Let’s start by identifying the two parts to this question, as explained by Heizer & Render (2009):
1) Objective Function: “A mathematical expression in linear programming that maximizes or minimizes some quantity (often profit or cost, but any goal may be used)”
2) Constraints: “Restrictions that limit the degree to which a manager can pursue an objective”
Circumstances where the objective function is more important than the constraints:
A scenario where the objective function is more important can be where the objective is critical to the success of the project in the development phase. Heizer & Render’s (2009, p.591) example of “OM in Action” for Homart Development company illustrates such a situation (book not required, continue reading). For Homart to develop their new mall their objective of attaining 3 “anchor” stores is an important factor in the success of the mall. The anchor stores are the largest stores that will no doubt attract the most customers. It is based on these anchor stores that many other stores will decide to rent in the mall and where they will position themselves in relation to the types/positions of the anchor stores. If the objective function is to get 3 anchor stores as tenants and the constraints are the required square meters floor size and required monthly rental income then if a highly popular anchor store offers to be a tenant with a higher required floor size and a lower required rental, Homart may very well consider taking the offer due to the popularity/benefits of having that anchor store in their new mall.
To state this rule in general terms I would say that where the benefit of the objective outweighs the constraints, or the future success of the project relies upon the objective being met then the constraints may be overlooked.
Circumstances where the constraints are more important than the objective function:
A situation where I would consider the constraints being more important than the objective function would be where the constraints are a valuable commodity that has a limitation that can be considered not-optional. Referring to Heizer & Render (2009, p.599) in their example of Cohen Chemicals (book not required, continue reading) where the organisation had an inventory of highly perishable raw materials that had to be used within the next 30 days to avoid wastage. While there may be situations where wastage of inventory is not an option it should be made high priority to use the raw materials the organisation has already purchased; more so than achieving the objective. In this situation, I am assuming that the outstanding orders have a certain leeway in which production is able to extend beyond the deadline; whereas the deadline for the raw materials to perish is non-negotiable.
The above example can be generally explained as; where the project has constraints that constitute to a greater loss than if the objective is not fully met (eg: needing to get rid of existing stock/inventories).
Heizer, J. & Render, B. (2009) Operations Management. Ninth Edition. Prentice Hall: New Jersey.