Unfortunately, when you expect the better part of people's waking hours each day, even responsible, loyal employees will experience times in their life where the demands of coping with the situations on-hand will interfere with their abilities to operate at 100% performance. This is especially true with young children. They aren't in school, many get sick often, and childcare providers don't want to take sick children (and forcing sick children to go to daycare instead of getting rest can prolong the problem or lead to additional or more severe illnesses). Daycare can be completely unaffordable for someone with two children who are not yet school-aged, so it is not uncommon for them to seek an alternative (such as someone who provides care in their home). This can lead to additional reliability issues because your employee is now subject to a third party getting sick (or having sick kids themselves) and cancelling.
In US culture, many people do not have support systems robust enough to handle these types of challenges. They don't have parents or relatives to step in when kids get sick because they either live too far away or have jobs themselves. The result is that, while this might be an isolated incident in a small company, it is not an isolated problem in general. Even in a committed relationship, there is generally a lot of absence due to sick children. The differences is that, when in a relationship, you can generally divide the responsibilities among two people so it is not as much of an issue. There is also a better chance that someone in the relationship will have a greater degree of flexibility so they can more effectively deal with these types of situations.
That is the human component of the equation. If this is an otherwise-good employee that is going through a rough patch in life, I'd be inclined to try and be creative and understanding. Regarding the business quotient, finding good employees is expensive and time-consuming. You could easily replace a temporary problem with another larger temporary problem, so unless you know you have eager, capable replacements available, I again think that trying to work through the problem makes sense from a financial standpoint as well.
The first step in resolving the problem is to open up the discussion with the employee. Focus on the result instead of the cause. For example, approach the employee and say, "the frequent, unplanned absences are causing the company not to be able to meet its commitments," instead of hitting the employee over the head with, "you're calling off 5 times a month because either you or your kids are sick." Acknowledge the employee's difficulties ("I am sure your situation has been incredibly difficult and we want to try and do our best to be understanding,") but don't remove the responsibility ("but it is also important that we're able complete orders when we've promised them."). Then let them know you'd like to find a solution that works for everyone ("You've been a good employee for a long time, and we'd like to find an arrangement that will work for both you and us.").
Now that you're (hopefully) having a productive conversation I'd explain the specific impact (we've had N orders fulfilled late due to unplanned absences in the last X months), focusing again on measurable details and the end result. Explain that it simply isn't possible, due to customer requirements, to put jobs off until another shift. Then, ask the employee to give suggestions on how they think they can help be part of the solution. You might be surprised how empowering someone to impact their own life can lead to solutions. They may suggest that they reduce their hours, or realize they need more support and find it. If the employee is well-respected by their peers, you might even suggest tackling the problem as team and seeing if others would be willing to step in and help by working a couple extra hours, or know someone that wouldn't mind helping with the kids, etc. You could also suggest resources for the employee (many larger employers offer an employee assistance plan specifically for these types of challenges).
A single parent dealing with a difficult break-up and suddenly finding themselves raising two young, demanding children, may very well be burnt out. Chronic stress (the kind that comes with these situations) will cause these issues. They need a break, and yet never get one because all of their time, PTO, etc. is spent caring for others. They desperately need to care for themselves, but are terrified to ask because they are afraid of losing their income, appearing less-than-dedicated, or don't want to admit they need help. The result is that they are in a constant state of fatigue from which they can't recover. It might be as simple as offering to allow the employee to take a little time off (paid or unpaid) and letting them know you support them and want to see them improve.
Put the details of the personal situation aside. Human employees are all going to hit rough patches. Is this an otherwise good employee that has suddenly come across a challenge they can't quite conquer alone? Is it temporary (most likely is)? Is it creating undue hardship, or is it an inconvenience (albeit a frustrating one)? If it is creating actual hardship, then it might be best to accept that the business simply doesn't have the resources to handle the situation and help the employee find something better suited. If it is an inconvenience, maybe accept the situation and instead focus on ways that can make it less inconvenient?