { name:'sandwich', weight:50, value:60, pieces:2 }, { name:'glucose', weight:15, value:60, pieces:2 }, { name:'tin', weight:68, value:45, pieces:3 }, { name:'banana', weight:27, value:60, pieces:3 }, { name:'apple', weight:39, value:40, pieces:3 }, { name:'cheese', weight:23, value:30, pieces:1 }, { name:'beer', weight:52, value:10, pieces:3 }, { name:'suntan, cream', weight:11, value:70, pieces:1 }, { name:'camera', weight:32, value:30, pieces:1 }, { name:'T-shirt', weight:24, value:15, pieces:2 }], 300)` should return `755`.

```js
assert.equal(
  findBestPack(
    [
      { name: 'map', weight: 9, value: 150, pieces: 1 },
      { name: 'compass', weight: 13, value: 35, pieces: 1 },
      { name: 'water', weight: 153, value: 200, pieces: 2 },
      { name: 'sandwich', weight: 50, value: 60, pieces: 2 },
      { name: 'glucose', weight: 15, value: 60, pieces: 2 },
      { name: 'tin', weight: 68, value: 45, pieces: 3 },
      { name: 'banana', weight: 27, value: 60, pieces: 3 },
      { name: 'apple', weight: 39, value: 40, pieces: 3 },
      { name: 'cheese', weight: 23, value: 30, pieces: 1 },
      { name: 'beer', weight: 52, value: 10, pieces: 3 },
      { name: 'suntan, cream', weight: 11, value: 70, pieces: 1 },
      { name: 'camera', weight: 32, value: 30, pieces: 1 },
      { name: 'T-shirt', weight: 24, value: 15, pieces: 2 }
    ],
    300
  ),
  755
);
```

`findBestPack([{ name:'map', weight:9, value:150, pieces:1 }, { name:'compass', weight:13, value:35, pieces:1 }, { name:'water', weight:153, value:200, pieces:2 }, { name:'sandwich', weight:50, value:60, pieces:2 }, { name:'glucose', weight:15, value:60, pieces:2 }, { name:'tin', weight:68, value:45, pieces:3 }, { name:'banana', weight:27, value:60, pieces:3 }, { name:'apple', weight:39, value:40, pieces:3 }, { name:'cheese', weight:23, value:30, pieces:1 }, { name:'beer', weight:52, value:10, pieces:3 }, { name:'suntan, cream', weight:11, value:70, pieces:1 }, { name:'camera', weight:32, value:30, pieces:1 }, { name:'T-shirt', weight:24, value:15, pieces:2 }], 400)` should return `875`.

```js
assert.equal(
  findBestPack(
    [
      { name: 'map', weight: 9, value: 150, pieces: 1 },
      { name: 'compass', weight: 13, value: 35, pieces: 1 },
      { name: 'water', weight: 153, value: 200, pieces: 2 },
      { name: 'sandwich', weight: 50, value: 60, pieces: 2 },
      { name: 'glucose', weight: 15, value: 60, pieces: 2 },
      { name: 'tin', weight: 68, value: 45, pieces: 3 },
      { name: 'banana', weight: 27, value: 60, pieces: 3 },
      { name: 'apple', weight: 39, value: 40, pieces: 3 },
      { name: 'cheese', weight: 23, value: 30, pieces: 1 },
      { name: 'beer', weight: 52, value: 10, pieces: 3 },
      { name: 'suntan, cream', weight: 11, value: 70, pieces: 1 },
      { name: 'camera', weight: 32, value: 30, pieces: 1 },
      { name: 'T-shirt', weight: 24, value: 15, pieces: 2 }
    ],
    400
  ),
  875
);
```

`findBestPack([{ name:'map', weight:9, value:150, pieces:1 }, { name:'compass', weight:13, value:35, pieces:1 }, { name:'water', weight:153, value:200, pieces:2 }, { name:'sandwich', weight:50, value:60, pieces:2 }, { name:'glucose', weight:15, value:60, pieces:2 }, { name:'tin', weight:68, value:45, pieces:3 }, { name:'banana', weight:27, value:60, pieces:3 }, { name:'apple', weight:39, value:40, pieces:3 }, { name:'cheese', weight:23, value:30, pieces:1 }, { name:'beer', weight:52, value:10, pieces:3 }, { name:'suntan, cream', weight:11, value:70, pieces:1 }, { name:'camera', weight:32, value:30, pieces:1 }, { name:'T-shirt', weight:24, value:15, pieces:2 }], 500)` should return `1015`.

```js
assert.equal(
  findBestPack(
    [
      { name: 'map', weight: 9, value: 150, pieces: 1 },
      { name: 'compass', weight: 13, value: 35, pieces: 1 },
      { name: 'water', weight: 153, value: 200, pieces: 2 },
      { name: 'sandwich', weight: 50, value: 60, pieces: 2 },
      { name: 'glucose', weight: 15, value: 60, pieces: 2 },
      { name: 'tin', weight: 68, value: 45, pieces: 3 },
      { name: 'banana', weight: 27, value: 60, pieces: 3 },
      { name: 'apple', weight: 39, value: 40, pieces: 3 },
      { name: 'cheese', weight: 23, value: 30, pieces: 1 },
      { name: 'beer', weight: 52, value: 10, pieces: 3 },
      { name: 'suntan, cream', weight: 11, value: 70, pieces: 1 },
      { name: 'camera', weight: 32, value: 30, pieces: 1 },
      { name: 'T-shirt', weight: 24, value: 15, pieces: 2 }
    ],
    500
  ),
  1015
);
```

`findBestPack([{ name:'map', weight:9, value:150, pieces:1 }, { name:'compass', weight:13, value:35, pieces:1 }, { name:'water', weight:153, value:200, pieces:2 }, { name:'sandwich', weight:50, value:60, pieces:2 }, { name:'glucose', weight:15, value:60, pieces:2 }, { name:'tin', weight:68, value:45, pieces:3 }, { name:'banana', weight:27, value:60, pieces:3 }, { name:'apple', weight:39, value:40, pieces:3 }, { name:'cheese', weight:23, value:30, pieces:1 }, { name:'beer', weight:52, value:10, pieces:3 }, { name:'suntan, cream', weight:11, value:70, pieces:1 }, { name:'camera', weight:32, value:30, pieces:1 }, { name:'T-shirt', weight:24, value:15, pieces:2 }], 600)` should return `1120`.

```js
assert.equal(
  findBestPack(
    [
      { name: 'map', weight: 9, value: 150, pieces: 1 },
      { name: 'compass', weight: 13, value: 35, pieces: 1 },
      { name: 'water', weight: 153, value: 200, pieces: 2 },
      { name: 'sandwich', weight: 50, value: 60, pieces: 2 },
      { name: 'glucose', weight: 15, value: 60, pieces: 2 },
      { name: 'tin', weight: 68, value: 45, pieces: 3 },
      { name: 'banana', weight: 27, value: 60, pieces: 3 },
      { name: 'apple', weight: 39, value: 40, pieces: 3 },
      { name: 'cheese', weight: 23, value: 30, pieces: 1 },
      { name: 'beer', weight: 52, value: 10, pieces: 3 },
      { name: 'suntan, cream', weight: 11, value: 70, pieces: 1 },
      { name: 'camera', weight: 32, value: 30, pieces: 1 },
      { name: 'T-shirt', weight: 24, value: 15, pieces: 2 }
    ],
    600
  ),
  1120
);
```

`findBestPack([{ name:'map', weight:9, value:150, pieces:1 }, { name:'compass', weight:13, value:35, pieces:1 }, { name:'water', weight:153, value:200, pieces:2 }, { name:'sandwich', weight:50, value:60, pieces:2 }, { name:'glucose', weight:15, value:60, pieces:2 }, { name:'tin', weight:68, value:45, pieces:3 }, { name:'banana', weight:27, value:60, pieces:3 }, { name:'apple', weight:39, value:40, pieces:3 }, { name:'cheese', weight:23, value:30, pieces:1 }, { name:'beer', weight:52, value:10, pieces:3 }, { name:'suntan, cream', weight:11, value:70, pieces:1 }, { name:'camera', weight:32, value:30, pieces:1 }, { name:'T-shirt', weight:24, value:15, pieces:2 }], 700)` should return `1225`.

```js
assert.equal(
  findBestPack(
    [
      { name: 'map', weight: 9, value: 150, pieces: 1 },
      { name: 'compass', weight: 13, value: 35, pieces: 1 },
      { name: 'water', weight: 153, value: 200, pieces: 2 },
      { name: 'sandwich', weight: 50, value: 60, pieces: 2 },
      { name: 'glucose', weight: 15, value: 60, pieces: 2 },
      { name: 'tin', weight: 68, value: 45, pieces: 3 },
      { name: 'banana', weight: 27, value: 60, pieces: 3 },
      { name: 'apple', weight: 39, value: 40, pieces: 3 },
      { name: 'cheese', weight: 23, value: 30, pieces: 1 },
      { name: 'beer', weight: 52, value: 10, pieces: 3 },
      { name: 'suntan, cream', weight: 11, value: 70, pieces: 1 },
      { name: 'camera', weight: 32, value: 30, pieces: 1 },
      { name: 'T-shirt', weight: 24, value: 15, pieces: 2 }
    ],
    700
  ),
  1225
);
```

# --seed--

## --seed-contents--

```js
function findBestPack(data, maxweight) {

}
```

# --solutions--

```js
function findBestPack(data, maxweight) {
  var m = [[0]]; // maximum pack value found so far
  var b = [[0]]; // best combination found so far
  var opts = [0]; // item index for 0 of item 0
  var P = [1]; // item encoding for 0 of item 0
  var choose = 0;
  for (var j = 0; j < data.length; j++) {
    opts[j + 1] = opts[j] + data[j].pieces; // item index for 0 of item j+1
    P[j + 1] = P[j] * (1 + data[j].pieces); // item encoding for 0 of item j+1
  }
  for (var j = 0; j < opts[data.length]; j++) {
    m[0][j + 1] = b[0][j + 1] = 0; // best values and combos for empty pack: nothing
  }
  for (var w = 1; w <= maxweight; w++) {
    m[w] = [0];
    b[w] = [0];
    for (var j = 0; j < data.length; j++) {
      var N = data[j].pieces; // how many of these can we have?
      var base = opts[j]; // what is the item index for 0 of these?
      for (var n = 1; n <= N; n++) {
        var W = n * data[j].weight; // how much do these items weigh?
        var s = w >= W ? 1 : 0; // can we carry this many?
        var v = s * n * data[j].value; // how much are they worth?
        var I = base + n; // what is the item number for this many?
        var wN = w - s * W; // how much other stuff can we be carrying?
        var C = n * P[j] + b[wN][base]; // encoded combination
        m[w][I] = Math.max(m[w][I - 1], v + m[wN][base]); // best value
        choose = b[w][I] = m[w][I] > m[w][I - 1] ? C : b[w][I - 1];
      }
    }
  }

  var best = [];
  for (var j = data.length - 1; j >= 0; j--) {
    best[j] = Math.floor(choose / P[j]);
    choose -= best[j] * P[j];
  }

  var wgt = 0;
  var val = 0;
  for (var i = 0; i < best.length; i++) {
    if (0 == best[i]) continue;
    wgt += best[i] * data[i].weight;
    val += best[i] * data[i].value;
  }

  return val;
}
```
