___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry Cutting Stock Problem Linear Programming

___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry Applications Cutting Stock Problem Raw product Final product Loss

Linear Programming ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry Applications Cutting Stock Problem Raw product 2 final products Cutting patterns

Linear Programming ___________________________________________________________________________ Quantitative Methods of Management  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

Linear Programming ___________________________________________________________________________ Quantitative Methods of Management  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

Linear Programming ___________________________________________________________________________ Quantitative Methods of Management  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)

Linear Programming ___________________________________________________________________________ Quantitative Methods of Management  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 Screw Time 30 min 60 min 8 hours per day Wooden board 1.1 m Wooden board 1.4 m --150

Linear Programming ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry Applications Cutting Stock Problem Example – John Starling & Sons Board 1.1 m Board 1.4 m Pattern No Board 30 cm Board 25 cm Loss Variable x1x1x1x1 x2x2x2x2 x3x3x3x3 x4x4x4x4 x5x5x5x5 x6x6x6x6 x7x7x7x7 x8x8x8x8 x9x9x9x9 Cutting patterns & Decision variables Bird table Bird box x 10 x 11

Linear Programming ___________________________________________________________________________ Quantitative Methods of Management  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 Optimal solution Total revenue CZK

Linear Programming ___________________________________________________________________________ ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry Quantitative Methods of Management  Jan Fábry Applications  Cutting Stock Problem  Production Process Models  Portfolio Selection Problem  Marketing Research  Blending Problems  Transportation Problem  Assignment Problem Distribution Problems

___________________________________________________________________________ ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry Quantitative Methods of Management  Jan Fábry Transportation Problem Linear Programming

___________________________________________________________________________ ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry Quantitative Methods of Management  Jan Fábry Applications Transportation Problem SourcesDestinations Shipments

Linear Programming ___________________________________________________________________________ ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry Quantitative Methods of Management  Jan Fábry Applications Transportation Problem SourcesDestinations  supply =  demand Balanced Unbalanced

Linear Programming ___________________________________________________________________________ ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry Quantitative Methods of Management  Jan Fábry Applications Transportation Problem SourcesDestinations  supply >  demand Dummy destination  supply <  demand Dummy source

Linear Programming ___________________________________________________________________________ ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry Quantitative Methods of Management  Jan Fábry Applications Transportation Problem Benešov Humpolec Jihlava Pelhřimov Tábor Example – Star Chips, Inc. Warehouses Humpolec Pelhřimov Subsidiaries Benešov Jihlava Tábor

Linear Programming ___________________________________________________________________________ ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry Quantitative Methods of Management  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

Linear Programming ___________________________________________________________________________ ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry Quantitative Methods of Management  Jan Fábry Applications Transportation Problem Example – Star Chips, Inc. BenešovJihlavaTábor Weekly supply Humpolec Pelhřimov Weekly demand Supply, demand, unit shipping cost

Linear Programming ___________________________________________________________________________ ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry Quantitative Methods of Management  Jan Fábry Applications Transportation Problem Example – Star Chips, Inc. BenešovJihlavaTáborDummy Weekly supply Humpolec Pelhřimov Weekly demand Dummy destination 150

Linear Programming ___________________________________________________________________________ ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry Quantitative Methods of Management  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

Linear Programming ___________________________________________________________________________ ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry Quantitative Methods of Management  Jan Fábry Applications Transportation Problem Example – Star Chips, Inc. BenešovJihlavaTáborRemain Humpolec Pelhřimov Optimal solution Total weekly cost CZK