Travel Itinerary Optimizer Time Windows Costs

The underlying problem is a routing question that operations researchers call the travelling salesman problem with time windows, or in its multi-vehicle form the vehicle routing problem with time windows. You have a set of sites, a travel cost between each pair, an opening and closing time at each site, and a limited working day, and you want an order of visits that fits. What makes it hard is that visiting n sites admits (n-1)! divided by 2 distinct tours, so 10 stops already give over 181,000 possibilities and 15 give more than 43 billion. This tool does not solve that problem. It uses the far simpler time-budget identity that n stops require n-1 transits, multiplies by your average transit figure, and subtracts the result from days times active hours to leave usable activity time. The 2.5 hour per site figure behind the recommended count reflects a planning heuristic for a substantial museum or landmark rather than a quick photo stop. Because transit cost is averaged rather than modelled per pair, the model is blind to the largest real saving available to a traveller, which is ordering. Clustering stops by neighbourhood and walking between adjacent ones routinely cuts total transit by a third compared with a plan that zigzags. Nor are actual opening hours inputs despite time windows appearing in the name, so a plan that looks feasible on total hours can still fail because three of your sites close at 17:00 and two shut on Mondays. The budget split of 35 percent accommodation, 25 percent food, 20 percent transport, 15 percent activities, and 5 percent miscellaneous shifts sharply by destination, with lodging often exceeding half the total in Tokyo, Zurich, or New York and falling well below a third across much of Southeast Asia. Treat efficiency above roughly 75 percent as the practical target.

Karl Menger posed the routing question formally in a 1930 Vienna colloquium, calling it the Botenproblem or messenger problem and noting that the obvious rule of always going to the nearest unvisited point does not generally give the shortest tour. The problem acquired its familiar name and its reputation at the RAND Corporation in the late 1940s. The landmark result came in 1954, when George Dantzig, Delbert Ray Fulkerson, and Selmer Johnson solved a 49-city instance covering one city in each of the then 48 US states plus Washington DC, introducing the cutting-plane method that still underpins modern solvers. Richard Karp placed the problem among his 21 NP-complete problems in 1972, confirming that no efficient exact algorithm was likely, and Nicos Christofides published a heuristic in 1976 guaranteeing a tour no worse than 1.5 times optimal for metric instances. The practical branch began with George Dantzig and John Ramser's 1959 paper The Truck Dispatching Problem, which generalised the single-tour question to a fleet serving many customers from a depot and effectively founded vehicle routing as a field. Time windows entered as a distinct variant in the 1980s, driven by delivery, school bus, and dial-a-ride scheduling; Marius Solomon published the benchmark instance sets in 1987 that researchers still use to compare algorithms. Computation caught up steadily, with the Concorde solver of David Applegate, Robert Bixby, Vasek Chvatal, and William Cook proving optimality for an 85,900-point instance in 2006. Consumer trip planning inherited all of this only indirectly, through routing engines in mapping applications, which is why most travellers still plan itineraries by neighbourhood clustering and rough time budgets rather than by optimisation.