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Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203

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BuyFindarrow_forward

Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203
Textbook Problem

Inventory Control The Tubular Ride Boogie Board Company has manufacturing plants in Tucson, Arizona, and Toronto, Ontario. You have been given the job of coordinating distribution of the latest model, the Gladiator, to outlets in Honolulu and Venice Beach. The Tucson plant, when operating at full capacity, can manufacture 620 Gladiator boards per week, while the Toronto plant, beset by labor disputes, can produce only 410 boards per week. The outlet in Honolulu orders 500 Gladiator boards per week, while Venice Beach orders 530 boards per week. Transportation costs are as follows:

Tucson to Honolulu: $10 per board; Tucson to Venice

Beach: $5 per board.

Toronto to Honolulu: $20 per board; Toronto to Venice Beach: $10 per board.

a. Assuming that you wish to fill all orders and ensure full-capacity production at both plants, is it possible to meet a total transportation budget of $10,200? If so, how many Gladiator boards are shipped from each manufacturing plant to each distribution outlet?

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

(a)

To determine

To calculate: The number of gladiator boards shipped from each manufacturing plant to each outlet, if the transportation budget is of $10200 with the help of provided information.

Explanation

Given Information:

The transportation budget of Board Company is of $10200.

Plant in Tucson and Toronto manufacture 620 and 410 boards per week respectively.

The transportation cost from Tucson to Honolulu and Venice is $10 and $5 per board.

The transportation cost from Toronto to Honolulu and Venice beach is $20 and $10 per board.

Also, the outlet in Honolulu and Venice beach orders 500 and 530 Gladiators boards each respectively.

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 boards sent from Tucson to Honolulu.

y be the number of boards sent from Tucson to Venice Beach.

z be the number of boards sent from Toronto to Honolulu.

w be the number of boards sent from Toronto to Venice Beach.

The given information can be depicted from the diagram drawn.

For shipment of boards from Tucson,

x+y=620

For shipment of boards from Toronto,

z+w=410

For the ordered quantity by Honolulu,

x+z=500

For the ordered quantity by Venice Beach,

y+w=530

Also, the transportation budget is $10200.

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

10x+5y+20z+10w=10200

Consider the system of equation,

x+y=620z+w=410x+z=500y+w=530

And

10x+5y+20z+10w=10200

The augmented matrix for the given system of equations is,

[1100620001141010105000101530105201010200]

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

Perform the operation R515R5,

[1100620001141010105000101530105201010200][110062000114101010500010153021422040]

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

Perform the operation R3R3R1, R5R52R1

[110062000114101010500010153021422040][11006200011410011012001015300142800]

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

Perform the operation, R3R3R2 and R5R54R2

[11006200011410011012001015300142800][11006200011410010153001015300102840]

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

(b)

To determine

To calculate: The number of gladiator boards shipped from each manufacturing plant to each outlet to minimize the transportation cost.

Plant in Tucson and Toronto manufacture 620 and 410 boards per week respectively.

The transportation cost from Tucson to Honolulu and Venice is $10 and $5 per board.

The transportation cost from Toronto to Honolulu and Venice beach is $20 and $10 per board.

Also, the outlet in Honolulu and Venice beach orders 500 and 530 Gladiators boards each respectively.

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