9 Location Allocation

9.1 Introduction

Aims

The aims of this practical are to:

1. Understand the aims and theory of location allocation models.
2. Understand the methods and criteria used for location allocation.
3. Investigate the sensitivity of location allocations to those methods-criteria.

Application

To that end, we’ll use location allocation models to optimise locations for new defibrillators, using population demand and the transport network as inputs.

Data

• Defibrillator locations listed on the National Defibrillator Network, managed by the British Heart Foundation and associates (The Circuit), available in [data/bhf] [Source]
• Ordnance Survey (OS) Open Roads transport network, available in [data/os/open-roads] [Source]
• UK Centre for Ecology and Hydrology (UKCEH) gridded population at 1 km resolution (2021), available in [data/ukceh] [Source]
• ONS boundaries data, filtered to the Western Isles, available in [data/ons/national] [Source]

Tools

Create Network Dataset (Network Analyst Tools), Build Network (Network Analyst Tools), Make Location-Allocation Analysis Layer (Network Analyst Tools), Near (Analysis Tools), Snap (Editing Tools), Round Up (Spatial Analyst Tools), Int (Spatial Analyst Tools), Create Feature Dataset (Data Management Tools), Raster to Point (Conversion Tools), Feature Class To Geodatabase (Conversion Tools)

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9.2 Practical

Location allocation methods are used to determine the optimum location for a facility relative to a measure of spatial demand. Typically these methods are designed to increase benefits e.g., finding a location for a new hospital that maximises the number of people within a time or distance threshold (Church and ReVelle, 1974), or relocating a fire station to minimise the average travel time to all households (Hakimi, 1965). Location allocation methods can also be used to reduce costs, for example if the facility in question is “obnoxious” (Church and Drezner, 2022) and would be expected to have a negative impact on its surroundings e.g., a hazardous waste disposal facility.

Location allocation is therefore an optimisation problem, which seeks to find the best solution from all feasible solutions within the search space. Taking our example above, location allocation methods could be used to evaluate a set of candidate locations for a new hospital, based on the land that is available, planning regulations, or access restrictions.

Thus, location allocation is often used alongside a constraints analysis, which excludes locations based on a set of user-specified constraints¹⁰⁶.

Our analysis is focused on location allocation of defibrillators, using the Western Isles of Scotland as a case study, given links between socioeconomic deprivation, urban-ruralness and defibrillator access (Burgoine et al., 2024). For every minute that someone’s in cardiac arrest without receiving CPR and having a defibrillator used on them, their chance of survival decreases by 10%. Optimising defibrillator locations to minimise response times or maximise coverage could therefore save lives.

9.2.1 Pre-processing

To begin:

Open ArcGIS Pro and create a new project practical-9 in the correct directory, establish a connection to data, and then load the road network vector [road_link_westcoast.shp], population raster [uk_population_2021_1km.tif] and island vector [west-coast-islands.shp]. All these layers utilise EPSG: 27700, and are subset to the Western Isles of Scotland.

These layers are essential for our location allocation analysis, representing the potential demand for defibrillators (i.e., population), and the routes by which those defibrillators can be accessed (i.e., the road network).

Load the defibrillator table (.csv), which contains data extracted from the National Defibrillator Network (The Circuit), and use XY Table to Points to utilise the geometry information that is present (lat, long, EPSG: 4326). Finally, Project to the British National Grid (EPSG: 27700).

How would you characterise the spatial distribution of defibrillators? Remember the key measures of Centrography, Randomness and Clustering from Practical 5.

Which key concept do we need to consider when interpreting the point pattern?

Suggestion

One important concept is the Observation Window, the region within which point patterns are observed and analysed. This can have a significant impact on our statistical appraisal of randomness e.g., via Clark-Evans \(R\). For example, you will get different results if you analysed the point pattern for the whole study region compared to a single island.

Our next step is to clean the defibrillator dataset, keeping valid entries and ensuring the points are spatially aligned to the road network.

Use Near to calculate the planar distance between each defibrillator (“Input Features”) and the nearest vertex or edge on the road vector (“Near Features”). When complete, inspect the distances (NEAR_DIST), for example via “Explore Statistics” and “Visualise Statistics”.

What can we infer from the distance values?

Suggestion

One observation is that most defibrillator points are within \(\approx100 \:\text{m}\) of the nearest road However, there are a small number \(>100\:\text{m}\) away (n=15). This could represent recording errors (e.g., FID=206), or perhaps additional access routes which are not present in the OS Open Roads dataset, but which could be present in data generated at a larger scale e.g., OS Transport Network. Either way, we can be justified in “snapping” most of the defibrillator points to the nearest road, and removing “inaccessible” defibrillators (\(>100\:\text{m}\)) from the subsequent analysis.

Use Select by Attributes to select the “inaccessible” defibrillators (NEAR_DIST > 100) and delete the selected rows (n=15).

When complete:

Use Snap, which modifies the input features, to snap points to the closest edge in the road vector, using a 100 m tolerance (“Features” = road_link_westcoast, “Type” = Edge, “Distance” = 100). Check the output for validity.

We could subset our dataset further, for example by restricting our analysis to those with 24-hour access, but for now we’ll continue with the full dataset.

We have cleaned the defibrillator dataset, but we also need to clean and prepare our population demand points. Currently this is in a raster format, where the raster values represent the modelled population within each 1 km grid cell:

While we could use this for analysis, this would require a more complex approach than that introduced here, for example modelling movement across the raster surface¹⁰⁷. Instead, we will use our population raster to generate representative demand points.

Use Raster to Point to convert the population raster to a point dataset (e.g., pop_points), where points are generated at cell centroids.

Next use Near to calculate the planar distance between each population point (“Input Features”) and the nearest vertex or edge on the road vector (“Near Features”), and then inspect the distances (NEAR_DIST).

Many of the calculated distances are extremely large, which has resulted in a highly skewed distribution (Mean = \(\approx10\:\text{km}\), Median = \(\approx1.6\:\text{km}\)). This reflects the differing extents of the datasets, with the road vector restricted to the islands, while the population raster covers a much larger region.

Discounting the measurements outside of our study region, there are still many population points far from the road vector, and many “offshore”, as shown below.

Why has this occurred?

Answer

This is because of the generalisation of the population raster, which aggregates population within a 1 km² area. This differs from the actual spatial distribution of people i.e., the location of individual households.

Our next step is to align the population points to the road vector. Given the spatial resolution of the raster (1 km), we can be confident that the population are within \(\approx700\:\text{m}\) of each cell centroid due to the Pythagorean theorem i.e., \(\sqrt{0.5\:\text{km}^2+0.5\:\text{km}^2}=0.7071\:\text{km}\), as illustrated below:

Use Select by Attributes to remove population points where NEAR_DIST > 700, and then use Snap to snap population points to the closest edge on the road vector, using a 700 m tolerance (“Features” = road_link_westcoast, “Type” = Edge, “Distance” = 700). Check the output for validity.

It is recognised that this is at best an approximation of the population demand. While we have used a reasonable approach, ensuring the population total (grid_code field) is snapped to the closest edge of the road vector, discounting points greater than half of the cell diagonal from the centroid (\(>700\:\text{m}\)), the actual distribution of population would look very different.

In the absence of household-level population estimates, an alternative approach would be to simulate more realistic spatial distributions, generating random points within each 1 km² area, and constraining those points to building geometries, perhaps obtained from the OS MasterMap Topography Layer or OpenStreetMap. While this would represent an improvement on our approach, which uses a single population demand point per 1 km² area, it would be much more complex to implement, and would require some sensitivity testing, evaluating the location allocation outputs based on many different randomly generated population distributions.

For efficiency, we are going to focus on a single island, the Isle of Mull.

Use Select by Attributes to select Mull in the west-coast-island layer (“Name” = Mull) and then use Clip to clip the following datasets to the geometry of the selected feature (Use the _mull suffix to denote clipped layers). This includes the road vector as well as the defibrillator points and the population demands points, both of which should have been snapped to the nearest road.

Location Allocation methods are often built upon a road or path layer, which is used to model distance or travel time. This is represented by our OS Open Roads layer, subset to the Isle of Mull. However, the ArcGIS implementation of location allocation will not work with a standard feature layer e.g., road_link_mull, but requires data to be stored in a Feature Dataset which has stricter rules regarding the CRS: all layers must share a common coordinate system, unlike a geodatabase gdb or a collection of files in your project directory.

Use Create Feature Dataset, setting the “Output Geodatabase” to practical-9.gdb, choosing EPSG: 27700 for the “Coordiante System” and with a suitable name e.g., mull_features. An empty Feature Dataset will be initialised, and data can be imported using Feature Class to Geodatabase, with road_link_mull as the “Input Features”, taking care to the select the Feature Dataset mull_features for the “Output Geodatabase”, rather than the project geodatabase.

9.2.2 Network building

To perform location allocation in ArcGIS, we need to create a network dataset from our OS Open Roads layer. Unlike road_link_mull, which is simply a line vector consisting of edges and vertices, a network dataset can include additional information, for example classified junctions (connections between edges), topological connectivity, and if present, relevant network information, such as speed limits or access restrictions.

We can achieve this by:

First creating an inital default network dataset using Create Network Dataset, choosing our Feature Dataset mull_features as the “Target Feature Dataset”, selecting road_link_mull as the “Source Feature Class”, and choosing a suitable output name e.g., road_model¹⁰⁸.

If we have additional rules to apply (e.g., speed limits), those can be defined at this stage, as described here, otherwise we can build our network, which defines the topological connectivity (e.g., junctions).

Run Build Network to generate the network structure needed for routing, using road_model as the input.

When complete, the only difference should be the removal of the “Dirty Areas” polygon, which is an esri term to flag areas of the network dataset which have not been built, or have changed since the last build.

We have now processed our input data to produce a network dataset that could be used for routing, calculating the distance or travel time between population demand points and current and candidate defibrillator locations, as well as a whole range of other tasks.

9.2.3 Optimality

When evaluating a set of candidate locations, the analyst has to make a decision on what criteria are being used to define optimality. Changing these criteria can produce different optimum locations, even with the same inputs for supply (e.g., defibrillator locations) and demand (e.g., population). One commonly used criterion is to select the \(p\) facilities which minimise the demand-weighted average (or total) distance between demand points and the nearest of the selected facilities. This is known as the \(p\)-median problem¹⁰⁹ (Hakimi, 1965; Daskin and Maass, 2015a), as illustrated below:

In short, \(p\)-median places facilities to minimise the total travel distance (or cost or time) to the nearest demand point. Sub-optimal locations would increase the total travel distance, as illustrated here:

Demand points can also be weighted, as illustrated below. For example, Leung et al. (2021) used weighted demand points to produce socioeconomically equitable defibrillator access, incorporating population deprivation, rather than the total population alone.

Another commonly used criterion for location allocation is to place facilities to maximise the total amount of demand covered within a cutoff (e.g., a distance or time threshold). This is known as the Maximise Coverage problem (Church and ReVelle, 1974).

In the example above, four of the six demand points are covered (i.e., accessible within the threshold distance), giving a coverage of \(\approx66\text{%}\) when \(p=1\). Moving the facility might provide coverage for the other demand points, but this would be sub-optimal, as the total coverage would be reduced.

In summary, optimality can be defined and evaluated using many different approaches, including \(p\)-median, Maximise Coverage, or other criteria, which you read about more fully in Laporte et al. (2015). None of these is “correct”, but your choice will depend on the location allocation problem you are addressing.

9.2.4 Hueristics

Having selected a criterion (e.g., \(p\)-median), the next step is to evaluate the candidate locations.

When \(p=1\) and/or when there is only a small set of \(n\) candidate facilities, this can be completed deterministically i.e., evaluate each candidate, calculate it’s score (e.g., total travel distance for \(p\)-median, or coverage % for Maximise Coverage), and return the optimum.

For most location allocation problems, this is impossible, because the number of evaluations increases dramatically as both \(p\) and \(n\) increase. For example, let’s say we have just five candidates \(n\) (\(A,B,C,D,E\)) and set \(p=2\) i.e., we want to return the two candidates which in combination return the highest score. We could calculate the total number of combinations to be evaluated as:

\[\frac{n!}{p!(n-p)!}=\frac{5!}{2!(5-2)!}=\frac{120}{12}=10\:\text{combinations}\]

This is a very simple location allocation problem¹¹⁰. However, if the task was to identify the optimium 25 facilities \(p\) from 10,000 candidate facilities \(n\) e.g., candidates spaced every 100 m in a 10 km² area, the total number of combinations¹¹¹ would be:

\[\frac{n!}{p!(n-p)!}=\frac{10000!}{25!(10000-25)!}=6.26 × 10^{74}\: \text{combinations}\]

This is an extremely large number.

As a result, a deterministic approach to find the optimum solution is often impossible and instead location allocation methods typically use a heuristic approach to guide the search for a near-optimal (“good enough”) solution¹¹².

A heuristic is a strategy that ignores part of the information, with the goal of making decisions more quickly, frugally, and/or accurately than more complex methods.
Gigerenzer and Gaissmaier (2011)

It is outside the scope of this unit to delve into these heuristic approaches in detail but if you are interested you may want to explore Cooper (1964), Reese (2006), Daskin and Maass (2015b), Mladenović et al. (2007) and Gwalani et al. (2021). The key thing to remember is that for higher values of \(n\) and \(p\), the modelled solution may not be the globally optimal solution but instead is more accurately described as the heuristic solution.

9.2.5 P-median

In the remaining sections, we are going to run some allocation location models, using the \(p\)-median and Maximise Coverage approaches, where the former minimises the demand-weighted average (or total) distance between demand points and the nearest of the selected facilities.

Use the Geoprocessing tool Make Location-Allocation Analysis Layer, with “Network Data Source” = road_model¹¹³, “Name” = la_pmedian_mull and “Problem Type” = Minimize impedance (P-median). Leave all other settings as the default. This may take ~5 minutes to complete…

When complete, the location allocation layer should be loaded to the Contents Pane, with groups for facilities and demand points (currently empty) and other network rules that the user might want to specify (e.g., barriers to movement).

Our next task is to add the current defibrillator locations and the demand points to the location allocation layer:

With the location allocation layer selected in the Contents Pane, use the Location-Allocation Layer tab → Input Data → Import Facilities. On the following input screen, use defib_points_mull for “Input Locations”. For “Property” → “FacilityType”, set “Default Value” to Required. As these are current defibrillators, we are requiring that they are included in the location allocation solution.

Repeat the above process, but this time use Input Data → Import Demand Points, selecting pop_points_mull for “Input Locations”. For “Property” and “Weight”, use “Field Name” = grid_code. As discussed above, here we are using a weighted approach, incorporating the modelled population for each point (stored in the grid_code field).

Before we begin to allocate new facilities, a good first analysis would be to evaluate the current distribution of defibrillators relative to the weighted-demand points, using the \(p\)-median criterion:

To run the location allocation analysis, first navigate to “Travel Settings” in the Location-Allocation Layer tab and specify the number of facilities to allocate. You should use \(p=38\) to match the current number of defibrillators on the Isle of Mull. When complete, press Analysis → Run.

When complete, we should be able to inspect the allocation of demand points to the nearest facility.

Inspect the output messages for “Solve”, which you can access via History → Geoprocessing.

How should we interpret the “Sum of allocated weight” and the “Sum of valid unallocated weight” statistics?

Answer

The sum of allocated weight in “Demand Points” is 3050.685136, which is the total weight of valid demand points that were assigned to a facility.

The sum of valid unallocated weight in “Demand Points” is 0, which is the total weight of valid demand points that were not assigned to a facility.

For the \(p\)-median criterion, neither of these statistics are particularly useful, because this criterion assigns all demand points to the nearest facility. In turn, and assuming that all demand points are “accessible” (e.g., aligned to the network), the sum of allocated weights should equal the total modelled population across all of the demand points. You can check this via the Attribute Table → Explore Statistics → grid_code.

What is the sum of allocated weighted impedance in “Demand Points” and how do we interpret it?

Answer

The sum of allocated weighted impedance (3685274.499361) is the key analysis metric i.e., the total population-weighted travel cost across all demand points to their assigned nearest facility. This is calculated by multiplying each demand points weight (i.e., population) by the travel impedance to the assigned facility (i.e., distance). The total weighted impedance is the sum of these values, where the optimal solution, as assessed using \(p\)-median, is the set of facilities with the lowest total weighted impedance, as illustrated above.

We will use this first model as our baseline for comparison. Our next step is to generate a set of candidate defibrillator locations across the island, aligned to the road network, before assessing the optimum configuration using \(p\)-median.

Use Dissolve for road_link_mull, choosing a suitable output name (e.g., road_link_mull_dissolve) and leaving all other settings as default. Next, use Create Random Points to generate 100 random points (random_mull_n100), using the dissolved road vector as the “Constraining Feature Class”. If we don’t use the dissolved output, we would generate \(n\) points per feature.

Return to the Location-Allocation Layer tab → Import Facilities, using the random points as the “Input Locations” and setting “Property” → “FacilityType” as Candidate.

To evaluate these facilities, we need to set the desired number of facilities under “Travel Settings”. If we leave the value as it was before (\(p=38\)), then the output would be identical, because we have required the current defibrillator facilities to be included in the solution. The Scottish Government has recently announced additional investment for new defibrillators, so let’s set \(p=50\) i.e., there is funding for 12 more defibrillators on the island. When complete, press Analysis → Run.

This time there are some more interesting visual outputs to investigate, with some of the candidate facilities now reassigned as “Chosen” i.e., these are the optimum 12 candidate facilities, based on the \(p\)-median criterion.

Remember that we generated our candidate locations using Create Random Points, so your candidates, chosen candidates, and location allocation statistics will differ from mine.

Why haven’t the “Sum of allocated weight” and the “Sum of valid unallocated weight” statistics changed?

Answer

These statistics quantify the total weight of valid demand points that were or were not assigned to a facility. \(p\)-median assigns all demand points to a facility, and assuming that all demand points are “accessible” and assuming that we haven’t changed the network, these statistics should remain the same across different optimisation models.

How has the sum of allocated weighted impedance changed and what is your interpretation?

Answer

The sum of allocated weighted impedance is 2720895.446077 for \(p=50\), compared to 3685274.499361 for \(p=38\). This is a significant reduction, equivalent to \(-964,379\) “person meters” of impedance (i.e., weights are population, distances are meters) and a \(\approx26.2\%\) reduction in the total population-weighted travel cost.

As a final test of the \(p\)-median criterion, we could evaluate an idealised scenario i.e., the island with no existing defibrillators.

Use Create Random Points to generate 1000 random points (random_mull_n1000), using the dissolved road vector as the “Constraining Feature Class”. An alternative approach would be to use regular spacing to ensure full coverage.

Next open the Attribute Table for the Facilities group in la_pmedian_mull and delete all rows, thereby removing all previous required, candidate, and chosen facilities. Return to the Location-Allocation Layer tab → Import Facilities, using the 1000 random points as the “Input Locations” and setting “Property” → “FacilityType” as Candidate.

To facilitate comparison with the current distribution of defibrillators, set \(p=38\) under “Travel Settings”, and then press Analysis → Run i.e., we are now evaluating the optimum locations (based on the \(p\)-median criterion) for our existing 38 defibrillators, of the 1000 candidate facilities (not all possible locations).

It is worth reiterating here that the solution is not necessarily the globally optimal solution but instead is the heuristic solution. The ArCGIS heuristic is described here, with the original formulation in Teitz and Bart (1968). For \(k=1000\) and \(p=38\), the total number of combinations to evaluate is:

\[\frac{n!}{p!(n-p)!}=\frac{1000!}{38!(1000-38)!}=9.38 × 10^{68}\: \text{combinations}\]

While this is smaller than the example provided above (\(p=25, k=10000\)), it is still an extremely large number, therefore requiring a heuristic approach, rather than a deterministic one.

How does the sum of allocated weighted impedance compare to the original distribution of defibrillators (\(p_{\text{original}}\)), and the model with \(p=50\)?

Interpretation

For my location allocation model, the allocated weighted impedance is 2430836.519422 for \(p=38\), compared to 2720895.446077 for \(p=50\) and 3685274.499361 for \(p_{\text{original}}=38\). The absolute difference in impedance compared to \(p_{\text{original}}\) is \(-1,254,438\) “person meters”, equivalent to a \(\approx34\%\) reduction in the total population-weighted travel cost.

One interpretation of this result is that we can improve access to defibrillators, defined as the total demand-weighted impedance between demand points and the nearest of facility (i.e., \(p\)-median), simply through location optimisation, without having to increase the total number!

Before we move on, make sure you understand what \(p\)-median is evaluating. While we have not investigated the underlying heuristics, you should have an understanding of the criterion, and the key choices involved (e.g., selection of \(p\), weighting).

9.2.6 Maximise coverage

Next we are going to evaluate the Maximise Coverage criterion, which maximises the total amount of demand (e.g., population) covered within a cutoff (e.g., a distance threshold).

Use the Geoprocessing tool Make Location-Allocation Analysis Layer, with “Network Data Source” = road_model, “Name” = la_maximise_mull and “Problem Type” = Maximise Coverage. This time, set “Cutoff” to 1000 m i.e., defibrillators should be located within 1 km of demand points, which is justified based on the rapid-response required in cases of Out-of-Hospital Cardiac Arrest (OHCA). Leave all other settings as the default. This may take ~5 minutes to complete…

In ArcGIS, there is the option to switch the problem type for an existing Location-Allocation Analysis Layer i.e., from P-Median to Maximise Coverage. From my testing, this doesn’t work, hence why we’re creating a separate layer. James can you reproduce?

We now need to load our current defibrillator locations and population demand points to the location allocation layer:

Use the Location-Allocation Layer tab → Input Data → Import Facilities. On the following input screen, use defib_points_mull for “Input Locations”. For “Property” → “FacilityType”, set “Default Value” to Required. Repeat this process for the demand points, but this time use Input Data → Import Demand Points, selecting pop_points_mull for “Input Locations”. For “Property” and “Weight”, use “Field Name” = grid_code.

As before, lets first evaluate the current distribution of defibrillators relative to the weighted-demand points, but this time using the Maximise Coverage criterion:

Set \(p=38\) for “Travel Settings” and then Analysis → Run.

Unlike \(p\)-median, which allocates all demand points to a facility, Maximise Coverage prioritises the overall demand coverage given the \(p\) facilities and the selected threshold distance, which can result in unallocated demand:

What is your interpretation of the “Sum of allocated weight” and the “Sum of valid unallocated weight” statistics?

What is your interpretation of the sum of allocated weighted impedance?

Interpretation

Unlike \(p\)-median, which allocates all demand points to the nearest facility, thereby ensuring that the sum of allocated weight equals the total weight of the demand, the Maximise Coverage criterion maximises the total amount of demand (e.g., population) covered within a cutoff (e.g., a distance threshold). As a result, some of the demand weights can be unallocated, if they are greater than the threshold distance from the nearest facility.

For the 38 required defibrillator facilities, the sum of allocated weight is 2008.805185, with an unallocated weight of 1041.879952. Here we are interested in the ratio of allocated to unallocated weights. Given the selected distance threshold (1 km) and the current distribution of facilities and demand, we can conclude that 1041 people are not served by current distribution of defibrillators. This equates to coverage of \(\approx62\%\).

The sum of allocated weighted impedance is 981885.557754, which is simply the distance between each allocated demand point and its assigned facility (Total_Length field in the “Facilities” Attribute Table), multiplied by its weighting (DemandWeight). You can check this value by using “Explore Statistics” on the TotalWeighted_Length field.

It is worth iterating that while this is a representation of the distance to facilities, this is not the criterion by which Maximise Coverage is evaluating and optimising location allocation.

Now let’s evaluate the idealised scenario again i.e., the island with no existing defibrillators.

Next open the Attribute Table for the Facilities group in la_maximise_mull and delete all rows, thereby removing all previous required, candidate, or chosen facilities. Return to the Location-Allocation Layer tab → Import Facilities, using the 1000 random points we generated earlier as the “Input Locations” and setting “Property” → “FacilityType” as Candidate.

To facilitate comparison with the current distribution of defibrillators, set \(p=38\) under “Travel Settings”, and then press Analysis → Run i.e., we are now evaluating the optimum locations (based on the Maximise Coverage criterion) for our existing 38 defibrillators from a set of 1000 candidate facilities.

How has the sum of allocated weighted impedance changed? Does this matter?

Interpretation

For this new scenario, and for my set of random candidate locations, the sum of allocated weighted impedance is 1222049.489314, which has increased compared to \(p_{\text{original}}\). If we were evaluating using the \(p\)-median criterion, this would be a problem, as we have increased the demand-weighted total distance!

However, Maximise Coverage is optimising for coverage, so the ratio of allocated to unallocated weights is more important. In turn, the increase in the total allocated weighted impedance is simply a reflection of the fact that more demand points have now been allocated i.e., more people can access a defibrillator within the 1 km cutoff.

What is your interpretation of the “Sum of allocated weight” and the “Sum of valid unallocated weight” statistics?

Interpretation

For this new scenario, the sum of allocated weights has increased to 2541.885593, compared to 2008.805185 for \(p_{\text{original}}\). As a result, the sum of unallocated weight has fallen by the same amount.

In turn, optimising the location of defibrillators has meant that an additional \(\approx500\:\text{people}\) can now access a defibrillator within the 1 km threshold distance, and population coverage is now \(\approx83\%\).

To finish:

Modify the “Cost Cutoff” under “Travel Settings”, for example comparing the coverage % when the distance threshold is 500 m or 5000 m.

How do the results change and which threshold do you think is most appropriate for our topic?

Interpretation

When the threshold distance is large (5000 m), all demand points can be connected to defibrillator facilities (Coverage = \(100\%\)), so the sum of allocated weight equals the total population weight (3050.685136).

When the threshold distance is reduced to 500 m, some demand points are unallocated and there is a reduction in coverage (\(\approx64\%\)). However, this coverage is still comparable to the baseline model, when the threshold distance was 1 km. This is a good illustration of the value of optimisation.

Considering our focus on defibrillators, and the importance of rapid CPR and defibrillator use, we would be justified in using a small a threshold as possible, balanced against the costs of purchasing and maintaining a defibrillator network.

To confirm your understanding:

What are the key differences between the \(p\)-median and Maximise Coverage criteria?

Which do you think is most appropriate for our topic?

Why is a heuristic approach necessary?

If you’re happy with your understanding… congratulations! You have completed the practical and should now have an understanding of the key problem types in Location Allocation models, and be able to produce the typical inputs (e.g., demand points, networks) and implement a Location Allocation analysis in ArcGIS.

Finished!

9.3 Extra

Using the Maximum Coverage approach, how many facilities do we need to ensure 95% of the population are within 1 km of a defibrillator?

Answer

Compared to a baseline coverage (\(p_{\text{original}}\)) of \(\approx62\%\), and an optimised coverage of \(\approx83\%\), we would need approximately 100 defibrillators to ensure 95% coverage within 1 km (Sum of allocated weight in “Demand Points” = 2937.332799 of 3050.685136, \(\approx96\%\)).

How does defibrillator accessiblity compare to other islands? Repeat the analysis for Jura…

Which key population metric are we not including?

Answer

The non-resident population i.e., tourists! The actual demand for defibrillators might be significantly higher at certain points of the year, while the spatial distribution of the non-resident population may differ from the resident population e.g., at beauty spots.

What are some of the limitations of our approach?

Suggestions

A few suggestions, by no-means exhaustive:

• Population generalisation: How representative are our demand points of the “true” spatial distribution of population?
• Use of distance as an impedance measure: While this is a reasonable approach, would travel time be more effective?
• Absence of constraints: Our analysis has evaluated the optimum locations for defibrillators without consideration of feasibility i.e., is defibrillator installation possible at our candidate locations?

9.4 Resources

• Hakimi, S.L. (1965). Optimum distribution of switching centers in a communication network and some related graph theoretic problems. Operations research, 13(3), pp.462-475.
• Church, R.L., and ReVelle, C. (1974). The maximal covering location problem. In Papers of the regional science association (Vol. 32, No. 1, pp. 101-118). Berlin/Heidelberg: Springer-Verlag.
• Church, R.L. (2002). Geographical information systems and location science. Computers & Operations Research, 29(6), 541–562.
• Church, R.L., & Murray, A. (2018). Location covering models: History, applications and advancements (Advances in Spatial Science). Springer.
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