Optzaton Methods: Integer Prograng Integer Lnear Prograng Module Lecture Notes Integer Lnear Prograng Introducton In all the prevous lectures n lnear prograng dscussed so far, the desgn varables consdered are supposed to take any real value. However n practcal probles lke nzaton of labor needed n a proect, t akes lttle sense n assgnng a value lke 5.6 to the nuber of labourers. In stuatons lke ths, one natural dea for obtanng an nteger soluton s to gnore the nteger constrants and use any of the technques prevously dscussed and then round-off the soluton to the nearest nteger value. However, there are several fundaental probles n usng ths approach:. The rounded-off solutons ay not be feasble.. The obectve functon value gven by the rounded-off solutons (even f soe are feasble) ay not be the optal one. 3. Even f soe of the rounded-off solutons are optal, checkng all the rounded-off solutons s coputatonally epensve ( for an n varable proble) n possble round-off values to be consdered Types of Integer Prograng When all the varables n an optzaton proble are restrcted to take only nteger values, t s called an all nteger prograng proble. When the varables are restrcted to take only dscrete values, the proble s called a dscrete prograng proble. When only soe varable values are restrcted to take nteger or dscrete, t s called ed nteger or dscrete prograng proble. When the varables are constraned to take values of ether zero or, then the proble s called zero one prograng proble. Integer Lnear Prograng Integer Lnear Prograng (ILP) s an etenson of lnear prograng, wth an addtonal restrcton that the varables should be nteger valued. The standard for of an ILP s of the for, D Nagesh Kuar, IISc, Bangalore ML
Optzaton Methods: Integer Prograng Integer Lnear Prograng a subect to c T X AX b X 0 X ust be nteger valued The assocated lnear progra droppng the nteger restrctons s called lnear relaaton LR. Thus, LR s less constraned than ILP. If the obectve functon coeffcents are nteger, then for nzaton, the optal obectve for ILP s greater than or equal to the rounded-off value of the optal obectve for LR. For azaton, the optal obectve for ILP s less than or equal to the rounded-off value of the optal obectve for LR. For a nzaton ILP, the optal obectve value for LR s less than or equal to the optal obectve for ILP and for a azaton ILP, the optal obectve value for LR s greater than or equal to that of ILP. If LR s nfeasble, then ILP s also nfeasble. Also, f LR s optzed by nteger varables, then that soluton s feasble and optal for IP. A ost popular ethod used for solvng all-nteger and ed-nteger lnear prograng probles s the cuttng plane ethod by Goory (Goory, 95). Goory s Cuttng Plane Method for All Integer Prograng Consder the followng optzaton proble. Maze subect to Z = 3 3, + 9 + 0 6 45 and are ntegers The graphcal soluton for the lnear relaaton of ths proble s shown below. D Nagesh Kuar, IISc, Bangalore ML
Optzaton Methods: Integer Prograng Integer Lnear Prograng 3 6 5 A 4 3 3 + 9 45 B ( 4 5,3 3 ) Z = 4 6 0 D C 0 3 4 5 6 It can be seen that the soluton s = 5, 4 = 3 3 and the optal value of Z = 4. The feasble solutons accountng the nteger constrants are shown by red dots. These ponts are called nteger lattce ponts. The orgnal feasble regon s reduced to a new feasble regon by ncludng soe addtonal constrants such that an etree pont of the new feasble regon becoes an optal soluton after accountng for the nteger constrants. The graphcal soluton for the eaple prevously dscussed takng and as ntegers are shown below. Two addtonal constrants (MN and OP) are ncluded so that the orgnal feasble regon ABCD s reduced to a new feasble regon AEFGCD. Thus the soluton for ths ILP s =, 3 and the optal value s Z = 5. 4 = D Nagesh Kuar, IISc, Bangalore ML
Optzaton Methods: Integer Prograng Integer Lnear Prograng 4 Addtonal constrants M 6 5 A O 4 3 E F (4,3) B ( 4 5,3 3 ) Z = 4 G Z = 5 P D C N 0 0 3 4 5 6 Goary proposed a systeatc ethod to develop these addtonal constrants known as Goory constrants. Generaton of Goory Constrants: Let the fnal tableau of an LP proble consst of n basc varables (orgnal varables) and non basc varables (slack varables) as shown n the table below. The basc varables are represented as (=,,,n) and the non basc varables are represented as y (=,,,). D Nagesh Kuar, IISc, Bangalore ML
Optzaton Methods: Integer Prograng Integer Lnear Prograng 5 Bass Z Table Varables n y y y y br Z 0 0 0 0 c c c c b 0 0 0 0 c c c c b 0 0 0 0 c c c c b 0 0 0 0 c 3 c 3 c 3 c 3 b n 0 0 0 0 c 4 c 4 c 4 c 4 b n Choose any basc varable wth the hghest fractonal value. If there s a te between two basc varables, arbtrarly choose any of the as th. Then fro the equaton of table, = b c = y..() Epress both b and c as an nteger value plus a fractonal part. b c = b = c + β + α...()...(3) where b, c denote the nteger part and β, α denote the fractonal part. β wll be a strctly postve fracton ( 0 < β < ) and α s a non-negatve fracton ( 0 < ) Substtutng equatons () and (3) n (), equaton () can be wrtten as = α. β α y = b c y.(4) = For all the varables and y to be ntegers, the rght hand sde of equaton (4) should be an nteger. β α y = nteger (5) = D Nagesh Kuar, IISc, Bangalore ML
Optzaton Methods: Integer Prograng Integer Lnear Prograng 6 Snce α are non-negatve ntegers and y are non-negatve ntegers, the ter α wll always be a non-negatve nuber. Thus we have, β < α y β.(6) = Hence the constrant can be epressed as β α y 0.() = = y By ntroducng a slack varable s (whch should also be an nteger), the Goory constrant can be wrtten as s = α y = β.(8) General procedure for solvng ILP:. Solve the gven proble as an ordnary LP proble neglectng the nteger constrants. If the optu values of the varables are ntegers tself, then there s nothng ore to be done.. If any of the basc varables has fractonal values, ntroduce the Goory constrants as dscussed n the prevous secton. Insert a new row wth the coeffcents of ths constrant, to the fnal tableau of the ordnary LP proble (Table ). 3. Solve ths by applyng the dual sple ethod. Snce the value of y = 0 n Table, the Goory constrant equaton becoes s = β whch s a negatve value and thus nfeasble. Dual sple ethod s used to obtan a new optal soluton that satsfes the Goory constrant. 4. Check whether the new soluton s all-nteger or not. If all values are not ntegers, then a new Goory constrant s developed fro the new sple tableau and the dual sple ethod s appled agan. 5. Ths process s contnued untl an optal nteger soluton s obtaned or t shows that the proble has no feasble nteger soluton. D Nagesh Kuar, IISc, Bangalore ML
Optzaton Methods: Integer Prograng Integer Lnear Prograng Thus, the fundaental dea behnd cuttng planes s to add constrants to a lnear progra untl the optal basc feasble soluton takes on nteger values. Goory cuts have the property that they can be generated for any nteger progra, but has the dsadvantage that the nuber of constrants generated can be enorous dependng upon the nuber of varables. D Nagesh Kuar, IISc, Bangalore ML