Equivalence lets you call two cash-flow streams at different times equal in value once you adjust for the time value of money. $1,000 today is equivalent to $1,100 a year from now at a 10% interest rate. Same economic value, different points in time.
Three flavours get distinguished:
Mathematical equivalence is the calculation: and friends. Two amounts at different times have equal present value if the formula says so at the given rate. Pure arithmetic. Given the rate and the times, the equivalence is unambiguous.
Decisional equivalence is the assumption that the decision-maker is genuinely indifferent between the two amounts. Offer you $1,000 today or $1,100 a year from now, and if at a 10% rate you have no preference, you’re operating under decisional equivalence. In practice people have preferences for liquidity, certainty, current spending that distort this, but for engineering analysis we assume it holds.
Market equivalence is the practical condition that you can actually make the exchange at zero cost. The market gives you the mechanism to convert money across time: bank accounts, bonds, loans, savings instruments. For it to hold you need to transact at the interest rate frictionlessly. In reality the lending rate (where you deposit) differs from the borrowing rate (where you take loans), and both carry fees. We assume these frictions are negligible.
When all three hold, moving cash flows around in time is meaningful and the equivalent values are comparable. The two assumptions, decisional and market, are idealisations engineering analysis uses to keep the math tractable. Rarely exactly true, but usually close enough for the resulting decision to be sound.
For project evaluation: pick a single point in time (usually for PW, or end-of-life for FW), use the equivalence formulas to move every cash flow to that point, then add or compare. Present worth method, Future worth method, Annual worth method, and Internal rate of return all rest on equivalence; Compound interest factor supplies the mechanics.