Centers of Gravity Calculator

Demo mode

In demo mode you can adjust input tables (filled with demo data), parameters & constraints, and run the software, to check out how it works.
However, changes to input table Customers are ignored. To run with your own customer data you need to create an account.

Demo guide

Open section Calculate Center of Gravity locations. Change number of Centers of Gravity. Press Calculate. See map above, Reports, and Service level distances.
Rerun with sensitivity analysis activated. How stable is the solution?
Set number of Centers of Gravity to 3, deactivate sensitivity analysis. Activate constraint "predefined warehouses". Two predefined warehouses will become part of the solution, one fixed at Birmingham (UK), one at Madrid (Spain) that may move within a range of 200 km and ends up in Guadalajara.^* Activate constraint "suppliers". Suppliers will pull Center of Gravity locations towards them. Guadalajara becomes Soria. Constraints and links to input tables are explained in user manual section Constraints. Check out the other constraints.

^* In fact, the demo contains 22 more predefined warehouses. As these can all move freely around (9999 km), they are not really predefined from a Centers of Gravity calculation perspective, and could have been left out for the demo so far. They have been added for the next part of the demo.
Deactivate constraints. Open section Select best warehouses. Press button Select best warehouses. The best 3 out of 24 predefined warehouses are selected. The solution resembles 3 Centers of Gravity. But transport costs are higher, 468867 versus 464759 (Centers of Gravity goal value), because predefined warehouses are not ideally located (and the selection routine does not relocate warehouses). Repeat comparison for another number of warehouses. Or reactivate constraints.
The selection changes if fixed/variable warehousing costs become relevant. As these are set equal for all warehouses, the selection still resembles Centers of Gravity (based on transport costs only). Increase fixed costs of Madrid to at least 106140. It will no longer get selected. In general, differentiated warehousing costs make the map look less logical/intuitive.

To learn more about how to use the software and its other (advanced) features, read the user manual, starting with section Logical steps.

This section describes how to use the software and explains user input. Throughout the app you will also see

- hover mouse over it for info.

To run a basic greenfield Center of Gravity analysis, you only need to enter customers. Reading section Customers will get you started.

Using the software

If where to open (a) new warehouse(s)? is the main question, then use Calculate Center of Gravity locations.

Summary

Step 1 is running a greenfield Center of Gravity analysis. You only need to enter demand.
Step 2 is replicating the current situation, to get figures to compare scenario's against, such as average customer distance. You enter current warehouses and fix customer assignments.
Step 3 is creating a realistic scenario that sits somewhere between the current situation and greenfield. Constraints of step 2 are partly relaxed, like "only some of the current warehouses remain as-is, but others may move to a Center of Gravity".
Step 4 (optional) is adding suppliers to include supplier transport costs in the optimization.
Step 5 is selecting a proper warehouse location for each Center of Gravity, once calculated.

1. Greenfield analysis

100% optimization freedom

Free amount of new warehouses at Center of Gravity locations
All customers reassignable
No warehouse capacity limits

3. Realistic scenario

Some optimization freedom

Some warehouses as is
Some warehouses closed
Some warehouses extended
Some warehouses added
Some customers reassignable
Some warehouse capacity limits applicable

2. Current situation

Zero optimization freedom

All warehouses as is
All customer assignments as is (so warehouse capacity limits are respected without defining those explicitly)

100% ←------ Optimization freedom ------→ 0%

Before setting up your own model, it may be helpful to play around with the demo data first, to get an idea of how the software works, and to check out - for instance - what data relates to what constraint.

A first run

Fill out table Customers under section Input tables - Customers.
- You can start with only the 4 obligatory fields filled: Customer_ID, Latitude, Longitude, Demand.
- All fields are explained in user manual section Customers.
Select the number of Center of Gravity locations under section Calculate Center of Gravity locations
Press Calculate to run the greenfield Center of Gravity analysis. That's it!
Main output - including Center of Gravity locations - is found in report section Output - Reports. All reports can easily be copied into Excel.
Always validate your input data! For example, check the (demand) map. Does it show large unknown customers? Does it show gaps where you expected to see customers? In general, check outliers, such as an extremely small or extremely large demand quantity. If your input data is invalid, then outcomes of the Center of Gravity analysis are (potentially) invalid as well.

Assess service levels

See service level distance metrics under report section Output - Service level distances showing percentage of demand and customers reached within a certain distance (estimated road distance between warehouse and customer). This indicates if more warehouses should be opened, or if some could be closed.
Check option "distance based customer coloring" under section Map settings below map. Customers beyond acceptable lead time distance (set in section Parameters & Constraints - Service level) become black. Based on this map you may decide to add/relocate warehouses. Or accept that some customers will have a longer lead time - which is probably already the case in reality. A few far-away customers should not be driving your supply chain network design!
Check option "Improve service level" in section Parameters & Constraints - Service level and rerun. Service level will be improved at the expense of increased transport costs.

Run sensitivity analyses

Go to section Parameters & constraints - Sensitivity analysis.
Check option "Randomize each demand quantity" and rerun. Each customer demand temporarily changes between −X% and +X%. See how it affects Center of Gravity locations.
Check option "Display cost-increase-areas" and rerun. A grey area around each Center of Gravity appears. If the warehouse would be located at the dotted border, then its transport costs would increase X%.

Enter current warehouses

To enter current warehouses:
- Fill input table Predefined warehouses with current warehouses.
- Set field Move_limit to 0, to fix the warehouse location.
- Check constraint "Predefined warehouses" in section Parameters & constraints - Constraints to activate those predefined warehouses, and rerun.

Fix customer-warehouse assignments

To assign customers to their current warehouse:
- Enter field Warehouse_ID in table Customers
- Check constraint "Fixed customer warehouse assignments" in section Parameters & constraints - Constraints, and rerun.
- Displaying current warehouse-customer links may already reveil some room for improvement.
- Fixing current customer-warehouse assignments automatically means that current capacity limits will be respected, so there is no need to enter field Capacity in Predefined warehouses table, yet.

Gradually relax some constraints

Investigate if current locations are optimal - given fixed customer assignments:
- Increase value of field Move_limit of (some) predefined warehouses, and rerun.
Investigate if current customer assignments are optimal - given fixed warehouse locations:
- Empty field Warehouse_ID of (some) customers, and rerun
- Or uncheck constraint "Fixed customer-warehouse assignments" in section Parameters & constraints - Constraints, and rerun.
Investigate where to open a next new warehouse:
- Increase the number of Center of Gravity locations beyond the number of predefined warehouses, and rerun.
- Make sure you allow (some) customers to get (re)assigned to this new warehouse.

Activate other types of constraints

Limit warehouse throughput capacities:
- Enter field Capacity in the Predefined warehouses table.
- Check constraint "Capacity limits" in section Parameters & constraints - Constraints, and rerun.
- Keep in mind that current limits may not apply in future, if additional space can be rented or the owned facility expanded!
Group customers for single sourcing:
To enforce all customers within a group to be assigned to the same, single warehouse:
- Enter field Group in table Customers.
- Activate constraint "Customer groups" in section Parameters & constraints - Constraints, and rerun.

Intro

A Center of Gravity analysis focuses on the demand side with its relatively high transport costs and lead time requirements. But you can add suppliers to include supply transport costs. Adding suppliers becomes more relevant in a Center of Gravity analysis if there are a few dominant supply locations (such as a large plant or entry harbour), in which case a warehouse should move towards that location. Note that a supplier may pull all warehouses, whereas a customer only pulls one.

Enter suppliers

Enter suppliers - at least obligatory fields Supplier_ID, Latitude, Longitude, Supply - in input table Suppliers.

Enter supplier transport rate

Supplier shipments are usually larger than customer shipments, on average. This makes supplier transport relatively cheaper. The lower transport rate makes suppliers pull Center of Gravity relatively less than customers do.
Enter supplier transport rate in section Parameters & Constraints - Transport rates.

Activate suppliers

Check "Suppliers" in section Parameter & constraints - Constraints.

Select supply option

Select 1 out of 4 supply options in section Parameter & constraints - Supply options. It determines how supply flows are constructed.

Non-unique products

Each supplier delivers to closest warehouse only
Used for a scenario in which local suppliers supply non-unique products to local warehouses. Closest warehouse (auto-detected) can be overruled by specifying a predefined warehouse in the supplier table. This option will neither create inter-warehouse flows nor rescale supply, in order to balance supply with demand, as it gives the user full control over supplier-warehouse flows.
LP optimized supply flows - with auto-sized suppliers
Used for a scenario in which suppliers supply non-unique products to warehouses. This Linear Programming procedure creates supplier-warehouse flows that minimize supplier-warehouse transport costs, while adhering to available supply quantities (and fixed supplier-warehouse assignments^**). You may give suppliers "infinite" supply, as this LP procedure makes supply match demand exactly. It auto-sizes suppliers.
^** If fixed supplier-warehouse assignments make it impossible for supply to reach demand, then these assignments are (partly) ignored. A supplier will then also supply warehouses other than the one you assigned it to. This causes a cost penalty. The LP procedure avoids penalties, as long as possible. If a penalty was unavoidable, then you will get a message. Penalties are excluded from costs outcomes as reported.

Unique products

Each supplier delivers to each warehouse → This is the default supply option.
Used for a scenario in which suppliers supply their unique products to each warehouse directly. The supplier volume to each warehouse is pro rata the customer demand assigned to each warehouse.
Each supplier delivers to each warehouse via closest warehouse
Used for a scenario in which suppliers supply their unique products to each warehouse - via the closest warehouse (2-echelon) - pro rata demand assigned.
Closest warehouse (auto-detected) can be overruled by specifying a predefined warehouse in the supplier table.

Notes

You can specify the supply option at individual supplier level. It will overrule the generic setting. This way you can create a mix of options 1/3/4. Option 2 cannot be combined with other options.
If you have specified a fixed warehouse at a supplier (= supply single warehouse) but also selected option 3 (= supply all warehouses), this causes a logical inconsistency. In that case option 3 is converted into option 4 'in background'.
Under options 2/3/4 each supplier - unlike a customer - pulls several/all warehouses.

Demand side only Only customers pull warehouses.	Supply option 3 Customers pull warehouses. Also, each supplier pulls all warehouses.	Supply option 4 Customers pull warehouses. Also, each supplier pulls all warehouses, and especially the closest one.

Select a proper location for each Center of Gravity

Once the Center of Gravity locations have been calculated (based on road distance estimations), select a proper warehouse location for each Center of Gravity (that may have ended up on top of a mountain or in the middle of a lake).

Under Map Settings, select ESRI map and decrease Center of Gravity size factor or opacity to see the road network.

Consider locations within the 1%-costs-increase-range around the Center of Gravity - the visual outcome of the sensitivity analysis as described under step 1.

If the Center of Gravity sits on top of a dominant customer or supplier, then search for a highway/expressway junction close by. In this case, the 1%-costs-increase-range will be relatively small.
Else, search for a location where two expressways cross each other. At such crossing, transport in any direction will not cause a detour. Whereas if a location is only near an east-west expressway, then transport to the north or south means a detour to reach the north-south expressway. Moving the location to the crossing will remove that detour for 3 directions, but add it to 1 direction (east). If demand is almost equally divided across all 4 directions - which is usually true for a Center of Gravity - then this move lowers the sum of weighted road distances.

You may want to double check your selected location by testing also locations around it using actual road distances (instead of estimations). These can be retrieved with Batch Driving Distances Caclulator. Note that Google Maps returns the road distance of the fastest route, not shortest. Which is fine. But fastest route may depend on time of the day, day of the week, time of the year, roadworks, or vary between routes that are nearly-equivalent. So, even actual road distance might still be an estimation, to some extent.

Even better would be to score locations using transport tariffs. But - most likely - you will not have all tariffs. You can obtain those by organizing an RFQ/RFP process - asking logistics service providers to quote for the transport operation (and warehousing operation, covering other/qualitative factors as well). Of course, there may be no warehouse space for rent/sale at your intended location.

If which warehouse(s) to open and which to close? is the main question, then use Select best warehouses.

Mandatory

Enter your data in the input tables. Predefined warehouses is no longer optional input, but mandatory.
Under section Select best warehouses: Press button "Read input data". Specify the number of warehouse to be selected (per group) and make your solver selection from 3×3 matrix. Press button "Select best warehouses".

Optional

Enter warehouse capacity limits. Keep in mind that current limits may not apply in future, if additional space can be rented or the owned facility expanded!
Enter fixed customer-warehouse assignments, if some customers definitely need to be delivered from a specific warehouse.
You may override the transport costs input as calculated by the software with your own transport costs based on your own costing method / on actual rate tables / on actual road distances that can be retrieved using Batch Driving Distances Calculator (Google Maps based).
Transport rates for new warehouses may be unknown. Or transport rates from a current warehouse to customers that are currently delivered from another warehouse may be unknown. Unknown rates is often a cumbersome part in supply chain network design, and reason to use a simpler costing method that is - more or less - similar to the one embedded in this software.
Enter warehouse fixed/variable costs, so these can be taken into account as well in the optimization.

How to activate constraints

Constraint	Supported	How to activate
Predefined warehouses	yes	Always active (regardles constraint setting).
Fixed customer-warehouse links	yes	By checking constraint in section Parameters & constraints - Constraints
Capacity limits	yes	By using Greedy solver or LP solver (not by checking constraint).
Customer groups	no
Suppliers	yes	By checking constraint in section Parameters & constraints - Constraints. Supply option 3 is activated in background. Other options not supported.
Border crossing points	yes	By checking constraint in section Parameters & constraints - Constraints.

User input

Customers

Customer_ID is a free format identifier.
Latitude and Longitude are both decimal numbers and the result of converting a textual ship-to address (not sold-to adress) into geographic coordinates - a process called geocoding. You can use:
- Batch Geocoder - Free
- Batch Geocoder - Paid
- Excel interfaces with Google Maps / TomTom / BING geocoding api
- or any other geocoder
- Double check postal codes on missing leading zeroes. If - for example - you notice German or French postal codes with only 4 digits instead of 5, then a leading zero might be missing. This issue is quickly caused in/by Excel if you have not set your postal code column to type "text". Incorrect postal codes may cause geocoding problems. And if you would aggregate customer demand at 2-digit postal code level (often seen in European supply chain models), you will end up with demand at the wrong location!
- Put aggregated demand of far-away-customers delivered by sea/air (on overseas island) on exit harbour/airport (latitude/longitude), to ensure that Center of Gravity locations are pulled in the right direction, and to avoid that one of the Center of Gravity locations ends up in such remote customer area. Or simply remove them, if their demand is marginal.
Demand is the total yearly future demand of a customer, ideally expressed in your transport cost driver: volume or (volumetric) weight.
- Collect a full year of demand data if your business is highly seasonal. Otherwise, one month × 12 may do.
- Design for the future: include demand growth percentages.
- If available, then use shipment data with volumes or weights. Otherwise, demand can be calculated based on sales order quantities × item dimensions. But beware: item masters are often not 100% clean, which may cause large calculation errors. Check at least for shipments that exceed truck capacity or are extremely small.
- If a sales budget is the only info (new market entry), then use demographics to create a geographical demand proxy.
- Though not obligatory, it is advised to enter demand figures as whole numbers (easier to read).

Remarks

To run a basic greenfield Center of Gravity analysis: after having entered obligatory data as described above, select the number of Centers of Gravity in section Calculate Center of Gravity locations, and press button Calculate.
Always validate your data! Check top-X customers list. Are major customers missing? Or do unknown customers appear? Check the demand map. Are gaps seen? Or do large unknown locations appear?
Default customer opacity is 0.9 (see map settings under the map), to make customers sitting behind each other still visible. You may also sort your customers from large to small, by clicking on field header Demand, so small customers are plotted on top of large ones (not behind). But still. Customers with the same latitude/longitude are plotted on top of each other. Only a relatively small dot will be seen on the map, that does not properly reflect total demand on that location. By aggregating demand, you may get a more insightful demand map.
Try to keep your model small. If you have 5,000‑20,000 customers, then - most likely - this amount can be greatly reduced by aggregation, while keeping the model valid. It makes your model run (much) faster.
- You may simply round each customer latitude and longitude to two decimals, then aggregate demand per rounded latitude, rounded longitude. By rounding a customer may shift 0.78 km (0.5 mile) from its original position, at most. This 0.78 km is the as-the-crow-flies distance between (latitude, longitude) point (0.000, 0.000) and (0.005, 0.005). The further away from the Equator, the smaller this maximum shift becomes. At a 'European' latitude of 44.000 it is 0.68 km. On average, customers shift approx. 0.37 km, which is unnoticeable on the map. Moreover, all customer shifts together will counterbalance each other to a large extent, so Center of Gravity locations shift a negligable 0.0-0.1 km.
  
  This aggregation remains more precise if using weighted average latitude and longitude within each aggregation category as customer coordinate (instead of rounded coordinates as such). This also means that if there is only one customer in a category (or several at exactly the same location), then it does not shift at all.
  
  Button Aggregate (below input table Customers) creates an aggregated customers input table, and shows the reduction of customers. You may also download this Excel aggregation file that may provide you some more calculations insight. You can also choose to round to nearest 0.02, 0.025, 0.05, or 0.1 instead of 0.01. Of course, the higher this number, the (slightly) less accurate the aggregation becomes.
- Or you may aggregate demand per 2-digit postal code, per country if creating a supply chain model for total Europe. Each country becomes 100 customers, at most. For example, customer DE-13 stands for all German customers with postal code 13XXX (ISO Alpha-2 country code for Germany is DE). (Side note: transport rate tables often use aggregated postal code zones, so are only accurate to that level as well.)
  Add 2-digit postal code to each customer. Create a pivot table to sum demand per 2-digit postal code. Use geocodes of largest customer with that 2-digit postal code to locate aggregated demand (no need to geocode all customers). Or use geocodes from
  geocodes of EU countries 2-digit postal codes
  (use file at your own risk).
  
  If your case is about a single country, then aggregating at 3-digit postal code level may be more appropriate. This will result in a model with 1000 customers, at most.
- Or you may (if the above would not have helped enough to reduce the amount of customers)
  - Remove UK customers from your European dataset, if you know upfront that UK will have its own warehouse(s), and run your Center of Gravity analysis for UK and mainland Europe separately.
  - Split your data into 20% top customers - 80% tail customers (Pareto), and aggregate tail customers only.
  - Use a higher level of aggregation for (the hinterland of) areas that have fixed warehouses.
  - ...

Optional fields

Warehouse_IDs is used to assign a customer to a predefined warehouse, so it should match to a Warehouse_ID as defined in table Predefined warehouses. If left empty, then no fixed assignment applies.
- Create a fixed assignment by entering a single warehouse_ID.
- Create a semi-fixed assignment by entering a subset of warehouses_IDs that the solver is allowed to select from.
  Use / as separator between the warehouse_IDs. For example, Birmingham/Madrid/WH05.
Check constraint "Fixed customer-warehouse assignments"
to activate, in section Parameters & constraints - Constraints
Group - free format - is used to force all customers within the same group to be delivered by the same single warehouse only (single-sourcing). What group stands for, is for you to decide. For example, a country. The group is assigned to the warehouse that comes with the lowest sum of weighted distances for this group of customers. Underlying cost calculation is still done at individual customer level. Read user manual section Center of Gravity algorithms - Customer grouping for more info.
A fixed customer-warehouse assignment takes precedence over a customer group. Warehouse capacity limits may cause a customer group to get broken up.
Check constraint "Customer groups"
to activate, in section Parameters & constraints - Constraints.
Shipment_size translates into an individual customer transport rate. If specified, then it overrules the general customer shipment size. Read user manual section Transport rates for more info. Make sure it does not exceed truck capacity. Shipment size can be expressed in KG, M3, pallets, loading metres, ... which - of course - should relate to how you have expressed demand.
Lead_time_distance is the individual maximum allowed distance from the warehouse. If specified, then it overrules the general lead time distance as set in section Parameters & constraints - Service level. Customers are colored in black if the leadtime distance is exceeded, indicating where service is compromised. Enter the distance in the correct distance unit, miles or kilometers, as you have selected in section Parameters & constraints - Distance.
Border_group links the customer to a border group. A border group has specific border crossing points to enter or exit the bordered area. Those border crossing points are defined in table Border Crossing Points. A flow from a warehouse to a customer will go via one of the border crossing points, unless the warehouse and customer are both inside the bordered area.
Check constraint "Border crossing points"
to activate, in section Parameters & constraints - Constraints.

Optional fields for "Select best warehouses" functionality only

Qty_1 to 5 are the warehouse cost driver quantities of a customer, such as #orders, #order lines, #unit_picks, #carton_picks, #pallet_picks. You may decide yourself what each qty stands for.

Predefined warehouses (optional)

Check constraint "Predefined warehouses"
to activate and include predefined warehouses in the solution, in section Parameters & constraints - Constraints.
Warehouse_ID - free format, for example "Birmingham" - is used in the Customers and Suppliers table to assign customers or suppliers to this warehouse (instead of having the software assign customers and suppliers).
Latitude, Longitude is the result of converting a textual address into geographic coordinates - a process called geocoding. For more info, read the help text for fields Latitude, Longitude of Customers.

Optional fields

Move_limit = 0 (default) means a fixed location, > 0 means a location that may move away from its initial coordinates but no further than its move limit. To be expressed in either miles or kilometers, as you have selected in section Parameters & constraints - Distance.
Capacity is used to limit the demand that can be assigned to a warehouse. It is total throughput capacity (in relation to demand data period), not storage capacity. Yearly througput capacity = storage capacity × yearly stock turns. If your demand data is about one quarter, then divide yearly throughput capacity by four, or else multiply all demand (and supply) by four.
Warehouse capacity limits may cause customer groups to 'break'.
Check constraint "Capacity limits"
to activate, via section Parameters & constraints - Constraints.
Transport_rate_index is used to adjust customer transport rates of all customers supplied from this warehouse. An index below 100 means transport from this warehouse to customers is relatively cheaper, an index larger than 100 means more expensive. Customer transport rates are multiplied by the warehouse Transport rate index/100, for all customers assigned to the warehouse. The lower this index, the more customers a warehouse will get assigned (unless other constraints prevent this).
IsSatellite = 0 (no) or 1 (yes) and is used to distinguish main warehouses from satellite warehouses. This is relevant if suppliers are activated and parameter "Link supplier to closest Main warehouse" is checked. Even if suppliers have not been activated, main→satellite warehouse flows are generated.
Color overwrites the default warehouse color. Format: #rrggbb (red,green,blue hexidecimal values)
Pick color → Color =

Optional fields for "Select best warehouses" functionality only

Whs_group (opt)
- Valid entries are 'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i', or 'j'. If left empty, default 'a' is used.
- Group 'j' is a special 'must-be-included' group. All warehouses in group 'j' will always be included in the selection (which of course may affect the selection of warehouses from other groups).
- Group can be interpreted as geographical region. By grouping warehouses per region you can ensure that each region gets a warehouse, if you set the number to be selected from that group higher than 0.
Fixed_costs are in fact binary-fixed costs, the fixed costs apply if a warehouse is selected.
Rate_1 to 5 are the warehousing cost per unit throughput for up to 5 cost drivers (and linked to your customer table). Example: if customer X has Qty_1 = 10 and Qty_2 = 5, and warehouse Y has Rate_1 = 2.50, and Rate_2 = 3.75, then the variable costs of this customer X at warehouse Y are 10×2.50 + 5×3.75.

Suppliers (optional)

Check constraint "Suppliers"
to activate and include suppliers, in section Parameters & constraints - Constraints.
Supplier_ID: free format identifier.
Latitude, Longitude is the result of converting a textual address into geographic coordinates - a process called geocoding. For more info, read the help text for fields Latitude, Longitude of Customers.
Supply is the total yearly future supply of a supplier. Usually, the supply of all suppliers should - more or less - match the demand of all customers. Unless supply is about the dry ingredients of liquid finished products, for example.

Optional fields

Warehouse_IDs (= link to table Predefined warehouses) is used to assign a supplier to a warehouse or subset of warehouses (similar to assigning a customer to a warehouse or subset of warehouses). If left empty, the software decides which Center of Gravity location(s) is/are supplied by the supplier. If specified, then this supplier will not supply all warehouses directly, even if supply option "Each supplier delivers to each warehouse directly" has been selected.
Check constraint "Fixed customer-warehouse assignments"
to activate, in section Parameters & constraints - Constraints.
Shipment_size translates into an individual supplier transport rate.
Supply_option (1, 3, or 4) can be set to overrule the general supply option (1, 3, or 4) as selected in section Parameters & constraints - Supply options. General supply option 2 (LP optimized supply flows) can not be overruled at individual supplier level (individual settings are ignored).

For more info about suppliers, transport rates, and supply options, read user manual section Logical steps - Step 4. Add suppliers

Border crossing points (optional)

Check constraint "Border crossing points"
to activate, in section Parameters & constraints - Constraints.
The use of border crossing points may improve distance accuracy and direction in which a Center of Gravity is pulled.
If an area can only be entered/exited at some specific points, you may enter these border crossing points.
A transport flow has to go via a border crossing point if
- a warehouse is outside the bordered area and the customer inside ("entry")
- a warehouse is inside the bordered area and the customer outside ("exit")
A flow via a border crossing point means a detour. The border crossing point causing the smallest detour is auto-detected. It also means that the warehouse is pulled towards this point instead of the customer, until the border has been crossed.
Border_group - free format - is the name of the bordered area with all its crossing points (latitude, longitude) entered in this table. Field border_group is also found in the customers table and used to specify which customers belong to the border group (thus defining its geographic shape). To function properly at least some customers of a bordered area should be located within a 50 km range of the border point.
In the picture on the right the border crossing points in the northern part of Italy prevent extreme shortcuts across the Adriatic Sea. If ferries are a valid transport option, then you may remove all border crossing points or add ferry harbours as points.
Current implementation handles only one border crossing point per flow, not multiple. A warehouse area exit point takes priority over a customer area entry point: "exit before entry".

Intro

By (un)checking constraints (optional) in section Parameters & constraints - Constraints you (de)activate parts of your input data, so you can quickly investigate how a constraint affects goal value and supply chain network structure.

The following explains the links between constraints and input tables. Similar info is provided in previous sections. But this section is more focused. It also describes demo data, so you can check how a constraint is set up.

Predefined warehouses table

Ad 1. Check constraint "Predefined warehouses" to include predefined warehouses in the solution.
Demo: contains one predefined warehouse in Birmingham (UK) and one in Madrid (Spain), with field Move_limit set to 0 for the Birmingham warehouse (it cannot move), and set to 200 for the Madrid warehouse (it can move within a 200 km range). If the number of Centers of Gravity is set higher than two, then these two locations will be part of the solution (Birmingham remains Birmingham, Madrid is moved to Guadalajara if running for 3 Centers of Gravity). If you make Madrid's Move_limit 0, it will remain Madrid.

Note
In fact, the demo contains 22 more warehouses. As these can all move freely around (9999 km), they can not really be considered predefined for the Centers of Gravity calculation, and they might just as well have been left out. They have been added to demonstrate the "Select best warehouses" functionality (that selects only, and does not relocate warehouses).
Ad 4. Check constraint "Capacity limits" to limit warehouse throughput capacities (more precise: total customer demand that can be assigned to the warehouse).
Demo: field Capacity is set for both predefined warehouses Birmingham and Madrid.
Ad 5. For more info about capacity limits and LP optimization, read user manual section Center of Gravity algorithms - Warehouse capacity limits.

Customers table

Ad 2. Check constraint "Fixed customer-warehouse assignments" to assign customers to a predefined warehouse (or subset of predefined warehouses).
Demo: field Warehouse_IDs is set to "Birmingham" for all customers in UK/Ireland to assign them to the predefined Birmingham warehouse. It is set to "Madrid" for all customers in Spain/Portugal.
Ad 6. Check constraint "Customer groups" to have all customers within the same group delivered by the same single warehouse that can deliver the group against lowest transport costs. These costs are still calculated at individual customer level, see example in user manual section Center of Gravity algorithms - Customer grouping.
Demo: field Group is filled with the applicable 2-letter ISO country code for all customers. So - in the demo - group stands for country. This prevents customers in Western France to get delivered by the UK warehouse.
Ad 8. Check constraint "Border crossing points" to activate border crossing points.
Demo: field Border_group is filled for customers in UK, IE, Italy, and Iberia.

Suppliers table

Ad 7. Check "Suppliers" to include suppliers. Also select a supply option.
Demo: contains one supplier in Rotterdam (The Netherlands - Entry harbour supplying 71% total demand), and one in Bologna (Italy).
Ad 3. Check constraint "Fixed customer-warehouse assignments" to assign suppliers to a predefined warehouse. It links to field Warehouse_ID. If filled, then the supplier will only supply this warehouse. And via this warehouse to each other warehouse, if supply option "Each supplier delivers to each warehouse via closest warehouse" is selected. (If field Warehouse_ID is filled, then a supplier will never supply all warehouses directly, even if supply option "Each supplier delivers to each warehouse directly" has been selected.)

Border crossing points table

Ad 8. Check constraint "Border crossing points" to activate border crossing points.
Demo: field Border_group relates to the same field found in the customer table and specifies which border crossing points (if any) are applicable for which customers. The demo contains a limited amount of border crossing points in Italy, Iberia, UK/Ireland, just to show you how this concept works.

Intro

In section Parameters & Constraints - Transport rates you set transport rates to differentiate between customer and supplier transport. You may also adjust the cost curve that expresses how much - relatively - more expensive smaller shipments (LTL-like transport) are compared to larger shipments (FTL-like transport).

Transport costs curve

Smaller shipments being relatively more expensive than larger shipments is expressed in the transport cost curve. The default cost curve shows that a shipment size of 50% FTL brings 72.4% FTL costs, meaning that - per each unit transported - it is 1.45 times more expensive than FTL transport. You can adjust the curve by dragging control point handles or entering control point values in the cost curve settings below the chart.
Center of Gravity transport costs curve

Small customer shipments versus large supplier shipments

Smaller shipments to customers are relatively more expensive than larger shipments from suppliers. Average customer and supplier shipment sizes - as set in the transport rates table - are translated into rates by applying the cost curve.

Shipment size can be expressed in KG, M3, pallets, loading metres, ... It should relate to how you have expressed customer demand, as a single transport unit of measurement applies to the whole model.

The ratio supplier/customer rate is important as it expresses how relatively less suppliers pull warehouses than customers do. This ratio is independent of cost/km. Therefore, high accuracy of cost/km is of lesser importance.

Ratio supplier rate/customer rate

The ratio supplier rate/customer rate expresses how relatively less suppliers pull warehouses than customers do. To minimize transport costs a Center of Gravity should move closer towards a customer than towards a supplier, if their total transport volumes would be equal, regardless distance!

Above chart based on solver outcomes can be verified by geometry that brings the following formula for the Center of Gravity x‑position = max(0, 100 + 50 × F / √(1 - F²) ), with F = (1 - 3 × ratio)/2. F stands for pull force in horizontal direction - of both customer C1 and C3 - required to reach an equilibrium state.
It is easy to understand that x‑position becomes exactly 100 at ratio 1/3, as supply quantity of 3 × 1/3 equals C2 quantity of 1.
For a ratio ≥0.93 the supplier pull force outweighs customer pull forces, so the Center of Gravity ends up on top of the supplier. This 0.93 is in fact (1+2×1/√(1²+0.5²))/3 =
(1+4/√5)/3 = 0.9296...

Keep in mind that a supplier is not just a small customer as they can pull all warehouses. Read user manual section Logical steps - Step 4. Add suppliers for more info.

Small customers shipments versus large customers shipments

Similarly, shipments to small customers are relatively more expensive than shipments to large customers. To differentiate you can enter individual shipment sizes at individual customer (and also at individual supplier level). The impact of individual shipment size on Center of Gravity locations may be small, if smaller and larger customers are scattered all around.

Auto-size shipments per individual customer

Calculates individual shipment sizes, replacing the average customer shipment size as set in transport costs table, assuming shipments to larger customers are larger and shipments to smaller customers are smaller. The chart shows the assumed relation between demand volume and shipment size.

As this method is rough, it is better to enter actual individual shipment sizes in the customer table than to activate auto-sizing.

Transport-rated volumes on the map

Under map settings you can choose to display location sizes as transport-rated volumes (volume × rate ratio), instead of plain volumes (default setting). Suppliers will then appear relatively smaller than customers, which better reflects the Center of Gravity 'pull forces' of suppliers and customers.

Two options to model warehouse → warehouse flows

A default model looks like: (suppliers →) warehouses → customers
But you can also model: (suppliers →) warehouses → warehouses → customers

There are two options to model warehouse → warehouse flows.

In a model with suppliers:
Select supply option "Each supplier delivers to each warehouse via closest warehouse".
Closest warehouse can be overruled by specifying a predefined warehouse in field Warehouse_IDs of the supplier table. Supply quantities (that are related to demand quantities) determine warehouse-warehouse flow quantities.
In a model without suppliers:
Set up predefined warehouses, some as main warehouse (field IsSatellite=0), some as satellite (IsSatellite=1).
Each satellite gets supplied by its closest warehouse. Satellite demand quantities determine warehouse-warehouse flow quantities.

You can combine it with forcing a link between a customer/supplier and a (subset of) warehouse(s) by entering field Warehouse_IDs.

In general, try to keep your model simple. Keep in mind that the purpose of a model is to help answering questions. That does not necessarily mean that the model should reflect reality as closely as possible. For example, if all suppliers deliver to a single plant (main warehouse), and the plant delivers to warehouses that deliver to customers, you can create a 2-echelon plant (main warehouse) → warehouses model that reflects reality. But if that plant location is fixed, then there is no possibility to optimize suppliers→plant flows. So, you may consider replacing all suppliers and plant by a single fake supplier at (previously) plant location, and revert to a simple 1-echelon model.

Note: a two-echelon warehouses → warehouses structure is not supported by the "Select best warehouses" routine.

Calculate Center of Gravity locations

Center of Gravity analysis - goal value

Goal value to be minimized is the sum of weighted distances. In a basic approach distances are weighted by quantities only. Centers of Gravity Calculator uses two weights. Distances are weighted by transport quantity (1^st weight) × transport rate (2^nd weight). The transport rate converts distances and quantities into costs. Primary reason for adding transport rate as a second weight is to make suppliers (larger shipments, FTL-like transport) pull Center of Gravity locations relatively less than customers (smaller shipments, LTL-like transport) do, rather than to come up with highly accurate transport costs.

The percentual change in goal value of future scenario compared to baseline can be applied to your actual baseline costs (derived from your financial reports) to estimate the transport costs effect of the future scenario. Because transport costs partly consist of (un)loading costs that will not change, you may add some factor to dampen the costs effect.

	Baseline	Future scenario
Baseline transport costs from your financials	USD 12,000,000
Goal value Centers of Gravity Calculator	10,000,000	9,000,000
Goal value change (as %)		-10%
Dampening factor (value assumed at 0.80)		0.80
Goal value change × dampening factor		-8%
Transport costs effect		- USD 960,000
Future scenario costs		USD 11,040,000

Read user manual section Transport rates for more info about transport rate differentiation between customers and suppliers.
By the way, a supplier pulls one/several/all warehouses (dependent on selected supply option), whereas a customer pulls only one warehouse, so a difference in transport rate is not the only difference between a customer and supplier.

In case of an owned truck operation, see section Parameters & Constraints - Yearly mileage of own truck operation. It supports the estimation of transport mileage. That mileage can be used to estimate transport costs effects more precisely.

Center of Gravity analysis - algorithms - How to calculate Center of Gravity locations

This section explains in detail how to calculate Center of Gravity locations, and includes several (interactive) animations.

A Center of Gravity is a location that minimizes transportation costs / Center of Gravity locations are indicative warehouse locations that minimize transportion costs. Center of Gravity locations are locations that minimize the sum of weighted distances (acting as an indicator for transport costs). In a basic model, weighted distance is the distance from warehouse to customer multiplied by customer demand. If a customer has a demand of 10 and is located 25 kilometers from its warehouse, then its weighted distance is 250 kilometers. In an extended model, also distances from suppliers, and from/to other warehouses are included.

Single Center of Gravity algorithm - Introductory thoughts

Upfront: there exists no mathematical formula that can calculate the Center of Gravity.

Imagine a case with two customers only. Customer A has a demand of 10 and customer B of 1. Where is the Center of Gravity? Somewhere on line A-B, closer to A?

Well, you could say that customer A pulls 10 times harder than customer B. If the 'Center of Gravity' * moves a distance d towards A the sum of weighted distances decreases (10 × d) – (1 × d) = 9 × d.

* If put between quotes then it refers to a location under investigation which is not yet the true Center of Gravity.

So, the Center of Gravity is right on top of customer A, not somewhere in between A and B!

The resultant pull force - sum of all customer pull forces - is leading. It always points in the direction that the 'Center of Gravity' should move into to decrease the sum of weighted distances. This resultant pull force is constructed by linking all customer pull forces together, head to tail.

However, the sum of weighted distances will only decrease if the 'Center of Gravity' moves the right distance! The resultant pull force does not tell how far to move. Move this far?

Or this far?

The smaller the move distance, the less chance of 'overshooting' the Center of Gravity, but the many more moves to be made. Therefore, it is better start with big moves. In case of 'extreme overshooting' the sum of weighted distances will have increased instead of decreased, and the resultant force arrow changed - or even completely reversed - its direction. By then, the move size needs to be reduced in order to get closer to the Center of Gravity. Else, the next move would end up at the previous position, followed by getting stuck in a loop of going back-and-forth without ever getting any closer. By reducing the move size each time the sum of weighted distances increased the 'Center of Gravity' ends up on top of the Center of Gravity.

Of course, the above was an extreme example. Usually, there is no single dominant customer, and the Center of Gravity will be somewhere in the center of all customers. Also then, the Center of Gravity is found by moving in the direction of the resultant pull force, but not too far. The resultant pull force becomes smaller the closer the location gets to the Center of Gravity (which implicates that the length of the resultant pull force somehow does relate to the distance to be moved). It becomes zero on top of the Center of Gravity, with all underlying pull forces (F) in a state of equilibrium in both x and y direction: ΣFx=0, ΣFy=0. All forces linked together start and finish at exactly the same location = zero resultant force.
Center of Gravity as equilibrium of customer pull forces

Center of Gravity as equilibrium of customer pull forces

Except if one dominant customer outweighs all other customers. Then the Center of Gravity is on top of this dominant customer, but the resultant force is not zero.
Another exceptional case, rather theoretical, is one in which there are only two customers, A and B, with equal demand. Then the resultant force is zero for all points between A and B on line A-B. Each of these points is Center of Gravity.

Calculating the Center of Gravity is not so much about distances as such, as it is about pull forces and directions. If in the above example customer 2 would be located 10 times further away in the same direction, this would not affect the Center of Gravity at all.

Single Center of Gravity location algorithm

There exists no mathematical formula that can calculate the Center of Gravity. It takes an iterative approach.

The Single Center of Gravity algorithm works like this on an x,y plane:

Initialization: locate the 'Center of Gravity' at an initial (random) position, and give the move distance a large value.
Calculate the resultant pull force. Move the 'Center of Gravity' in that direction, at move distance.
Calculate the sum of weighted distances. If it increased, then divide the move distance by two.
Repeat steps 2 and 3 until the move distance becomes very small.

This algorithm will always find the global optimum with a single run. With this type of problem, there is no chance of getting stuck in local optimum. The algorithm relates to the gradient descent method (wikipedia), also used by Excel Solver's method GRG Nonlineair (GRG stands for Generalized Reduced Gradient).

Determining the Center of Gravity is like finding the x and y coordinate of the lowest point (x,y,z) in a bowl-shaped 3D‑chart - visualizing the sum of weighted distances z = f(x,y) for each x and y - by going into the direction of steepest descent as pointed to by the resultant pull force. During this descent the direction gets adjusted after each move made/step taken.

There are methods to determine a proper move distance (a.k.a. step size) per iteration. But these are rather complex and (thus) time-consuming. Luckily, there is no urgent need for those, because the method of step 3 works and is fast.
Center of Gravity gradient descent and step size

Center of Gravity gradient descent and step size

Single Center of Gravity algorithm - Summary

Initialization: locate the 'Center of Gravity' at an initial (random) position, and give the move distance a large value.
Calculate the resultant pull force. Move the 'Center of Gravity' in that direction, at move distance.
Calculate the sum of weighted distances. If it increased, then divide the move distance by two.
Repeat steps 2 and 3 until the move distance becomes very small.

Single Center of Gravity algorithm - Steps more detailed

Initialization: locate the 'Center of Gravity' at an initial (random) position, and give the move distance a large value.
- A good initial position is the weighted average x and y coordinates of its customers. It is often near-optimal, but not the Center of Gravity, though you may have read elsewhere it is. Theoretically, it can be even 100% worse!
  Often only 0-5% worse...
  Often, the sum of weighted distances of the weighted average x,y position versus that of the true Center of Gravity will be 0-5% worse, but more than 5% worse is also possible, as sometimes will happen/can be seen in the simulation further below.
  
  ...but theoretically even 100% worse is possible!
  Imagine a case with only two customers, customer A with demand 1000 at x,y position (0, 0) and B with demand 1 at position (100, 0). The weighted average x,y position is (0.0999, 0), with 0.0999 calculated as (1000×0 + 1×100)/(1000+1). If the 'Center of Gravity' moves a distance d towards customer A the goal value improves 1000×d (closer to A) − 1×d (further from B) = 999×d. So the Center of Gravity is on top of customer A, not at the initial weighted average x,y position! The Center of Gravity has a sum of weighted distances of 1000×0 (customer A) + 1×100 (customer B) = 100, whereas the initial position has a value of 1000×0.0999 (customer A) + 1×99.9 (customer B) = 199.8, which is 99.8% worse!
- A random location within the customer cloud can be used just as well, as the remainder of the algorithm will always find the Center of Gravity, in almost equal time.
- It is better to start with a bit too large move distance (100 kilometers) than with a bit too small, because if too small then it will take many iterations before the Center of Gravity is reached.
Calculate the resultant pull force. Move the 'Center of Gravity' in that direction, at move distance.
- Sum up all pull forces (vectors) to get the resultant pull force (vector) with a certain size (less relevant) and direction (most relevant) by summing up x and y components separately. For each customer pull force F: Fx = length demand vector × (customer_x − CoG_x)/(distance from customer to CoG), or shorter: Fx = demand × cosinus(angle F). Fy = demand × sinus(angle F).
- Normalize this resultant pull force (a vector with origin at 0,0 and end point at x,y) by dividing x by its length √(x² + y²) and y by its length. The vector keeps pointing in the same direction, but its length becomes 1, which makes it a unit vector.
- Multiply x and y values of this unit vector with the step size to get the move vector representing move distance in x and y direction.
- If the move vector's tail starts at the current 'Center of Gravity' position, then its head ends at the next 'Center of Gravity' position. So, add the move distance in x direction to the current x position of the 'Center of Gravity', and add distance in y direction to its current y position. This brings the next 'Center of Gravity' position.
Calculate the sum of weighted distances. If it increased, then divide the move distance by two.
- This approach originates from the binary search algorithm - a.k.a. half interval search - to find the position of a target value within a sorted array. See half-interval search (wikipedia).
- Optional: move the 'Center of Gravity' back to its previous position if the sum increased, so with each 'accepted move' the solution always improves (or remains equal, but never worsens).
Repeat steps 2 and 3 until the move distance becomes very small.
- Then the Center of Gravity hardly changes anymore, and the sum of weighted distances converges.
- The Center of Gravity may be in the middle of a lake or on top of a mountain.

Single Center of Gravity algorithm - Visual simulation

Press button Next process step to start the simulation.

Earth is a globe - Required adjustments translating Cartesian into Spherical coordinate system

X,y on a flat plane (Cartesian coordinate system) becomes latitude,longitude on a globe (Spherical coordinate system). The algorithm involves summing up vectors with a length and an angle. The distance between two points on a globe is calculated with the Haversine formula, and the angle with a Bearing formula. Demand vectors are mapped to a flat tangent space before summing up, and the result is translated back to the globe - see also the picture of section 'Weiszfeld's algorithm as alternative'.		"No, it's not a flat plane."

Latitudes and longitudes in both formulas need to be expressed in radians = degrees × 𝜋/180. Haversine formula - in 2 steps a = sinus²((latitude_customer − latitude_Center of Gravity)/2) + cos(latitude_Center of Gravity) × cos(latitude_customer) × sinus²((longitude_customer − longitude_Center of Gravity)/2) As-the-crow-flies-distance = 2 × atan2(√a , √(1-a)) × 6371. 6371 kilometers is Earth's mean radius. See Excel - As-the-crow-flies distance function for an Excel implementation. Excel's atan2 function takes its arguments in reversed order! Bearing formula Bearing = atan2(sinus(longitude_customer − longitude_Center of Gravity) × cosinus(latitude_customer) , cosinus(latitude_Center of Gravity) × sinus(latitude_customer) − sinus(latitude_Center of Gravity) × cosinus(latitude_customer) × cosinus(longitude_customer − longitude_Center of Gravity) Bearing is measured against the vertical longitude axis, not the horizontal latitiude axis, so you need to swap the use of cosinus/sinus when calculating forces in x and y direction.

Weiszfeld's algorithm as faster alternative

Weiszfeld's algorithm - see also Geometric median (wikipedia) - is simpler (but less intuitive) and works like this on an x,y plane:

Initialization: locate the 'Center of Gravity' at an initial (random) position.
Calculate each customer weight as demand divided by distance to 'Center of Gravity'.
Calculate the weighted average x,y of customers as the next 'Center of Gravity'.
Repeat steps 2 and 3 until the sum of weighted distances converges.

Applying Weiszfeld's algorithm to Earth is done via a flat tangent space. For more info, read section 2.3 of scientific article Generalized Weiszfeld Algorithms for Lq Optimization by K. Aftab, R. Hartley, J. Trumpf.

Differences between Single Center of Gravity and Weiszfeld's algorithm

Major difference between Single Center of Gravity algorithm and Weiszfeld's algorithm is that the Single Center of Gravity algorithm uses an explicitly controlled move distance that - indirectly - enables an advanced feature: you can limit the move range of predefined warehouses. The optimal location may then be somewhere at the border of this limited range and Weiszfeld's algorithm cannot handle this.

That difference is also the reason that Weiszfeld's algorithm converges approx. 10% faster.. Because, if the initial move distance is set too low, it will take many iterations to move a Center of Gravity to its final destination, so it is better to initialize it too high. But as a result, the first few iterations will often be useless overshoots (ignored - after which step size is divided by two - and then it quickly catches up with Weiszfeld's algorithm).
Center of Gravity convergence speed of COG algorithm versus Weiszfelds algorithm

Center of Gravity convergence speed of COG algorithm versus Weiszfelds algorithm

Centers of Gravity Calculator uses both algorithms

Centers of Gravity Calculator uses both algorithms: the faster Weiszfeld's algorithm as its default, and Single‑Center of Gravity algorithm in case of predefined warehouses with limited move range (and some other specific cases).

Multiple Center of Gravity locations algorithm

Below, 'Center of Gravity' and warehouse are used interchangeably.

A single run of the Multiple Center of Gravity locations algorithm does the following:

Initialization: locate each warehouse at a random location within the customer cloud.
(Re)assign each customer to its closest warehouse.
Move each warehouse to the Center of Gravity of its assigned customers.
= apply Single Center of Gravity location algorithm / Weiszfeld's algorithm per warehouse
Repeat steps 2 and 3 until the sum of weighted distances converges.

Multiple runs are done to find the global optimum, as a run may get stuck in a local optimum. The more runs, the more likely the global optimum is found. Usually, this global optimum will then have been found multiple times as best solution. The warehouse locations of the best solution found are considered to be the Center of Gravity. But - especially with a higher number of warehouses - it may happen that two different warehouse structures bring an almost similar sum of weighted distances.

Similarity with K-means Clustering

The above algorithm is similar to K-means Clustering (wikipedia) used for cluster analysis in data mining and machine learning.

Animations of the Multiple Center of Gravity locations algorithm

The animations show what happens during a single run of the Multiple Center of Gravity locations algorithm. Customers are assigned to their closest warehouse, warehouses move to the Center of Gravity of their assigned customers, customer are reassigned, warehouses move, et cetera, until the final situation is reached where none of the customer assignments change anymore, and none of the warehouse positions.

Although - in the first three examples below - each run starts with different initial random locations, the solution found is the global optimum with a minimal sum of weighted distances.

Run 1 - global optimum (50 iterations)	Run 2 - global optimum (33 iterations)	Run 3 - global optimum (21 iterations)

But a run may get stuck in a local optimum, as shown below. Therefore, multiple runs need to be done to find the global optimum.

Run 4 - local optimum (32 iterations)

Beware of an overseas customer, far, far away

As seen in the previous section, calculating a single Center of Gravity location is not so much about distances as it is about pull forces and directions, finding the spot where all forces summed up together become zero.
An exceptional multiple Center of Gravity locations case in which distance may become "directly relevant" is a case in which one customer is located thousands of miles away from all others, on some overseas island that belongs/belonged to a country. Then one of the Center of Gravity locations may end up on top of that customer. But locating a warehouse over there would make no sense. So you better remove such customer from your input data, or relocate it to exit harbor or airport.

Underlying assumptions

A major underlying assumption is that transport cost = rate/kilometer × distance. This assumption is only partly valid. For instance, parcel rates are often distance independent within a region. Macro-economic imbalances cause direction-dependent rates. Different transport companies may have different rates for the same route because they have a different customer base that may enable efficiencies (or not). From that perspective, it would be better to calculate with costs based on actual carrier rates. But most often you will not have transport rates for each and every possible warehouse location to every customer location. So then you have to rely on transport costs estimators.

Distances used are not actual road distances, but estimated road distances. Estimated road distance = as-the-crow-flies distances × distance circuity factor. An important reason for using as-the-crow-flies distances is because run-time would become too long when having to calculate (extremely) large amounts of road distances.
Note that from a cost perspective the fastest route is more relevant than the shortest route, as driver and truck cost are more time-related than distance-related. Google Maps retuns fastst route by default. Which is fine. But fastest route may depend on time of the day, day of the week, time of the year, roadworks, or vary between routes that are nearly-equivalent. So, even actual road distance might still be an estimation, to some extent.
Calculating Center of Gravity locations is not so much about distances as it is about pull forces and directions, finding the spot where all forces summed up together become zero, as explained in the section about the Single Center of Gravity location algorithm.
As distance estimation errors occur in each and every direction (one customer distance might be a bit too large, another a bit too small), errors tend to cancel each other out, and the Center of Gravity locations (indicative warehouse locations) become pretty accurate. The use of as-the-crow-flies distance is usually not an issue, if the road network is dense. Approximations are acceptable, and commonly used in network design software.

Broader supply chain network design perspective

Of course, transport costs are only a part of supply chain costs. The optimal number of warehouses and their locations are driven by many quantitative and qualitative factors such as (future) transport and warehousing space and labor rates, future demand (and supply), lead time requirements, inventory effects, supply chain risk/redundancy, contractual obligations, taxes, distance to parcel or logistics hubs, workforce availability/public transport connections, et cetera.

Nevertheless, it is common practice to run a Center of Gravity analysis to get a view on which logical warehouse locations to consider in a supply chain network design.

A completely different alternative approach

A completely different alternative approach is one in which a large set of possible warehouse locations is pre-generated, after which the k best ones are selected (see Combination - wikipedia).

Advantages are that the warehouse-customer cost matrix can be based on various transport rate structures and accurate road distances, and that it may take into account binary-fixed and variable warehousing costs as well.
Disadvantages are that (much) more input is required, that there is a risk that the set of possible warehouse locations does not contain optimal locations, and that runtime may become an issue.

See section Select best warehouses to apply this approach.

Add-ons

Greedy algorithm (default)

Centers of Gravity Calculator's default algorithm to handle capacity constraints is a superfast, greedy algorithm. Roughly described, it works like this.

Assign each customer (group) to a warehouse as if warehouse capacities are unlimited
Loop through all warehouse, and while the capacity of a warehouse is exceeded shift that customer (group) to that other warehouse that will increase the costs the least - and only if the customer (group) still fits in that warehouse if it is a warehouse earlier in the loop.

Greedy means that the solution may be sub-optimal, cost-wise. Capacities will not exceed 100% unless all set super-tight.

LP optimization = Vogel's approximation + MODI + Stepping stone

Alternatively, LP optimization can be activated.

If predefined warehouses have limited capacities, then a - so called - transportation problem arises. A transportation problem can be solved cost-optimally with the well-known Simplex method (LP). But it can be solved 50-100 times faster with a combination of Vogel's approximation + MODI + Stepping Stone method that generates a cost optimal solution while respecing capacity constraints. This combination has been implemented in Centers of Gravity Calculator. Detailed technical descriptions of each separate method can be found on the internet. For a demo case that shows the speed difference, see Transportation problem solver

However, this method may create a cost-optimal solution in which a single customer receives from multiple warehouses (which happens as soon as a warehouse is utilized 100.00%). But Center of Gravity Calculator assumes 'single sourcing'. So, if a customer receives from multiple warehouses in the LP solution, then this customer gets assigned 'in full' to the warehouse that supplies the most. This non-optimal, post-processing-step to avoid 'multiple-sourcing' may cause utilization to exceed 100%, often by a marginal percentage only, as it involves few customers only.

But it may become worse, especially if your case contains few customers only per warehouse, or some exceptionally large customers. For example, if capacity of Birmingham is adjusted to 41470 in the demo case (only 8.7% of original capacity, so only few customers can be assigned to it), and the move limit of Madrid is adjusted to 9999 (instead of 200), then Birmingham's utilization becomes 131%. This is caused by a single large customer, as shown below. In the underlying optimal LP solution customer 551 receives a quantity of 19480 from Birmingham and 13020 from WH03. As most comes from Birmingham, customer 551 is assigned to Birmingham. Birmingham capacity is now exceeded with 13020, making its utilization 131% (= 54760/41740) instead of 100.00% in LP solution. (If that customer is removed from Birmingham, then its utilization drops to 53%.)

		Underlying optimal LP Solution
Customer_ID	Demand	Birmingham	Madrid	WH03
336	3200	3200	0	0
337	2880	2880	0	0
551	32500	19480	0	13020	Assigned in full to Birmingham
555	12840	12840	0	0
557	3340	3340	0	0
Total	54760	41740	0	13020

Centers of Gravity calculator reports which customers cause warehouse utilization to exceed 100%, so you can decide if you want to assign those manually to another warehouse, or split them in two.

Also note that LP optimization ignores customer groups. But LP optimization will only get activated if warehouses still exceed capacity, after first having tried to solve capacity issues at group level.

As the LP optimization truly minimizes costs, the solution may show less illogical/inefficient/crossing customer flows.

LP optimization: continuous optimization or post-optimization only

You can choose between:

Continuous optimization:	LP continuously active, during whole process, for all runs. May increase total run time considerably.
Post‑optimization only:	LP applied for best run only, afterwards. It will not affect Center of Gravity locations, but may prevent illogical customer flows.

Simply run with and without LP optimization

Of course, you can simply run with and without LP optimization, and compare solutions. Usually, they do not differ that much.

Including supply-side

Suppliers come with supplier transport costs. Including suppliers means that Center of Gravity locations should move towards suppliers. But only as long as total transport costs decrease.

Besides customers, suppliers also pull Center of Gravity locations, but in 'opposite' direction, and usually relatively less hard, as supplier transport is relatively less expensive than customer transport. See section Parameters & Constraints - Transport rates.

Centers of Gravity Calculator supports 4 supply options that determine how supply flows are constructed.

Each supplier delivers to closest warehouse only
i
LP optimized supply flows - with auto-sized suppliers
i
Each supplier delivers to each warehouse
i
Each supplier delivers to each warehouse via closest warehouse
i

Each supplier causes (multiple) flows to warehouses. Each flow translates into a pull force (supplier or interdepot transport rate weighted) comparable to a customer pull force (customer tranport rate weighted) but pulling in 'opposite' direction. All forces are fed into the Single Center of Gravity algorithm that determines the position that minimizes total transport costs.

Besides, the customer assignment procedure of the Multiple Center of Gravity locations algorithm (step 2) takes into account (expected) supply-side transport costs when determining to which warehouse a customer gets assigned to.

Customer grouping (single-sourcing)

A customer group is used to force all customers within a group to be delivered by a single warehouse only (single-sourcing). The group is assigned to the warehouse that comes with the lowest sum of weighted distances for this group of customers. Underlying cost calculation is still done at individual customer level.

What group stands for, is up to you to decide. For example, a group can be used to group customers per country, so each country is delivered from a single warehouse only.

A fixed customer-warehouse assignment takes precedence over a customer group. Warehouse capacity limits may cause a customer group to 'break'.

Select best warehouses

Select best warehouses goal value

The "Select best warehouses" routine selects those warehouses from a predefinced set that minimize the goal value, being the sum of customer plus supplier transport costs + warehouse binary‑fixed costs + warehouse variable handling costs.

Customer transport cost if customer c is delivered from warehouse w = demand qty of customer c × as-the-crow-flies-distance between warehouse w and customer c × distance circuity factor × customer transport rate (= transport cost/distance unit/qty unit).
- Distance is based on coordinates of the predefined warehouse. (So, not the coordinates of Center of Gravity in case a warehouse was allowed to move. You may want to copy Center of Gravity coordinates back into your predefined warehouse table.)
- Distance includes detour via a border crossing point if applicable.
- You can adjust cost input via section
  "Override warehouse-customer transport costs input (optional)"
  .
Supplier transport cost (optional) for each supplier s = supply qty of supplier s × as-the-crow-flies-distance between warehouse w and supplier s × distance circuity factor × supplier transport rate * warehouse qty/total demand. This is conform supply option 3.
Warehouse binary-fixed costs (optional) of warehouse w are its fixed costs and apply only if warehouse w is opened.
Warehouse variable handling costs (optional) if customer c is delivered from warehouse w = productsum(customer c's whs quantities × warehouse w's variable rates). Note that customer field Demand (= transport qty) is irrelevant in this formula.

How to activate constraints

Constraint	Supported	How to activate
Predefined warehouses	yes	Always active (regardles constraint setting).
Fixed customer-warehouse links	yes	By checking constraint in section Parameters & constraints - Constraints
Capacity limits	yes	By using Greedy solver or LP solver (not by checking constraint).
Customer groups	no
Suppliers	yes (partly)	By checking constraint in section Parameters & constraints - Constraints. Supply option 3 is activated in background. Other options not supported.
Border crossing points	yes	By checking constraint in section Parameters & constraints - Constraints.

Select best warehouses algorithms - Summary

The selection of best warehouses is handled by a combination of two separate procedures.

A procedure that generates valid warehouse combinations = open/close decision for each warehouse, with valid meaning that the number of warehouses to select (per group) is respected and each must-be-included warehouse is indeed opened.
- K-means Clustering solver and Superfast Selection solver generate only "promising" combinations, so solve fast but may bring sub-optimal open/close decisions, though this rarely happens.
- Enumeration solver generates all possible combinations, so solves relatively slow, but optimally.
A procedure that assigns each customer to an open warehouse (given open/close decisions) while respecting warehouse capacity limits and minimizing costs.
- Ignore / not applicable: each customer can simply be assigned to the open warehouse that brings the lowest costs.
If capacity constraints are applicable, then a transportation problem must be solved:
- Greedy solver solves fast, but may produce sub-optimal customer assignments.
- LP solver solves relatively slow, but optimally.

Those two procedures have each 3 (solver) options, resulting in 3×3 combinations you can select from.

Best is to use Enumeration Solver (plus LP Solver if capacity limits still apply in future). If the run takes too long, then stop it and

use K-means Clustering Solver with a limited number of runs or use Superfast Selection Solver to generate warehouse combinations
and/or
use Greedy Solver to handle warehouse capacity limits. Especially LP Solver runs slow.

Warehouse combinations generation

K-means Clustering Solver

Problem-specific solver. Performs a K-means-Clustering-like procedure.

Single run

Initialisation: for each group, select a random warehouse within the group and open it, until the amount of warehouses to be opened for this group has been reached.
As long as costs still decrease, keep looping through all open warehouses, and
- Close this open warehouse.
- Find the closed warehouse within this group that will decrease costs the most (or increase the least), if it is opened.
- Open that warehouse. (It might be the same warehouse as the one that just has been closed.)

Multiple runs

Perform the single run procedure multiple times (each starting with different randomly selected warehouses).
Often, this best solution will have been found in several times of those runs. (Just like the Center of Gravity calculation procedure that will find the same solution several times doing N runs.).

This solver found optimal solutions in almost all tests, or else 0-2% worse. The chance of missing out the optimum becomes higher, if the number of runs is set lower.

Speed is usually in the same ballpark as Superfast Selection Solver. But if a high number of warehouses and (relatively) high number to be opened is applicable, then the limit on number of runs makes it finish faster than Superfast Selection Solver.

Superfast Selection Solver

Problem-specific solver (custom-built). Roughly described, it works like this:

Open 1 by 1 the next best warehouse, until the amount of opened warehouses exceeds the amount to be opened with X. Then close X worst warehouses 1 by 1.
Swap N open warehouses with N closed ones (within 1 to N groups). Run for all possible swaps, enumeration-wise. Revert swap if solution got worse.

For example, it solved a 145 million combinations case optimally in 0.1 second, versus 5 minutes Enumeration Solver. It found optimal solutions in almost all tests, or else 0-2% worse.

Enumeration Solver

Generates all possible valid warehouse combinations, so it will always find the optimum. But the higher the number of permutations, the longer the run time. It may even take hours. Reason to develop the other two solvers, that run quickly and will almost always find the optimal solution, however without guarantee. Because all combinations are generated and evaluated, the Enumeration Solver can produce a "top 20" list of warehouse combinations.

Warehouse capacity limits handling

Ignore / not applicable

If warehouse capacities are to be ignored / not applicable then each customer can simply be assigned to the open warehouse that comes with the lowest costs. (There is no need for a solver.)

Greedy Solver

Greedy Solver runs a fast, greedy algorithm that works likes this:

Assign customers to open warehouses as if warehouse capacities are unlimited
Loop through all open warehouse, and while warehouse capacity is exceeded shift that customer to that other open warehouse that will increase the costs the least (and only if the customer still fits in that warehouse if it is a warehouse earlier in the loop).

The solution may become sub-optimal.

LP Solver

If open warehouses have limited capacity, a transportation problem arises. Such problem can be solved optimally with the Simplex method (LP). It can also be solved optimally, and 50-100 times faster, with a combination of Vogel + MODI + Stepping Stone method, therefor this combination has been implemented. Detailed descriptions of each separate method (standard as such) can be found on the internet. For a demo case that shows the speed difference, see Transportation problem solver

Note that if a customer receives from two warehouses in the optimal transportation problem solution (in which warehouse utilization never exceeds 100%), then this customer gets assigned in full to the warehouse that supplies the most. This non-optimal, post-step may cause utilization to exceed 100%, usually a little only as it will involve few customers only.

Download Excel template

Or click button 'Import | Export' above each input table, enter data (tab or semicolon separated) in textarea, click button again.

You have not logged in. Changes to input table Customers are ignored. Create an account to run with your own customers data. In this demo mode you can adjust all other input tables, parameters & constraints, and run the software. See section Demo guide.

Customers

1	41.1533	20.1683	3430		AL		172
2	46.6333	14.3128	4270		AT		214
3	46.9894	15.5817	4200		AT		210
4	47.252	11.42	4690		AT		235
5	47.5784	12.7305	3550		AT		178
6	47.9911	13.8878	11530		AT		577
7	48.0671	13.1775	3300		AT		165
8	48.2025	16.4186	4600		AT		230
9	48.2049	16.2734	6100		AT		305
10	48.2297	16.4692	2620		AT		131
11	48.3656	13.9771	3340		AT		167
12	43.3319	17.7513	5900		BA		295
13	43.8784	18.3593	2460		BA		123
14	50.2458	4.8519	2720		BE		136
15	50.3753	4.3819	3420		BE		171
16	50.4172	4.447	3220		BE		161
17	50.4555	3.9662	3500		BE		175
18	50.461	4.8498	2940		BE		147
19	50.4797	4.1698	4600		BE		230
20	50.588	5.8602	2840		BE		142
21	50.5882	4.3253	3120		BE		156
22	50.59	3.3922	2960		BE		148
23	50.5942	5.4331	3200		BE		160
24	50.5978	6.1325	3280		BE		164
25	50.6473	5.5832	6550		BE		328
26	50.7237	4.5543	3140		BE		157
27	50.7416	3.2238	2420		BE		121
28	50.7886	4.8612	2980		BE		149
29	50.8137	5.2089	2460		BE		123
30	50.8185	3.2785	2440		BE		122
31	50.849	4.4297	5360		BE		268
32	50.8521	5.6878	3520		BE		176
33	50.8832	4.3551	9200		BE		460
34	50.8903	4.7381	8200		BE		410
35	50.8989	4.391	3200		BE		160
36	50.9316	5.3368	4880		BE		244
37	50.9406	3.1181	6900		BE		345
38	50.9439	4.0442	2920		BE		146
39	50.9676	5.4886	3160		BE		158
40	50.9836	4.821	3280		BE		164
41	50.9894	3.5106	3380		BE		169
42	51.0077	3.368	3080		BE		154
43	51.0293	4.094	2540		BE		127
44	51.0353	4.4665	3280		BE		164
45	51.0532	3.6307	9480		BE		474
46	51.12	4.5734	2780		BE		139
47	51.1462	4.3383	4120		BE		206
48	51.16	4.8277	3440		BE		172
49	51.1695	4.1647	4240		BE		212
50	51.2062	5.3798	2480		BE		124
51	51.212	4.4686	3670		BE		184
52	51.2147	2.919	3280		BE		164
53	51.2211	4.3664	3130		BE		157
54	51.2329	3.2118	3610		BE		181
55	51.2384	5.1949	3280		BE		164
56	51.2706	4.5028	4450		BE		223
57	51.3369	3.3234	3850		BE		193
58	51.3531	4.771	2780		BE		139
59	42.687	23.3273	4660		BG		233
60	42.7266	23.3158	18640		BG		932
61	53.1232	26.5796	7000		BY		350
62	46.0338	8.9719	2940		CH		147
63	46.1258	7.2274	2620		CH		131
64	46.3407	6.1974	3670		CH		184
65	46.5902	6.6716	4600		CH		230
66	46.9374	7.4345	2700		CH		135
67	46.96	9.0552	5800		CH		290
68	46.9652	8.2183	2640		CH		132
69	47.1921	7.5615	3360		CH		168
70	47.3092	7.6031	4560		CH		228
71	47.3475	9.4081	2520		CH		126
72	47.3776	8.5381	5900		CH		295
73	47.4036	8.0374	2820		CH		141
74	47.5567	7.5979	3730		CH		187
75	47.5777	9.0899	3340		CH		167
76	35.0309	33.1374	5960		CY		298
77	49.1812	16.6769	2520		CZ		126
78	49.8397	18.3248	2800		CZ		140
79	49.886	16.471	12610		CZ		631
80	49.8879	14.6555	5000		CZ		250
81	50.0446	14.5616	7790		CZ		390
82	50.1368	14.4601	3000		CZ		150
83	50.224	14.0776	2840		CZ		142
84	47.7487	10.464	2420		DE		121
85	47.8441	12.2004	4060		DE		203
86	47.8484	11.2849	4480		DE		224
87	47.8948	8.0721	3700		DE		185
88	47.9083	9.5446	2560		DE		128
89	47.9821	8.6801	3240		DE		162
90	48.125	11.6005	7370		DE		369
91	48.1827	11.5495	6400		DE		320
92	48.3254	8.9595	2620		DE		131
93	48.3371	10.8477	4870		DE		244
94	48.4342	12.4117	3240		DE		162
95	48.4461	10.1133	3880		DE		194
96	48.4891	11.5684	3560		DE		178
97	48.7264	13.0673	2800		DE		140
98	48.7557	9.1817	3520		DE		176
99	48.7875	9.8422	4630		DE		232
100	48.8183	9.3529	4300		DE		215
101	48.9686	8.2909	6120		DE		306
102	49.0929	12.3527	7240		DE		362
103	49.2842	9.5062	4300		DE		215
104	49.341	7.1115	3370		DE		169
105	49.3578	10.9545	5520		DE		276
106	49.4069	10.9999	5080		DE		254
107	49.425	11.961	3380		DE		169
108	49.4288	8.9278	2440		DE		122
109	49.4619	8.5093	4750		DE		238
110	49.5131	7.989	4630		DE		232
111	49.7477	8.7663	3260		DE		163
112	49.8457	7.7122	5360		DE		268
113	49.9622	10.0166	15520		DE		776
114	50.0564	9.0616	8740		DE		437
115	50.0891	11.9207	2960		DE		148
116	50.113	11.0084	2500		DE		125
117	50.1152	8.6563	4800		DE		240
118	50.2084	8.1902	4600		DE		230
119	50.3019	7.5317	4840		DE		242
120	50.5261	10.6956	3180		DE		159
121	50.5498	12.4075	2980		DE		149
122	50.6064	6.9274	4900		DE		245
123	50.6781	9.7457	2800		DE		140
124	50.7328	11.7088	4840		DE		242
125	50.7774	13.1232	3940		DE		197
126	50.7986	8.6866	3480		DE		174
127	50.8085	6.2872	4160		DE		208
128	50.9001	6.7587	5900		DE		295
129	50.9315	7.9958	2860		DE		143
130	50.9609	7.3826	5360		DE		268
131	51.1753	6.4773	3520		DE		176
132	51.1773	13.7897	5600		DE		280
133	51.1868	10.8303	4840		DE		242
134	51.2019	14.5207	3280		DE		164
135	51.214	6.8136	4280		DE		214
136	51.2161	9.1977	4360		DE		218
137	51.3013	7.5981	3100		DE		155
138	51.3441	12.9357	3840		DE		192
139	51.4942	8.0154	4630		DE		232
140	51.4967	12.0202	4360		DE		218
141	51.5245	9.9208	3520		DE		176
142	51.532	7.3695	5800		DE		290
143	51.5434	6.3875	7500		DE		375
144	51.5557	7.1372	8980		DE		449
145	51.717	6.5946	2400		DE		120
146	51.7568	14.0347	2780		DE		139
147	51.7898	8.6824	5400		DE		270
148	52.1235	7.4302	4200		DE		210
149	52.1595	8.8232	2760		DE		138
150	52.1875	10.683	4420		DE		221
151	52.2455	14.0142	3250		DE		163
152	52.3179	12.7886	2740		DE		137
153	52.3235	10.099	4060		DE		203
154	52.3851	11.6495	2560		DE		128
155	52.4013	9.6957	5000		DE		250
156	52.4594	13.4892	3220		DE		161
157	52.5117	13.4079	3370		DE		169
158	52.5316	7.7278	5700		DE		285
159	52.5878	13.3392	8560		DE		428
160	52.9493	13.1648	3280		DE		164
161	53.0899	8.8441	3060		DE		153
162	53.2082	9.3574	2500		DE		125
163	53.2934	7.7666	4900		DE		245
164	53.3574	11.4806	3360		DE		168
165	53.456	9.808	3520		DE		176
166	53.5625	9.9914	4540		DE		227
167	53.59	13.3337	3060		DE		153
168	53.6327	10.0874	7780		DE		389
169	53.955	10.956	6050		DE		303
170	54.0318	12.4273	4570		DE		229
171	54.3167	9.8918	3550		DE		178
172	54.3256	9.1257	3340		DE		167
173	55.6147	12.3188	7300		DK		365
174	55.6689	12.55	3250		DK		163
175	59.2217	24.187	6810		EE		341
176	35.2931	-2.9492	2420	Madrid	ES	Iberia	121
177	35.8917	-5.3305	3000	Madrid	ES	Iberia	150
178	36.1408	-5.3536	2440	Madrid	ES	Iberia	122
179	36.5295	-5.7554	11200	Madrid	ES	Iberia	560
180	36.8099	-4.6632	12760	Madrid	ES	Iberia	638
181	37.3103	-2.3428	6040	Madrid	ES	Iberia	302
182	37.3968	-3.23	11000	Madrid	ES	Iberia	550
183	37.5034	-6.7991	5800	Madrid	ES	Iberia	290
184	37.5264	-5.5807	14280	Madrid	ES	Iberia	714
185	37.9667	-3.3442	5800	Madrid	ES	Iberia	290
186	37.9718	-4.7691	7300	Madrid	ES	Iberia	365
187	38.073	-1.4702	12160	Madrid	ES	Iberia	608
188	38.3649	-0.4237	18480	Madrid	ES	Iberia	924
189	38.7208	-5.9683	8840	Madrid	ES	Iberia	442
190	38.7359	-1.865	6000	Madrid	ES	Iberia	300
191	38.979	-3.7994	5700	Madrid	ES	Iberia	285
192	39.3672	-0.7361	25920	Madrid	ES	Iberia	1296
193	39.6178	2.9067	6920	Madrid	ES	Iberia	346
194	39.7771	-6.2981	4800	Madrid	ES	Iberia	240
195	39.8069	-4.1161	5000	Madrid	ES	Iberia	250
196	39.9488	-2.1217	2880	Madrid	ES	Iberia	144
197	40.2532	-0.1381	5300	Madrid	ES	Iberia	265
198	40.5295	-3.7669	77840	Madrid	ES	Iberia	3892
199	40.6066	-0.7299	3320	Madrid	ES	Iberia	166
200	40.6391	-4.9062	2540	Madrid	ES	Iberia	127
201	40.7418	-2.5093	5250	Madrid	ES	Iberia	263
202	40.7762	-6.0096	4690	Madrid	ES	Iberia	235
203	41.0536	0.9218	5000	Madrid	ES	Iberia	250
204	41.1183	-3.9573	3340	Madrid	ES	Iberia	167
205	41.693	-6.1506	3160	Madrid	ES	Iberia	158
206	41.7153	-4.6488	4600	Madrid	ES	Iberia	230
207	41.747	2.0723	57700	Madrid	ES	Iberia	2885
208	41.8449	-0.8662	12800	Madrid	ES	Iberia	640
209	42.0671	1.0973	4720	Madrid	ES	Iberia	236
210	42.0734	2.5063	9820	Madrid	ES	Iberia	491
211	42.1097	-0.0637	2960	Madrid	ES	Iberia	148
212	42.1882	-7.5237	5000	Madrid	ES	Iberia	250
213	42.2716	-2.4001	6040	Madrid	ES	Iberia	302
214	42.3387	-3.5794	4520	Madrid	ES	Iberia	226
215	42.3728	-8.3689	8020	Madrid	ES	Iberia	401
216	42.4117	-4.4814	3480	Madrid	ES	Iberia	174
217	42.6085	-1.6054	11160	Madrid	ES	Iberia	558
218	42.63	-5.8945	6850	Madrid	ES	Iberia	343
219	42.8386	-2.7489	4840	Madrid	ES	Iberia	242
220	43.0522	-7.3717	5800	Madrid	ES	Iberia	290
221	43.1264	-3.9952	6100	Madrid	ES	Iberia	305
222	43.1369	-8.4757	15040	Madrid	ES	Iberia	752
223	43.1534	-2.1568	4600	Madrid	ES	Iberia	230
224	43.2274	-2.9412	7930	Madrid	ES	Iberia	397
225	43.254	-5.8255	7320	Madrid	ES	Iberia	366
226	60.1616	24.4844	27040		FI		1352
227	42.17	9.0596	2400		FR		120
228	42.6311	2.4388	3520		FR		176
229	43.0339	2.4754	3340		FR		167
230	43.1379	0.164	2700		FR		135
231	43.1855	-0.8778	7150		FR		358
232	43.3118	1.2479	11440		FR		572
233	43.4189	6.295	5440		FR		272
234	43.5424	5.0287	7510		FR		376
235	43.5903	3.363	6760		FR		338
236	43.7836	2.2493	2440		FR		122
237	43.9002	7.1858	10900		FR		545
238	43.951	4.0774	3520		FR		176
239	43.9988	5.216	3790		FR		190
240	44.0016	-0.6842	3300		FR		165
241	44.0659	1.3696	2520		FR		126
242	44.2961	2.6416	2920		FR		146
243	44.3561	0.4815	3240		FR		162
244	44.7237	5.2495	2480		FR		124
245	44.8705	-0.4731	15430		FR		772
246	45.1283	0.7113	2760		FR		138
247	45.2843	5.5725	4600		FR		230
248	45.3313	1.8835	3520		FR		176
249	45.4821	6.4061	2960		FR		148
250	45.6645	0.251	2400		FR		120
251	45.707	-0.6045	3160		FR		158
252	45.7517	4.2417	5480		FR		274
253	45.7648	3.1922	7250		FR		363
254	45.8667	4.7158	8040		FR		402
255	45.9005	1.2659	3640		FR		182
256	46.0596	5.4672	2520		FR		126
257	46.0701	6.4298	3160		FR		158
258	46.3615	3.1238	2760		FR		138
259	46.475	-0.3132	3220		FR		161
260	46.5871	0.5476	3140		FR		157
261	46.6602	4.5432	2800		FR		140
262	46.6798	-1.3315	3400		FR		170
263	46.7656	5.7251	3080		FR		154
264	46.8094	1.5497	3520		FR		176
265	47.0122	2.4347	2680		FR		134
266	47.0558	6.3897	3080		FR		154
267	47.2181	0.7264	4360		FR		218
268	47.3432	-1.7285	6680		FR		334
269	47.3935	-0.5369	3730		FR		187
270	47.4617	4.8003	4840		FR		242
271	47.6285	6.9587	2760		FR		138
272	47.6607	1.4362	2400		FR		120
273	47.8153	-2.8675	3430		FR		172
274	47.8505	3.6084	3460		FR		173
275	47.867	7.2316	3280		FR		164
276	47.9159	2.2866	4090		FR		205
277	48.0176	0.2363	4330		FR		217
278	48.142	-0.6382	3580		FR		179
279	48.1548	6.3145	3240		FR		162
280	48.1559	-1.6391	9960		FR		498
281	48.2314	-4.069	5840		FR		292
282	48.3211	4.147	3440		FR		172
283	48.4526	-2.7761	4090		FR		205
284	48.4557	1.3917	3020		FR		151
285	48.5286	2.2616	7000		FR		350
286	48.5721	0.0739	2920		FR		146
287	48.5815	7.5971	6350		FR		318
288	48.6138	2.9691	5050		FR		253
289	48.763	1.8421	7840		FR		392
290	48.7731	2.4515	6700		FR		335
291	48.8409	2.235	34540		FR		1727
292	48.8608	2.3273	8740		FR		437
293	48.9057	2.4519	10500		FR		525
294	48.9459	5.9818	5760		FR		288
295	48.9684	4.2424	6850		FR		343
296	49.0086	5.3814	3340		FR		167
297	49.0145	6.7953	3250		FR		163
298	49.0663	1.0732	2500		FR		125
299	49.0729	2.1066	6200		FR		310
300	49.0903	-1.3387	3580		FR		179
301	49.0927	-0.3537	3910		FR		196
302	49.4129	2.4279	4600		FR		230
303	49.4657	3.6021	3400		FR		170
304	49.6714	0.9306	5650		FR		283
305	49.9746	2.3059	2980		FR		149
306	50.531	3.1643	14560		FR		728
307	50.5375	2.3571	4750		FR		238
308	41.9186	44.3628	2940		GE		147
309	35.3225	25.0298	3490		GR		175
310	37.5319	23.3407	4760		GR		238
311	37.8935	23.7765	2640		GR		132
312	37.938	23.7179	3360		GR		168
313	37.9867	23.7207	6160		GR		308
314	37.9894	23.7376	9750		GR		488
315	38.0038	23.636	6160		GR		308
316	38.0182	23.8641	4640		GR		232
317	38.0829	23.8314	3300		GR		165
318	38.2264	21.8153	3340		GR		167
319	39.5679	20.8495	3520		GR		176
320	39.5716	22.4681	3020		GR		151
321	40.5811	23.0107	3480		GR		174
322	40.6752	22.9383	3340		GR		167
323	40.6986	23.2985	3700		GR		185
324	40.7725	22.9897	3580		GR		179
325	45.7921	16.0251	4600		HR		230
326	47.4768	19.1392	6220		HU		311
327	47.5492	19.0032	3700		HU		185
328	47.6316	19.3811	4360		HU		218
329	51.8969	-8.4863	7960	Birmingham	IE	IE	398
330	52.2523	-7.1272	3520	Birmingham	IE	IE	176
331	52.2713	-9.6999	2800	Birmingham	IE	IE	140
332	52.6541	-7.2448	2640	Birmingham	IE	IE	132
333	52.668	-8.6305	4390	Birmingham	IE	IE	220
334	53.2706	-9.0567	4150	Birmingham	IE	IE	208
335	53.3498	-6.2603	25990	Birmingham	IE	IE	1300
336	53.7179	-6.3561	3200	Birmingham	IE	IE	160
337	54.2766	-8.4761	2880	Birmingham	IE	IE	144
338	36.9078	14.6821	2600		IT	IT	130
339	37.0372	15.0897	3520		IT	IT	176
340	37.4942	14.8494	9080		IT	IT	454
341	37.8751	12.7816	3260		IT	IT	163
342	37.8969	13.6601	3920		IT	IT	196
343	38.0551	14.913	4600		IT	IT	230
344	38.3917	16.1125	2860		IT	IT	143
345	38.9736	16.6502	4560		IT	IT	228
346	39.5872	16.3783	3370		IT	IT	169
347	39.6113	9.0092	6530		IT	IT	327
348	40.164	18.1712	3960		IT	IT	198
349	40.2283	9.0985	2460		IT	IT	123
350	40.4187	15.151	4510		IT	IT	226
351	40.5298	15.8913	3120		IT	IT	156
352	40.5583	17.2642	4090		IT	IT	205
353	40.6411	17.7568	2680		IT	IT	134
354	40.7877	14.4594	9910		IT	IT	496
355	40.8184	8.9029	4390		IT	IT	220
356	40.99	15.0829	2860		IT	IT	143
357	41.0328	16.6632	7930		IT	IT	397
358	41.2059	14.142	2420		IT	IT	121
359	41.2296	14.7779	2540		IT	IT	127
360	41.4621	13.1919	2460		IT	IT	123
361	41.5281	15.5969	4280		IT	IT	214
362	41.6198	13.4899	3320		IT	IT	166
363	41.7225	14.5424	3380		IT	IT	169
364	41.8548	12.5014	27680		IT	IT	1384
365	42.1057	14.4617	3480		IT	IT	174
366	42.1738	13.6418	2820		IT	IT	141
367	42.3187	14.0083	3240		IT	IT	162
368	42.5055	11.9724	3500		IT	IT	175
369	42.6706	12.4041	2820		IT	IT	141
370	42.6787	13.7479	3600		IT	IT	180
371	42.7657	11.2306	2960		IT	IT	148
372	43.0065	13.5481	3480		IT	IT	174
373	43.1048	12.6	4440		IT	IT	222
374	43.163	11.4531	4060		IT	IT	203
375	43.1698	13.2772	3360		IT	IT	168
376	43.4761	10.622	9560		IT	IT	478
377	43.4872	13.2056	2460		IT	IT	123
378	43.5243	11.919	3520		IT	IT	176
379	43.6923	12.6369	3120		IT	IT	156
380	43.8281	11.2139	5400		IT	IT	270
381	43.9536	7.8298	3200		IT	IT	160
382	44.0325	12.1991	7100		IT	IT	355
383	44.2232	9.7594	2920		IT	IT	146
384	44.2582	8.3252	3380		IT	IT	169
385	44.3448	11.9549	4000		IT	IT	200
386	44.442	11.2898	5760		IT	IT	288
387	44.451	9.0572	4640		IT	IT	232
388	44.4667	7.5591	4600		IT	IT	230
389	44.542	10.9065	4330		IT	IT	217
390	44.6048	10.5067	3670		IT	IT	184
391	44.6927	9.9567	3910		IT	IT	196
392	44.7542	11.7822	2560		IT	IT	128
393	44.8288	9.5974	3380		IT	IT	169
394	44.8353	8.1888	2720		IT	IT	136
395	44.8537	8.647	3100		IT	IT	155
396	44.9671	11.8537	3600		IT	IT	180
397	45.0264	9.0348	4850		IT	IT	243
398	45.1356	10.8501	3260		IT	IT	163
399	45.1736	7.379	12700		IT	IT	635
400	45.1926	9.8936	4270		IT	IT	214
401	45.3923	11.7787	6550		IT	IT	328
402	45.422	11.0567	8620		IT	IT	431
403	45.4588	12.4754	4870		IT	IT	244
404	45.4808	9.1197	23500		IT	IT	1175
405	45.5522	8.196	4540		IT	IT	227
406	45.6177	11.4649	3820		IT	IT	191
407	45.7362	9.8421	4720		IT	IT	236
408	45.7573	10.3516	7930		IT	IT	397
409	45.7798	13.6245	3120		IT	IT	156
410	45.8094	12.2113	3940		IT	IT	197
411	45.8571	8.7999	4840		IT	IT	242
412	45.8655	8.3388	4760		IT	IT	238
413	45.9305	9.1572	3610		IT	IT	181
414	46.1299	11.202	2760		IT	IT	138
415	46.1357	9.9182	3180		IT	IT	159
416	46.1933	13.0143	7150		IT	IT	358
417	46.2907	12.1646	2780		IT	IT	139
418	46.6645	11.3893	3380		IT	IT	169
419	54.8917	23.9221	2620		LT		131
420	55.1508	22.955	3220		LT		161
421	55.1694	23.8813	4750		LT		238
422	55.9302	25.3559	3480		LT		174
423	56.9874	24.1059	4750		LV		238
424	49.8153	6.1296	3940		LX		197
425	47.0252	28.8509	3440		MD		172
426	41.9876	21.4902	3480		MK		174
427	35.9375	14.3754	6120		MT	IT	306
428	50.8866	5.9619	3440		NL		172
429	51.1076	5.8626	5720		NL		286
430	51.3712	6.1711	2420		NL		121
431	51.4204	5.401	3420		NL		171
432	51.4375	5.4817	3160		NL		158
433	51.4924	4.0378	3120		NL		156
434	51.5886	4.7813	2900		NL		145
435	51.598	5.0923	2440		NL		122
436	51.686	5.3027	13100		NL		655
437	51.8064	5.2585	12640		NL		632
438	51.8471	5.8635	4450		NL		223
439	51.9176	4.4887	3790		NL		190
440	51.9805	5.9059	2680		NL		134
441	52.023	5.0361	4920		NL		246
442	52.024	5.557	2840		NL		142
443	52.0787	4.3172	4870		NL		244
444	52.0879	5.1162	2680		NL		134
445	52.1326	5.2913	13740		NL		687
446	52.1518	5.0451	2700		NL		135
447	52.1554	4.4916	2760		NL		138
448	52.2208	6.8993	2800		NL		140
449	52.2244	5.1764	7900		NL		395
450	52.3081	4.9442	4600		NL		230
451	52.3728	4.9067	6700		NL		335
452	52.5113	6.0903	2760		NL		138
453	52.6307	4.7505	3120		NL		156
454	52.7308	6.4723	5740		NL		287
455	52.7847	6.9009	2860		NL		143
456	53.218	6.5707	2620		NL		131
457	59.7326	10.7913	6450		NO		323
458	59.9342	10.7994	7780		NO		389
459	50.218	19.002	4160		PL		208
460	51.8135	19.445	10450		PL		523
461	52.1077	21.0815	29210		PL		1461
462	52.2658	17.6252	2860		PL		143
463	52.282	21.0386	3000		PL		150
464	37.0213	-7.9338	2680	Madrid	PT	Iberia	134
465	38.525	-8.8986	6120	Madrid	PT	Iberia	306
466	38.7367	-9.1339	19060	Madrid	PT	Iberia	953
467	39.2322	-8.686	2620	Madrid	PT	Iberia	131
468	39.3999	-8.2245	2720	Madrid	PT	Iberia	136
469	39.7425	-8.8064	2760	Madrid	PT	Iberia	138
470	40.2108	-8.4221	5640	Madrid	PT	Iberia	282
471	40.6266	-8.6479	2840	Madrid	PT	Iberia	142
472	40.6447	-7.3044	2780	Madrid	PT	Iberia	139
473	40.6592	-7.9147	3460	Madrid	PT	Iberia	173
474	41.1515	-8.6051	10180	Madrid	PT	Iberia	509
475	41.5453	-8.4199	3940	Madrid	PT	Iberia	197
476	44.3902	26.1161	26080		RO		1304
477	44.4403	26.0342	5960		RO		298
478	44.4531	26.1286	7580		RO		379
479	44.5207	26.1358	9080		RO		454
480	45.7945	25.2265	3040		RO		152
481	45.9303	24.3044	18170		RO		909
482	44.0165	21.0059	20720		RS		1036
483	44.8262	20.4571	9300		RS		465
484	52.2320	21.0158	145900		PL		7295
485	57.387	12.2792	22750		SE		1138
486	46.0563	14.5047	10090		SI		505
487	46.5565	15.6499	4900		SI		245
488	48.0498	18.2532	3140		SK		157
489	48.0665	17.155	3490		SK		175
490	48.1976	17.1478	4570		SK		229
491	48.674	20.7849	3120		SK		156
492	48.7536	19.2941	5640		SK		282
493	49.1356	21.1867	3190		SK		160
494	36.8995	30.7059	2780		TR		139
495	37.0369	37.2875	2620		TR		131
496	37.2867	35.7333	3040		TR		152
497	37.8698	32.3994	4320		TR		216
498	38.489	27.2261	22280		TR		1114
499	38.7097	35.8409	6160		TR		308
500	39.6242	37.3257	2440		TR		122
501	39.9191	32.753	8560		TR		428
502	40.1914	29.0555	2460		TR		123
503	40.3689	36.5431	3260		TR		163
504	40.9071	39.8005	2480		TR		124
505	41.0105	29.0025	55600		TR		2780
506	41.2007	36.1377	2900		TR		145
507	48.3794	31.1656	24400		UA		1220
508	48.4371	34.9045	4900		UA		245
509	50.4424	30.5221	7780		UA		389
510	49.2044	-2.137	2860	Birmingham	UK	UK	143
511	50.4536	-3.7099	2960	Birmingham	UK	UK	148
512	50.4769	-4.4283	4040	Birmingham	UK	UK	202
513	50.8499	-0.9419	19360	Birmingham	UK	UK	968
514	50.8647	-0.1239	2940	Birmingham	UK	UK	147
515	50.9034	-3.7971	2820	Birmingham	UK	UK	141
516	50.9553	-1.3683	4090	Birmingham	UK	UK	205
517	51.0893	-0.2995	3180	Birmingham	UK	UK	159
518	51.097	0.5201	2840	Birmingham	UK	UK	142
519	51.1703	-0.7554	6450	Birmingham	UK	UK	323
520	51.3326	-0.3905	3440	Birmingham	UK	UK	172
521	51.3775	0.0853	9240	Birmingham	UK	UK	462
522	51.4156	0.2693	2480	Birmingham	UK	UK	124
523	51.4249	-2.6571	3160	Birmingham	UK	UK	158
524	51.4389	-0.4269	3460	Birmingham	UK	UK	173
525	51.4477	0.0123	2800	Birmingham	UK	UK	140
526	51.4535	-0.1893	4960	Birmingham	UK	UK	248
527	51.5144	-0.1785	3140	Birmingham	UK	UK	157
528	51.5168	-0.1805	3020	Birmingham	UK	UK	151
529	51.5313	-0.0258	3100	Birmingham	UK	UK	155
530	51.5491	-0.3638	3580	Birmingham	UK	UK	179
531	51.5559	-0.1155	2600	Birmingham	UK	UK	130
532	51.5678	0.2725	3280	Birmingham	UK	UK	164
533	51.5699	-0.1035	3280	Birmingham	UK	UK	164
534	51.5805	-0.2073	3910	Birmingham	UK	UK	196
535	51.5844	-0.3479	4120	Birmingham	UK	UK	206
536	51.6029	0.0771	3360	Birmingham	UK	UK	168
537	51.622	-3.3992	8700	Birmingham	UK	UK	435
538	51.6633	-0.3879	2740	Birmingham	UK	UK	137
539	51.7784	-0.3156	8840	Birmingham	UK	UK	442
540	51.7924	0.4802	3440	Birmingham	UK	UK	172
541	51.8347	-1.2995	3910	Birmingham	UK	UK	196
542	51.8808	-4.4348	2780	Birmingham	UK	UK	139
543	51.9679	0.8749	2560	Birmingham	UK	UK	128
544	51.9687	-0.153	2660	Birmingham	UK	UK	133
545	52.0941	-0.6851	48300	Birmingham	UK	UK	2415
546	52.2252	0.2378	2400	Birmingham	UK	UK	120
547	52.273	-0.8661	3060	Birmingham	UK	UK	153
548	52.3304	-1.4821	2800	Birmingham	UK	UK	140
549	52.3767	-1.8368	3520	Birmingham	UK	UK	176
550	52.4658	-1.9879	3420	Birmingham	UK	UK	171
551	52.4664	-1.8886	32500	Birmingham	UK	UK	1625
552	52.6388	-1.0236	4720	Birmingham	UK	UK	236
553	52.6639	1.2331	14430	Birmingham	UK	UK	722
554	52.7268	0.128	3260	Birmingham	UK	UK	163
555	52.9482	-3.8144	12840	Birmingham	UK	UK	642
556	53.0054	-0.801	3100	Birmingham	UK	UK	155
557	53.2034	-3.0312	3340	Birmingham	UK	UK	167
558	53.2582	-1.4047	2800	Birmingham	UK	UK	140
559	53.3608	-1.4674	2600	Birmingham	UK	UK	130
560	53.374	-2.5496	2720	Birmingham	UK	UK	136
561	53.4236	-1.4576	3260	Birmingham	UK	UK	163
562	53.4301	-1.4092	2420	Birmingham	UK	UK	121
563	53.4669	-2.2238	3460	Birmingham	UK	UK	173
564	53.51	-2.2427	3280	Birmingham	UK	UK	164
565	53.5178	-1.4647	2940	Birmingham	UK	UK	147
566	53.5278	-0.6153	2440	Birmingham	UK	UK	122
567	53.5478	-2.6538	3400	Birmingham	UK	UK	170
568	53.6686	-1.458	11700	Birmingham	UK	UK	585
569	53.7526	-0.3007	2760	Birmingham	UK	UK	138
570	53.7726	-2.7852	2900	Birmingham	UK	UK	145
571	53.8316	-2.9804	3340	Birmingham	UK	UK	167
572	53.8482	-2.3786	7000	Birmingham	UK	UK	350
573	53.859	-1.5386	3130	Birmingham	UK	UK	157
574	53.9762	-2.0178	5920	Birmingham	UK	UK	296
575	54.1184	-0.7891	3220	Birmingham	UK	UK	161
576	54.5486	-1.1267	5760	Birmingham	UK	UK	288
577	54.7576	-2.9054	6200	Birmingham	UK	UK	310
578	54.8088	-1.7722	2820	Birmingham	UK	UK	141
579	54.8533	-1.394	2760	Birmingham	UK	UK	138
580	55.2237	-2.0276	6250	Birmingham	UK	UK	313
581	55.3781	-3.436	16000	Birmingham	UK	UK	800
582	55.4331	-4.509	3020	Birmingham	UK	UK	151
583	55.6095	-3.7723	3360	Birmingham	UK	UK	168
584	55.7945	-4.2076	6600	Birmingham	UK	UK	330
585	55.8453	-4.3112	6500	Birmingham	UK	UK	325
586	56.7077	-2.7597	3040	Birmingham	UK	UK	152
587	57.2367	-2.7821	2780	Birmingham	UK	UK	139
588	57.8682	-4.4448	2620	Birmingham	UK	UK	131

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2;46.6333;14.3128;4270;;AT;;;;214
3;46.9894;15.5817;4200;;AT;;;;210
4;47.252;11.42;4690;;AT;;;;235
5;47.5784;12.7305;3550;;AT;;;;178
6;47.9911;13.8878;11530;;AT;;;;577
7;48.0671;13.1775;3300;;AT;;;;165
8;48.2025;16.4186;4600;;AT;;;;230
9;48.2049;16.2734;6100;;AT;;;;305
10;48.2297;16.4692;2620;;AT;;;;131
11;48.3656;13.9771;3340;;AT;;;;167
12;43.3319;17.7513;5900;;BA;;;;295
13;43.8784;18.3593;2460;;BA;;;;123
14;50.2458;4.8519;2720;;BE;;;;136
15;50.3753;4.3819;3420;;BE;;;;171
16;50.4172;4.447;3220;;BE;;;;161
17;50.4555;3.9662;3500;;BE;;;;175
18;50.461;4.8498;2940;;BE;;;;147
19;50.4797;4.1698;4600;;BE;;;;230
20;50.588;5.8602;2840;;BE;;;;142
21;50.5882;4.3253;3120;;BE;;;;156
22;50.59;3.3922;2960;;BE;;;;148
23;50.5942;5.4331;3200;;BE;;;;160
24;50.5978;6.1325;3280;;BE;;;;164
25;50.6473;5.5832;6550;;BE;;;;328
26;50.7237;4.5543;3140;;BE;;;;157
27;50.7416;3.2238;2420;;BE;;;;121
28;50.7886;4.8612;2980;;BE;;;;149
29;50.8137;5.2089;2460;;BE;;;;123
30;50.8185;3.2785;2440;;BE;;;;122
31;50.849;4.4297;5360;;BE;;;;268
32;50.8521;5.6878;3520;;BE;;;;176
33;50.8832;4.3551;9200;;BE;;;;460
34;50.8903;4.7381;8200;;BE;;;;410
35;50.8989;4.391;3200;;BE;;;;160
36;50.9316;5.3368;4880;;BE;;;;244
37;50.9406;3.1181;6900;;BE;;;;345
38;50.9439;4.0442;2920;;BE;;;;146
39;50.9676;5.4886;3160;;BE;;;;158
40;50.9836;4.821;3280;;BE;;;;164
41;50.9894;3.5106;3380;;BE;;;;169
42;51.0077;3.368;3080;;BE;;;;154
43;51.0293;4.094;2540;;BE;;;;127
44;51.0353;4.4665;3280;;BE;;;;164
45;51.0532;3.6307;9480;;BE;;;;474
46;51.12;4.5734;2780;;BE;;;;139
47;51.1462;4.3383;4120;;BE;;;;206
48;51.16;4.8277;3440;;BE;;;;172
49;51.1695;4.1647;4240;;BE;;;;212
50;51.2062;5.3798;2480;;BE;;;;124
51;51.212;4.4686;3670;;BE;;;;184
52;51.2147;2.919;3280;;BE;;;;164
53;51.2211;4.3664;3130;;BE;;;;157
54;51.2329;3.2118;3610;;BE;;;;181
55;51.2384;5.1949;3280;;BE;;;;164
56;51.2706;4.5028;4450;;BE;;;;223
57;51.3369;3.3234;3850;;BE;;;;193
58;51.3531;4.771;2780;;BE;;;;139
59;42.687;23.3273;4660;;BG;;;;233
60;42.7266;23.3158;18640;;BG;;;;932
61;53.1232;26.5796;7000;;BY;;;;350
62;46.0338;8.9719;2940;;CH;;;;147
63;46.1258;7.2274;2620;;CH;;;;131
64;46.3407;6.1974;3670;;CH;;;;184
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439;51.9176;4.4887;3790;;NL;;;;190
440;51.9805;5.9059;2680;;NL;;;;134
441;52.023;5.0361;4920;;NL;;;;246
442;52.024;5.557;2840;;NL;;;;142
443;52.0787;4.3172;4870;;NL;;;;244
444;52.0879;5.1162;2680;;NL;;;;134
445;52.1326;5.2913;13740;;NL;;;;687
446;52.1518;5.0451;2700;;NL;;;;135
447;52.1554;4.4916;2760;;NL;;;;138
448;52.2208;6.8993;2800;;NL;;;;140
449;52.2244;5.1764;7900;;NL;;;;395
450;52.3081;4.9442;4600;;NL;;;;230
451;52.3728;4.9067;6700;;NL;;;;335
452;52.5113;6.0903;2760;;NL;;;;138
453;52.6307;4.7505;3120;;NL;;;;156
454;52.7308;6.4723;5740;;NL;;;;287
455;52.7847;6.9009;2860;;NL;;;;143
456;53.218;6.5707;2620;;NL;;;;131
457;59.7326;10.7913;6450;;NO;;;;323
458;59.9342;10.7994;7780;;NO;;;;389
459;50.218;19.002;4160;;PL;;;;208
460;51.8135;19.445;10450;;PL;;;;523
461;52.1077;21.0815;29210;;PL;;;;1461
462;52.2658;17.6252;2860;;PL;;;;143
463;52.282;21.0386;3000;;PL;;;;150
464;37.0213;-7.9338;2680;Madrid;PT;;;Iberia;134
465;38.525;-8.8986;6120;Madrid;PT;;;Iberia;306
466;38.7367;-9.1339;19060;Madrid;PT;;;Iberia;953
467;39.2322;-8.686;2620;Madrid;PT;;;Iberia;131
468;39.3999;-8.2245;2720;Madrid;PT;;;Iberia;136
469;39.7425;-8.8064;2760;Madrid;PT;;;Iberia;138
470;40.2108;-8.4221;5640;Madrid;PT;;;Iberia;282
471;40.6266;-8.6479;2840;Madrid;PT;;;Iberia;142
472;40.6447;-7.3044;2780;Madrid;PT;;;Iberia;139
473;40.6592;-7.9147;3460;Madrid;PT;;;Iberia;173
474;41.1515;-8.6051;10180;Madrid;PT;;;Iberia;509
475;41.5453;-8.4199;3940;Madrid;PT;;;Iberia;197
476;44.3902;26.1161;26080;;RO;;;;1304
477;44.4403;26.0342;5960;;RO;;;;298
478;44.4531;26.1286;7580;;RO;;;;379
479;44.5207;26.1358;9080;;RO;;;;454
480;45.7945;25.2265;3040;;RO;;;;152
481;45.9303;24.3044;18170;;RO;;;;909
482;44.0165;21.0059;20720;;RS;;;;1036
483;44.8262;20.4571;9300;;RS;;;;465
484;52.2320;21.0158;145900;;PL;;;;7295
485;57.387;12.2792;22750;;SE;;;;1138
486;46.0563;14.5047;10090;;SI;;;;505
487;46.5565;15.6499;4900;;SI;;;;245
488;48.0498;18.2532;3140;;SK;;;;157
489;48.0665;17.155;3490;;SK;;;;175
490;48.1976;17.1478;4570;;SK;;;;229
491;48.674;20.7849;3120;;SK;;;;156
492;48.7536;19.2941;5640;;SK;;;;282
493;49.1356;21.1867;3190;;SK;;;;160
494;36.8995;30.7059;2780;;TR;;;;139
495;37.0369;37.2875;2620;;TR;;;;131
496;37.2867;35.7333;3040;;TR;;;;152
497;37.8698;32.3994;4320;;TR;;;;216
498;38.489;27.2261;22280;;TR;;;;1114
499;38.7097;35.8409;6160;;TR;;;;308
500;39.6242;37.3257;2440;;TR;;;;122
501;39.9191;32.753;8560;;TR;;;;428
502;40.1914;29.0555;2460;;TR;;;;123
503;40.3689;36.5431;3260;;TR;;;;163
504;40.9071;39.8005;2480;;TR;;;;124
505;41.0105;29.0025;55600;;TR;;;;2780
506;41.2007;36.1377;2900;;TR;;;;145
507;48.3794;31.1656;24400;;UA;;;;1220
508;48.4371;34.9045;4900;;UA;;;;245
509;50.4424;30.5221;7780;;UA;;;;389
510;49.2044;-2.137;2860;Birmingham;UK;;;UK;143
511;50.4536;-3.7099;2960;Birmingham;UK;;;UK;148
512;50.4769;-4.4283;4040;Birmingham;UK;;;UK;202
513;50.8499;-0.9419;19360;Birmingham;UK;;;UK;968
514;50.8647;-0.1239;2940;Birmingham;UK;;;UK;147
515;50.9034;-3.7971;2820;Birmingham;UK;;;UK;141
516;50.9553;-1.3683;4090;Birmingham;UK;;;UK;205
517;51.0893;-0.2995;3180;Birmingham;UK;;;UK;159
518;51.097;0.5201;2840;Birmingham;UK;;;UK;142
519;51.1703;-0.7554;6450;Birmingham;UK;;;UK;323
520;51.3326;-0.3905;3440;Birmingham;UK;;;UK;172
521;51.3775;0.0853;9240;Birmingham;UK;;;UK;462
522;51.4156;0.2693;2480;Birmingham;UK;;;UK;124
523;51.4249;-2.6571;3160;Birmingham;UK;;;UK;158
524;51.4389;-0.4269;3460;Birmingham;UK;;;UK;173
525;51.4477;0.0123;2800;Birmingham;UK;;;UK;140
526;51.4535;-0.1893;4960;Birmingham;UK;;;UK;248
527;51.5144;-0.1785;3140;Birmingham;UK;;;UK;157
528;51.5168;-0.1805;3020;Birmingham;UK;;;UK;151
529;51.5313;-0.0258;3100;Birmingham;UK;;;UK;155
530;51.5491;-0.3638;3580;Birmingham;UK;;;UK;179
531;51.5559;-0.1155;2600;Birmingham;UK;;;UK;130
532;51.5678;0.2725;3280;Birmingham;UK;;;UK;164
533;51.5699;-0.1035;3280;Birmingham;UK;;;UK;164
534;51.5805;-0.2073;3910;Birmingham;UK;;;UK;196
535;51.5844;-0.3479;4120;Birmingham;UK;;;UK;206
536;51.6029;0.0771;3360;Birmingham;UK;;;UK;168
537;51.622;-3.3992;8700;Birmingham;UK;;;UK;435
538;51.6633;-0.3879;2740;Birmingham;UK;;;UK;137
539;51.7784;-0.3156;8840;Birmingham;UK;;;UK;442
540;51.7924;0.4802;3440;Birmingham;UK;;;UK;172
541;51.8347;-1.2995;3910;Birmingham;UK;;;UK;196
542;51.8808;-4.4348;2780;Birmingham;UK;;;UK;139
543;51.9679;0.8749;2560;Birmingham;UK;;;UK;128
544;51.9687;-0.153;2660;Birmingham;UK;;;UK;133
545;52.0941;-0.6851;48300;Birmingham;UK;;;UK;2415
546;52.2252;0.2378;2400;Birmingham;UK;;;UK;120
547;52.273;-0.8661;3060;Birmingham;UK;;;UK;153
548;52.3304;-1.4821;2800;Birmingham;UK;;;UK;140
549;52.3767;-1.8368;3520;Birmingham;UK;;;UK;176
550;52.4658;-1.9879;3420;Birmingham;UK;;;UK;171
551;52.4664;-1.8886;32500;Birmingham;UK;;;UK;1625
552;52.6388;-1.0236;4720;Birmingham;UK;;;UK;236
553;52.6639;1.2331;14430;Birmingham;UK;;;UK;722
554;52.7268;0.128;3260;Birmingham;UK;;;UK;163
555;52.9482;-3.8144;12840;Birmingham;UK;;;UK;642
556;53.0054;-0.801;3100;Birmingham;UK;;;UK;155
557;53.2034;-3.0312;3340;Birmingham;UK;;;UK;167
558;53.2582;-1.4047;2800;Birmingham;UK;;;UK;140
559;53.3608;-1.4674;2600;Birmingham;UK;;;UK;130
560;53.374;-2.5496;2720;Birmingham;UK;;;UK;136
561;53.4236;-1.4576;3260;Birmingham;UK;;;UK;163
562;53.4301;-1.4092;2420;Birmingham;UK;;;UK;121
563;53.4669;-2.2238;3460;Birmingham;UK;;;UK;173
564;53.51;-2.2427;3280;Birmingham;UK;;;UK;164
565;53.5178;-1.4647;2940;Birmingham;UK;;;UK;147
566;53.5278;-0.6153;2440;Birmingham;UK;;;UK;122
567;53.5478;-2.6538;3400;Birmingham;UK;;;UK;170
568;53.6686;-1.458;11700;Birmingham;UK;;;UK;585
569;53.7526;-0.3007;2760;Birmingham;UK;;;UK;138
570;53.7726;-2.7852;2900;Birmingham;UK;;;UK;145
571;53.8316;-2.9804;3340;Birmingham;UK;;;UK;167
572;53.8482;-2.3786;7000;Birmingham;UK;;;UK;350
573;53.859;-1.5386;3130;Birmingham;UK;;;UK;157
574;53.9762;-2.0178;5920;Birmingham;UK;;;UK;296
575;54.1184;-0.7891;3220;Birmingham;UK;;;UK;161
576;54.5486;-1.1267;5760;Birmingham;UK;;;UK;288
577;54.7576;-2.9054;6200;Birmingham;UK;;;UK;310
578;54.8088;-1.7722;2820;Birmingham;UK;;;UK;141
579;54.8533;-1.394;2760;Birmingham;UK;;;UK;138
580;55.2237;-2.0276;6250;Birmingham;UK;;;UK;313
581;55.3781;-3.436;16000;Birmingham;UK;;;UK;800
582;55.4331;-4.509;3020;Birmingham;UK;;;UK;151
583;55.6095;-3.7723;3360;Birmingham;UK;;;UK;168
584;55.7945;-4.2076;6600;Birmingham;UK;;;UK;330
585;55.8453;-4.3112;6500;Birmingham;UK;;;UK;325
586;56.7077;-2.7597;3040;Birmingham;UK;;;UK;152
587;57.2367;-2.7821;2780;Birmingham;UK;;;UK;139
588;57.8682;-4.4448;2620;Birmingham;UK;;;UK;131

Click on cell to change data and/or highlight a customer on the map. Click on header to sort data.
Set customer size factor in Map settings to change customer sizes on the map.

To run a greenfield Center of Gravity analysis, you only need to enter Customer_ID, Latitude, Longitude, and Demand.
All fields are explained in user manual section Customers.

Try to limit the number of customers.
Most models have (far) less than 5,000 customers. Read user manual section Customers for data aggregation suggestions.

Round latitudes and
longitudes to nearest

Aggregated customers

Excel aggregation file does the same and may provide you some more calculations insight.

Predefined warehouses (optional)

Birmingham;52.5056;-1.6711;0;482500;100;0;a;100000;1
Madrid;40.42896;-3.67909;200;550000;100;0;a;100000;1
WH03;49.88661;16.46538;9999;;;0;a;100000;1
WH04;48.84086;2.237;9999;;;0;a;100000;1
WH05;46.11951;14.3278;9999;;;0;a;100000;1
WH06;45.14897;9.91068;9999;;;0;a;100000;1
WH07;51.21531;6.81972;9999;;;0;a;100000;1
WH08;48.04984;18.25211;9999;;;0;a;100000;1
WH09;51.90353;13.07948;9999;;;0;a;100000;1
WH10;49.87173;9.02881;9999;;;0;a;100000;1
WH11;44.01766;21.00458;9999;;;0;a;100000;1
WH12;52.21854;5.17088;9999;;;0;a;100000;1
WH13;50.88846;4.29655;9999;;;0;a;100000;1
WH14;52.10933;21.08253;9999;;;0;a;100000;1
WH15;47.10398;-0.77346;9999;;;0;a;100000;1
WH16;47.54028;7.6329;9999;;;0;a;100000;1
WH17;44.4484;26.09114;9999;;;0;a;100000;1
WH18;51.68498;5.30275;9999;;;0;a;100000;1
WH19;53.54222;10.00315;9999;;;0;a;100000;1
WH20_SAT;42.61115;-1.60723;9999;;;1;a;100000;1
WH21_SAT;42.6902;23.30746;9999;;;1;a;100000;1
WH22_SAT;55.81751;37.49929;9999;;;1;a;100000;1
WH23_SAT;48.2597;11.76837;9999;;;1;a;100000;1
WH24_SAT;45.76480;3.19220;9999;;;1;a;100000;1

All fields are explained in user manual section Warehouses.

Demo data contains 24 warehouses for Select best warehouses demo. For Calculate Center of Gravity locations demo, only warehouses Birmingham and Madrid are relevant. Others have 9999 move limit, so are not really predefined from a Centers of Gravity perspective, and could have been left out.

Suppliers (optional)

All fields are explained in user manual section Suppliers.

Border crossing points (optional)

All fields are explained in user manual section Border crossing points.

After data entry, check relevant constraints, otherwise your input data will be partly ignored!
This is done in section Parameters & constraints - Constraints, as is also explained in user manual section Constraints.

Kilometers

Miles

Distance circuity factor

Color customers black if warehouse distance exceeds lead time distance of

Improve service level

Display city closest to Center of Gravity location

Only consider cities with a population larger than

Only consider cities within country

Closest city is based on a database containing cities with at least 15,000 inhabitants.

Display

-costs-increase-areas around Center of Gravity

Randomize each demand quantity ±

; using a

distribution

Read user manual section Logical steps - Add suppliers for more info. Options 1/3/4 can be mixed if set at individual supplier level.

Forbid supply shortage Ignore supply side costs

Non-unique products

1. Each supplier delivers to closest warehouse only

2. LP optimized supply flows - with auto-sized suppliers

Unique products

3. Each supplier delivers to each warehouse (default option)

4. Each supplier delivers to each warehouse via closest warehouse

Link suppliers to closest main warehouse (not satellite)

Also set Transport rates

Read user manual section Transport rates for more info.

Auto-size shipments per customer

	i Shipment size	% FTL	See curve % FTL costs	Cost ratio	i Costs/km	Costs/km/qty
Full Truck Load		100.00%	100.00%	1.000000
Customer
Supplier
Interdepot
Center of Gravity locations depend on ratio supplier rate/customer rate =

Transport costs curve i
Cost factor as % FTL cost
	Shipment size as %FTL

Point	X (as %)	Y (as %)
Start point (P₀)
Start control point (P₁)
End control point (P₂)
End point (P₃)
Some point on the curve
Animation of Bezier curve construction
You may also drag the red control points in chart itself.

This section estimates the yearly customer delivery mileage of an owned truck operation. Calculate Center of Gravity locations first, before using this section. Center of Gravity locations and customer assignments are used as input for its calculations. It determines an average roundtrip per warehouse - based on a.o. delivery area shape and size per warehouse (based on customer locations) - which then brings yearly mileage per warehouse, thus total yearly mileage.

Own truck operation

Working days per year

Distribution hours per workday(s)

Unloading minutes per delivery

Average speed (km/hour)

Average speed (miles/hour)

Maximum speed (km/hour)

Truck fill rate

Sum of area size(s) (square km)

Area scanning level

Display area(s) on map

Avg Inter Drop Distance

Manual overrule IDD (km)

Manual overrule IDD (miles)

Avg deliveries per roundtrip

Avg roundtrip length

Sum of yearly roundtrip kilometers

Sum of yearly roundtrip miles

Demand beyond reach

Customers beyond reach

Above fields remain empty, because an infeasibility error is encountered at a warehouse.

To calculate more precisely, on a daily basis, with actual roundtrips per warehouse being constructed, use Fleet Mileage Calculator

Centers of Gravity Calculator goal value assumes that transport costs are linearly, perfectly positive correlated to weighted distance, as if 1 customer delivery means 1 trip with 1 drop only, and as if costs are fully determined by distance driven. In reality, multiple drops are done in a single roundtrip, and trip costs are not only costs of driving, but also costs of loading and unloading (see picture on the right).
Center of Gravity cost model versus transport reality

Center of Gravity cost model versus transport reality

Centers of Gravity Calculator minimizes the sum of weighted distances. This is the same as minimizing the weighted average distance. That distance is the distance from warehouse to the first customer in a trip, and also from last customer back to warehouse. It does not equal the distance between customers on a trip. That inter-drop distance depends on the number of drops and delivery area size. So, trip length - and costs of driving - is only partly determined by weighted average distance. And costs of (un)loading is not related to distance. Because transport costs are only partly determined by weighted average distance, X% change of goal value does not mean X% change of transport costs. To estimate future scenario transport costs you may combine your actual baseline transport costs with goal value change and some dampening factor, as shown below.

	Baseline	Future scenario
Baseline transport costs from your financials	USD 12,000,000
Goal value Centers of Gravity Calculator	10,000,000	9,000,000
Goal value change (as %)		-10%
Dampening factor (value assumed at 0.50)		0.50
Goal value change × dampening factor		-5%
Transport costs effect		- USD 600,000
Future scenario costs		USD 11,400,000

Or use the sum of yearly roundtrip kilometers of above section Own truck operation to estimate cost effects.

Constraints settings

Centers of Gravity

runs

Minimum goal value		i
Current goal value
Weighted distances		i

Scenario

Name

memory extended

Scenario to reload

Show outcomes of Center of Gravity

Service level distance metrics

Service level distance table

Distance

Demand cum

Demand

Customers cum

Customers

# Customers

Service level metrics and table are also found in the output reports.

Service level distance chart

Copy chart

Distance = estimated road distance = as-the-crow-flies distance x circuitry factor, set in section Parameters & constraints - Distance.

Copy reports	Tab separated	Decimal point
Copy reports	Semicolon separated	Decimal comma

Original input data and parameters are also copied in, at the end.

Summary












i




i




i

Center of Gravity locations

Customer assignments

Service level distances

The "Select best warehouses" routine selects those predefined warehouses that minimize supplier plus customer transport costs + warehouse binary‑fixed costs + warehouse variable handling costs.

You may override the warehouse-customer transport costs input as calculated by the software with your own transport costs based on your own costing method / on actual rate tables / on actual road distances (see Batch Driving Distances Calculator).
Enter a C × W matrix (C customers top to bottom × W warehouses left to right) in below text area with each matrix cell containing the total transport costs to deliver the total demand of customer c from warehouse w.
Uncheck constraint "Fixed customer-warehouse assignments" in section Constraints if you no longer want to constrain assignments.
Press Clear if you no longer want to override.

Number of warehouses to select										Must‑be‑included
Warehouse group	a	b	c	d	e	f	g	h	i	j
Number of warehouses in each group
Number of warehouses to select

Solvers selection		Warehouse combinations generation (open/close decision for each warehouse)
Solvers selection		K-means Clustering runs i	Superfast Selection solver	Enumeration solver
Warehouse capacities (and assignment of each customer to an open warehouse)	Ignore / not applicable
	Greedy solver
	LP solver i

Best is to use Enumerations Solver (plus LP Solver if warehouse capacities are applicable). Solving a case with many warehouses with constrained capacities may take long. Hence the other solvers that run faster, but may produce a sub-optimal solution. For more info, read user manual section Select best warehouses algorithms.

Display selected warehouses only

Create "Top X" report, with X =

Copy reports	Tab separated	Decimal point
Copy reports	Semicolon separated	Decimal comma

Summary

Selected warehouses

Customer assignments

Enumeration solver - Top X warehouse combinations

Reports appear after running.

Your data stays on your computer. However, you can collaborate by exchanging scenario files.

Export

First, save your scenario. Reload the scenario you want to share. Press button Export scenario file. Email file to your coworker.

Import

First, save your scenario. Save file as received by email from your coworker on your computer. Press button Import scenario file.

Imported data is saved in local memory under a scenario name, once you press button Save under section Calculate.

Stelling Consulting

✉

support@stellingconsulting.nl

"Suggestions for improvement are always welcomed!"

Alrik Stelling
Owner Stelling Consulting

You want Support to investigate your case?

Open section Collaborate, export a scenario file, and email your scenario file (not a filled Excel template^* ) with your question or issue description.

^* Excel template lacks your parameter settings.

Demo mode

Demo guide

Next

Using the software

Summary

1. Greenfield analysis

3. Realistic scenario

2. Current situation

A first run

Assess service levels

Run sensitivity analyses

Enter current warehouses

Fix customer-warehouse assignments

Gradually relax some constraints

Activate other types of constraints

Intro

Enter suppliers

Enter supplier transport rate

Activate suppliers

Select supply option

Non-unique products

Unique products

Notes

Demand side only

Supply option 3

Supply option 4

Select a proper location for each Center of Gravity

Mandatory

Optional

How to activate constraints

User input

Customers

Remarks

Optional fields

Optional fields for "Select best warehouses" functionality only

Predefined warehouses (optional)

Optional fields

Optional fields for "Select best warehouses" functionality only

Suppliers (optional)

Optional fields

Border crossing points (optional)

Intro

Predefined warehouses table

Customers table

Suppliers table

Border crossing points table

Intro

Transport costs curve

Small customer shipments versus large supplier shipments

Ratio supplier rate/customer rate

Small customers shipments versus large customers shipments

Auto-size shipments per individual customer

Transport-rated volumes on the map

Two options to model warehouse → warehouse flows

Calculate Center of Gravity locations

Center of Gravity analysis - goal value

Center of Gravity analysis - algorithms - How to calculate Center of Gravity locations

Single Center of Gravity algorithm - Introductory thoughts

Single Center of Gravity location algorithm

Single Center of Gravity algorithm - Summary

Single Center of Gravity algorithm - Steps more detailed

Single Center of Gravity algorithm - Visual simulation

Earth is a globe - Required adjustments translating Cartesian into Spherical coordinate system

Weiszfeld's algorithm as faster alternative

Differences between Single Center of Gravity and Weiszfeld's algorithm

Centers of Gravity Calculator uses both algorithms

Multiple Center of Gravity locations algorithm

Similarity with K-means Clustering

Animations of the Multiple Center of Gravity locations algorithm

Beware of an overseas customer, far, far away

Underlying assumptions

Broader supply chain network design perspective

A completely different alternative approach

Add-ons

Greedy algorithm (default)

LP optimization = Vogel's approximation + MODI + Stepping stone

LP optimization: continuous optimization or post-optimization only

Simply run with and without LP optimization

Including supply-side