Integer Programming. All Integer Linear Programming

Transcription

1 Integer Programmng All Integer Lnear Programmng

2 Objectves To dscuss the need for Integer Programmng (IP) To dscuss about the types of IP To explan Integer Lnear Programmng (ILP) To dscuss the Gomory Cuttng Plane method for solvng ILP Graphcally Theoretcally 2

3 Introducton 3 In many practcal problems, the values of decson varables are constraned to take only nteger values For example, n mnmzaton of labor needed n a project, the number of labourers should be an nteger value By roundng off a real value to an nteger value have several fundamental problems lke Rounded solutons may not be feasble Even f the solutons are feasble, the objectve functon gven by the rounded off solutons may not be the optmal one Fnally, even f the above two condtons are satsfed, checkng all the rounded-off solutons s computatonally expensve (2 n possble solutons to be consdered for an n varable problem) Ths demands the need for Integer Programmng

4 Types of IP 4 All Integer Programmng: All the varables are restrcted to take only nteger values Dscrete Programmng: All the varables are restrcted to take only dscrete values Mxed Integer or Dscrete Programmng: Only some varables are restrcted to take nteger or dscrete values Zero One Programmng: Varables are constraned to take values of ether zero or

5 Integer Lnear Programmng (ILP) An extenson of lnear programmng (LP) Addtonal constrant: Varables should be nteger valued Standard form of an ILP: subject max to c T X AX b X 0 X must be nteger valued Assocated lnear program, droppng the nteger restrctons, s called lnear relaxaton (LR) 5

6 Checks for ILP: Mnmzaton: Optmal objectve value for LR s less than or equal to the optmal objectve for ILP Maxmzaton: Optmal objectve value for LR s greater than or equal to that of ILP If LR s nfeasble, then ILP s also nfeasble If LR s optmzed by nteger varables, then that soluton s feasble and optmal for IP 6

7 All Integer Programmng 7 Most popular method: Gomory s Cuttng Plane method Orgnal feasble regon s reduced to a new feasble regon by ncludng addtonal constrants such that all vertces of the new feasble regon are now nteger ponts Thus, an extreme pont of the new feasble regon becomes an optmal soluton after accountng for the nteger constrants Consder the optmzaton problem Maxmze subjectto Z = 3x 2x 3x x, x + 9x 2 x 2 + x ; x and x 2 are ntegers

8 Graphcal Illustraton Graphcal soluton for the lnear approxmaton (neglectng the nteger requrements) s shown n fgure 8

9 Graphcal Illustraton contd. x = 45, x2 Optmal value of Z =7 4 7 and the soluton s 7 7 Red dots n the fgure show the feasble solutons accountng for the nteger requrements These ponts are called nteger lattce ponts Now to reduce the orgnal feasble regon to a new feasble regon (consderng x and x 2 as ntegers) s done by ncludng addtonal constrants Graphcal soluton for the IP s shown n fgure below = Two addtonal constrants (MN and OP) are ncluded so that the orgnal feasble regon ABCD s reduced to a new feasble regon AEFGCD 3 3 9

10 Graphcal Illustraton contd. Optmal value of ILP s Z =5 and the soluton s x = 4, x 2 = 3 0

11 Generaton of Gomory Constrants Consder the fnal tableau of an LP problem consstng of n basc varables (orgnal varables) and m non basc varables (slack varables) The basc varables are represented as x (=,2,,n) and the non basc varables are represented as y j (j=,2,,m).

12 Generaton of Gomory Constrants contd. x Pck the varable havng the hghest fractonal value. In case of a te, choose arbtrarly any varable as x From the th equaton, x = b m j= c j y Expressng both b j and c j as an nteger part plus a fractonal part, b c j = b = c j + β + α j j 2

13 Generaton of Gomory Constrants contd. b, c denote the nteger part and β j, denote the fractonal part for whch 0 < and α j Thus, the equaton becomes, β m j= α j y j = x b m j= ( ) c j y β ( 0 α < ) < j j 3

14 Generaton of Gomory Constrants contd. Consderng the nteger reurements, the RHS of the equaton also should be an nteger. Thus, we have m β < αj y j β j= 4 Hence, the constrant can be expressed as, β m j= α j y j 0 After ntroducng a slack varable s, the fnal Gomory constrant can be m wrtten as, s j= α j y j = β

15 Procedure for solvng All-Integer LP Solve the problem as an ordnary LP problem neglectng the nteger requrements. If the optmum values of the varables are not ntegers, then choose the basc varable whch has the largest fractonal value, and generate Gomory constrant for that varable. Insert a new row wth the coeffcents of ths constrant, to the fnal tableau of the ordnary LP problem. Solve ths by applyng the dual smplex method 5

16 Procedure for solvng All-Integer LP contd. Check whether the new soluton s all-nteger or not. If all values are not ntegers, then a new Gomory constrant s developed for the non-nteger valued varable from the new smplex tableau and the dual smplex method s appled agan. The process s contnued untl An optmal nteger soluton s obtaned or It shows that the problem has no feasble nteger soluton. 6

17 Thank You 7