Figure Q$4$ shows three residential zones $(A,B$ & C$),5km$ apart among each other, that are linked

by straight highways. There are $10000,20000$ and $5000$ households in zone A$,$ B and C$,$

respectively. It is known that the trip productions per week follow this relationship:

$P_i={\displaystyle \underset{j}{\overset{?}{}}}(A_j{H}_i-\frac{H_i}{20}{x}_{ij}^2)$

Where $P_i,H_i,$ are, respectively, the trip productions and number of households in zone $i.$ The

variable $A_j$ is the Relative Attractiveness of the shopping centre; $x_{ij}$ is the shortest distance $($in

$km)$ between zone $\frac{?}{?}$ and shopping centre $j$ along the highways.

$($a$)$ A developer plans to build a new shopping centre $($called S$1)$ with a Relative Attractiveness

of $6$ units along the highway between A & B$.$ Where should the developer build S$1$ such that the

number of customers per week is maximized? What is the maximum number of customers per

week?

$($b$)$ After the first developer has built S$1$ at the location as determined in $($a$),$ a second

developer decides to open another shopping centre $($called S$2)$ with a relative attractiveness

of $12$ units along the highway between A & B$.$ He has two options for the location of S$2,$ either

$($i$)$ In the middle between A & B$,$ or

$($ii$)$ At the same location of $S1$ as determined in $($b$)($right next to S$1)$

Between the two options, what is the best location for $$2$ if the second developer wants to

maximize S$2\text{'}$s customers?

You may assume that the gravity model calibrated for this situation has these coefficients: $C=2$

and all socioeconomic adjustment factors equal $1.$