Add the three lines from above and a tank-fed cycle costs roughly 23 to 27 cents to run: about 2 cents of water, 8.5 cents of electricity, and 13 to 17 cents of detergent and rinse aid. The honest question isn't whether that's cheap — it obviously is — but what it's cheap compared with, and the answer depends entirely on how the dishes would otherwise get washed.
Against a running tap (assumption: 1.8 gal/min faucet flow, about 9 minutes for a comparable load, half the flow hot at a 45 °F rise):
9 min × 1.8 gal = 16.2 gallons ≈ 12× the machine's 1.32 gallons
Water + sewer at $0.015/gal: 16.2 × $0.015 = $0.243 · hot-water electricity: ≈ $0.15 · a squirt of dish soap: ≈ $0.03
Total ≈ $0.42 per equivalent wash — well above the machine's ~$0.25
Against a filled, tap-off basin (assumption: 5 gallons total, half hot at the same 45 °F rise):
Water + sewer: 5 × $0.015 = $0.075 · hot-water electricity: ≈ $0.05 · dish soap: ≈ $0.03
Total ≈ $0.16 per equivalent wash — below the machine's ~$0.25, once detergent is priced in
Both boxes use the same rates and physics as the full water-and-cost model on this site, laid out side by side rather than re-derived. Against a running tap — how a lot of people actually wash — the machine is clearly cheaper. Against a careful, disciplined basin-and-tap-off routine, dishwasher detergent is expensive enough to erase the water and electricity savings, and the machine becomes the pricier way to get the same plate clean.
That split matters more than either number alone: the honest answer to "is a portable dishwasher cheaper than washing by hand" is it depends how you currently wash, and manufacturer claims of 75–80% water savings implicitly compare against the running-tap case, not the basin case.