There is one rule for profit maximization that applies to all firms in all market structures.  The rule is that the firm must produce at the quantity where the marginal cost of producing the last unit is equal to the marginal revenue that is received when that unit is sold.

This means that it is not always advisable to pass on savings to the customer.  Instead, the firm must find the price at which its marginal costs will equal its marginal revenues.  That is the price it must charge if it is to maximize its profits.

To maximize profits, then, a firm should pass on whatever amount of savings will get it to the price and quantity where MR = MC.

I would not say that it is necessary for a manufacturer to pass on all the savings due to an increased efficiency in manufacturing. One important thing that needs to be considered is the price elasticity of demand. If a reduction in price can boost sales to a large extent, it may be advisable to reduce the price. The price at which a manufacturer makes the highest profit is that beyond which a reduction in price does not increase profits even with higher revenues.

If maintaining the same price does not reduce sales, and reducing the price does not increase profits, there is no point in passing on the savings. Any manufacturer needs to be able to offer a better deal than its competitor; if the price of both is the same, and they are also able to manage aspects like advertising, promotion, distribution, etc. in the same way, it shouldn't be necessary that the prices be brought down.