Principal Engineer building route optimisation, geospatial services and AI agents for logistics. Writing about solvers, systems and AI-native engineering.https://shubhampaliwal.vercel.app/https://shubhampaliwal.vercel.app/blog/vrp-primer-d-probability-and-statistics/https://shubhampaliwal.vercel.app/blog/vrp-primer-d-probability-and-statistics/Random numbers, seeds, expectation, spread and percentiles: the statistics you need to read randomized routing results and compare methods fairly.Sat, 17 Oct 2026 00:00:00 GMThttps://shubhampaliwal.vercel.app/blog/vrp-11-neighborhoods-and-the-2-opt-move/https://shubhampaliwal.vercel.app/blog/vrp-11-neighborhoods-and-the-2-opt-move/How local search improves a tour one small change at a time: the 2-opt move, neighborhoods, delta evaluation, and how far 2-opt gets on berlin52.Sat, 10 Oct 2026 00:00:00 GMThttps://shubhampaliwal.vercel.app/blog/vrp-10-approximation-algorithms/https://shubhampaliwal.vercel.app/blog/vrp-10-approximation-algorithms/How the double-tree and Christofides algorithms guarantee a tour within 2× and 1.5× of optimal, why that needs the triangle inequality, and its limits.Sat, 12 Sep 2026 00:00:00 GMThttps://shubhampaliwal.vercel.app/blog/vrp-09-construction-heuristics/https://shubhampaliwal.vercel.app/blog/vrp-09-construction-heuristics/Build a tour in milliseconds with nearest neighbour, three insertion rules and a space-filling curve, and measure each against the berlin52 optimum.Sat, 04 Jul 2026 00:00:00 GMThttps://shubhampaliwal.vercel.app/blog/vrp-08-constraint-programming-for-routing/https://shubhampaliwal.vercel.app/blog/vrp-08-constraint-programming-for-routing/How constraint programming models a route as variables, domains and rules, lets propagation cut the search, and how OR-Tools builds on it.Sat, 09 May 2026 00:00:00 GMThttps://shubhampaliwal.vercel.app/blog/vrp-07-branch-and-cut-and-concorde/https://shubhampaliwal.vercel.app/blog/vrp-07-branch-and-cut-and-concorde/How branch-and-cut adds subtour and comb constraints only when the LP breaks them, and how Concorde uses that idea to solve huge TSPs exactly.Sat, 21 Mar 2026 00:00:00 GMThttps://shubhampaliwal.vercel.app/blog/vrp-06-integer-programming-formulations/https://shubhampaliwal.vercel.app/blog/vrp-06-integer-programming-formulations/How to write the TSP as an integer program: one variable per leg, degree constraints, and the two classic ways to forbid subtours, DFJ and MTZ.Sat, 07 Feb 2026 00:00:00 GMThttps://shubhampaliwal.vercel.app/blog/vrp-primer-c-linear-and-integer-programming/https://shubhampaliwal.vercel.app/blog/vrp-primer-c-linear-and-integer-programming/Variables, constraints and objectives; corners and the simplex method; LP relaxations, shadow prices and reduced costs, on a tiny fleet problem.Sat, 24 Jan 2026 00:00:00 GMThttps://shubhampaliwal.vercel.app/blog/vrp-05-branch-and-bound/https://shubhampaliwal.vercel.app/blog/vrp-05-branch-and-bound/How branch and bound proves a tour optimal: branch on edges, bound with assignment, 1-tree and Lagrangian relaxations, prune, and pick a search order.Sat, 20 Dec 2025 00:00:00 GMThttps://shubhampaliwal.vercel.app/blog/vrp-04-dynamic-programming-held-karp/https://shubhampaliwal.vercel.app/blog/vrp-04-dynamic-programming-held-karp/How the Held-Karp algorithm solves the TSP exactly by building answers over subsets of cities, worked by hand on six stops, and why its memory explodes.Sat, 15 Nov 2025 00:00:00 GMThttps://shubhampaliwal.vercel.app/blog/vrp-03-the-tsp-family/https://shubhampaliwal.vercel.app/blog/vrp-03-the-tsp-family/Five relatives of the TSP, the asymmetric, multiple, time-window, prize-collecting and orienteering versions, each solved on the same eight stops.Sat, 11 Oct 2025 00:00:00 GMThttps://shubhampaliwal.vercel.app/blog/vrp-02-why-is-the-tsp-hard/https://shubhampaliwal.vercel.app/blog/vrp-02-why-is-the-tsp-hard/Count the tours, brute-force eight stops by hand and in code, and get an honest intuition for NP-hardness and for 'optimal' versus 'good enough'.Sat, 13 Sep 2025 00:00:00 GMThttps://shubhampaliwal.vercel.app/blog/vrp-primer-b-algorithms-and-complexity/https://shubhampaliwal.vercel.app/blog/vrp-primer-b-algorithms-and-complexity/Big-O notation, recursion, arrays, linked lists and heaps, and why exponential growth decides which routing methods are usable at all.Sat, 06 Sep 2025 00:00:00 GMThttps://shubhampaliwal.vercel.app/blog/vrp-01-what-is-the-tsp/https://shubhampaliwal.vercel.app/blog/vrp-01-what-is-the-tsp/What a tour is, how to price one, why it is a permutation, symmetric vs. asymmetric, and why Dijkstra's shortest path does not solve it.Sat, 23 Aug 2025 00:00:00 GMThttps://shubhampaliwal.vercel.app/blog/vrp-primer-a-graphs/https://shubhampaliwal.vercel.app/blog/vrp-primer-a-graphs/The graph words the whole series uses: vertices, edges, weights, paths, cycles, tours, spanning trees and matchings, worked by hand on eight stops.Tue, 19 Aug 2025 00:00:00 GMThttps://shubhampaliwal.vercel.app/blog/vrp-00-history-and-series-map/https://shubhampaliwal.vercel.app/blog/vrp-00-history-and-series-map/Routing research from 1930 to today in one timeline, the 13 parts of this series and how they connect, the running examples, and where to start.Sat, 16 Aug 2025 00:00:00 GMThttps://shubhampaliwal.vercel.app/blog/vrp-glossary-and-notation/https://shubhampaliwal.vercel.app/blog/vrp-glossary-and-notation/Every term and symbol the routing series uses, in one place, each with a short meaning and a link to the post that defines it.Sat, 16 Aug 2025 00:00:00 GMThttps://shubhampaliwal.vercel.app/blog/starting-this-blog/https://shubhampaliwal.vercel.app/blog/starting-this-blog/Eight years of logistics engineering, and the 70-post series on vehicle routing I'm starting here.Sat, 09 Aug 2025 00:00:00 GMT