Inventory Control Red Bookstore wants to ship books from its warehouses in Brooklyn and Queens to its stores, one on Long Island and one in Manhattan. Its warehouse in Brooklyn has 1,000 books, and its warehouse in Queens has 2,000. Each store orders 1,500 books. It costs $5 to ship each book from Brooklyn to Long Island and $1 to ship each book from Brooklyn to Manhattan. It costs $4 to ship each book from Queens to Long Island and $2 to ship each book from Queens to Manhattan.

a. If Red bas a transportation budget of $9,000 and is willing to spend all of it, bow many books should Red ship from each warehouse to each store in order to fill all the orders?

b. Is there a way of doing this for less money? [HINT: See Example 4.]

To calculate: The number of books should red ship from warehouse to each store in order to fulfil all the orders, if red has transportation budget of $9000 with the help of the provided information.

Given Information:

Red has transportation budget of $9000.

Warehouse in Brooklyn has 1000 books and warehouse in queens has 2000.

Both tsores in long island and Manhattan orders 1500 books.

It costs $5 to ship each book from Brooklyn to long island and $1 to ship each book from Brooklyn to Manhattan.

It cost $4 to ship each book from queens to long island and $2 to ship each book from queens to Manhattan.

Formula Used:

Elementary row operations

Type 1: Replacing the row Ri by aRi, where a is a nonzero number.

Type 2: Replacing the row Ri by aRi±bRj, where a is a nonzero number.

Gauss Jordan reduction method:

Step 1: First clear the fractions or decimals if any, using operations of type 1.

Step 2: Select the first nonzero element of the first row as pivot.

Step 3: Use the pivot to clear its column using operations of type 2.

Step 4: Select the first nonzero element in the second row a pivot and clear its column.

Step 5: Turn all the selected pivot elements into a 1 using operations of type 1.

Calculation:

Let,

x be the number of books sent from Brooklyn to Long Island.

y be the number of books sent from Brooklyn to Manhattan.

z be the number of books sent from Queens to Long Island.

w be the number of books sent from Queens to Manhattan.

The given information can be depicted from the diagram drawn.

Firstly, total number of books is 3000 and the order is of 3000 books thus both the warehouses need to clear their stocks.

Therefore, for Brooklyn stock

x+y=1000

For Queens stock

z+w=2000

For long island store,

x+z=1500

For Manhattan store,

y+w=1500

Also, the transportation budget is $9000.

Therefore, summation of product of shipment cost per book and quantity of book is the total budget.

5x+y+4z+2w=9000

Consider the system of equation,

x+y=1000x+z=1500z+w=2000y+w=1500

And

5x+y+4z+2w=9000

The augmented matrix for the given system of equations is,

[1010150001011500110010000011200051429000]

Apply Gauss Jordan reduction method to get the solution of the given system of equation.

Next pivot the first nonzero element of the first row and clear its column.

Perform the operation R3→R3−R1, R5→R5−5R1

[1010150001011500110010000011200051429000]→[101015000101150001−10−5000011200001−121500]

Next pivot the first nonzero element of the second row and clear its column.

Perform the operation, R3→R3−R2 and R5→R5−R2

[101015000101150001−10−5000011200001−121500]→[101015000101150000−1−1−20000011200000−110]

Next pivot the first nonzero element of the third row and clear its column

To calculate: The number of books should red ship from warehouse to each store in order to fulfil all the orders, to minimize the cost with the help of provided information.