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Facility location models for small businesses

Problem Statement:

Let say, you want to start a restaurant business in a city. For a successful restaurant business, location is everything. A restaurant’s location can make or break it! You need a spot that draws crowds, is easily accessible, and has the potential for growth. As part of the strategic decisions, you want to know the answers to the following questions:

1. What is the minimum number of spot/facilities required to cover all of the demand points?

2. What if we can only establish a predetermined number of restaurants, say P from a set of given candidate sites and we want to utilize our limited resources maximally by covering maximum demand points?

3. What if we want to minimize the average distance or total demand-weighted distance between restaurants and demand points?

4. Let say, we don’t have any candidate locations and need to find the best location to site a new restaurant. Where should we establish a new station?

This post will answer your questions using location allocation optimization models. Typically, the location models are two categories: discrete location model and continuous location model. The Discrete location model selects the best locations from a set of given candidate sites, whereas the continuous location model selects the best locations with an open search.

We will illustrate this problem by taking Louisville, Kentucky, USA, as an example. Louisville has 42 zip codes and for the simplicity we assumed the centroid of each zip code as demand nodes. Gurobi optimization solver is used to solve the models.

Models and Implementations:

Notations:

iϵI:Index and set of demand points

jϵJ:Index and set of candidate facility locations

dᵢⱼ:Distance between demand point i and facility location j

D:Maximum distance threshold people are willing to travel

hᵢ:Number of populations at demand point i

P: Number of facility to be opened within the budget limit.

Model-1: To answer your first question, we used well-known general set covering model.

Decision variable:

xⱼ: A binary variable xⱼ ϵ {0,1} indicating whether the restaurant should be sited in the location j (xⱼ=1) or not (xⱼ=0).

Objective function: Locate the minimum number of facility required to cover all the demand points.

Subject to:

Constraint 1 states that, all demand points must be covered by at least 1 facility. Constraint 2 states that either we will build or not a restaurant at each candidate location.

Model-2: To answer the second question, we used maximal covering model.

Decision variables:

xⱼ: A binary variable xⱼ ϵ {0,1} indicating that whether the restaurant should be sited in the location j (xⱼ=1) or not (xⱼ=0).

zᵢ: A binary variable zᵢ ϵ {0,1} indicating that whether the demand location i is covered(zᵢ=1) or not (zᵢ=0).

Objective function: Maximize the total demand to be covered.

Subject to:

Constraint 1states that the demand node is not covered unless at least one facility is within the required distance. Constraint 2is our budget constraint. Total number of selected candidate locations has to be equal to our budget. Constraints 3and 4states that for each candidate locations may or may not be selected and for each demand points may or may not be covered respectively.

Model-3: To answer the third question, we used p-median model.

Decision variables:

xⱼ: A binary variable xⱼ ϵ {0,1} indicating that whether the restaurant should be sited in the location j (xⱼ=1) or not (xⱼ=0).

yᵢⱼ: A binary variable yᵢⱼ ϵ {0,1} , if demand point i is covered by restaurant j (yᵢⱼ=1) otherwise (yᵢⱼ=0)

Objective function: Minimizes the demand-weighted or simply total distance between demand nodes and the facilities to which they are assigned.

Subject to:

Constraint 1 states that, total number of selected candidate locations has to be equal to our budget. Constraint 2 ensure that, each demand location is covered. Constraint 3 states that, we cannot assign a facility to a demand node unless we open that facility.

Model-4:

Let’s say, we don’t know the candidate locations. We need to do an open search to find the best location or locations to site the facility. This is a continuous location problem. It can be a single or multifacility problem. This post illustrated the single facility location problem.

Several methods are available to solve this problem — simple methods such as the center of gravity method and median method, or iterative method such as Weiszfeld method. Here, weiszfeld method is used to find the optimal solution.

Weiszfeld Method: An iterative procedure

2. Defines a set of weights that are inversely proportional to the distances from the current estimate to the samples.

3. Create a new estimate that is the weighted average of the samples according to these weights.

4. Repeat the adjustment process above until two consecutive estimates are close to each other (set stopping criteria).

Mathematically the iterations look like below:

xᵢ: Longitude of demand nodes

yᵢ: Latitude of demand nodes

wᵢ: Number of population at each demand node, read as weights

Data Exploration

Optimization Models

Model-1 implementation:

Decision from Model-1:

Conclusion: In model-1, our assumption was people are not willing to travel more than 10 mile to access a restaurant. In that case, we need 3 restaurants in 3 red pentagon locations.

Model-2 implementation:

Decision from Model-2

Conclusion: In model-2, we had a budget to open at most 2 restaurants and people are willing to travel at most 10 miles to access your restaurants. To cover the maximum demands, red two pentagon are best locations among the 5 candidates locations. Note: as we have budget for 2 and maximum distance threshold 10 miles, some demand may not cover.

Model-3 implementation:

Decision from Model-3

Conclusion: In Model-3, our objective was to minimize the demand weighted travelling distance. To do so, we have to select locations in 2 red pentagon. Assignment demand to restaurants are shown below (codes are available in github, reference at the end)

As the food tastes of people from different geographical area may different, now the restaurant can customize their recipes based on the demand.

Model-4 implementation:

Conclusion: If we are able to open only one restaurant and don’t know any candidate locations, we can search for best location using Weiszfeld method. According to this iterative method the best location is on the red pentagon location.

References

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