To calculate the optimal way to get to a value given a denomination set

Question

So for example to get to a value of 237 given a denomination set of 1, 2, 5, 10, 20, 50 the optimal way to make up the value would be 4x50 + 1x20 + 1x10 + 1x5 + 1x2. But how to achieve that programatically in PHP?

Answer 1

You could do something like this :

$number = 237;

$values = array(1, 2, 5, 10, 20, 50);
arsort($values); // greater to lower sorting
$numberOf = array();

foreach ($values as $v) {
    $numberOf[$v] = floor($number / $v);
    $number -= $numberOf[$v] * $v;
}

var_dump($numberOf);

This will produce :

array(6) {
  [50]=>
  int(4)
  [20]=>
  int(1)
  [10]=>
  int(1)
  [5]=>
  int(1)
  [2]=>
  int(1)
  [1]=>
  int(0)
}

the demo

Edit : Denominations with finit amounts

If you don't have infinit denominations, you can try this :

$number = 237;

$values = array(1, 2, 5, 10, 20, 50);
$quantities = array(7, 1, 3, 5, 2, 3);
// sort by greater value to lower value
// but keeping the related quantities
array_multisort($values, SORT_DESC, $quantities);

$numberOf = array();

foreach ($values as $i => $v) {
    $quantity = $quantities[$i]; // get the corresponding quantity
    // if we have less than the required value, we put the max we can
    $numberOf[$v] = floor($number / $v) > $quantity ? $quantity : floor($number / $v);
    $number -= $numberOf[$v] * $v;
}

var_dump($numberOf);

In this example, you only have a quantity of 3 for 50. So you will have an output like :

array(6) {
  [50]=>
  int(3)
  [20]=>
  int(2)
  [10]=>
  float(4)
  [5]=>
  float(1)
  [2]=>
  float(1)
  [1]=>
  float(0)
}

3 * 50 + 2 * 20 + 4 * 10 + 5 * 1 + 2 * 1 + 1 * 0 = 237

Yippee !
The demo

See also array_multisort() to understand this key line :

array_multisort($values, SORT_DESC, $quantities);

And even better, a ready to go function :

function getDenominations($amount, $denominations, $quantities = null) {
    if (is_array($quantities) && count($denominations) != count($quantities)) return false;
    array_multisort($denominations, SORT_DESC, $quantities);

    $numberOf = array();

    foreach ($denominations as $i => $v) {
        $quantity = $quantities[$i]; 
        $numberOf[$v] = floor($amount/ $v) > $quantity ? $quantity : floor($amount / $v);
        $amount -= $numberOf[$v] * $v;
    }

    return $amount == 0 ? $numberOf : false;
}

To use like this :

$result = getDenominations(
    237,
    array(1, 2, 5, 10, 20, 50),
    array(7, 1, 3, 5, 0, 4)
);

var_dump($result);

Return values : array|bool

If no quantity is specified, it returns the denominations with infinite amount. If specified, with finite amount.
Returns "false" whether there's not enough quantities or if the sizes of the arrays are different.