Here's the simplest explanation I've seen:
COST INDEX= ratio of FUEL COST ($ per kilogram)/ TIME COST (fixed operating cost $ per minute)
Where TIME COST includes the amortization of the aircraft, crew, maintenance, management, depreciation, etc.
For example, a CI=30 means the cost of 30 kilograms of fuel is equivalent to the cost of 1 minute of time cost.
Therefore, it is management who eventually determines which base reference cost index to use. As flight deck crew, we should initially set the CI according to what management instructs us then revise strategically as required to adjust for time savings bearing in mind the consequential effect on the EFOB or fuel remaining predictions at destination and/or alternate.