We present interval-flow networks, network flow modelsin which the flow on an arc may be required to be eitherzero or within a specified range. The addition of suchconditional lower bounds creates a mixed-integer programthat captures such well-known restrictions as timewindows and minimum load sizes. This paper builds onprevious research [1] that described the mathematicalproperties of interval-flow networks and based on thosedesigned an efficient new heuristic approach thatincorporated the conditional bounds into the simplexpivoting process and exploits the efficient, specializedpure-network simplex technologies. The algorithm isapplied to interval-flow transportation problems with avariable conditional lower bound and tested on problemswith up to 100 nodes and 2000 arcs. Empiricalcomparisons with CPLEX demonstrate the effectivenessof this methodology, both in terms of solution quality andprocessing time.