Spatial interaction is motivated by a deficit (or demand) in one place and a surplus (or supply) in another, right?  Well, what better example of supply than all the "stuff" in a retail store?  We can use our simple model of spatial interaction to model the amount of demand around a retail outlet, on the assumption that people:  (a) have to pay the costs (in money and time) to get to and from a retail outlet, and (b) want to minimize those costs, so long as they can buy what they need.  We can even model the shape of the market area around an outlet, if we know the locations of competing outlets.  Then, we can do things like tailor our marketing plan to characteristics and needs of people in the market area -- or figure out where to locate a new retail outlet to be in the midst of the kind of market area we think will best suit the kinds of things we're selling!
 

MARKET  POTENTIAL  (see the Hanink text, page 245-246)
Recall our model for the interaction between two places, i and j:

Well, if we’re concerned with the total flow to j from all points i, we would create a formula like

which is the sum of all the interactions between all i’s and a particular j.

If we are establishing a retail store or shopping center at j, and we want to estimate how many shoppers will use it, we care about this kind of total flow from all i’s to j.  We would expect to get more shoppers from nearby places, and from places with larger populations.  These “places” would be sub-areas of a region — census tracts, for example.

In the equation above, since P_j is a constant, we can multiply it by  k, the other constant, to yield a new constant M_j = kP_j, resulting in a simpler formula

which is the same as

This is a measure of the potential interaction between point j and all the surrounding points i.  If we think of j as a retail store or a shopping center, and each i as a zone nearby (each zone has a particular population and a particular distance from j), then I_j is a measure of the potential number of shoppers destined to j.  This is sometimes called the market potential of j.
You can see that Ij is high for places j that are close to heavily populated zones, and low for places j that are far from heavily populated zones.  While this is intuitive, this measure gives you a systematic way to compare two locations that seem quite similar to one another (downtown Everett versus downtown Tacoma).
See Figure 6.16 in the S&deS text, which implicitly uses this formula (though in that simple example, k = 1 and a  = 1).

Now, if we’re talking about the potential for retail sales at a point j, there are better variables than the population of surrounding zones or tracts.  We need some measure of purchasing power, such as total personal income in each zone i.Even better would be a measure of general purchasing power or personal income, multiplied by the proportion of that income that is spent on the kinds of things our store would be selling.

This is precisely what you’ll do in the second assignment.  You’ll be estimating the amount of retail sales you’d get in a retail store at a particular location j, by allowing the computer to compute the sales that the store should get from customers in each of several zones i.  You’ll tell the computer the total household income in each zone, and the proportion of total income spent on groceries.  The computer will compute the distance from each i to j, and the consequent level of sales (S) expected from all the zones in the market area of the store.

where C_i =  food or drug or furniture expenditures in i  =  (household income in i) (proportion of income spent on food or drugs or furniture).  To make this clearer, this is the parameter k multiplied by:  each zone's food (etc.) purchases divided by the distance of that zone from location j, summed across all the zones i.
 

COMPETING  MARKET  CENTERS
A further elaboration is to recognize that

• there are competing retail outlets,
• and that the likelihood that a consumer in zone i will shop at a store in zone j depends on the distance ij relative to the distance from i to other zones where there are similar stores.

• larger stores (or shopping centers) are more attractive to consumers than smaller stores (or shopping centers) — for reasons of variety and, usually, lower prices.

where
B_C = "breaking point" between the primary market areas of centers A and C, expressed as distance from C
d_AC = distance between centers A and C
P_A = population of city A or square footage of shopping center A
P_C = population of city C or square footage of shopping center C

This results from "Reilly's law of retail gravitation,"  which suggests that R, the retail attractiveness of a central place (or shopping center) j to a potential customer at i increases proportionately with the population P (or size in square feet) of j, and increases inversely with the square of the distance ij:

Note that whether we're using population or square footage, the implication is that the greater possibility of finding more of the goods/services needed, with only one trip, is a powerful attraction of a customer to a particular place.