Lp and ip programming cp 9
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This document provides an introduction to linear and integer programming. It defines key concepts such as linear programs (LP), integer programs (IP), and mixed integer programs (MIP). It discusses the complexity of different optimization problem types and gives examples of LP and IP formulations. It also covers common techniques for solving LPs and IPs, including the simplex method, cutting plane methods, branch and bound, and heuristics like beam search.
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Lp and ip programming cp 9
- 1. 1 Linear & Integer Programming : Introduction Engg System Design Optimization M S Prasad
- 2. General Optimization Program • Standard form: • where, • Too general to solve, must specify properties of X, f,g and h more precisely.
- 3. Complexity Analysis • (P) – Deterministic Polynomial time algorithm • (NP) – Non-deterministic Polynomial time algorithm, – Feasibility can be determined in polynomial time • (NP-complete) – NP and at least as hard as any known NP problem • (NP-hard) – not provably NP and at least as hard as any NP problem, – Optimization over an NP-complete feasibility problem
- 4. Optimization Problem Types – Real Variables • Linear Program (LP) – (P) Easy, fast to solve, convex • Non-Linear Program (NLP) – (P) Convex problems easy to solve – Non-convex problems harder, not guaranteed to find global optimum
- 5. Let: X1, X2, X3, ………, Xn = decision variables Z = Objective function or linear function Requirement: Maximization of the linear function Z. Z = c1X1 + c2X2 + c3X3 + ………+ cnXn …..Eq (1) subject to the following constraints: Problems
- 6. 6 Linear programming (LP) model • LP: • Matrix form: )n,,1j(0x bxa...xaxa bxa...xaxa bxa...xaxa :tosubject xc...xcxcmin j mnmn22m11m 2nn2222121 1nn1212111 nn2211 =≥ ≤+++ ≤+++ ≤+++ +++ 0x bAx xcmin T ≥ ≤ objective function constraints variable restrictions where: x, c: n-vector A: m,n-matrix b: m-vector
- 8. 8 Linear programming example: graphical solution (2D) 1 2 3 4 5 6 x1 1 2 3 4 5 6x2 )objective( 6xx 21 =+ 4xx2 21 ≤+ 12x4x3 21 ≤+ )2,1j(0x 12x4x3 4xx2 :tosubject xxmax j 21 21 21 =≥ ≤+ ≤+ + 0 solution space
- 9. 9 Linear programming (cont.) • Decision variables can take the fractional values. ( May not suitable in some areas such as Resource mgmt , allocation etc) • Solution techniques: – (dual) simplex method – interior point methods (e.g. Karmarkar algorithm)
- 10. 10 Integer programming (IP) models • Integer variable restriction – IP: integer variables only – MIP: part integer, part non-integer variables – BIP: binary (0-1) variables • General IP-formulation: • Complex solution space )Zx(integerx bAx xcmin T + ∈ ≤
- 11. 11 Integer programming example: graphical solution (2D) 1 2 3 4 5 6 x1 1 2 3 4 5 6x2 )objective( 6xx 21 =+ )2,1j(Zx 12x4x3 4xx2 :tosubject xxmax j 21 21 21 =∈ ≤+ ≤+ + + 0 2 optimal solutions!
- 12. 12 Total unimodularity property for integer programming models Suppose that all coefficients are integer in the model: i.e. Example: transportation problem if A has the total unimodularity property (i.e. every square submatrix has determinant 0,1,-1) ⇒ there is an optimal integer solution x* & the simplex method will find such a solution 0x bAx xcmin T ≥ ≤ j,i,b,a iij ∀Ν∈
- 13. 13 Integer programming tricks PROBLEM: x = 0 or x ≥ k use binary indicator variable y= restrictions: kfor x, 0for x, 1 0 ≥ = { }0,1y ykx x)onupperboundanis(MyMx ∈ ⋅≥ ⋅≤
- 14. 14 PROBLEM: fixed costs: if xi>0 then costs C(xi) use indicator variable yi= restrictions : Integer programming tricks (2) 0x bAx )x(Cminimize ≥ = 0.for x 0,for x xck 0 )x(C :where i i iii i > = + = 0for x, 0for x, 1 0 i i > = { }0,1y xcyk)x(C yMx i iiiii ii ∈ ⋅+⋅= ⋅≤ )i(∀
- 15. 15 • Hard vs. soft restrictions – hard restriction: must hold, otherwise unfeasibility for example: – soft restriction: may be violated, with a penalty for example: (Integer) programming tricks (3) 0Y,0x Y5xx 100Yxcminimize 21 T ≥≥ +≤+ ⋅+ 5xx 21 ≤+
- 16. 16 • Absolute values: solution: (Integer) programming tricks (4) −+ −+ +=⇒ −= ttt ttt yyy yyy freey,0x ybxa ymin tt,j ttt,j j j j t ≥ +=∑ ∑ ( ) 0y,0y,0x yybxa yymin ttt,j tttt,j j j t tt ≥≥≥ −+= + −+ −+ −+ ∑ ∑ goal variation
- 17. 17 • Conjunctive/disjunctive programming - conjunctive set of constraints: must all be satisfied - disjunctive set of constraints: at least one must be satisfied • example (Appendix A.4): Integer programming tricks (5) jkj kjk j jj pxx or pxx xwmin ≥− ≥− ∑ }1,0{y )y1(Mpxx yMpxx xwmin 2jkj 1kjk j jj ∈ −⋅−≥− ⋅−≥− ∑
- 18. 18 IP example nonpreemptive single machine, total weighted completion time (App. A.3) function)(objectivextwimizemin jjoboftimecompletionxt jjoboftimecompletionxt n 1j 1-Cmax 0t jtj 1-Cmax 0t jt jt ∑ ∑ ∑ = = = ⋅⋅⇒ =⋅⇒ ==⋅⇒ 1xif jt otherwise0andt,timeatjjobif,1xjt completes= objective function: minimize weighted completion time: model definition:
- 19. 19 IP example (cont.) Restriction: all jobs must be completed once: 1x 1-Cmax 0t jt =∑= Restriction: only one job per time t: t)timeperjoboneexactly:on(restricti1x t)duringprocessinisjjob(if1x n 1j pt ts js pt ts js j j =⇒ =⇒ ∑∑ ∑ = + = + = if job j is in process during t, it must be completed somewhere during [t,t+pj]
- 20. 20 IP example (cont.) Complete IP-model: { } 1)-Cmax,0,tn,,1,j(for1,0x 1)-Cmax,0,t(for1x n),1,j(for1x :tosubject xtwinimizem jt n 1j pt ts js 1-Cmax 0t jt n 1j 1-Cmax 0t jtj j ==∈ == == ⋅⋅ ∑∑ ∑ ∑ ∑ = + = = = = n×Cmax integer variables
- 21. 21 IP example (cont.) Additional restriction: precedence constraints Model definition: SUCC(j) = successors of job j ⇒ job j must be completed before all jobs in SUCC(j): n),1,jSUCC(j),k(fortxtx kjoboftimestartptx jjoboftimecompletiontx 1maxC 0t kt 1maxC 0t jt k 1maxC 0t kt 1maxC 0t jt =∈≤⇒ =− = ∑∑ ∑ ∑ − = − = − = − =
- 22. 22 Integer programming solution techniques • Heuristic vs. explicit approach: – trade-off between solution quality and computation time – trade-off between implementation effort/costs and yield (i.e. profits gained from solution quality improvement) • Heuristic methods; for example: – local search (e.g. simulated annealing, tabu search, k-opt) – (composite) dispatching rules (e.g. EDD, SPT, MS) – adaptive search – rounding fractional solutions – beam search
- 23. 23 • Explicit methods; 3 categories: 1. dynamic programming 2. cutting plane (polyhedral) methods 3. branch and bound or: hybrid methods (combination of the above) • Commercial IP solvers usually use a combination of heuristics and 2, 3 Integer programming solution techniques (cont.)
- 24. 24 Dynamic programming • Problem divided into stages • Each stage can have various states • A recursive objective function is used to iterate through all states and all stages (forwards or backwards) ))}x,i(i(F)x,i(c{min)i(F tt1t1tttt x tt t −−+= T),0,(txt = ti (constant)c)i(F 000 =
- 25. 25 Cutting plane methods STEP 0: Create a relaxation of the problem by omitting restrictions (e.g. the integrality restrictions) STEP 1: Solve the current problem STEP 2: If solution is infeasible then generate a restriction that cuts of the solution, and add it to the problem ⇒ STEP 1 Otherwise: DONE
- 26. 26 Branch and bound • Enumeration in a search tree • each node is a partial solution, i.e. a part of ... ... root node child nodes child nodes Level 0 Level 1 Level 2
- 27. 27 Branch and bound example 1 Disjunctive programming (appendix A.4): disjunctive set of constraints: at least one must be satisfied xj = completion time of job j restriction: )Ik,j(pxxorpxxeither jkjkjk ∈∀≥−≥− solve LP without disjunctive restrictions (= LP relaxation) if disjunct. restr. violated for j & k Level 0 Level 1 ... kjk pxx ≥− jkj pxx ≥−
- 28. 28 Branch and bound (cont.) • Upper bound: e.g. a feasible solution • Lower bound: e.g. a solution to an “easier” problem • Node elimination (fathom/discard nodes): when lower bound >= upper bound
- 29. 29 Branch and bound (cont.) • Branching strategy: how to partition solution space • Node selection strategy: – sequence of exploring nodes: • depth first (tries to obtain a solution fast) • breadth/best bound first (tries to find the best solution) – which nodes to explore (filter and beam width) • filter width: #nodes selected for thorough evaluation • beam width: #nodes that are branched on (≤ filter width) ⇒ Beam search
- 30. 30 Branch and bound example 2 • Single machine, maximum lateness, release and due dates lower bound: EDD + preemption Jobs 1 2 3 4 p(j) 4 2 6 5 r(j) 0 1 3 5 d(j) 8 12 11 10 (?,?,?,?) (1,?,?,?) (4,?,?,?)(2,?,?,?) Level 0 Level 1(3,?,?,?)
- 31. 31 Branch and bound example 2 • Lower bound for: (1,?,?,?) Lower bound: Lmax = max(0,17-12,15-11,0)=5 Jobs 1 2 3 4 p(j) 4 2 6 5 r(j) 0 1 3 5 d(j) 8 12 11 10 t r(2) r(3) r(4) d(4)<d(3) d(3)<d(2) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 3 3 21 4
- 32. 32 (?,?,?,?) (1,?,?,?) (4,?,?,?)(2,?,?,?) Level 0 Level 1(3,?,?,?) Branch and bound example 2 (cont.) LB=5 LB=7* =UB (1,2,?,?) (1,3,?,?) Jobs 1 2 3 4 p(j) 4 2 6 5 r(j) 0 1 3 5 d(j) 8 12 11 10 infeasible: (1,3,4,3,2) LB=6* =UB (1,2,4,3) LB=5*=UB (1,3,4,2) DONE (1,2,4,3) (1,3,4,2)
- 33. 33 Branch and bound example 3 1 2 3 4 5 6 x1 1 2 3 4 5 6x2 )objective( 6xx 21 =+ 0 4.2x ,8.0x 2 1 = = 0x1 ≤ 1x1 ≥ 3x ,0x 2 1 = = 2x ,1x 2 1 = = 2x2 ≤ 3x2 ≥ 2x ,1x 2 1 = = 3x ,0x 2 1 = = obj: 3 obj: 3 obj: 3 obj: 3 LP solution:
- 34. 34 Beam search example 1 single-machine, total weighted tardiness Upper bound: ATC rule (apparent tardiness cost): • schedule 1 job at a time • every time a machine comes available, determine ranking of jobs: ⋅ −− − = pK )0,tpdmax( j j j jj e p w )t(I MS rule WSPT rule look-ahead parameter: K = 4.5 + R (R ≤ 0.5) K = 6 - 2R (R ≥ 0.5) = due date range factor( ) maxminmax C/ddR −=
- 35. 35 Beam search example 1 (cont.) single-machine, total weighted tardiness (?,?,?,?) (1,?,?,?) (4,?,?,?)(2,?,?,?) (3,?,?,?) 1,2,3)(j p w e p w )t(I j jpK )0,tpdmax( j j j jj ===⇒ ⋅ −− − Upper bound by ATC rule: )3,2,1j(0)0,tpdmax( jj ==−− Jobs 1 2 3 4 p(j) 10 10 13 4 d(j) 4 2 1 12 w(j) 14 12 1 12 w(j)/p(j) 1.4 1.2 0.1 3
- 36. 36 Beam search example 1 (cont.) single-machine, total weighted tardiness (?,?,?,?) (1,?,?,?) (4,?,?,?)(2,?,?,?) (3,?,?,?) Jobs 1 2 3 4 p(j) 10 10 13 4 d(j) 4 2 1 12 w(j) 14 12 1 12 w(j)/p(j) 1.4 1.2 0.1 3 Jobs C(j) d(j) T(j) w(j)*T(j) 1 10 4 6 84 2 24 2 22 264 3 37 1 36 36 4 14 12 2 24 Upper bound by ATC rule: Total = 408
- 37. 37 Beam search example 1 (cont.) single-machine, total weighted tardiness (?,?,?,?) (1,?,?,?) (4,?,?,?)(2,?,?,?) (3,?,?,?) UB=408 UB=436 UB=814 UB=440 discardedexplored further (beam width = 2) 4 nodes analyzed (filter width=4)
- 38. 38 Beam search example 1 (cont.) (?,?,?,?) (1,?,?,?) (4,?,?,?)(2,?,?,?) (3,?,?,?) UB=408 436 814 440 (1,2,?,?) (1,3,?,?) (1,4,?,?) UB=480 706 408 (1,4,2,3) (1,4,3,2) UB=408 554 best solution (2,1,?,?) (2,3,?,?) (2,4,?,?) 436 (2,4,1,3) (2,4,3,1) 436 608