Introduction to linear programming

Transcription

1 Introducton to lnear programmng

2 Overvew Lnear programmng Mathematcal and lnear programs Lnear programmng and resoluton methods; Modelng n (nteger) lnear programmng; Schedulng problems; Optmzaton problems n logstcs and transportaton Examples exercses. Integer lnear programmng: arborescent methods examples exercses.

3 A lnear program A mathematcal program s an optmzaton problem wth an obectve (or optmzaton) functon of n-varables satsfyng m constrants. If at least one constrant or the obectve functon are not lnear t s called a non-lnear program otherwse t s a lnear program. Un program s convex f the optmzaton functon s convex and the doman defned by the constrants s also convex : then any local optmum s also a global optmum and t s acheved at some extreme pont of the doman. A real-valued functon f(x) defned on an nterval s called convex f the graph of the functon les below the lne segment onng any two ponts of the graph. A lnear program s a convex program.

4 A lnear program There can be dstngushed: Decson varables Obectve (or optmzaton) functon Constrants; Doman of varables;

5 A lnear program General case Suppose that a lnear program s composed of : - The obectve functon of n varables x (maxmzaton) ; - All varables take postve values. - The constrants are lnear functons bounded by some constant That s : Interpretaton : n actvtes m resources Max s. c. A... A x 21 m1 1 x x x 1 2 A 1 11 A 1 A x m2 n 1 A 2 n z c x x x 12 2 x 0. 2 x... A... A 2n... A x mn n x 1n b n x n 2 b b ; m ; 1 ;

6 Geometrcal nterpretaton max: Z = 2x 1 + x 2 Subect to: x 1 +2x 2 7 x 1 +x 2 5 x 1 1 x 1 4 x 0

7 Introducton to lnear programmng Lnear programmng (LP) s a technque for optmzaton of a lnear obectve functon of varables x 1 x 2 x n subect to lnear equalty and lnear nequalty constrants. How to solve lnear programmng : The smplex algorthm ( ) developed by George Dantzg solves LP problems by constructng an admssble soluton at a vertex of the polyhedron and then walkng along edges of the polyhedron to vertces wth successvely hgher values of the obectve functon untl the optmum s reached. (CPLEX EXPRESS-MP etc.). Alternatve methods : the ellpsod method by Leond Khachyan n 1979 In 1984 N. Karmarkar proposed a new nteror pont proectve method for lnear programmng. (Karmarkar's algorthm)

8 Smplex algorthm : bascs The doman formed by lnear constrants gves a polyhedron and the search for optmal solutons can be restrcted to extreme ponts. The smplex method explores extreme ponts and mproves the value of optmzaton functon n each step. Advantages : Effcency (not polynomal but polynomal-tme n average). Nce theoretcal propertes (dualty).

9 Smplex algorthm : dualty Prmal : Max c 1 x 1 + c 2 x 2 s.t.: A 11 x 1 + A 12 x 2 b 1 ; A 21 x 1 + A 22 x 2 = b 2 ; x 1 0 and x 2 unsgned; Dual : Mn b 1 y 1 + b 2 y 2 s.t.: A 11 y 1 + A 21 y 2 c 1 ; A 12 y 1 + A 22 x 2 = c 2 ; y 1 0; y 2 unsgned; Transformaton rules : Take the transposed matrx A the rght hand terms become coeffcents of the obectve functon etc.

10 Smplex algorthm: dualty Proposton. If x s feasble for the prmal (maxmum) problem and f y s feasble for ts dual then ctx ytb. Proof. ctx ytax ytb. The frst nequalty follows from x 0 and ct yt. The second nequalty follows from y 0 and Ax b. Economc nterpretaton: - Estmate the cost/prce of resources - Estmate the proft ncreasng when changng the rght hand terms; Sensblty analyss

11 A smple example (I) Example. Determne the quanttes to be produced such that all the producton constrants are satsfed and the beneft s maxmzed. We suppose that two products A and B can be produced each of them passng through cuttng and packng stages respectvely (C) and (P) : Cuttng Packng Necessary tme to produce 1 unty of A 2 hours 3 hours Necessary tme to produce 1 unty of B 2 hours 1 hours Avalablty n workng hours 200 hours 100 hours The untary benefts for A et B are respectvely 20 and 10.

12 A smple example (II) Prmal PL : Max 20A + 10B Subect to : 2A + 2B 200; 3A + B 100; A 0 et B 0; Dual : Mn 200a + 100b Subect to : 2a + 3b 20; 2a + b 10; a 0 et b 0; Economc nterpretaton : a wll gve the cost/prce of one hour for the cuttng process and b the prce for the packng one.

13 Lnear program modelng General cases : lower and upper bounds; render nequaltes to equaltes; express unsgned varables as nonnegatve ones;

14 Lnear program modelng Expressng the obectve functon: maxmn or mnmax functons; Let be gven varables x 1 x2 xn. The obectve functon conssts n maxmzng the mnmum of them. Intro duce a new varable y whch s a lower bound of x; The obectve functon becomes: Max y; multplcatve functons Mn 1/x => Max x; y=ax where x bnary varable and a [0.. M]. y <= a; y <= Mx; y >= a + (x-1)m. how to do when both bnary varables?

15 Lnear program modelng Expressng constrants : Capacty constrants Usually they gve rse to upper bound constrants; Demand constrants Usually they gve rse to lower bound constrants; Mass balance constrants Proporton constrants a b and c gve all and a s exactly 30%: a = 03 (a+b+c)

16 LP modelng technques R x A s t x D x N Mn.. ; 1;... ; ; 1 : dr y A y D s t y N Mn Functon Ob x d y x D d Set

17 Integer Lnear Programmng

18 Introducton to nteger lnear programmng Integer Lnear Programmng (ILP) An nteger lnear program s a lnear programmng problem wth varables takng values n Z. Bnary or 0-1 lnear programmng problems are a specal case. How to solve nteger lnear programmng : Cuttng planes Branch and bound methods branch and cut

19 Modelng by bnary varables

20 Lnear programmng modelng Modelng constrants n schedulng problems: dsunctve constrants : task A before task B or task B before task A; conunctve constrants; task A before task B; or task A before B and C Expressng dsunctve constrants:

21 Schedulng problems Problem formalzaton: Data: tasks (ready tme processng tme due tme) number of machnes precedence constrants preempton etc Problem modelng data: r (ready tme) p (processng tme) d (due tme); Decson varables : t (start date of executon) x (executon order between and ) c (completon executon tme). Precedence/conunctve constrants: task before task : t + p <= t: Dsunctve constrants: t + p <= t or t + p <= t : t + p <= t + M(1-b) ; t + p <= t + Mb ; b {01} Obectve functon: mnmze the completon tme (makespan) total lateness number of delayed tasks etc. 21

22 The one machne schedulng problem We consder the n-ob one-machne schedulng problem wth ready tme r processng tme p and due tme d for each ob. Preempton s not allowed and precedence constrants among obs are not consdered. Decson varables: x (boolean varables): task executed at the -th order ; t (>= 0) start tme for task placed at order ; LP formulaton f.ob. : Mn tn + xn*p ; s.t. for all : x=1 ; for all : x=1 ; for all : t >= x*r ; for all : t+1 >= t + x*p ; for all : t + x*p <= x*d ;

23 Examples n LP modelng 2 3 Problem. Computer producton plannng The company ORDI produces laptops. Sellng prevsons (n thousands of untes) for the next sx months are reported n the followng table. The producton capacty of the company s per month. It s possble to produce more at hgher cost 300 Euros per pece nstead of 250. anuary february march aprl may une uly Sellng prevsons for the next sx months There are 2000 laptops n stock. The storng cost s of 25 Euros per unty for what s n stock at the end of the month. We suppose that the stockng capactes are unlmted. We begn at the 1 st January. What s the producton plan for the next sx months such that all demands are satsfed and the cost s mnmzed.

24 LP model 2 4 Decson varables : x : quantty produced wth nomnal tarff durng month ; y : quantty produced wth secondary tarff durng month ; s : stock of month and s0 ntal stock (=2000) ; Mass conservaton constrants : For from 1 to 6 : s-1 + x + y = D + s ; Capacty constrant: x <= ; Postvty and ntegralty constrants: x y s >= 0 ; x y s are ntegers Obectve functon: Mn ( x*250 + y*300 + s*25).

25 Modelng optmzaton problems n logstcs and transportaton

26 Modelng optmzaton problems n logstcs and transportaton Related problems: Parttonng generalzed matchng coverng faclty locaton problems; Lp models for transportaton problems: Transportaton programs Mnmal cost flows and multflows vehcle routng problems etc.

27 Parttonng coverng Parttonng problems ask whether s possble to splt a gven set n subsets satsfyng some crtera. (b)partton problem : the problem s to decde whether a gven a set of ntegers can be parttoned nto two "halves" that have the same sum. Coverng problems ask whether a certan combnatoral structure 'covers' another or how large the structure has to be to do that. A cover of a set X s a collecton of sets such that X s a subset of the unon of sets n the collecton. An exact cover s a subcollecton such that each element n X s contaned n exactly one subset n t. The vertex cover problem : a vertex cover of a graph s a set of vertces such that each edge of the graph s ncdent to at least one vertex of the set. Propose an ILP model for the vertex cover problem.

28 Coverng problems total cover - Data: n canddate stes (ndexed by ) m demands (ndexed by ) D c maxmum cover (reachng) dstance. constants (01) noted as a ndcatng f can be covered by. c nstallaton cost of. d dstances between any and. Goal: compute whch stes wll be open such that all clents are covered at mnmal nstallaton cost of stes. PL01 Formulaton: Mn c x a x 1 x 0 1.

29 maxmal cover - It occurs when one wants to maxmze the area/demand covered when the number of allowed stes s bounded. addtonal coeffcents and varables: p maxmum number of stes allowed to be nstalled; d demand (known data) z bnary varable ndcatng f some s covered. PL01 formulaton: Max z x z x d z a x p

30 p-center and p-medan problems Data: - A set of demands/clents to satsfy - A set of n stes such that only p are allowed to open - Dstance d between demands/clents and stes; p-center problems Goal: Where placng stes n order to mnmze the maxmal dstance; Example : placng frestatons etc. p-medan problems Goal: Mnmze the average dstance Example : dstrbuton centers...

31 p-center problem LP model: ; 1; p x y 0; z y x z Mn z y d x y x -> ste ; y -> demand covered by

32 LP model: ; 1; p x y y x y d q Mn x y q -> quantty of demand ; x -> ste ; y -> demand covered by p-medan problem

33 Faclty locaton problems Faclty locaton s concerned wth computng optmal placement of facltes n order to mnmze transportaton costs satsfy clent demands outperform compettors' facltes Examples : medcal urgency centers dstrbuton centers placng concentrators n a network... etc.

34 Warehouse faclty locaton problem Mnmzng the total cost ncludng nstallaton cost and the transport cost. Determne whch stes/warehouse facltes have to be nstalled and what quanttes they should delver to each clent. LP model : Mx y d y y c x f Mn y x Deduce the sgnfcaton of varables used n the above model.

35 A transportaton program The transportaton problem deals wth sources where a supply of some commodty s avalable and destnatons where the commodty s demanded. Example. Cars to rent A company specalzed n car s rent has two garages wth respectvely 12 et 8 cars n stock and three rent shops demandng for respectvely 8 7 and 5 cars. The untary costs of transport are gven n the followng table. What would be the transport plan of mnmal cost? Garages Shops Supply Demand

36 LP model Mnmze 3x11 + 5x12 +3x13 + 2x21 +7x22 + x23 Subect to: x11 + x21 8; x12 + x22 7; x13 + x23 5; x11 + x12 + x13 12; x21 + x22 + x23 8; x Z+; x : decson varables : number of cars to be sent from to. Exercse : Gve the general LP model for the transportaton program : There are a1 a2... am unts n the warehouses m. We want to transport these unts to the destnatons n where the requests b1 b2... bn occur wth Σa = Σb. 36

37 Transportaton program transport untary cost matrx supply demand Mn x a 1... m x b 1... n x 0 c x.

38 The TSP In 1859 Sr W. R. Hamlton bult a puzzle dodecahedron n wood. Ths dodecahedron has 20 vertces and 12 faces: Queston. Fnd an hamltonen cycle passng through the vertces of dodécahédron.

39 TSP : LP formulaton To each arc () we assocate some varable x takng 1 f () s ncluded n the hamltonan cycle and 0 otherwse. ) ( a n x d x Mn. 01 x ) ( b n x 1 _ S S x ) ( ; 1 1 c N u n n nx u u

40 Path problems LP formulatons LP model for the shortest path problem (!) x x ; X s t. succ prec (!!) x s x succ s Mn w x t prect 01. x The drected cycle wth mnmum weght/length s called the mnmum cycle mean. The LP formulaton : 1; (!) x x ; X. (!!) Mn w x succ prec x 1; all( ) 01. x

41 Maxmum flow problem Data: let be G=(XUCst) where: X a set of n nodes U a set of m arcs C capacty set on arc () source s and snk t. A flow s an applcaton de U to R+ : Satsfy capacty constrants (1) Krchoff constrants (2) ( 1) C ; ( ) U. ( 2) ; X s t. succ prec ( 3) s succ s Max F t F prect ;

42 Mn cost flow problem Data: let be gven G=(XUCwst) wth: X a set of n nodes U a set of m arcs C capacty of arc () and w untary cost of use of arc (). Let s and t be respectvely the source and the snk and F the flow value to be satsfed. (!) C ; ( ) U. (!!) ; X s t. succ prec (!!!) s succ s Mn w t F prect ;

43 Mult-commodty flow problem Qute smlar to the flow problem: Satsfy capacty constrants (!) Flow conservaton (!!) Flow values to satsfy (!!!) d (!) C ; ( ) U. d d (!!) d ; X d D succ d (!!!) d s succ s d prec prect d t F d s d t d.

44 A smple applcaton to telecommuncatons (I) Mnmum cost flow allocaton problem Data: Goal: Network represented by G(XUD) where : X the set of nodes; U the set of arcs () wth capacty C D the traffc demand where T d gve the demand traffc value to be satsfed for commodty d. Determne a routng scheme n the network such that the capacty constrants and traffc demands are both satsfed whle the use of resources (bandwdth) over the network s mnmzed.

45 A smple applcaton to telecommuncatons (II). ) ( 0; D d U d. ) ( ; U C d d. D d F d prect t d succ s s d d d. ; d d prec d succ d t s D d X U ) ( LP formulaton : Mn

46 The art gallery problem The art gallery or museum problem orgnates from a realworld problem of guardng an art gallery wth the mnmum number of guards whch together can observe the whole gallery. Propose an ILP model.

47 The far loadng problem Let be gven N tran wagons wth a load capacty lmted to P. We have m boxes 12.. m wth weghts p 1 p 2.. p m. Queston 1. Propose an ILP model for the assgnment of boxes to wagons such that mnmzng the total weght of the most loaded wagon. Queston 2. Study the case wth only two wagons and propose a smplfed ILP model.

48 Exercse Employment plannng for a restaurant The manager of a restaurant needs to ensure permanence and servce n hs restaurant on the bass of some statstcs (say for day (1 7) are needed a employees): He needs to fnd the mnmal number of employees when any of them should work 5 consecutve days and next takes two days of break. Gve an LP model for ths problem.

49 Resoluton methods for ILP

50 A nave method: roundng the relaxed LP soluton Let P=Ax b (x 0 et A 0) be a maxmzaton problem (cx c 0) and x 0 a non-ntegral soluton of the relaxed LP problems. x 0 s a feasble soluton but non x 0. In general the rounded soluton can gve as well a feasble and a non feasble soluton. Example: Maxmze x 1 + 2x 2 Subect to: 4x 1 + 3x 2 12; -2x 1 + x 2 1; 3x 1 5; x 1 0 et x 2 0 x 1 x 2 ntegers. Relaxed LP soluton ( ) Integral soluton (1 2);

51 TU matrxes (totally unmodular) Let be gven the followng ILP: P = Max z=cx s.t. Ax b; x 0 x take nteger values; Queston : n what condtons t happens that the optmal soluton of the relaxed P s an optmal soluton for ntal P? Defnton : a totally unmodular matrx (TU matrx) s a matrx for whch every square submatrx s wth determnant 1 0 or +1. Examples : the shortest path problem mnmum cost flow problems etc..

52 An example of a TU matrx u u u u u u u 4 u 2 u u

53 Cuttng plane algorthm General descrpton of the cuttng plane algorthm : Prncple: an ILP s solved through a sequence of LP problems augmented wth vald nequaltes. Solve the relaxed problem.e. removng the ntegralty constrants from the ntal problem. If the soluton say x* contans only ntegers t s optmal for the ILP problem. If not fnd a vald nequalty volated by x* and add t to the LP and solve t agan (ths s a cuttng plane). Ths procedure contnue lke ths untl an ntegral soluton s obtaned or t s not possble to fnd a vald nequalty. Vald nequalty : An nequalty ctx c0 s vald for a set S Rn f ct x c0 for all x S. A vald nequalty s called a vald cut nequalty when the soluton x* of the relaxed LP problem doesn t satsfy the nequalty.e. f ct x* > c0.

54 Cuttng plane algorthm

55 Cuttng plane algorthm

56 Cuttng plane algorthm

57 Cuttng plane algorthm

58 Cuttng plane algorthm

59 Exact methods branch and bound method Intally proposed to solve ILP problems by Lang e&dog (1960) and Dakn (1965). Prncple. It combnes the prmal and dual smplex algorthm wth Branch and Bound prncple. Consder the maxmzaton problem: Max cx = z Ax = b x N =1..n Intalzaton : the relaxed problem gves the root of the tree S0 solve the relaxed problem f the soluton s nteger END. otherwse we obtan a default evaluaton; contnue wth splttng.

60 Branchng and boundng Branchng: Splt on x k that s not nteger : x k x k and x k x k +1 whch gves two subsets (nodes) S et S ; Evaluaton: Solve the assocated relaxed LP problems : PL S = PL S + x k x k ; (prmal smlex algorthm) PL S = PL S + x k x k +1 ; (dual smplex algorthm)

61 Branchng and boundng Node examnaton choce Breadth frst Depth frst Default evaluaton : the best feasble (nteger) soluton currently obtaned (z 0 ). Prunng rule: v(s) v(z 0 ).

62 Branchng and boundng

63 Branchng and boundng

64 Branchng and boundng

65 Example of B&B on a mnmzaton problem.

66 Example

67 The general branch and bound method Prncple. Branch and bound (B&B) conssts of a complete enumeraton of all canddate solutons where subsets of frutless canddates are dscarded by usng upper and lower estmated bounds of the quantty beng optmzed.

68 The general Branch and bound method Consder a mnmzaton problem: Prncple: buld a tree structure (the root could be the assocated relaxed problem). Splt the set of feasble solutons n subsets that cover the ntal set (gvng subproblems): Any nfeasble subproblem s dscarded; If possble compute the optmal soluton for the subproblem or a lower bound (boundng); f t s larger than the current best soluton dscard the subproblem (prunng). Otherwse splt agan and repeat the procedure (branchng). Combned wth cuttng planes t gves the Branch and Cut algorthm (B&C).

69 An example Solve the followng bnary LP: Max 4x1 + 4x2-3x3 4x1 + 2x2 - x3 7 6x1 + 2x2 + 3x3 10 x1 + x2 - x3 2 x1 x2 x3n.

70 Arborescent method : exercse Fve people wth dfferent natonaltes lve n the frst fve houses of a street all on the same sde of the street (numbered from 1 to 5). These people have each a dfferent ob as well as a favourte drnk and anmal (each dfferent for each person).the fve houses are each of a dfferent colour. 1 : The englshman lves n the red house. 2 : The spanard has a dog. 3 : The apanese man s a panter. 4 : The talan drnks tea. 5 : The nowegan lves n the frst house on the left. 6 : The owner of the green houses drnks coffee. 7 : Ths green house s mmedately on the rght of the whte house. 8 : The sculptor rases snals. 9 : The dplomat lves n the yellow house. 10 : The person who lves n the mddle house drnks mlk. 11 : The norwegan lves next to the blue house. 12 : The volnst drnks frut uce. 13 : The fox s n a house next to the doctor's house. 14 : The horse s next to the dplomat's house. Who has a zebra and drnks water? (Attrbuted to Lews Carrol)

71 Solvers and modelers There are two types of products: solvers et modelers Solvers : specalzed computer programs desgnated to solve mathematcal programs (lnear or not) Freewares: Lp-solve Lp_optmser SoPlex Con PCx Ms-excel solver The others: CPLEX XPRESS-MP AIMMS MINOS OSL LINDO MPSIII Modelers : desgnated to model the problem by non specalsts: (MPS AMPL) recognzed by most mportant solvers allow also to analyze and nterpret the results Other modelers: AIMMS GAMS LINGO LP-TOOLKIT MathPro MIMI MINOPT MPL OMNI OPL Studo.

72 END