What the demo measures, and what it assumes
A yard with more freight than the fleet can carry, and every package worth a different amount. The morning question is not "what is the best route" but "what is worth carrying, and what does earning it cost". This page is every number behind the answer, and exactly which part of it a quantum chip touched.
THE PROBLEM
Forty packages on the dock, four trucks of 1,400 kg. The freight weighs 9,603 kg, about seven trucks' worth, so not everything goes out. A plan is two decisions made together: pick, which packages leave today, and pack, which of them share a truck. Pack is where the money hides: a pallet rides deepest, so delivering it while cases are still aboard means digging the cases out and putting them back.
There is no single best plan; there is a curve. Ask for $2,600 of revenue and the cheapest plan is three trucks. Ask for $3,600 and it is four trucks and much more digging. The page traces that curve with seven revenue floors: for each floor, the cheapest plan that clears it. Truck count is not a setting. A truck's day is priced, and the plan rolls it only when the packages it would carry are worth more than it costs. Floors the fleet cannot reach are marked unreachable, not returned as a near miss.
THE ARITHMETIC
trucks rolled × (day rate you set + yard round trip)
+ each package's road cost to its truck's patch
+ re-handling for every pallet–case pair sharing a truck
Three terms, and only the last one couples two packages. That is the whole reason the problem is more than a sorting job, and the only part of it a chip can take.
metres × (diesel $1.19/L at 0.35 L/km + $34/h at 11.2 m/s)
Each package is priced by its distance to its truck's patch on the real street graph. Priced once, per package, per truck, so the road term is linear.
rate you set (s/kg) × case kg on the same truck × $34/h
A pallet is dug out from behind case freight only; a pallet never adds to the pile a later pallet has to move. At 0 s/kg the term disappears and the digging column reads $0. That is the control, and you can slide it there.
driving at the speed limit + 7 min and 2 s/kg at each stop
+ digging time + the yard round trip per truck rolled
What a dispatcher actually buys. Profit per truck-hour is revenue minus cost, divided by this.
same plans, re-priced: each truck routed yard to stops to yard on the real streets
(nearest-neighbour + 2-opt), speed and fuel rising with the load aboard,
trucks billed per hour worked, re-handling last-in-first-out
Nothing is re-solved. Every plan on the grid was found under the simple bill above; this button prices the same plans with a bill a dispatcher would recognise and asks whether the ranking survives. Costs roughly halve, mostly because a truck back by noon is charged half a day, and the order of the rows is the test. The best-alternative line and the pack-only chip diamonds have no saved routes and are hidden in this view.
THE TWO PROBLEMS, WRITTEN OUT
Variables: xik = 1 if package i rides truck k (no truck means it stays in the yard), yk = 1 if truck k rolls. Rk, cik and wij are the three cost terms above; pi is what package i pays.
min Σk Rkyk + ΣiΣk cikxik + ΣkΣi<j wijxikxjk
s.t. ΣiΣk pixik ≥ ε revenue floor
Σk xik ≤ 1 for every i one truck or none
Σi mixik ≤ C·yk, Σi uixik ≤ U·yk weight, volume
All 40 packages are candidates: 40 × 4 truck choices plus "stay home", plus four truck bits. Solved seven times per cell, once per floor ε, by simulated annealing and by the three classical alternatives. Every dot and circle on the chart is one solve of this.
min Σk Rkyk + Σi∈SΣk cikxik + ΣkΣi<j∈S wijxikxjk
+ A Σi∈S (Σk xik − 1)² + A Σi,k xik(1 − yk)
Same R, c and w; the sliders set them once for both. Gone: the revenue floor (already met by the pick), the "stay home" option (S is fixed), and capacity (repaired afterwards). Added: a one-hot penalty so every package lands on exactly one truck, and a link so an empty truck cannot carry anything. |S| × 4 + 4 variables, 48 to 96 on this cell, one-hot the only rule. This is the matrix the chip is handed; neal solves the identical matrix classically. Every diamond on the chart is one solve of this.
THE SLIDERS ARE COEFFICIENTS
Neither slider is a decision the solver makes. Each one retunes the matrix the solver then solves. Re-handling sets the pairwise term, the only quadratic coefficient in the model; at 0 the problem is a plain assignment, at 12 the pairwise term dominates. Truck day rate sets the fixed cost of rolling a truck; it decides whether a fourth truck is worth it. Four day rates by six re-handling rates is 24 matrices, each solved at seven floors: 168 solves, all classical, all done in advance. Pressing SUBMIT opens the precomputed cell. Nothing is solved on your phone.
WHAT IS ON THE CHART
Black dots, solid line. One floor's cheapest joint plan, pick and pack together, found by simulated annealing. Classical, in JavaScript, on a laptop. The label 3T or 4T is the truck count the price chose. Click one and the numbers below switch to that plan.
Grey circles, dashed line. The same floor solved by the best of three other classical methods, greedy, hill-climb and tabu, at exactly the simulated annealer's evaluation budget. Also pick and pack, so its revenue can differ a little. It is there so the solid line is measured against something honest, and on most of the grid it wins by a hair.
Green diamonds. Only at re-handling 12 and $150 a day, the one cell that ran on hardware. Each diamond is a D-Wave chip's pack of the simulated annealer's pick for that floor: same packages, so the same revenue by construction; only the cost can differ. They are not joined, because the chip did not trace a curve; it answered four separate packing questions.
THE HARDWARE PAGE
It is that one cell, and only its four reachable floors. For each floor the package list is taken from the frontier as given, and every row, greedy, neal, hill-climb, tabu, the simulated annealer's own partition and the chip, re-does only the pack. The chip solves a small QUBO: those packages by four trucks, one-hot as its only rule, 48 to 96 variables. Capacity is repaired afterwards, classically. Every row is then priced by the arithmetic above, never by the QUBO's own energy, so a row there and a dot here at the same revenue differ only in how the packages were grouped.
SOLVE LIVE reruns the four floors on a chip in about seventy seconds and saves the result; that saved result is what draws the diamonds. On today's pick, cheapest first: tabu $3,175, hill-climb $3,204, the simulated annealer $3,209, the chip $3,220, neal $3,269, greedy $3,635. Behind the best classical packer by 1.4%, on a synthetic stress instance, measured on Advantage_system6.
EVERY ASSUMPTION
28 m³What each truck holds. Set low on purpose so the fleet binds: 9,603 kg of freight against 5,600 kg of trucks. Weight is the limit that bites on this freight; volume never does.ASSUMED, CHOSEN SO THE FLEET IS SHORT
40 packagesThe yard. Forty packages drawn from a wholesale distribution round, seed 23, on real San Francisco streets.ASSUMED, A WHOLESALE DISTRIBUTION ROUND
■ REAL OR MEASURED■ YOURS TO SET, OR REASONED■ ASSUMED, CHANGE IT FOR YOUR OPERATION
WHAT IS CLAIMED, AND WHAT IS NOT
Claimed. A curve a dispatcher can read: for each revenue target, the cheapest plan, with the truck count chosen by price. Built classically, checked against a second classical method at equal budget, and priced by one cost function throughout.
Claimed. A bare quantum chip, with classical pre-processing, packs a frozen list of packages close to the best classical packer we have. Measured on one cell, at a stress rate, on four chips: Advantage2_system1, Advantage2_system4, Advantage_system4 and Advantage_system6. Its worst total was 1.3% behind the best classical packer, its best 0.2% behind. It was cheapest on 4 of the 16 floors measured, and level to the dollar on 1 more. Through the Quip network, the same partition posted as paid job orders came back 0.2% behind the frontier partition from Cpu miners, a classical result (aglais) and 5.4% behind the frontier partition from a D-Wave miner (aglais QpuDwave).
Not claimed. That the chip picks, draws the curve, or wins. Selection, the revenue floor, fleet size and capacity are classical everywhere. The frontier chart is classical at every slider position; the diamonds are the chip's four answers laid on it.
Not claimed. Advantage at a real dock. At light-parcel rates the simulated annealer and tabu find the same plans and there is nothing for any solver, quantum or classical, to find. At 4 s/kg, heavy freight, the chip matches the annealer and trades floors with tabu. The stress tier is marked in coral for that reason.
Not claimed. That this is your network. One city, one freight profile, one fleet, synthetic prices. The mechanism is general; the numbers belong to this setup.
HOW TO BREAK IT
Set re-handling to 0 s/kg and the digging column reads $0 and the two lines lie on top of each other; there is nothing manufactured underneath. Set the day rate to $300 and watch the plans hold trucks back; set it to $75 and they roll all four. Leave the stress tier and the diamonds disappear, because the chip only ran there. Every one of those is in the demo on purpose.
SAN FRANCISCO DIRECTED STREET GRAPH, OSM
CHIP RESULTS: D-WAVE ADVANTAGE2_SYSTEM1, ADVANTAGE2_SYSTEM4, ADVANTAGE_SYSTEM4 AND ADVANTAGE_SYSTEM6; VIA THE QUIP NETWORK: AGLAIS AND AGLAIS QPUDWAVE, ONE CELL, PACK ONLY