Question

Joe buys 10 gallons of milk at $3.00 per gallon at Publix. He plans to buy more milk at other grocery stores later this week and then compute the weighted mean cost of milk at the end of the week. What number will Joe plug in as his first "w" value when making this computation?

Answer 1

The require weighted mean formula for this problem will be as under:

x-bar = ? (x_i*w_i) / ?w_{i ,} where i=1 to n denoting the number of grocery stores.

Now it is given that at i=1, x_i= $3.00 and w_i= 10 gallons (This is milk purchased by Joe from grocery store Publix.)

Now assuming that daily milk consumption by Joe is on average constant at 10 gallons. This is normal to assume as there is nothing given in the question to mean that daily consumption would vary across week. Here we also assume that this is first day of the week and Joe needs to buy more milk from other grocery stores later this week for another 6 days.

Since goal of this problem is to find the weekly weighted average cost of Milk and since it is given that Joe plans to buy Milk from other grocery stores as well. It is wise and prudent to assume that per gallon price of milk would vary across stores on different days of the week.

Now Joe's goal for the week should be to either minimize the mean cost of milk for the week or to at best keep it constant at $3.00 per gallon.

Now at second store, when Joe will go to buy Milk on second day, following three scenarios are possible:

Scenario 1: Milk is available at same rate as that of store Publix i.e. $3.00 per gallon at i=2.

This situation is easy for Joe as he can safely put w_i=2 = 10 gallons without adversely affecting the weighted average mean cost of milk for two days.

Scenario 2: Milk is available at higher rate as that of store Publix i.e. say $4.00 per gallon at i=2.

In this situation, Joe will have to reduce consumption to keep the mean cost of milk at same level of $3.00 per gallon. So the required weight Wi-₂ may be determined by following equation:

$3 = ($3 * 10) / (10 + Wi-₂) + ($4* Wi-₂) / (10 + Wi-₂)

Solving this gives Wi-₂ = 0, i.e. Joe should refrain from purchasing from this store and look other stores that matches the price.

Scenario 3: Milk is available at lower rate as that of store Publix i.e. say $2.75.00 per gallon at i=2.

In this situation, Joe will have to keep consumption same to keep the mean cost of milk at lower level in relation to previous price of $3.00 per gallon. So the required weight Wi-₂ may again be determined by following equation:

min($3) = ($3 * 10) / (10 + Wi-₂) + ($2.75* Wi-₂) / (10 + Wi-₂)

So if he keeps consumption same i.e. Wi-₂ = 10, his weighted average cost of milk would reduce to $2.875 per gallon for two days.

In this way Joe may choose to select store for purchase of Milks, each time keeping W = 0 or 10 gallon for store selling Milk at higher or lower rate then $3.00 per gallon respectively. With this strategy, at the end of the week, Joe is likely to minimize his weighted average cost of Milk for the week.