Question

A hardware company has  different retail stores in which  different products are sold. write a paragraph or two explaining why matrices provide an efficient method of inventory control, and what matrix operations would be use

Answer 1

let a, b, c be quantities of products x, y, and z respectively.

total quantities of products available from the store 1, then total availability:
R1 = $A_{1}\ x + B_{1}\ y + B_{1}\ z \ is\ =\ [\ A_1\ B_1\ C_1\ ]$

Same way quantities of products sold from stores 1 is given by:
S1 = $d_1\ x\ +\ e_1\ y\ +\ f_1\ z\ \ =\ [\ d_1\ e_1\ f_1\ ]$

The quantities of products remaining in stores 1 is given by:
 = $R1 - S1$              -  Row R1 - row S1

If X, Y, Z are their prices , then
         $[d_1 e_1 f_1 ] [X Y Z]^{T}$      gives the earnings from sales at stores1

If $P_x, P_y, P_z$are the profits for each products, then
         $[d_1 e_1 f_1 ] [P_x P_y P_z]^{T}$  gives the profits from sales at stores1

IF You have one row for each stores, the it will be a matrix :

$|\ d_1\ \ e_1\ \ f_1\ \ |\\ |\ d_2\ \ e_2\ \ f_2\ \ | \ \ = Matrix\ A \\ |\ d_3\ \ e_3\ \ f_3\ \ |$

So matrix $A [X Y Z]^{T}$ gives overall profits for all stores.

$|\ \ X\ \ X\ \ X\ \ | \\ |\ \ Y\ \ Y\ \ Y\ \ |\ \ \ =\ PROF \\ |\ \ Z\ \ Z\ \ Z\ \ |$

Also,   $A\ \ PROF^{T} [ Identity ]$ gives profit from all stores FROM sales of ONE product. So we can find which product is giving maximum profit.

we can find sales of products each day from each stores and overall total. We can find stocks remaining also. So the quantities for replenishment will be know.

We can compare and find similarity or correlation between the customer purchases at one stores and another stores.

If one matrix is maintained for one stores, with different rows representing stock, or sales then average stock or sales on each day, can be found by adding all rows and dividing by number of rows.