1. ___________________________________________________________________________ Operations Research  Jan Fábry Cutting Stock Problem Linear Programming

2. ___________________________________________________________________________ Operations Research  Jan Fábry Applications Cutting Stock Problem Raw product Final product Loss

3. Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry Applications Cutting Stock Problem Raw product 2 final products Cutting patterns

4. Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry Applications Cutting Stock Problem  Raw products  Objective - pipes - roles of paper or textile - bars - wooden sticks or bars - steel plates  minimization of total loss or number of cut raw products  maximization of number of assembled products

5. Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry Applications Cutting Stock Problem  Decision variables ~ patterns The value of the variable determines the number of the raw products being cut/sliced according to corresponding pattern  Finding of all possible cutting patterns

6. Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry Applications Cutting Stock Problem Example – John Starling & Sons  2 products:bird table bird box  Special collection for the exhibition (with possible sales) in 20 days  Inputs:wooden boards screws time Objective: maximize total revenue (all production will be sold)

7. Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry Applications Cutting Stock Problem Example – John Starling & Sons Bird table Bird box Available Price 260 CZK 570 CZK - Wooden board 30 cm 12unknown Wooden board 25 cm 14unknown Screw816 3 000 Time 30 min 60 min 8 hours per day Wooden board 1.1 m --500 Wooden board 1.4 m --150

8. Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry Applications Cutting Stock Problem Example – John Starling & Sons Board 1.1 m Board 1.4 m Pattern No 123456789 Board 30 cm 321-4321- Board 25 cm -234-2345 Loss20-51020-51015 Variable x1x1x1x1 x2x2x2x2 x3x3x3x3 x4x4x4x4 x5x5x5x5 x6x6x6x6 x7x7x7x7 x8x8x8x8 x9x9x9x9 Cutting patterns & Decision variables Bird table Bird box x 10 x 11

9. Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry Applications Cutting Stock Problem Example – John Starling & Sons Board 1.1 m Board 1.4 m Bird table Bird box x1x1x1x1 x2x2x2x2 x3x3x3x3 x4x4x4x4 x5x5x5x5 x6x6x6x6 x7x7x7x7 x8x8x8x8 x9x9x9x9 x 10 x 11 06500480001020160 Optimal solution Total revenue 91 200 CZK

10. Linear Programming ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Applications  Cutting Stock Problem  Production Process Models  Portfolio Selection Problem  Marketing Research  Blending Problems  Transportation Problem  Assignment Problem Distribution Problems

11. ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Transportation Problem Linear Programming

12. ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Applications Transportation Problem SourcesDestinations Shipments

13. Linear Programming ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Applications Transportation Problem SourcesDestinations  supply =  demand Balanced Unbalanced

14. Linear Programming ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Applications Transportation Problem SourcesDestinations  supply >  demand Dummy destination  supply <  demand Dummy source

15. Linear Programming ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Applications Transportation Problem Benešov Humpolec Jihlava Pelhřimov Tábor Example – Star Chips, Inc. Warehouses Humpolec Pelhřimov Subsidiaries Benešov Jihlava Tábor

16. Linear Programming ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Applications Transportation Problem Example – Star Chips, Inc. Warehouses Humpolec Pelhřimov Subsidiaries Benešov Jihlava Tábor Supply of potatoes Requirements Objective: satisfy requirements with minimal total shipping cost

17. Linear Programming ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Applications Transportation Problem Example – Star Chips, Inc. BenešovJihlavaTábor Weekly supply Humpolec33025035070 Pelhřimov30024025080 Weekly demand 456035 Supply, demand, unit shipping cost 140 150

18. Linear Programming ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Applications Transportation Problem Example – Star Chips, Inc. BenešovJihlavaTáborDummy Weekly supply Humpolec330250350070 Pelhřimov300240250080 Weekly demand 45603510 Dummy destination 150

19. Linear Programming ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Applications Transportation Problem Example – Star Chips, Inc. BenešovJihlavaTáborDummy Humpolec x 11 x 12 x 13 x 14 Pelhřimov x 21 x 22 x 23 x 24 Decision variables

20. Linear Programming ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Applications Transportation Problem Example – Star Chips, Inc. BenešovJihlavaTáborRemain Humpolec-60-10 Pelhřimov45-35- Optimal solution Total weekly cost 37 250 CZK