Question $2($Gravity Model$)$

A retail company wants to open a store in a city and needs to decide on the optimal location based on the potential customer flow. The city has several districts with varying populations and distances from potential store locations. The Gravity Model will be used to estimate the interaction $($customer flow$)$ to each potential store location from various districts.

Given Data:

Districts in the City $($with populations and distances to potential store locations$)$:

District A: Population $=50,000,$ Distance to Store Locations $=[2$ km$,5$ km$,10$ km$]$

District B: Population $=70,000,$ Distance to Store Locations $=[5$ km$,3$ km$,6$ km$]$

District C: Population $=60,000,$ Distance to Store Locations $=[7$ km$,8$ km$,2$ km$]$

District D: Population $=40,000,$ Distance to Store Locations $=[10$ km$,7$ km$,9$ km$]$

Potential Store Locations $($with associated costs$)$:

Location $1$

Location $2$

Location $3$

Fixed Costs for Store Locations:

Location $1$: $$30,000$

Location $2$: $$25,000$

Location $3$: $$35,000$

Objective:

Determine the optimal location for the new store to maximize the estimated customer flow based on the Gravity Model, while considering the fixed costs of each location.

$1-$ Carefully write the mathematical formulations including the constraints.

$2-$ Using Excel Solver, Minimize the total cost of operating factories, the distribution center, and transporting goods from factories to the DC and then to customers.