'Reducing complexity' is a reason frequently given by organizations when instigating an application rationalization initiative.  However, complexity can be challenging to define and measure in all but subjective terms.  For many organizations embarking on rationalization, the primary focus is the reduction or elimination of complexity.  Complexity is often associated with higher costs and reduced adaptability or agility.  However, where the value of complexity to the organization is high (due to the competitive advantage or ability to maintain/defend a market position and brand equity), these may be prices worth paying.  EAs play a critical role in defining those areas where complexity can add value and what the trade-offs are.  This enables organizations to take a more rounded view of complexity in the context of application rationalization.  

Complexity to Achieve Market Differentiation
When competitors use similar IT systems, their capabilities and efficiencies in serving customers probably will be similar to yours. In effect, this makes organizations more interchangeable. To mitigate this, organizations seek an edge in terms of service differentiation, capability or cost. This consideration is particularly important in highly populated marketplaces. For example, many services organizations use quality of service or customer reputation as key tools for online sales. This means that IT capabilities and efficiencies play a major part in defining these organizations' relative market positioning.  
Where an organization operates in a highly specialized marketplace, the nature of its applications can act as a barrier to prevent potential competitors from entering the market. This is partly because the organization sets the minimum performance requirements for entry, but also because it effectively defines the applicable "standards" for operation.  Through close coupling with the specific business processes employed by an organization, complexity in IT systems can become a key element in defining those processes and, hence, a constituent part of the value proposition of the organization deploying it.
Complexity to Achieve Time-Based Advantages
Any organization trying to be first to market with a product or service that is founded on new technology developments probably will need more complex IT to achieve that objective. Emerging technologies necessarily mean a lack of applicable open standards, few best practices in deployment or integration and little choice in suppliers.  The higher cost of deploying and using more complex IT can be defrayed against higher margins derived from a first-mover advantage in the company's sales area.
Complexity to Develop or Protect High Value Intellectual Property Assets
Information security is a critical concern for most organizations. Where data is stored in and accessed through highly standardized IT systems and processes, security can be more challenging to maintain.  Vulnerabilities in hardware, software and networks may be much more broadly known and, thus, more difficult to mitigate. Organizations dealing with highly sensitive data or business-critical IP assets may elect to use more complex IT to reduce their security exposure. 
Complexity to Support Established Systems
An organization with already highly complex IT systems and applications typically requires higher levels of complexity for new IT products and services. Although it may want to move toward more standards-based products and services, an organization may be unable to achieve such change rapidly because of the necessity to integrate the standardized "new" with the complex "old." Unless a new deployment approach is adopted, moving away from complex IT toward more standardized IT likely will be an evolutionary, not revolutionary, process that requires some time to achieve.
Complexity to Meet Regulatory Requirements
In some vertical industries (such as healthcare, government or pharmaceuticals), regulatory or legal requirements may require a certain degree of complexity.  This complexity can increase when organizations operate across geographies, where regulatory requirements and the associated mandates related to compliance and reporting differ.
There are potentially good business reasons to embrace complexity. EAs can help organizations by enabling them to understand the level of complexity needed, as well as the opportunity cost of making that selection when undertaking applciation rationalization. Higher levels of complexity typically mean higher costs (directly or indirectly), so they must be associated with a business return.