Linear Programming Cutting Stock Problem Operations Research Jan
Linear Programming Cutting Stock Problem ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Cutting Stock Problem Final product Raw product Loss ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Cutting Stock Problem Raw product 2 final products Cutting patterns ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Cutting Stock Problem Ø Raw products - pipes - roles of paper or textile - bars - wooden sticks or bars - steel plates Ø Objective minimization of total loss or number of cut raw products maximization of number of assembled products ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Cutting Stock Problem Ø Decision variables ~ patterns The value of the variable determines the number of the raw products being cut/sliced according to corresponding pattern Ø Finding of all possible cutting patterns ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Cutting Stock Problem Example – John Starling & Sons Ø 2 products: bird table bird box Ø Inputs: wooden boards screws time Ø Special collection for the exhibition (with possible sales) in 20 days Objective: maximize total revenue (all production will be sold) ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Cutting Stock Problem Example – John Starling & Sons Bird table Bird box Available Price 260 CZK 570 CZK - Wooden board 30 cm 1 2 unknown Wooden board 25 cm 1 4 unknown Screw 8 16 3 000 Time 30 min 60 min 8 hours per day Wooden board 1. 1 m - - 500 Wooden board 1. 4 m 150 ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Cutting Stock Problem Example – John Starling & Sons Cutting patterns & Decision variables Board 1. 1 m Board 1. 4 m Pattern No 1 2 3 4 5 6 7 8 9 Board 30 cm 3 2 1 - 4 3 2 1 - Board 25 cm - 2 3 4 5 Loss 20 - 5 10 15 Bird table box Variable x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Cutting Stock Problem Example – John Starling & Sons Optimal solution Board 1. 1 m Bird table box Board 1. 4 m x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 0 65 0 0 48 0 0 0 102 0 160 Total revenue 91 200 CZK ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Ø Production Process Models Ø Blending Problems Ø Marketing Research Ø Portfolio Selection Problem Distribution Problems Ø Cutting Stock Problem Ø Transportation Problem Ø Assignment Problem ______________________________________ Operations Research Jan Fábry
Linear Programming Transportation Problem ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Transportation Problem Shipments Sources Destinations ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Transportation Problem Sources Destinations supply = demand Balanced supply = demand Unbalanced ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Transportation Problem Sources supply Destinations > demand Dummy destination supply < demand Dummy source ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Transportation Problem Example – Star Chips, Inc. Benešov Humpolec Tábor Jihlava Pelhřimov Warehouses Humpolec Pelhřimov Subsidiaries Benešov Jihlava Tábor ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Transportation Problem Example – Star Chips, Inc. Warehouses Humpolec Pelhřimov Supply of potatoes Subsidiaries Benešov Jihlava Tábor Requirements Objective: satisfy requirements with minimal total shipping cost ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Transportation Problem Example – Star Chips, Inc. Supply, demand, unit shipping cost Benešov Jihlava Tábor Weekly supply Humpolec 330 250 350 70 Pelhřimov 300 240 250 80 Weekly demand 45 60 35 150 140 ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Transportation Problem Example – Star Chips, Inc. Dummy destination Benešov Jihlava Tábor Dummy Weekly supply Humpolec 330 250 350 0 70 Pelhřimov 300 240 250 0 80 Weekly demand 45 60 35 10 150 ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Transportation Problem Example – Star Chips, Inc. Decision variables Benešov Jihlava Tábor Dummy Humpolec x 11 x 12 x 13 x 14 Pelhřimov x 21 x 22 x 23 x 24 ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Transportation Problem Example – Star Chips, Inc. Optimal solution Benešov Jihlava Tábor Remain Humpolec - 60 - 10 Pelhřimov 45 - 35 - Total weekly cost 37 250 CZK ______________________________________ Operations Research Jan Fábry
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