What are the different ways to optimize costs in the HR function and the business/organization? Please provide a few examples.

Building on what has been already shared by other colleagues, I would advise start with audit of the HR activities that takes most time, also activities that are operative, repetitive, and most importantly perhaps - the impact on the organization and current/ future business goals. 
For the TA and recruitment functions:  shall it be in-house or maybe work with external vendor
For L&D - developing internal academy and internal trainers for some skills (I am a huge fan of internal learning academies for hard skills specifically, as this is a great way to keep and sustain the knowledge inside the organisation) 
For payroll & admin - all in-house or partially done though external vendor 
For first contact with employees - chatbot and AI, templates for most commonly asked questions; internal knowledge storage place (articles, policies, etc)
For Talent Management and Succession Planning - using handy reports to pull data, develop leadership programs and knowledge sharing, mentoring programs. It sounds (and is) investment of time, yet once it starts working and bringing results, the HR has rather a facilitation role. 

For me, having a regular check-in and monitoring the efficiency of the HR activities, is important not only for the organisation, but for development of HR people in the team - by minimizing the operative work, giving them the opportunity to grow and develop in the role, be creative and innovative. 

Hope it helps. 
 

If we take a modern stance it would be interesting for us to explore how can we leverage AI and GenAI to automate some of the manual tasks that the HR function manages. In the recent times we have been able to take the support of a primitive chatbot to support queries that can be solved via Knowledge articles, but in future we could also leverage an advanced support system that is able to have a conversation with the requestor and help them solve for the query that they might have.  

I notice that you say optimise rather than reduce. I concur with Amrita. Salary costs generally make up a large portion of HR costs so having the appropriate number of HR people for the size of the organisation and seeking efficiencies in how the HR function works is a useful first step. The use of external providers is another cost that can be examined.

Another step is to consider the return on investment of your activities. How is HR contributing to the strategic goals of the organisation? What can you stop or start doing to get a better return on the HR investment. Does the business understand how HR contributes? What are the business perceptions of the HR function? What tools can you use to streamline your admin tasks to ensure your highly specialised HR staff are fully utilising their skills?

There are several effective strategies to optimize costs within your HR budget. Start by examining each line item to identify opportunities for automation, managing third-party vendor related activities internally through slow transition, and reducing headcount through the deployment of AI tools, particularly in sourcing and recruitment functions.

For functions like learning and development, consider being creative, creating and facilitating in-house content instead of relying heavily on external providers.

For L1 employee queries, enable automated query resolution through bots so that HRBPs can focus on strategic, value-added activities. Automate HR reporting by integrating HRMIS system APIs with internal Power BI and using predictive analytics for a comprehensive, 360-degree view of reporting accessible to the management.

In specialized HR sub-functions like Talent Management (TM) and Talent Development (TD), where resource costs are typically high due to the specialized nature of these roles, continuous monitoring and control of salary expenses are important. Regularly review your resource list and adjust as necessary to ensure cost-effectiveness without compromising the quality of these essential functions.