Bookshelf Organizer

You are given an array of books where each book is represented as an object with two properties: the width (an integer) and the height (an integer). The books must be arranged in the given order on a bookshelf. You are also given an integer shelfWidth representing the maximum sum of widths that can be placed on any single shelf.

The goal is to place the books on a series of shelves such that the following rules are observed:

1. Books must be placed in order (i.e., you cannot rearrange the book order).
2. For each shelf, you can place consecutive books until adding another book would exceed shelfWidth.
3. The height of a shelf is the maximum height of any book on that shelf.

Your task is to determine the minimum total height of the bookshelf after arranging all the books.

Write a function/method that takes in the list of books and the integer shelfWidth, and returns the minimum total height achievable.

For example, if the input is:

books = [
  {"width": 100, "height": 30},
  {"width": 200, "height": 40},
  {"width": 300, "height": 20},
  {"width": 400, "height": 50}
]
shelfWidth = 500

One possible arrangement could be placing the first two books on the first shelf (total width 300, shelf height 40) and the remaining books on subsequent shelves, but the optimal arrangement might be different. Your solution should explore the possibilities and compute the minimum total height.

Note: You can assume that the number of books is reasonable and that all widths and heights are positive integers. Consider efficient algorithms, as some inputs might be large.