Question

John’s Construction has three projects under way. Each project requires a regular supply of gravel, which can be obtained from three quarries. Shipping costs differ from location to location, and are summarized in the table.

From:	Job 1	Job 2	Job 3	Tonnage allowance
Quarry A	$9	$8	$7	1500
Quarry B	$7	$11	$6	1750
Quarry C	$4	$3	$12	2750
Job Requirements (tonnes)	2000	3000	1000	6000

Formulate a transportation model (but do not attempt to solve it) which could be used to determine the amount of gravel to be shipped from each quarry to the various job sites.

Answer 1

Let the amount of gravel from Quarry A to Job 1 (In tonnes) =

Let the amount of gravel from Quarry A to Job 2 (In tonnes) =

Let the amount of gravel from Quarry A to Job 3 (In tonnes) = ​​​​​​​

Let the amount of gravel from Quarry B to Job 1 (In tonnes) =

Let the amount of gravel from Quarry B to Job 2 (In tonnes) = ​​​​​​​

Let the amount of gravel from Quarry B to Job 3 (In tonnes) = ​​​​​​​

Let the amount of gravel from Quarry C to Job 1 (In tonnes) =

Let the amount of gravel from Quarry C to Job 2 (In tonnes) = ​​​​​​​

Let the amount of gravel from Quarry C to Job 3 (In tonnes) =

Our objective is to minimise the shipping costs

Therefore we can formulate the objective as

Minimise

Subject to the constraints

(Tonnage allowance in quaary A)

(Tonnage allowance in quaary B)

​​​​​​​ (Tonnage allowance in quaary C)

(Job requirement in job 1)

(Job requirement in job 2)

(Job requirement in job 3)

​​​​​​​ (Non negativity)