ISE 754 (Fall 2026) 2.1 Types of Location Problems
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Overview
Michael G. Kay introduces a taxonomy of location problems and shows how modeling choices change solutions, from competitive hot dog stands to cooperative facility placement. Using a Durham-to-Raleigh commute example, he contrasts equitable squared-distance and efficient linear-distance objectives, then explains how weighted medians solve one-dimensional minisum problems by hand.
Key takeaways
- Two independently owned hot dog stands on a uniformly populated beach converge on the midpoint, while one coordinated owner minimizes average customer distance by placing the stands at the quarter points.
- The Durham-Raleigh couple’s equitable apartment location is 20 miles from Durham, but minimizing total weighted commute distance places them in Raleigh, illustrating that fairness and efficiency can produce different solutions.
- Minimizing weighted squared distance yields a weighted centroid, while minimizing weighted linear distance yields a weighted median in one dimension.
- A mixed-integer model can represent the second-car choice with a binary variable that permits giving up a car only when the apartment is located in Raleigh.
- For one-dimensional linear-distance problems, the optimal weighted median is the first location at which cumulative weight reaches at least half the total weight.
Chapters
- Location problems divide first into competitive models, where independent firms seek market share, and cooperative models, where one decision-maker coordinates locations.
- Cooperative problems divide into linear minisum models and other, often nonlinear, formulations.
- Transport-oriented location focuses on transport cost as the dominant factor, while local-input-oriented models also consider factors such as labor and taxes.
- Transport-oriented problems can be resource-oriented, placing facilities near inputs, or market-oriented, placing them near customers.
- On a beach scaled from 0 to 1, two independent stands selling identical hot dogs to uniformly distributed customers converge on the midpoint and split the market.
- For a single owner coordinating both stands, minimizing customers’ average distance instead places them at the 1/4 and 3/4 points.
- The comparison assumes customers buy from the nearest stand, products have the same price, and beachgoers are uniformly distributed.
- When customers choose between beaches, their decision depends on hot dog prices and the dollar value they assign to walking distance; higher walking costs favor the cooperative beach.
- For a couple commuting along the Durham-to-Raleigh corridor, one partner makes 1 trip per day to Durham and the other makes 2 trips per day to Raleigh.
- The equitable location equalizes each partner’s total daily travel distance, placing the apartment 20 miles from Durham, or two-thirds of the way to Raleigh.
- Squaring distance creates an objective whose first-order condition matches the fairness requirement; the solution is the weighted centroid.
- The resulting model minimizes total weighted squared distance and can be solved analytically as an unconstrained nonlinear optimization problem.
- Replacing squared distance with linear distance changes the couple’s objective to total travel, whose minimum is Raleigh, where the weighted commute distance is 30.
- A broader cost model combines travel valued in dollars per mile with the cost of owning vehicles, so unlike quantities can be compared in one objective.
- A second car can be given up only when the apartment is in Raleigh; elsewhere along the corridor, the couple needs two cars.
- The model uses a binary variable Z: Z = 1 forces the apartment location to Raleigh and counts one car, while Z = 0 permits other locations and counts two cars.
- For one-dimensional linear-distance location problems, add the demand weights and find the first location where cumulative weight equals or exceeds half the total.
- That weighted median minimizes the total weighted distance and can be found by hand, as illustrated with an I-40 location example.
- The method generally does not extend as simply to two-dimensional problems using great-circle distances.
- Linear distance is often a realistic logistics assumption because transportation companies commonly pay truck drivers per mile, aligning an easy-to-solve minisum model with efficiency.
Transcript excerpt
0:02 This lecture starts our discussion of location problems. Location is important to logistics engineering, but the reason we're starting with location, beyond just the importance of this topic, is that it is a very good way of learning modeling, in particular how it applies to logistics engineering. I think one of the reasons for that is that it's very graphic or physical. You can see the objective function, you can see results. So, it's a good way of getting feedback on modeling. Starting with a taxonomy of different types of location problems, the first major uh division would be between cooperative location problems and competitive location problems. And there's an
0:50 example in a minute that describes the difference between these two. We're not going to be covering competitive location problems to any great de- detail, so you can see there's nothing really beyond just a intro- introduction to competitive location problems. We're going to be spending most of our time on cooperative location problems. And then amongst cooperative location problems, the major division is between minisum location problems and then everything else is grouped as nonlinear location. So, that implies, which is true, minisum location is essentially a linear location problem. It turns out to be relatively easy to solve, but also very appropriate as a way of modeling the types of logistics problems that we'll be looking at.
1:40 Going further down the hierarchy, the next uh distinction is between location problems where the primary focus on deciding what's the best location is transport-oriented issues. You know, the cost where you locate, has a big impact on transport cost, and transport cost is by far the largest component into producing whatever this the logistics network is producing. If that's not the case, then all of that gets grouped into what can be termed
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Related notes
Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Michael G. Kay.