Gravity Model for Facility Location

Hello! Welcome to the next lesson in your MBA journey.

In our previous session, we evaluated different distribution network designs, deciding what kind of network architecture best aligns with a company's products and service goals. We saw that this is a strategic choice between efficiency and responsiveness.

Today, we address the logical next question: once you've chosen a network design, where do you physically place the facilities? This lesson focuses on the learning outcome: Apply a quantitative model, such as the gravity location model, to identify a potential facility location.

We will explore the Center of Gravity model, a foundational tool in logistics for determining an optimal location for a single new facility (like a distribution center or warehouse) that serves a set of existing locations. Given your background in applied mathematics and physics, you'll find the underlying principle quite familiar—it's essentially a calculation of the center of mass for a system of points, where the "mass" is a business metric like customer demand or shipping volume.

1. The Center of Gravity Model: Concept and Formula

Imagine a company needs to place a new central warehouse to supply several retail stores. The goal is to find a location that minimizes the total distance traveled to all stores, but not all stores are equal—those with higher sales should have a stronger "pull" on the warehouse's location. This is the core idea of the Center of Gravity model.

The problem is typically visualized on a coordinate grid, with each existing location plotted as a point.

This image illustrates a typical setup for a Center of Gravity problem. Existing locations (showrooms) are plotted on a map, and each has an associated demand (monthly sales). The task is to find the optimal coordinate for a new central facility.

The model calculates the coordinates for this new facility by taking the weighted average of the coordinates of the existing locations. The "weight" for each location is its demand, shipping volume, or another relevant metric.

The formulas are:

C_{x} = \frac{\sum _{i = 1}^{n} x _{i} W _{i}}{\sum _{i = 1}^{n} W _{i}} and C_{y} = \frac{\sum _{i = 1}^{n} y _{i} W _{i}}{\sum _{i = 1}^{n} W _{i}}

Where:

$C_{x}, C_{y}$ = The x and y coordinates of the new facility (the "center of gravity").
$x_{i}, y_{i}$ = The x and y coordinates of the i-th existing location.
$W_{i}$ = The weight of the i-th location (e.g., volume, demand, number of trips).

To see a clear explanation of this concept and the formulas, please watch the beginning of the following video.

Operations Management: Location Selection – Centre of Gravity Method

This video from The Business Doctor provides a concise introduction to the Center of Gravity method, explaining its purpose and breaking down the formulas.

Please watch from the beginning to 02:10. Focus on the conceptual explanation of what the model achieves and how the variables in the formulas (xi, yi, Wi) correspond to the elements of a location problem.

2. Applying the Model: A Worked Example

Now, let's walk through the application of the model using the case study from the video you just started. This will solidify your understanding of the mechanics.

The problem data for the company, Microserve, is as follows:

City Section	x-coordinate	y-coordinate	Number of Contracts (Weight, $W_{i}$ )
A	8.0	10.5	90
B	6.7	5.9	220
C	12.0	5.2	50
D	15.0	6.3	300
E	11.7	8.3	170

The goal is to find the optimal coordinates $(C_{x}, C_{y})$ for a new service center. Before watching the solution, take a moment to consider how you would plug these values into the formulas we just discussed.

Now, let's see the calculation in action.

Operations Management: Location Selection – Centre of Gravity Method

We'll continue with the same video, which now applies the formulas to the Microserve case study.

Watch from 02:10 to 05:09. Follow the step-by-step calculation for both the x-coordinate (Cx) and the y-coordinate (Cy). Note how the final location is influenced by the sections with more contracts (higher weights).

As the video demonstrates, the calculation is straightforward:

For the x-coordinate ( $C_{x}$ ):
$C_{x} = \frac{( 8.0 \times 90 ) + ( 6.7 \times 220 ) + ( 12.0 \times 50 ) + ( 15.0 \times 300 ) + ( 11.7 \times 170 )}{90 + 220 + 50 + 300 + 170}$
$C_{x} = \frac{720 + 1474 + 600 + 4500 + 1989}{830} = \frac{9283}{830} \approx 11.18$

For the y-coordinate ( $C_{y}$ ):
$C_{y} = \frac{( 10.5 \times 90 ) + ( 5.9 \times 220 ) + ( 5.2 \times 50 ) + ( 6.3 \times 300 ) + ( 8.3 \times 170 )}{830}$
$C_{y} = \frac{945 + 1298 + 260 + 1890 + 1411}{830} = \frac{5804}{830} \approx 6.99$

The optimal location for the service center is approximately (11.18, 6.99). As the video notes, this point is pulled towards the locations with higher demand (D and B), rather than being at the simple geographic center of all five points.

For a text-based reference of the formulas and another worked example, you can review the following resource.

Gravity model: balancing supply and demand

This article, 'Top Quantitative Models for Facility Location Decision-Making', provides a very clear and concise summary of the gravity model, its formulas, and a quick example.

Please read the section 'Gravity model: balancing supply and demand'. It will serve as a quick, consolidated review of what we've just covered.

3. Practice and Strategic Considerations

Now it's your turn to apply the model. This is a crucial step in moving from theory to practice.

Activity: Locate a Central Office

A healthcare firm needs to locate a central office for conducting internal audits. The team will travel from this new office to eight existing facilities. Most facilities will be visited 3 times a year, but the busy Toronto facility will be visited 5 times a year.

Your task is to calculate the coordinates for the new central office using the Center of Gravity method.

Here is the data:

Facility Location	x-coordinate	y-coordinate	Annual Visits (Weight, $W_{i}$ )
Oshawa	9.2	3.5	3
Richmond Hill	7.3	2.5	3
Toronto	7.8	1.4	5
Mississauga	6.5	1.8	3
Guelph	4.2	2.1	3
Waterloo	3.8	2.3	3
London	2.5	1.5	3
Sudbury	1.2	8.5	3

Take a few minutes to perform the calculations for $C_{x}$ and $C_{y}$ .

(Once you have your answer, you can check your work with the solution below.)

Solution:

The total number of visits is $(7 \times 3) + 5 = 26$ .

The weighted sum for the x-coordinates is:
$(9.2 \times 3) + (7.3 \times 3) + (7.8 \times 5) + (6.5 \times 3) + (4.2 \times 3) + (3.8 \times 3) + (2.5 \times 3) + (1.2 \times 3) = 154.8$

The weighted sum for the y-coordinates is:
$(3.5 \times 3) + (2.5 \times 3) + (1.4 \times 5) + (1.8 \times 3) + (2.1 \times 3) + (2.3 \times 3) + (1.5 \times 3) + (8.5 \times 3) = 113.2$

$C_{x} = \frac{154.8}{26} \approx 5.95$
$C_{y} = \frac{113.2}{26} \approx 4.35$

The optimal location is (5.95, 4.35). You can watch the following video to see this calculation performed step-by-step.

Location Selection: Centre of Gravity Method

This video from The Business Doctor walks through the exact problem you just solved.

Watch from the beginning to 03:00 to see the setup and the calculation for the optimal coordinates, confirming your own result.

Beyond the Coordinates: From Model to Reality

The coordinate (5.95, 4.35) is a mathematical answer, but it's not a business decision. An MBA-level analysis requires us to recognize the model's limitations and integrate its output into a broader strategic context. A company can't build a facility on a precise coordinate if it's in the middle of a lake or a residential area.

The quantitative result is a starting point that significantly narrows the search. The next step is to evaluate real-world sites near this optimal point, considering qualitative and other quantitative factors.

Location Selection: Centre of Gravity Method

Let's finish the video on the healthcare firm, as it raises these crucial real-world considerations.

Watch from 02:47 to 03:39. Pay attention to the list of qualitative factors that must be considered after the model gives its recommendation.

The Center of Gravity model makes several simplifying assumptions:

Transportation costs are a linear function of distance.
The model uses straight-line (Euclidean) distances, not actual road networks.
It doesn't account for geographical barriers, infrastructure, or regional cost differences.

Therefore, the final decision must also incorporate:

Infrastructure: Proximity to major highways, airports, or ports.
Real Estate: Availability and cost of suitable land or buildings.
Labor: Availability, skill, and cost of the local workforce.
Regulations: Zoning laws, environmental regulations, and local taxes.
Geographic Factors: Terrain, lakes, rivers, and other physical constraints.

The model provides the "where to start looking," and these factors determine the final selection.

Conclusion

In this lesson, you learned to apply a foundational quantitative model for facility location. This is a classic example of how managers use simplified models to bring structure and data to complex, open-ended business problems.

Key Takeaways:

The Center of Gravity model identifies a theoretically optimal location for a new facility by calculating the weighted average of the coordinates of existing demand/supply points.
The "weight" used in the calculation is a critical business variable, representing factors like demand volume, population, or transportation frequency.
The model's output is a starting point for analysis, not a final answer. The mathematically optimal coordinate must be evaluated against real-world qualitative factors like infrastructure, land cost, labor availability, and geography.

Preview of the Next Lesson:
We've now covered what kind of distribution network to build and a method for where to place a facility. A related strategic question is whether the company should own and operate these facilities itself or outsource these activities. In our next lesson, we will tackle this by learning to evaluate a strategic sourcing decision by comparing a vertically integrated 'make' option against a 'buy' option using the total cost of ownership framework.

Recommended resources

1. The Center of Gravity Model: Concept and Formula

2. Applying the Model: A Worked Example

3. Practice and Strategic Considerations

Beyond the Coordinates: From Model to Reality

Conclusion