1. Revision Linear Optimisation

2. A miller can buy wheat from three suppliers: Airey Farm, Berry Farm and Cherry Farm. In each case the wheat is contaminated with two things – bran and husks. When combined, the wheat must contain no more than 5% bran and no more than 4% husks. The miller wishes to make 50 tonnes of wheat in total, by purchasing from each farm. Each farm’s wheat contains the following amounts of bran and husks.

3. Identify the decision variables Let x = tonnes of wheat purchased from Airey Farm Let y = tonnes of wheat purchased from Berry Farm Let z = tonnes of wheat purchased from Cherry Farm Write z in terms of x and y

4. Define the constraint due to bran 3% of x + 5% of y + 7% of z is less than or equal to 5% of 50 tonnes.

5. Define the constraint due to husks 5% of x + 2% of y + 6% of z is less than or equal to 4% of 50 tonnes.

6. Non-negativity

7. Supply of wheat from Cherry Farm must be non-negative too.

8. Graph the solution region 100 50 25 50

9. Find the vertices of the solution region 100 50 25 50 (0, 50) (14.29, 21.43) (33.33, 16.67)

10. Define the objective function Cost = $70x + $60y + $40z Cost = $70x + $60y + $40(50 - x - y) Cost = 30x + 20y + 2000

11. Substitute each vertex’s coordinates into the objective function Cost = 30x + 20y + 2000 A(0, 50) Cost = $3000 B(14.29, 21.43) Cost = $2857.30 C(33.33, 16.67) Cost = $3333.30 B gives the minimum costs so purchase 14.29 from Airey, 21.43 from Berry and 14.28 from Cherry

13. Define the decision variables Let x = number of bicycles shipped to distributor X from factory A Therefore 70 - x is the number shipped to X from factory B Let y = number of bicycles shipped to distributor Y from factory A Therefore 90 - y is the number shipped to y from factory B

14. Summarise the decision variables, supply and demand in a table Factory AFactory BTotal demand from each distributor Distributor X Distributor Y Total supply from each factory

15. Summarise the decision variables, supply and demand in a table Factory AFactory BTotal demand from each distributor Distributor Xx70 - x70 Distributor Yy90 - y90 Total supply from each factory 100150

16. Define constraints based on the fact that all shipments must be non-negative amounts Factory A:

17. Define constraints based on the fact that all shipments must be non-negative amounts Factory B:

18. Define constraints based on the fact that all shipments must be non-negative amounts Non-negativity:

19. Graph the solution region 100

20. Determine the vertices 100 (0, 10) (0, 90) (10, 90) (70, 30) (70, 0) (10, 0)

21. Determine the objective function Factory A to distributor X: x bicycles at $6 Factory A to distributor Y: y bicycles at $3 Factory B to distributor X: 70 - x bicycles at $7 Factory B to distributor Y: 90 - y bicycles at $5 Total cost =

22. Determine the cost for each vertex A(0, 10) Cost = 920 B(0, 90) Cost = 760 C(10, 90) Cost = 750 D(70, 30) Cost = 810 E(70, 0) Cost = 870 F(10, 0) Cost = 930 Minimum costs occur when x = 10 and y = 90

23. Amount to ship from each factory Ship 10 from factory A to distributor X Ship 90 from factory A to distributor Y Ship 60 from factory B to distributor X Ship 0 from factory B to distributor Y