Tobler’s Law and Spatial Optimization

Abstract

In 1970, Tobler produced a movie simulating population growth in the City of Detroit. He argued that his model did not need to include terms for faraway places like Singapore, while still being relative accurate in his forecast by invoking what he called the first law of geography. In spatial optimization, like the general warehouse location problem (GWLP), it is assumed that all possible linkages need to be included, as arbitrarily dropping potential variables may prevent optimal solutions from being identified. In this article, it is demonstrated that it may be possible to meet such exacting standards in spatial optimization, while at the same time being guided by Tobler’s argument for being simple and frugal. This article gives a demonstration of how this might be achieved using the GWLP as an example. A new model form is proposed which distinguishes between “near” (low cost) allocations and “far” (high cost) allocations and uses both explicit and implicit variables for capacity allocations. Computational experience in using this new model is given which shows that optimal solutions can be identified and verified while eliminating a substantial number of allocation/transport variables and constraints. This article ends with a challenge for the reformulation and redevelopment of other spatial optimization problems in regional science.

Quantitative geography III: Future challenges and challenging futures

“…Instead, Tobler’s Law should be used as Waldo Tobler (1970) intended: to simplify problems. In this spirit, Church (2018) uses the Law to simplify warehouse location problems. Classical methods consider all warehouses when supplying each demand site, so odd decisions are considered but never made, like Detroit, Michigan, supplying Bakersfield, California.…”

“…Classical methods consider all warehouses when supplying each demand site, so odd decisions are considered but never made, like Detroit, Michigan, supplying Bakersfield, California. Church (2018) only explicitly models ‘near’ facilities, and finds optimal solutions by gradually increasing what is ‘near’ each facility. This is Tobler’s Law enforced, not obeyed.…”

“…If replication remains inhibited by the field’s theoretical issues discussed in Section II, it will remain elusive. For instance, is Church (2018) a ‘replication’ of Tobler’s Law because it proved useful in simplifying a problem? Or, is Tobler’s Law only replicated by distance decays in spatial auto-covariance functions?…”

Big data, spatial optimization, and planning

“…In the past two decades, there has been remarkable progress made in this field by scientists and scholars from a variety of disciplines, such as geography, ecology and environment, regional science, operational research, engineering, and urban planning. Most of these studies involving spatial optimization modeling have been directed toward a finer scale, with the support of more effective optimization models, as well as more efficient optimization problem solving capabilities from well-adapted heuristic approaches, high performance computing (HPC) and methods to trim superfluous features from spatial optimization models, making them easier to solve without compromising on the task of finding optimality (Church, 2018). Alongside the rapid development of information and communications technology as well as the empowered availability of (spatio-temporal) big data and analytics capabilities (Li et al, 2020), there have been a few notable shifts in the field of spatial optimization enabled planning in the past few years, which shed light on not only the opportunities and insights in this field, but also the challenges that are worth more attention from scientists and scholars in the future.…”

Learning Geographical Manifolds: A Kernel Trick for Geographical Machine Learning

“…Inspired by Church (2018)'s heuristic to simplify a family of facility location problems, we propose a kernel-mixing heuristic like that in Equation 2. This encourages "short hops" along the high-dimensional data surface that make geographical sense.…”