Piecewise Linear Models

Transcription

1 6-6 Applied Operaions Research Piecewise Linear Models Deparmen of Mahemaics and Saisics The Universi of Melbourne This presenaion has been made in accordance wih he provisions of Par VB of he coprigh ac for he eaching purposes of he Universi. For use of sudens of he Universi of Melbourne enrolled in he subjec 6-6. Coprigh 8 b Heng-Soon Gan Some conens of his presenaion are adaped from ear 5 course noes for 6-6 Applied Operaions Research Deparmen of Mahemaics and Saisics The Universi of Melbourne (compiled b Prof Naashia Boland and Dr Renaa Soirov) 66 Applied Operaions Research (Deparmen of Mahemaics & Saisics Universi of Melbourne)

2 Piecewise Linear LP Models cos c c c e.g. overime coss a effecs e.g. economies of scale minimise conve/maimise concave piecewise linear funcion minimise non-conve or concave/maimise non-concave or conve piecewise linear funcion EASY! LP-model HARD: Ineger Programming Model 66 Applied Operaions Research (Deparmen of Mahemaics & Saisics Universi of Melbourne) producion producion

3 Producion Scheduling Machine Ma. # boles able o produce per monh a a Labour uilizaion per bole per monh b b Planning period: Demand for boles: d Labour availabili: L Iniial invenor: I Objecive: Deermine producion levels on each machine in each monh so as o mee demand and minimise maimum monhl flucuaion in labour uilizaion. 66 Applied Operaions Research (Deparmen of Mahemaics & Saisics Universi of Melbourne)

4 Producion Scheduling Primar decision variables: # boles produced on machine in monh # boles produced on machine in monh Invenor variables: # boles in invenor a end of monh Labour uilizaion variables: l labour uilizaion in monh l ma flucuaion ime 66 Applied Operaions Research (Deparmen of Mahemaics & Saisics Universi of Melbourne) 4

5 Producion Scheduling min ma l l s.. l b b I d d l L a a Objecive funcion is NOT LINEAR! 66 Applied Operaions Research (Deparmen of Mahemaics & Saisics Universi of Melbourne) 5

6 Producion Scheduling Inroduce new variable: m ma l l i.e. min s.. l m l l b b m ec Sill NOT LINEAR! 66 Applied Operaions Research (Deparmen of Mahemaics & Saisics Universi of Melbourne) 6

7 66 Applied Operaions Research (Deparmen of Mahemaics & Saisics Universi of Melbourne) 7 Producion Scheduling Consider his: LP Model: v w v w v w and ec b b l l l m l l m s m.. min

8 66 Applied Operaions Research (Deparmen of Mahemaics & Saisics Universi of Melbourne) 8 Ma. a Piecewise Linear Concave Funcion Eample: ( ) ma s f ( ) ( ) ( ) < < where f ( ) f

9 Ma. a Piecewise Linear Concave Funcion We can re-model his problem as an LP b replacing in all consrains where f ( ) and replacing f( ) in he objecive b / / /4 66 Applied Operaions Research (Deparmen of Mahemaics & Saisics Universi of Melbourne) 9

10 66 Applied Operaions Research (Deparmen of Mahemaics & Saisics Universi of Melbourne) Ma. a Piecewise Linear Concave Funcion New model is: ma s

11 Ma. a Piecewise Linear Concave Funcion For he new model o be correc we need he opimal soluion ( ) o saisf:. if < hen and. if < hen Can we be sure of his? We will show b conradicion. 66 Applied Operaions Research (Deparmen of Mahemaics & Saisics Universi of Melbourne)

12 66 Applied Operaions Research (Deparmen of Mahemaics & Saisics Universi of Melbourne) Ma. a Piecewise Linear Concave Funcion Suppose ( ) is opimal and < and >. Se ε min{ }. Noe ε >. Se ˆ ˆ ˆ ˆ ε ε ( ) feasible. mus be ˆ ˆ So ˆ ˆ ˆ Now ε ε

13 Ma. a Piecewise Linear Concave Funcion The objecive value of > ( ε ) ( ε ) objecive value of ( ˆ ˆ ) 4 ( ) is : ε Therefore ( ) is no opimal! This is a conradicion! So i mus be ha if < hen. The oher properies needed can be shown o hold b similar argumens. Eercise: Show ha if < hen and ha if < hen. THUS THE NEW LP MODEL IS VALID! 66 Applied Operaions Research (Deparmen of Mahemaics & Saisics Universi of Melbourne)

14 Ma. a Piecewise Linear Concave Funcion The LP model has opimal soluion / i.e. 5 / is he opimal soluion o he original problem. 66 Applied Operaions Research (Deparmen of Mahemaics & Saisics Universi of Melbourne) 4

15 Ma. a Piecewise Linear Concave Funcion Wha if f was convex insead? f ( ) If we do he same kind of LP model wih in he consrains where 4 and 4 we ge he opimal LP soluion o be / wih LP value /. 66 Applied Operaions Research (Deparmen of Mahemaics & Saisics Universi of Melbourne) 5

16 Ma. a Piecewise Linear Concave Funcion Bu hen 5 / which has objecive value 8 / in he original problem! The LP model is no using he correc objecive! Eercise: Check his b solving he LP. Hard eercise: Wha is he opimal soluion of he original problem? 66 Applied Operaions Research (Deparmen of Mahemaics & Saisics Universi of Melbourne) 6

17 Ma. a Piecewise Linear Concave Funcion In general a problem maimising a conve funcion CANNOT be re-modelled as an LP. Similarl minimising conve funcion which are piecewise linear can be solved using an LP model bu minimising concave funcion CANNOT be. For general funcions: Minimising Maimising CONVEX EASY HARD CONCAVE HARD EASY NEITHER HARD HARD 66 Applied Operaions Research (Deparmen of Mahemaics & Saisics Universi of Melbourne) 7