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One-dimensional cutting problems: models and algorithms

a survey on complexity results, integer linear programs and algorithms for the one-dimensional cutting stock problem

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One-dimensional cutting problems: models and algorithms

1. 1. models and algorithms for one-dimensional cutting problems Fabrizio Marinelli fabrizio.marinelli@univpm.it Università Tor Vergata Roma, November 2013 Università Politecnica delle Marche Università Politecnica delle Marche
2. 2. Fabrizio Marinelli - Cutting problems 2 Road map  A typology of cutting problems  The one-dimensional cutting stock problem  Complexity: NP-completeness and polynomial cases  Four integer program formulations  Solution methods: heuristics, approx results, exact algorithms and topics packing
3. 3. Fabrizio Marinelli - Cutting problems 3 The problem
4. 4. Fabrizio Marinelli - Cutting problems 4 Cutting problem “large objects” (stock) have to be cut up into “small objects” (parts or part-types) in order to fulfill a given requirement of parts. Five sub-problems can be distinguished: 1. how select the stocks, 2. how select the parts, 3. how group the parts, 4. how allocate the groups of parts to stocks, 5. how lay out the parts on stocks.
5. 5. Fabrizio Marinelli - Cutting problems 5 An example supply of Stocks (warehouse) demand of Parts (Production bill)  4  15  6 cutting pattern: a (geometric) assignment of parts to stocks. activation level INPUT: geometric objects of given shape OUTPUT: (unordered) set of cutting operations
6. 6. Fabrizio Marinelli - Cutting problems 6  [definition] A cutting pattern is a vector p  Nn that describes how to cut a single stock: the i-th component indicates how many parts of type i are produced by adopting the pattern.  [definition] The activation level of a cutting pattern p is the number of stocks processed according to p. In math terms…
7. 7. Fabrizio Marinelli - Cutting problems 7 Cutting in industry Textile - Missoni (Italy),… Paper - Pine Falls (Canada) Stone Consolidated (Canada, UK), ... Industrial woodworking – IKEA (Sweden),… Steel bars – Hydro (Italy, Norway),… Glass – Pilkington (Italy, France), ... Rubber – Dayco (Italy, Belgium), ... Logistics – Pallet Loading, assortment, pallet layout,…
8. 8. Fabrizio Marinelli - Cutting problems 8 Other applications: Multiprocessor scheduling [problem] Multiprocessor Scheduling  n jobs, all released at time zero, each requiring a given processing time pi  m machines, each available for a time interval [0, D]  no due dates or precedence constraints; preemption is not allowed. Is there a schedule which ﬁts all n jobs onto the m machines? We are asking whether n bars, with lengths pi , can be bin-packed into m bins of length D.
9. 9. Fabrizio Marinelli - Cutting problems 9 Other applications: Assembly Line Balancing Task to workstation assignment (cutting plan) t1 t2 t6 t5 t8 t3 t10 t4 t7 t9 t11 Workstations (bins) Cycle time t2 t6 t8 Tasks (production bill) t1 t5 t3 t10 t4 t7 t9 t11 Generalized Bin Packing Problem
10. 10. Fabrizio Marinelli - Cutting problems 10 Geometrical characteristics  Dimensionality  Part Shape Regular Convex Irregular two-dimensional (sheet, panels, clothes)  Cut Shape one-dimensional (rods, bars) Rotation allowed Guillotine Nested
11. 11. Fabrizio Marinelli - Cutting problems 11 Combinatorial characteristics  Stocks and Parts Assortment  one Stock, many identical Stocks, distinct Stocks;  one type and many parts, few types and many parts per type, many types and one or few parts per type.  Parts to Stocks Assignment  All stocks are required to obtain a selection of parts;  All parts must be produced by using a selection of stocks.
12. 12. Fabrizio Marinelli - Cutting problems 12  Dyckhoff, H, A typology of cutting and packing problems, 1990  4-fields notation  /  /  /  Typology of cutting & packing problems 1. Dimensionality (1) one-dimensional (2) two-dimensional (3) three-dimensional (N) N-dimensional 2. Kind of assignment (B) all stocks and a selection of parts (V) a selection of stocks and all parts 3. Assortment of stocks (O) single stock (I) identical figure (D) different figures 4. Assortment of parts (F) few parts (of different types) (M) many parts of many different types (R) many parts of relatively few different parts  Some ambiguity problems, i.e., the well-known Vehicle Loading Problem is coded both as 1/V/I/F and 1/V/I/M
13. 13. Fabrizio Marinelli - Cutting problems 13 Typology of cutting & packing problems  Wäsher et al., An Improved typology of cutting and packing problems, 2007 Criteria:  dimensionality: 1D, 2D, 3D;  kind of assignment: output maximization, input minimization;  assortment of parts: identical, weakly heterogeneous, strongly heterogeneous;  assortment of stock: one, several;  shape of parts: regular, irregular. Basicproblems Intermediateproblems Refinedproblems
14. 14. Fabrizio Marinelli - Cutting problems 14 Typology: basic problems C&P problems Output maximization Input minimization kind of assignment identical weakly heterog. strongly heterog. all dimension fixed Assortment of parts identical item packing placement Knapsack Open dimension Cutting Stock (CSP) Bin Packing (BPP) basic problem Arbitrary weakly heterog. strongly heterog. variable dimension (s) all dimension fixed
15. 15. Fabrizio Marinelli - Cutting problems 15 Typology: intermediate problems Input minimization parts weakly heterogeneous strongly heterogeneous stocks identical weakly heterogeneous strongly heterogeneous alldimensionfixed One large stock Variable dimension(s) Single Stock Size CSP SSSCSP Multiple Stock Size CSP MSSCSP Residual CSP RCSP Single Bin Size BPP SSSBPP Multiple Bin Size BPP MBSBPP Residual BPP RBPP Open Dimension Problem ODP
16. 16. Fabrizio Marinelli - Cutting problems 16 Community  ESICUP (EURO Special Interest Group on Cutting and Packing)  Founded in 1988 and EURO Working Group since 2003  Around 500 members worldwide  Around 600 papers between 1995 and 2004 1D 2D 2D 3D tot regular irregular Input minimization 108 79 52 24 263 Output maximization 64 71 12 35 182 tot 172 150 64 59 445
17. 17. Fabrizio Marinelli - Cutting problems 17 The one-dimensional case
18. 18. Fabrizio Marinelli - Cutting problems 18 Cutting Stock Problem (CSP)  a sufficient number m of stocks of length L  a production bill Q consisting in n part-types. Part-type i is described by the length li and the number di of items to be produced. INPUT OUTPUT  A set {p1,…, pK} of cutting patterns and relevant activation levels.  The problem is feasible if: li < L for all i
19. 19. Fabrizio Marinelli - Cutting problems 19 An example L = 10 m Stocks Production bill Q l1 = 7 l3 = 4 l2 = 3 d1 = 5 d2 = 10 d3 = 5 (0,2,1) (1,0,0)  5  5 = (0,10,5) = (5,0,0) [Solution 1] Cutting patterns Activation level Production (5,10,5)10Trim loss = 3  5 = 15m
20. 20. Fabrizio Marinelli - Cutting problems 20 An example L = 10 m Stocks Production bill Q l1 = 7 l3 = 4 l2 = 3 d1 = 5 d2 = 10 d3 = 5 (0,2,1) (0,3,0)  1  1 = (0,2,1) = (0,3,0) [Solution 2] Cutting patterns Activation level Production (5,10,5)9 (1,1,0) (0,0,2)  5  2 = (5,5,0) = (0,0,4) Trim loss = 2  2 + 1  1 = 5m
21. 21. Fabrizio Marinelli - Cutting problems 21 Some equivalent variants TxLdltrimloss i i n i ii minminmin 11     If stocks are identical, number of stocks minimization and trim loss minimization are equivalent problems. Let T = xi be the number of used stocks in an optimal solution:  Satisfy exactly or at least the demand of parts are equivalent problems Clearly, the latter is a relaxation of the former and optimal solutions with the same value can always be obtained from an optimal solution of the latter by removing the surplus of parts.
22. 22. Fabrizio Marinelli - Cutting problems 22 Cutting Stock & Bin Packing  In principle, Bin Packing Problem (BPP) and CSP are equivalent.  The size of the input of a BPP instance is exponential in the size of the corresponding instance of CSP  polynomial-time algorithm for BPP is not necessarily polynomial- time for CSP  polynomial-size formulation for BPP is not necessarily polynomial- size for CSP
23. 23. Fabrizio Marinelli - Cutting problems 23 Problem complexity: preliminaries  [definition] A Polynomial Time Approximation Scheme (PTAS) is an algorithm which computes, for any given ε > 0, a solution zA within a factor 1 + ε of being optimal, i.e., zA < (1 + ε) z*. The running time of a PTAS is required to be polynomial in the size of the problem for every fixed ε.  [definition] A Fully Polynomial Time Approximation Scheme (FPTAS) is a PTAS algorithm having a running time polynomial in the size of the problem and in 1/ε for any ε > 0.  [definition] An Asymptotic Polynomial Time Approximation Scheme (APTAS) is a PTAS algorithm only when z* > C(ε) for a given function C of ε.
24. 24. Fabrizio Marinelli - Cutting problems 24 Problem complexity CSP is strongly NP-hard and is NP-hard to approximate within a factor less than 3/2 (i.e., there is no PTAS algorithms for CSP unless P = NP) The decision version of CSP is in NP (EisenbrandShmonin, 2006) CSP with n = 2 part-types is polynomial: the algorithm by McCormick et al. (2001) runs in O(log2 L) the algorithm by Agnetis  Filippi (2005) runs in O(log L) CSP with a fixed number n > 3 of part-types is polynomial (Goemans – Rothvoss, 2013)
25. 25. Fabrizio Marinelli - Cutting problems 25 Problem complexity: NP-hardness CSP is strongly NP-hard and is NP-hard to approximate within a factor less than 3/2 (i.e., there is no PTAS algorithms for CSP unless P = NP)  [Theorem] deciding if BPP has a solution with 2 bins is NP-complete [proof] Let {l1,…,ln} be an instance of PARTITION. Consider an instance of BPP with items {l1,…,ln} and 2 bins li / 2 long. l1 ln l2 li / 2 li / 2 The BBP instance is a yes-instance iff the PARTITION instance is a yes-instance.
26. 26. Fabrizio Marinelli - Cutting problems 26 Problem complexity: NP-hardness CSP is strongly NP-hard and is NP-hard to approximate within a factor less than 3/2 (i.e., there is no PTAS algorithms for CSP unless P = NP)  [Corollary] BPP is NP-hard to approximate with ratio < 3/2 [proof] an approximation algorithm with ratio  < 3/2 should solve a yes-instance of PARTITION in polynomial time (since  2 < 3 = 2 bins)
27. 27. Fabrizio Marinelli - Cutting problems 27 Problem complexity: NP-hardness CSP is strongly NP-hard and is NP-hard to approximate within a factor less than 3/2 (i.e., there is no PTAS algorithms for CSP unless P = NP)  [Theorem] deciding if BPP has a solution with m bins is strongly NP-complete [proof] Reduction from 3-DIMENSIONAL MATCHING Given  3 finite disjoint sets X = {1,…,}, Y = {1,…,}, Z = {1,…,},  and a set T  X × Y × Z of m triples, the set M  T is a 3-dimensional matching if  |M| =   xi ≠ xj, yi ≠ yj, and zi ≠ zj for any two distinct triples i = (xi, yi, zi)  M and j = (xj, yj, zj)  M.
28. 28. Fabrizio Marinelli - Cutting problems 28 Problem complexity: NP-hardness (cont’d) CSP is strongly NP-hard and is NP-hard to approximate within a factor less than 3/2 (i.e., there is no PTAS algorithms for CSP unless P = NP)  [Theorem] deciding if BPP has a solution with m bins is strongly NP-complete [proof] Reduction from 3-DIMENSIONAL MATCHING 4 3 2 1 4 3 2 1 4 3 2 1 ZYX  T = {(1,1,2), (2,2,1), (3,4,3), (3,3,3), (4,4,4)}  M = {(1,1,2), (2,2,1), (3,3,3), (4,4,4)}
29. 29. Fabrizio Marinelli - Cutting problems 29 Problem complexity: NP-hardness (cont’d)  Let (x1, y1, z1)…(xh, yh, zh)…(xm, ym, zm) be an arbitrary sequence of the triples in T  Define an instance I of BPP with  m bins  L = 404 + 15 ( is a large number, i.e., 100)  n = 4m items (4 items for each triple h = (xh, yh, zh) in T )Z h Y h X hh llll ,,,       111 110 4 4   h hX h x x l for the first occurrence of xh in the sequence for the subsequent occurrence of xh       211 210 24 24   h hY h y y l for the first occurrence of yh in the sequence for the subsequent occurrence of yh       48 410 34 34   h hZ h z z l for the first occurrence of zh in the sequence for the subsequent occurrence of zh 324 810  hhhh zyxl 
30. 30. Fabrizio Marinelli - Cutting problems 30 From a 3D-matching to a bin packing A sequence of triples such that the triples of M contain only first occurrences and the triples of T M contain only subsequent occurrences  T = {(1,1,2), (1,2,1), (2,2,1), (1,2,3), (3,4,3), (3,3,3), (4,4,4)}  M = {(1,1,2), (2,2,1), (3,3,3), (4,4,4)} (1,1,2), (2,2,1), (3,3,3), (4,4,4), (1,2,1), (1,2,3), (3,4,3) An arbitrary sequence of triples (1,1,2), (1,2,1), (2,2,1), (1,2,3), (3,4,3), (3,3,3), (4,4,4) T MM A packing into m bins is obtained by filling each bin with the items associated to each triple. Indeed…
31. 31. Fabrizio Marinelli - Cutting problems 31 Length of triples The length of a triple of all first occurrences equals the length of the bin 324 810  hhhh zyxl  + 110 4   h X h xl 210 24   h Y h yl 410 34   h Z h zl + + = 1540 4  L The length of a triple of all subsequent occurrences equals the length of the bin 111 4   h X h xl 211 24   h Y h yl 48 34   h Z h zl 324 810  hhhh zyxl  + + + = 1540 4  L
32. 32. Fabrizio Marinelli - Cutting problems 32 From a bin packing to a 3D-matching mLm  )1540( 4   Let u(k) be the number of occurrences of k in the triples of T. The total length of items is SUM( I ) = =  Zk kuk )(3 mm 48)1(10 44   )810( 4 m +   ZkYjXi kukjujiui )()()( 32  +mm  44 11)1(10    Xi iui )( +mm 211)1(10 44     Yj juj )(2  Property 1: any solution exactly fills the m bins
33. 33. Fabrizio Marinelli - Cutting problems 33 Properties of the item lengths  The largest items is 410 34   )(810 324   The smallest items is L 4 1 4 15 10 4   Property 2: in any solution, each bin contains exactly 4 items L 3 1 5 3 40 4  
34. 34. Fabrizio Marinelli - Cutting problems 34 Properties of the item lengths (cont’d)  From Property 1 (any solution exactly fills the m bins) and L = 404 + 15 S = L and S mod  =15  Let S be the sum of the item sizes in a bin.  15 must be obtained with the residuals 1, 2, 4, 8 of the item sizes.  There is only one way to obtain 15 by summing four numbers (Property 2: (each bin contains exactly 4 items), with repetition allowed, out of 1, 2, 4, 8. Each bin must contain (exactly) an item lh associated to a triple h = (xh, yh, zh), an item li X , an item lj Y and an item lk Z
35. 35. Fabrizio Marinelli - Cutting problems 35 Properties of the item lengths (cont’d) 14   i X i X i xCl + 810 324   hhhh zyxl + 224   j Y j Y j yCl +  S mod 2 = 15 434   k Z k Z k zCl = 15)()()( 324   hkhjhiS zzyyxxCS (xi – xh) + 15 = 15 xi = xh  S mod 3 = 15  S mod 4 = 15 (yj – yh)2 + 15 = 15 (zk – zh)3 + 15 = 15 yj = yh zk = zh The bin must contain (exactly) an item lh associated to a triple h = (xh, yh, zh), and the items lh X , lh Y , lh Z associated to the components of h
36. 36. Fabrizio Marinelli - Cutting problems 36 Properties of the item lengths (cont’d) 14   i X i X i xCl 224   j Y j Y j yCl 434   k Z k Z k zCl 810 324   hhhh zyxl + + + = 15)()()( 324   hkhjhiS zzyyxxCS 40:10  Z k Y j X iS CCCC 10 + 10 + 10 + 10 a triple of all first occurrences 10 + 11 + 11 + 8 a triple of all subsequent occurrences The bins containing triples of all first occurrences describe a 3D matching
37. 37. Fabrizio Marinelli - Cutting problems 37 A 3D matching From a bin packing to a 3D-matching  T = {(1,1,2), (1,2,1), (2,2,1), (1,2,3), (3,4,3), (3,3,3), (4,4,4)}  M = {(1,1,2), (2,2,1), (3,3,3), (4,4,4)} An arbitrary sequence of triples 1 1 2 1 3 3 4 1 2 2 2 4 3 4 2 1 1 3 3 3 4 A bin packing l1 l2 l3 l4 l5 l6 l7 l1 X l1 X l2 X l1 X l3 X l3 X l4 X l1 Y l2 Y l2 Y l2 Y l4 Y l3 Y l4 Y l2 Z l1 Z l1 Z l3 Z l3 Z l3 Z l4 Z
38. 38. Fabrizio Marinelli - Cutting problems 38 Problem complexity: CSP2 CSP with n = 2 part-types is polynomial: the algorithm by McCormick et al. (2001) runs in O(log2 L)  Given a CSP2 instance (decision version): l1, l2, d1, d2, m, L w.l.o.g.  N {0}  Let P be the convex hull of all the feasible cutting patterns: P = {p  N2 | l1 p1 + l2p2 < L}  [definition] A triangle is unimodular if has integral vertices and area equal to 1/2.
39. 39. Fabrizio Marinelli - Cutting problems 39 Problem complexity: CSP2  [theorem] If the point M = (d1/m, d2/m) is in a unimodular triangle contained in P then the CSP2 instance is feasible. If M is not in P the instance is infeasible.  Consider a unimodular triangle T = (p1, p2, p3) and the matrix            111 321 321 yyy xxx ppp ppp B  If M  T then M is a convex combination of (p1, p2, p3), i.e., the system B = [M, 1]T admits a non negative solution  .  Let x = m   Since the area of T is ½ |det(B)|, then T unimodular implies B unimodular.  [d1,d2, m] is integer.                               m d d x x x ppp ppp yyy xxx 2 1 3 2 1 321 321 111 x is integer
40. 40. Fabrizio Marinelli - Cutting problems 40 Algorithm by McCormick et al.: step 1  Step 1: compute the O(log L) vertices of P by Harvey’s Algorithm in O(logL) and draw conv(P) in O(logL loglogL) P = {p  N2 | 3 p1 + 4 p2 < 25}
41. 41. Fabrizio Marinelli - Cutting problems 41 Algorithm by McCormick et al.: step 2 P = {p  N2 | 3 p1 + 4 p2 < 25}  Step 2: if M is in P we have done.  [example] for [d1,d2, m] = [10, 7, 3] we have M = [10/3, 7/3] and M = [4, 3] M M p1 p2 p3
42. 42. Fabrizio Marinelli - Cutting problems 42 Algorithm by McCormick et al.: step 3 P = {p  N2 | 3 p1 + 4 p2 < 25}  Step 3: compute the axial triangles in O(log L). An axial triangle is prime if its sides are coprime integers  The point M is in one of such triangles.
43. 43. Fabrizio Marinelli - Cutting problems 43 Algorithm by McCormick et al.: step 4  Step 4: if M is in a not prime axial triangle T with sides a and b, then decompose T in prime axial triangles with sides a′ = a / gdc(a,b) and b′ = b / gdc(a,b) . a = 4, b = 2 gdc(a,b) = 2 a′ = 2, b′ = 1 a b T  The point M is in one of such triangles. If a′ = b′ = 1 we have done.
44. 44. Fabrizio Marinelli - Cutting problems 44 Algorithm by McCormick et al.: step 5  Step 5: Bézout’s identity: gdc(a,b) = pa  qb with p, q  N Consider the kS southeast step triangles: from (0, b) in the direction (q,− p)  a = 3, b = 8  1 = 3a – 1b p = 3, q = 1 (a, 0) (0, b)  p q (1, 5) (2, 2)
45. 45. Fabrizio Marinelli - Cutting problems 45 Algorithm by McCormick et al.: step 5  Step 5: Bézout’s identity: gdc(a,b) = pa  qb with p, q  N Consider the kN northwest step triangles: from (a, 0) in the direction (q − a, b − p)  a = 7, b = 19  1 = 11a – 4b p = 11, q = 4 (a, 0) (0, b) b  p q  a (4, 8) (1, 16)
46. 46. Fabrizio Marinelli - Cutting problems 46 Algorithm by McCormick et al.: step 5  [Proposition] Any steps triangle is unimodular.  [Proposition] min{kS , kN } = 1 and max{kS , kN } > 2  Step 5: w.l.o.g. suppose kS > 2  Case A: point M is in one of the steps triangles (the check is made indirectly, without drawing the triangles) (a, 0) (0, b) M
47. 47. Fabrizio Marinelli - Cutting problems 47 Algorithm by McCormick et al.: step 5  [Proposition] Any steps triangle is unimodular.  [Proposition] min{kS , kN } = 1 and max{kS , kN } > 2 (a, 0) (0, b)  Step 5: w.l.o.g. suppose kS > 2  Case B: point M is in one of the new axial triangles. Goto step 3 M
48. 48. Fabrizio Marinelli - Cutting problems 48 Algorithm by McCormick et al. draw conv(P)O(logL loglogL)Step 1 yes compute the axial triangles and find that containing M Step 3 O(logL) compute gdc(a,b) and decompose the triangle in prime axial triangles Step 4 O(logL) Step 5 M is in one of the steps trianglesO(logL) check trivial casesStep 2 O(logL) no O(logL)
49. 49. Fabrizio Marinelli - Cutting problems 49 Problem complexity The decision version of CSP is in NP (EisenbrandShmonin, 2006)  Not trivial since the number of cuts grows with the demand and therefore the size of the optimal value could be exponential in the problem size. The result is a direct consequence of the integer analogues of  [Carathéodory’s theorem] Let X  Rn be a finite set. If b  cone(X), there is a subset Y of X consisting of n points such that b  cone(Y)  [Theorem] Eisenbrand – Shmonin, 2006  Let X  Zn be a finite set of nonnegative integer points. If b  int.cone(X), there exists a subset Y of X with |Y| < size(b) such that b  int.cone(Y).  Let P  Rn be a convex set. If b  int.cone(X), there exists a subset Y of X with |Y| < 2n such that b  int.cone(Y).
50. 50. Fabrizio Marinelli - Cutting problems 50 Problem complexity: CSP with n > 3 CSP with a fixed number n > 3 of part-types is polynomial (2013)  decision version of CSP: can the demand d be expressed as an integer conic combination ixi of the points in the set X = {x  Nn | lTx < L}? P = {x  R2 | 3 x1 + 4 x2 < 20} l = (3, 4), d = (7, 9), and L = 20 d x1 x2 x3 1 = 2 2 = 1 3 = 1 Problems  Points in X are exponentially many in size(L)  Sizes of  can be exponential in size(d)
51. 51. Fabrizio Marinelli - Cutting problems 51 Polynomiality for fixed number of part-types 1. Problem: points in X are exponentially many in size(L) Solution: for fixed n, Eisenbrand and Shmonin showed that the integer cone of X can be generated by a constant number of integer points in P 2. Problem: size() can be exponential in size(d) Solution: uhm… The problem cannot be directly formulated as an ILP with a constant number of variables and then solved in polynomial time (Lenstra, 1983) … unless you provide a bound of size() which only depends on n
52. 52. Fabrizio Marinelli - Cutting problems 52 Polynomiality for fixed number of part-types  [Theorem] Goemans – Rothvoss, 2013 Let P, Q  Rn be two polytopes and X = P  Nn .  Any y  int.cone(X)  Q can be expressed as an integer conic combination of at most 22n + 1 points of X  The coefficients can be computed in time)1(2 )()( )( O QsizePsize nO   [Corollary] CSP can be solved in polynomial time for any fixed number n of part-types By choosing P = {(x1, …, xn, 1) | lTx < L} and Q = {d}  [0, z] we can decide in polynomial time whether z cuts suffice. Then use binary search on [0, z] P (d,0) Q (d,z)
53. 53. Fabrizio Marinelli - Cutting problems 53 Thm by Goemans and Rothvoss (proof sketch) Facts  X can be covered by at most N < mnnO(n)logn many integral parallelepipeds  in NO(1) time
54. 54. Fabrizio Marinelli - Cutting problems 54 Thm by Goemans and Rothvoss (proof sketch) Facts  X can be covered by at most N < mnnO(n)logn many integral parallelepipeds  in NO(1) time 3x x v1 v2 y1 = 1v1 + 6v2 + y1 2. Any integer point d = x, with   N and x integer point of , can be rewrite in terms of an integer conic combination of the vertices of  plus the sum of at most 2n integer points of 
55. 55. Fabrizio Marinelli - Cutting problems 55 Thm by Goemans and Rothvoss (proof sketch) Facts  X can be covered by at most N < mnnO(n)logn many integral parallelepipeds  in NO(1) time 2. Any integer point d = x, with   N and x integer point of , can be rewrite in terms of an integer conic combination of the vertices of  plus the sum of at most 2n integer points of  3. Any point d  int.cone(X) can be expressed as iyi with i  N and yi  Y  X, | Y | < 2n (Eisenbrand – Shmonin, 2006) For any yi with i > 0, take the parallelepiped  containing yi and rewrite iyi by using 2. d can be rewritten as an integer conic combination of at most 22n points of X (vertices of parallelepipeds) plus at most 22n point of X (extra points)
56. 56. Fabrizio Marinelli - Cutting problems 56 Thm by Goemans and Rothvoss (proof sketch) P = {x  Rn | Ax < b} Q = {x  Rn | Dx < f} A(m  n), D(m´  n) Choose Y  Ext() = vertices of parallelepipeds of P, |Y| < 22n, times  Coefficients of the integer conic combination and extra points are unknown but we can compute them by an ILP with a constant number of variables Extra points must be in P d = sum of extra pts + y  int.cone(Ext()) d must be in Q nn i i i Y n i i Y ib n 2 2 1 2 21 21 2        N N x v fDd dxv Ax v v v           n n N 2 2 2
57. 57. Fabrizio Marinelli - Cutting problems 57 Thm by Goemans and Rothvoss (proof sketch)  Coefficients of the integer conic combination and extra points are unknown but we can compute them by an ILP with a constant number of variables nn i i i Y n i i Y ib n 2 2 1 2 21 21 2        N N x v fDd dxv Ax v v v   Variables Constraints m22n n P = {x  Rn | Ax < b} Q = {x  Rn | Dx < f} A(m  n), D(m´  n) Choose Y  Ext() = vertices of parallelepipeds of P, |Y| < 22n, times m´ 22n 22n n22n n22n constant polynomial         n n N 2 2 2
58. 58. Fabrizio Marinelli - Cutting problems 58 Asymptotically exact algorithms  Karmarkar – Karp, 1982 zH < z* + O(log 2 n) AFPTAS  Jansen – Solis-Oba, 2011 zH < z* + 1 in O((log2 ∑di + log L)3 log5 ∑di)  Agnetis – Filippi, 2005 zH < z* + n – 2 in O(poly(log L))
59. 59. Fabrizio Marinelli - Cutting problems 59 Models
60. 60. Fabrizio Marinelli - Cutting problems 60 Literature  1960 - L. Kantorovich “Mathematical methods of organising and planning production”, Management Science  1961 – 1963 - Gilmore, Gomory “A Linear Programming Approach to the Cutting Stock Problem”, Operations Research  1981 - H. Dyckhoﬀ, “A new linear programming approach to the cutting stock problem”, Operations Research  1999 - J.M. Valério de Carvalho, “Exact solution of bin-packing problems using column generation and branch-and-bound”, Annals of Operation Research  2003 - G. Belov, R. Weismantel. “A class of subpattern formulations for 1D stock cutting”, tech. rep.
61. 61. Fabrizio Marinelli - Cutting problems 61 Notation m = number of available stocks L = stock length Q = production bill n = number of parts (n = |Q|) di = demand of parts of type i (1  i  n) li = length of part-type i (1  i  n) z* = optimal value: minimum number of required stocks P = set of all feasible cutting patterns (N = |P|)
62. 62. Fabrizio Marinelli - Cutting problems 62 Assignment model (Kantorovich, 1939) xij  N number of parts of type i cut from the stock item j  Decision variables yj  {0,1} 1 if the stock item j is used, 0 otherwise part-type requirements cut feasibility  [KAN] formulation: mjLyxl nidx yz j n i iji i m j ij m j j          1 1 min 1 1 1 * minimize the number of used stocks   n i idm 1               n i i i l L dm 1 /or
63. 63. Fabrizio Marinelli - Cutting problems 63 How many stocks? m must be not less than the minimum number of required stocks.   n i idm 1               n i i i l L dm 1 / max number of stocks required to cover the demand of part type i.  more precisely: Parts of type i achievable from a single stock  trivially, one stock per part:
64. 64. Fabrizio Marinelli - Cutting problems 64 An example m = 1 + 3 + 1 = 5 L = 10 Stocks Production Bill Q l1 = 7 l3 = 4 l2 = 3 d1 = 1 d2 = 8 d3 = 2 z* = min y1 + y2 + y3 + y4 + y5 Requirements x11+ x12+ x13 + x14 + x15  1 x21+ x22+ x23 + x24+ x25  3 x31+ x32+ x33 + x34+ x35  1 Cut feasibility 7 x11 + 3 x21 + 4 x31  10 y1 7 x12 + 3 x22 + 4 x32  10 y2 7 x13 + 3 x23 + 4 x33  10 y3 7 x14 + 3 x24 + 4 x34  10 y4 7 x15 + 3 x25 + 4 x35  10 y5 Variable domains xij  N i = 1,…,3 j = 1,…,5 y1 y2 y3 y4 y5  {0,1}
65. 65. Fabrizio Marinelli - Cutting problems 65 Pattern-based model (Gilmore-Gomory, 1961) Given the set P = {p1,…, pN} of all feasible cutting patterns, define the integer variables xj  N indicating the activation level of cutting pattern pj  Decision variables  [GG] formulation: nidxp i N j jij  1 1   N j jxz 1 * min part-type requirements minimize the number of used stocks )( L nON  Compact form: z* = min{1Tx: Ax > d, x > 0, integer} A(n  N), x  NN
66. 66. Fabrizio Marinelli - Cutting problems 66 An example L = 10 Stocks Production bill Q l1 = 7 l3 = 4 l2 = 3 d1 = 1 d2 = 8 d3 = 2 4 maximal cutting patterns:            0 1 1 1p            0 3 0 2p            1 2 0 3p            2 0 0 4p z* = min x1 + x2 + x3 + x4 requirements p1 x1 + p2 x2 + p3 x3 + p4 x4  d Variable domains x1 x2 x3 x4  N
67. 67. Fabrizio Marinelli - Cutting problems 67 An example L = 10 Stocks l1 = 7 l3 = 4 l2 = 3 d1 = 1 d2 = 8 d3 = 2            0 1 1 1p            0 3 0 2p            1 2 0 3p            2 0 0 4p z* = min x1 + x2 + x3 + x4 requirements x1  1 x1+ 3x2+ 2x3  8 x3 + 2x4  2 Variable domains x1 x2 x3 x4  N  [note] in case of requirement constraints in terms of equality, also non-maximal cutting patterns are required to guarantee feasibility. 4 maximal cutting patterns: Production bill Q
68. 68. Fabrizio Marinelli - Cutting problems 68 0 2 31 4 5 Arc-flow model (de Carvalho, 1999)  Oriented acyclic graph G(V, A)  V = {0,1,2,…, L};  an edge between i < j if there is a part-type j i long;  Additional edges (k, k + 1), 0 < k < L  1, and (L , 0). [example] L = 5, n = 2, l1 = 2, l2 = 3  Any path from 0 to L describes a valid cutting pattern  A circulation of one unit flow describes a single cut.
69. 69. Fabrizio Marinelli - Cutting problems 69 Arc-flow model (de Carvalho, 1999) xij  N flow through the edge ( i , j )  Decision variables  [deC] formulation: nkdx Ljf Lj jf xx fz k Alii lii Akj jk Aji ij k k                 1 1,...,10 0 min ),( , ),(),( * part-type requirements minimize the number of used stocks flow conservation, i.e., valid cutting patterns
70. 70. Fabrizio Marinelli - Cutting problems 70 An example z* = min f flow conservation node 0: f  x02  x03  x04 = 0 node 2: x02  x24  x25  x23 = 0 node 3: x35 + x34  x03  x23 = 0 node 4: x45  x24  x04  x34 = 0 node 5: f + x35 + x25 + x45 = 0 0 2 31 4 5 [Instance] L = 5, n = 3, l = (4,3,2) d = (1,8,2) Requirements x04 > 1 x03 + x25 > 8 x02 + x24 + x35 > 2
71. 71. Fabrizio Marinelli - Cutting problems 71 Sub-pattern model (BelovWeismantel, 2003)  Idea: patterns are not explicitly described. They are obtained by combining subpatterns                                             0 0 2 ... 0 0 4 0 0 2 0 0 1 1 11121110 K Kpppp phk = 2keh h = 1,…,n and k = 0,…,Kh    hh lLd /,minlog
72. 72. Fabrizio Marinelli - Cutting problems 72 Sub-pattern model (BelovWeismantel, 2003)  h k hkjhkj pp  4 0 0 0 2 0 0 0 0 0 2 0 0 0 0 2 0 0 0 1                                                                  j10 = j11= j21 = j41= j42 = 1 6 0 2 3             jp  Decision variables xj  N activation level of pattern pj j = 1 ,…, N´ jhk  {0,1} 1 if subpattern phk is used to form pattern pj, 0 otherwise yjhk  N activation level of subpattern phk as part of pattern pj
73. 73. Fabrizio Marinelli - Cutting problems 73 Sub-pattern model (BelovWeismantel, 2003)  [BW] formulation: nhdy h N j K k jhk k h     12 1 0     N j jxz 1 * min part-type requirements minimize the number of used stocks size(d) + size(z*) NjLl n h K k jhkh k h    12 1 0  jhkj yx  jhkk h jhk d y      2 cutting pattern feasibility act. level of pj cannot be smaller than the act. levels of forming subpatterns act. levels of phk in pattern j can be positive only if phk forms pj
74. 74. Fabrizio Marinelli - Cutting problems 74 What’s the best model?
75. 75. Fabrizio Marinelli - Cutting problems 75 some preliminaries : Implicit Enumeration
76. 76. Fabrizio Marinelli - Cutting problems 7676 Implicit enumeration [principle] Divide et Impera  The problem P is recursively decomposed (branch) in subproblems P1, …, Pk, smaller and easier to solve.  Not interesting subproblems, i.e., subproblems whose solutions are surely suboptimal for P, are removed (bounding) from the enumeration tree x1 x2 P x1 x2 P1 P2  Subproblems are arranged in an enumeration tree P P1 P2
77. 77. Fabrizio Marinelli - Cutting problems 77 Branch-and-bound algorithm 0. Initialization: l = P; x = ; zU =  1. Stop criteria: if l =  then x is an optimal solution. STOP. 2. Subproblem selection: choose a subproblem Pi from l and remove it from l 3. Subproblem evaluation: if Pi is infeasible go to 1. Let xi R be an optimal solution of Pi R (relaxation of Pi) 4. Bounding: if c(xi R) > zU then go to 1. if xi R is integer then zU = c(xi R); x = xi R; go to 1. 5. Branching: decompose Pi into subproblems Pi1,…,Pik such that Pi = jPij and add the subproblems to l; go to 1.
78. 78. Fabrizio Marinelli - Cutting problems 78 branch-and-bound: optimality gap 1 2 3 4 5 6  At a generic iteration, the list l contains the subproblem still to be solved (active subproblems); they are the leaves of the current enumeration tree. active subproblems zL = minil{c(xi R)}c(x1 R) c(x2 R) = c(x6 R) c(x4 R) c(x3 R) c(x5 R) zU optimality gap = zU  zL gap% = zU  zL zL
79. 79. Fabrizio Marinelli - Cutting problems 79 branch-and-bound: convergence  The gap decreases from above when the value zU is updated (by pruning for optimality or by heuristics).  The gap eventually increases from below when a subproblem defining zL (best bound) is selected from l. zL = c(x0 R) zU z t initial gap gap at time t
80. 80. Fabrizio Marinelli - Cutting problems 80 branch-and-bound efficiency The efficiency of the algorithm strongly depends on several factors: Search strategy  subproblem selection (step 2.),  Type of branching  branching variable (step 5.) Formulation quality (size, tightness, symmetries,…) Relaxations adopted to compute bounds Heuristics adopted to compute incumbents …
81. 81. Fabrizio Marinelli - Cutting problems 81 [KAN] model: discussion  Easy to handle.  The number of variables and constraints is exponential in the size of the input (the numbers m + n of constraints and m(n + 1) of variables grow with the total demand).  An optimal solution might be of exponential size.  The lower bound given by the continuous relaxation is very weak.  The model exhibits symmetries: solutions obtained by stock index permutations are equivalent; bounding is ineffective.
82. 82. Fabrizio Marinelli - Cutting problems 82 [KAN] model: continuous relaxation         L dl z ii R *  [Theorem] Martello – Toth (1990), originally derived for BPP The lower bound z* R given by the continuous relaxation of [KAN] is:  [proof for the BPP] The value of the solution xii = 1, xij = 0 (i  j), yi = li/L is li / L and clearly no solution can be lower than that. Round up, because the number of stocks must be integer.  For instances with large loss the bound can be very poor, i.e. 2 * * z zR 
83. 83. Fabrizio Marinelli - Cutting problems 83 continuous relaxation of [KAN] model zR * = min y1 + y2 x11+ x12  2 5.01 x11  10 y1 5.01 x12  10 y2 x11 , x12 > 0 0 < y1 , y2 < 1  [Instance] L = 10, n = 1, l1 = 5.01, d1 = 2. 2 stocks are required, therefore z* = 2. [KAN] model: continuous relaxation The optimality gap is nearly 50% Solution x11= x12 = 1 and y1= y2 = 0.501 is feasible and values 1.02, hence is optimal. Clearly 02.111*  L dl zR
84. 84. Fabrizio Marinelli - Cutting problems 84 [KAN] model: discussion  Easy to handle.  The number of variables and constraints is exponential in the size of the input (the numbers m + n of constraints and m(n + 1) of variables grow with the total demand).  An optimal solution might be of exponential size.  The lower bound given by the continuous relaxation is very weak.  The model exhibits symmetries: solutions obtained by stock index permutations are equivalent; bounding is ineffective.
85. 85. Fabrizio Marinelli - Cutting problems 85 L = 10 Stocks Production bill Q l1 = 7 l3 = 4 l2 = 3 d1 = 5 d2 = 14 d3 = 3 The optimal value is 10. Demands can be satisfied activating cutting patterns p1 and p2 both at level five.            1 2 0 1p            0 1 1 2p [KAN] model: symmetries
86. 86. Fabrizio Marinelli - Cutting problems 86 y1 = … = y10 = 1, yi = 0 i > 10 x11 = 0, x21 = 2, x31 = 1 x12 = 0, x22 = 2, x32 = 1 … x15 = 0, x25 = 2, x35 = 1 x16 = 1, x26 = 1, x36 = 0 x17 = 1, x27 = 1, x37 = 0 … x1,10 = 1, x2,10 = 1, x3,10 = 0 [KAN] model: symmetries An optimal solution y1 = … = y10 = 1, yi = 0 i > 10 x11 = 1, x21 = 1, x31 = 0 x12 = 0, x22 = 2, x32 = 1 … x15 = 0, x25 = 2, x35 = 1 x16 = 0, x26 = 2, x36 = 1 x17 = 1, x27 = 1, x37 = 0 … x1,10 = 1, x2,10 = 1, x3,10 = 0 All the permutations of stock indices 1,…,10 are (equivalent) optimal solutions... The same solution obtained by swapping the indices of stocks 1 and 6
87. 87. Fabrizio Marinelli - Cutting problems 87 … optimal solutions “evenly” spread among the leaves of the tree (red leaves in the figure):  Since zi L < zi * < zU for each subproblem i, a node can be fathomed if its sub-tree has not any optimal solution (independently from the adopted exploration strategy). Symmetries makes ineffective the bounding; in practice, the search tree is almost completely visited. [KAN] model: symmetries
88. 88. Fabrizio Marinelli - Cutting problems 88 [GG] model: discussion  The number of variables is exponential in the size of the input. It is O(nL/) where  is the length of the smallest part-type.  There exists an optimal solution with a polynomial number of activated patterns (Eisenbrand – Shmonin, 2006).  The lower bound given by the continuous relaxation dominate the combinatorial bound by MartelloToth (Vanderbeck, 1999) and is very tight.  The model does not exhibit symmetries.
89. 89. Fabrizio Marinelli - Cutting problems 89 The size of an optimal solution of [GG]  [Carathéodory’s theorem] Let X  Rn be a finite set. If b  cone(X), there is a subset Y of X consisting of n points such that b  cone(Y) cone(X) b X y1 y2
90. 90. Fabrizio Marinelli - Cutting problems 90 The size of an optimal solution of [GG]  [Theorem] EisenbrandShmonin (2005) Let X  Zn be a finite set of nonnegative integer points. If b  int_cone(X), there exists a subset Y of X with |Y| < size(b) such that b  int_cone(Y) an upper bound for |Y| cannot be given in terms of d (1,4) (1,2) (1,1) (3,7)
91. 91. Fabrizio Marinelli - Cutting problems 91 The size of an optimal solution of [GG]  W.l.o.g. let Y  X the smallest set s.t.   Yx xxb  x > 0, integer    Zx xb  Z  Y  Points are nonnegative, hence the number of distinct points resulting by the sum of points in Z  Y is bounded by    n i ib 1 1  If (by contradiction) than there exists a point p resulting by the sum of two distinct subsets A, B of Y, i.e.    n i i Y b1 12    BA xx xxp  Let A´ = A B and B´ = B A. Clearly A´ and B´ are disjoint and not empty, and    BA xx xx    BABBAA xxxx xxxx
92. 92. Fabrizio Marinelli - Cutting problems 92 The size of an optimal solution of [GG]  = min xA´{x}    AAYY x x x x x x xxxb     AAAY xx x x x xxx  )(    BAAY xx x x x xxx  )(    BABAY x x x x x x xxx )()( )(     BA xx xx one of these coefficients is 0, i.e. there exists Z  Y such that b  int_cone(Z)   )( 1log2log 1 bsizeY bY n i i        n i i Y b1 12 contradiction
93. 93. Fabrizio Marinelli - Cutting problems 93 The size of an optimal solution of [GG]  [Corollary] Let z* = min{cTx: Ax = b, x > 0} A(m  n), aij  N, x  Nn, c  Nn. If this integer program has a finite optimum with optimal value z*, then there exists an optimal solution x* with at most size(b) + size(z*) nonzero components. [Proof]                                                      m n mn n n mm b b z x a a c x a a c x a a c    1 * 1 2 2 12 2 1 1 11 1 The integer vector (z*, b)T is an integer conic combination of the column vectors of the matrix (c, A)T; the solution x corresponds to the coefficients of the combination.
94. 94. Fabrizio Marinelli - Cutting problems 94 The size of an optimal solution of [GG] [Corollary] The size of an optimal solution x* of the [GG] model is polynomial in the size of the problem. [Proof]  z* < di since li < L, i  size(z*) = log(z*+ 1) < log((di + 1)) < log(di + 1) = size(d)  By the previous corollary supp(x*) < size(z*) + size(d)  therefore supp(x*) < 2 size(d) [GG] z* = min{1Tx: Ax = d, x > 0} A(n  N), x  NN
95. 95. Fabrizio Marinelli - Cutting problems 95 [GG] model: discussion  The number O(nL) of variables is exponential in the size of the input.  There exists an optimal solution with a polynomial number of activated patterns (Eisenbrand – Shmonin, 2006).  The lower bound given by the continuous relaxation dominate the combinatorial bound by MartelloToth (Vanderbeck, 1999) and is very tight.  The model does not exhibit symmetries.
96. 96. Fabrizio Marinelli - Cutting problems 96  Given an integer 0 <  < L / 2 let Martello & Toth combinatorial bound I1 = {i : li > L –  }  I2 = {i : L / 2 < li < L –  } I3 = {i :  < li < L / 2} L / 2                           L dl d dd z IIi ii Ii i Ii i Ii i LB )( 32 1 21 max)(  each part in I1 and I2 requires a distinct bin; parts in I3 are neglected continuous lower bound of parts in I2 and I3
97. 97. Fabrizio Marinelli - Cutting problems 97  Given an integer 0 <  < L / 2 let Martello & Toth combinatorial bound I1 = {i : li > L –  } L / 2  I2 = {i : L / 2 < li < L –  } I3 = {i :  < li < L / 2}  iLlzz iLBLB somefor2/:)(max    The maximum value of zLB() is achieved for  = li < L / 2 
98. 98. Fabrizio Marinelli - Cutting problems 98  [Theorem] Vanderbeck, 1999 For any feasible solution x of [GG] and 0 <  < L / 2 Martello & Toth bound vs [GG] bound LB Pi i zx         [Proof] Let Pk = {p  P : pi > 0 for some i  Ik}. Notice that P1  P2 =  and P1  P3 =     )( 2121 PPPi i Pi i Pi i Pi i xxxx     11 Ii i Ii Pj jij dxp     22 Ii i Ii Pj jij dxp because 1 21  IIi ijp for all patterns pj in P1 and P2
99. 99. Fabrizio Marinelli - Cutting problems 99 Martello & Toth bound vs [GG] bound          2121 )( ,0max Pi i PPi i PPPi i xxx L dl IIi ii  )( 32  )( 32 IIi iidl  2Ii id     )( 32 IIi Pj jiji xpl   1PPj jxL            1 32 )(PPj j IIi iji xpl
100. 100. Fabrizio Marinelli - Cutting problems 100 [GG] continuous relaxation zR * = min x1 x1  2 x1 > 0 [GG] model: optimality gap The optimal solution of the continuous relaxation is x1 = 2 and integer. The unique maximal cutting pattern is  11 p  [Instance] L = 10, n = 1, l1 = 5.01, d1 = 2. 2 stocks are required, therefore z* = 2. The optimality gap is 0%
101. 101. Fabrizio Marinelli - Cutting problems 101 [GG] model: optimality gap  In most cases the optimal solutions of the continuous relaxation of [GG] satisfy the IRUP property (Integer Round Up Property): z*   zR *  = 0  [Theorem] Marcotte, 1986 deciding if a given CSP instance has the IRUP property is NP-hard  [Theorem] Filippi, 2007 z*   zR *  < (n  1)/3 + 1
102. 102. Fabrizio Marinelli - Cutting problems 102 [GG] model: optimality gap  [Theorem] Marcotte, 1985 IRUP holds for instances with n = 2 but not for n > 3 [Proof sketch] CSP: min{1Tx: Ax > d, x > 0, integer} A(2  N), x  NN  Columns of A are maximal cutting patterns, i.e., maximal integer points of the polyhedron P = conv(X) where X = {xN2 | l1x1 + l2x2 < L}.  P is integral;  P is lower comprehensive (x  P and 0 < y < x  y  P);  P satisfies the integral decomposition (kN > 2 and for any integral ykP = {kx | x P}, the point y can be expressed as the sum of k integral points of P) CSP has the IRUP property (Baum – Trotter, 1981).
103. 103. Fabrizio Marinelli - Cutting problems 103 [GG] model: optimality gap  [Theorem] Marcotte, 1985 IRUP holds for instances with L < 8, and for instances having the successive divisibility property  [Definition] A CSP problem is said to have the property of successive divisibility if there exists an order of part types such that: l1 | l2 | … | ln, i.e., li divides li + 1 for i = 1,…, n  [non IRUP instance] l = (91, 26, 14), d = (1, 4, 12), and L = 182 z* = 3 and zR * = 1.9945
104. 104. Fabrizio Marinelli - Cutting problems 104 [GG] model: MIRUP  [Conjecture] Scheithauer – Terno, 1996 The optimal solutions of [GG] satisfy the Modified IRUP: z*   zR *  < 1  [Theorem] The instances with n < 6 (Scheithauer – Terno, 1996) and n < 7 (Shmonin, 2008) satisfy the MIRUP  Largest gap known is z*  zR * = 7/6
105. 105. Fabrizio Marinelli - Cutting problems 105 [GG] model: discussion  The number O(nL) of variables is exponential in the size of the input.  There exists an optimal solution with a polynomial number of activated patterns (Eisenbrand – Shmonin, 2006).  The lower bound given by the continuous relaxation dominate the combinatorial bound by MartelloToth (Vanderbeck, 1999) and is very tight.  The model does not exhibit symmetries.
106. 106. Fabrizio Marinelli - Cutting problems 106 L = 10 Stocks Production bill Q l1 = 7 l3 = 4 l2 = 3 d1 = 5 d2 = 14 d3 = 3 The optimal value is 10. Demands can be satisfied activating cutting patterns p1 and p2 both at level five.            1 2 0 1p            0 1 1 2p Such solution is described by [GG] model in a unique way: x1 = 5, x2 = 5, xi = 0 per i {1,2} [GG] model: symmetries
107. 107. Fabrizio Marinelli - Cutting problems 107 [deC] model: discussion  The size is exponential in the size of the input (the numbers L + n + 1 of constraints and O(nL) of variables grow with the stock length).  The lower bounds given by the continuous relaxations of [deC] and [GG] are the same.  It is easy to rewrite a solution of [GG] (activation levels of cutting patterns) into a solution of [deC] (circulation on G) and vice-versa.  The model exhibits symmetries: several paths correspond to the same cutting pattern.
108. 108. Fabrizio Marinelli - Cutting problems 108 From solutions of [deC] to solutions of [GG] L = 10 Stocks l1 = 5 l3 = 2 l2 = 3 d1 = 1 d2 = 8 d3 = 2 Production bill Q 0 2 31 4 5 6 8 97 10  Flow decomposition property: any feasible flow can be decomposed into  a sum of flows in paths from node 0 to node L plus  a sum of flows around cycles
109. 109. Fabrizio Marinelli - Cutting problems 109 From solutions of [deC] to solutions of [GG] L = 10 Stocks l1 = 5 l3 = 2 l2 = 3 d1 = 1 d2 = 8 d3 = 2 Production bill Q 0 2 31 4 5 6 8 97 10 [GG] solution x1 = 1 x2 = 1 x3 = 2 x4 = 0            0 1 1 1p            0 3 0 2p            1 2 0 3p            2 0 0 4p 1 1 1+2 1+1+2 1 + 2 1 + 2 1 2  A solution with integer flow is decomposed into an integer solution of model [GG]
110. 110. Fabrizio Marinelli - Cutting problems 110 Models: pros & cons mjLyxl nidx yz j n i iji i m j ij m j jR          1 1 min 1 1 1 KAN [KAN] nkdx Ljf Lj jf xx fz k Alii lii Akj jk Aji ij R k k                 1 1,...,10 0 min ),( , ),(),( deC [deC] nidxp i N j jij  1 1   N j jR xz 1 GG min [GG] Size Tightness Symmetries
111. 111. Fabrizio Marinelli - Cutting problems 111 Model reformulation Dantzig-Wolfe decomposition 1960 - G. Dantzig, P. Wolfe, “Decomposition Principle for Linear Programs”, Operations Research 1958 - L. Ford, D. Fulkerson, “A suggested computation for maximal multi-commodity network flows”, Management Science
112. 112. Fabrizio Marinelli - Cutting problems 112 [Teorema] Resolution Theorem (Weyl-Minkowski, 1936) Let P be a non-empty polyhedron with at least one extreme point. Then P = conv(ext(P)) + rec(P) P v1 v2 v3 conv(Ext(P)) rec(P) x d u Inner representation of polyhedra
113. 113. Fabrizio Marinelli - Cutting problems 113 The decomposition principle fDx bAx xcT    s.t. min* z compact formulation A(m1  n) D(m2  n) Q(D, f) = {x  Rn | Dx > f} Hp: Q(D,f ) is a non-empty polytope Resolution Theorem (Weyl-Minkowski, 1936): A point x belongs to Q(D,f ) if and only if it can be written as a convex combination of the extreme points {v1,…,vq} of Q(D,f ) : Q(D,f )  x = i ivi with i i = 1 and i > 0
114. 114. Fabrizio Marinelli - Cutting problems 114 Dantzig-Wolfe decomposition qi z i q i i q i ii q i iiDW           10 1 )(s.t. )(min 1 1 1 *     bAv vcT extensive formulation (master problem)  Both formulations have the same optimal value: z* DW = z*  The extensive formulation (in variables ) has fewer constraints (only m1+1) but a huge number of variables (order of ) Substituting for x in the compact formulation we obtain an equivalent       2m n
115. 115. Fabrizio Marinelli - Cutting problems 115 Convexification of an integer program n z Z s.t. min*     x fDx bAx xcT Q(D, f) Q(D, f) = {x  Rn | Dx > f} X = Q(D, f)  Zn  Convexification (analogous of DW decomposition for IPs) X = Q(D, f)  Zn = conv(X)  Zn
116. 116. Q(D, f) Fabrizio Marinelli - Cutting problems 116 Convexification of an integer program n z Z s.t. min*     x fDx bAx xcT conv(X)  If conv(X)  Q(D,f) then the continuous relaxation of the convexification is stronger than the continuous relaxation of the original model. n C Z Xconv z     x x bAx xcT )( min*
117. 117. Fabrizio Marinelli - Cutting problems 117 An example aTx = b dTx = f P = {aTx < b, dTx < f, x > 0, x  Z2}
118. 118. Fabrizio Marinelli - Cutting problems 118 aTx = b dTx = f P = {aTx < b, dTx < f, x > 0, x  Z2} PR = {aTx < b, dTx < f, x > 0} An example
119. 119. Fabrizio Marinelli - Cutting problems 119 dTx = f X = {dTx < f, x > 0, x  Z2} P = {aTx < b, dTx < f, x > 0, x  Z2} PR = {aTx < b, dTx < f, x > 0} An example
120. 120. Fabrizio Marinelli - Cutting problems 120 X = {dTx < f, x > 0, x  Z2} P = {aTx < b, dTx < f, x > 0, x  Z2} PR = {aTx < b, dTx < f, x > 0} x  conv(X) An example
121. 121. Fabrizio Marinelli - Cutting problems 121 aTx = b X = {dTx < f, x > 0, x  Z2} P = {aTx < b, dTx < f, x > 0, x  Z2} PR = {aTx < b, dTx < f, x > 0} x  conv(X) PC = {aTx < b, x  conv(X) , x  Z2} An example
122. 122. Fabrizio Marinelli - Cutting problems 122 Convexification and lagrangian relaxation n ZX P   x bAx xcT s.t. min: convexification lagrangian relaxation provide the same bound lagrangian dual problem  )(minmax TT bAxuxc x0u   X L  )(min)( TT bAxuxcu x  X Llagrangian subproblem 0u ,  )(minmax TT )( bAxuxc x0u   Xconv L
123. 123. Fabrizio Marinelli - Cutting problems 123 Convexification and lagrangian relaxation Theorem of linear programming  )(minmax TT )( bAxuxc x0u   Xconv L Reformulation of min  )(minmax TT bAvuvc 0u   kk Kk L 0u bAvuvc    )( max TT Kkt tL kk index set of ext(conv(X)) 0u vcbAvu    )( max TT Kkt tL kk General form
124. 124. Fabrizio Marinelli - Cutting problems 124 Convexification and lagrangian relaxation Dual problem 0λ λ1 0bAv vc       1 )( )(min T T Kk kk Kk kk λ λ 0u vcbAvu    )( max TT Kkt tL kk lagrangian dual problem of P Convexification of P
125. 125. Fabrizio Marinelli - Cutting problems 125 Discretization of an integer program n z Z s.t. min*     x fDx bAx xcT Q(D, f) = {x  Rn | Dx > f} X = Q(D, f)  Zn Theorem (Nemhauser-Wolsey, 1988): X is generated by a finite number of integer points {p1,…,pq} of X X  x = i ipi with i i = 1 and i {0,1} Q(D, f)
126. 126. Fabrizio Marinelli - Cutting problems 126 discretization vs. convexification PD = {aTx < b x = i=1..8 ipi i=1..8 i = 1 i  {0,1} x  N2} PD = {i=1..8aTpii < b i=1..8 i = 1 i  {0,1}} conv(X) p1 p2 p3p4 p5 p6 p7 p8 aTx = b Discretization: X  x = i ipi with i i = 1 and i {0,1}
127. 127. Fabrizio Marinelli - Cutting problems 127 discretization vs. convexification PC = {aTx < b, x = i=1..4 vii i=1..4 i = 1 i > 0 x  N2} PC = {i=1..4 aTvi i < b, i=1..4 i = 1 i  N4} conv(X) v1 v2 v3v4 aTx = b Convexification: X  x = i ivi , integer with i i = 1 and i  [0,1]
128. 128. Fabrizio Marinelli - Cutting problems 128 discretization vs. convexification ►Bounds: same polytope and therefore same bounds. ►Branching:  convexification: must be performed on the original variables x  discretization: can be performed directly on binary variables  ►Convexification and discretization are equivalent only when all the original variables are binaries, i.e., when the set of integer points of conv(X) corresponds to the set of extreme points of conv(X).
129. 129. Fabrizio Marinelli - Cutting problems 129 Discretization of [KAN] model }1,0{ integer,0 1 1 min 1 1 1 *            j ij j n i iji i m j ij m j j y x mjLyxl nidx yz Each of the m polyhedra corresponding to the pattern feasibility constraints is discretized: }1,0{ 1 1 1        j k q k j k q k j k j kij j j x   p mjLplX n i ii nj ,...,1N 1          p qj = | X j |
130. 130. Fabrizio Marinelli - Cutting problems 130 Discretization of [KAN] model Each of the m polyhedra corresponding to the pattern feasibility constraints is discretized: }1,0{ 11 1)( min 1 1 1 1 1 *             j k q k j k i m j q k i j k j k m j q k j kD mj nid z j j j     p }1,0{ integer,0 1 1 min 1 1 1 *            j ij j n i iji i m j ij m j j y x mjLyxl nidx yz
131. 131. Fabrizio Marinelli - Cutting problems 131 Discretization of [KAN] model }1,0{ 11 1)( min 1 1 1 1 1 *              j k q k j k i q k m j j kik q k m j j kD mj nid z     p Notice that X1 = ,…, = X m = X pk m kk pp ,...,1 and q1 =,...,= qm = qTherefore replace with k > 0 integer summation m m j q k j k  1 1 
132. 132. Fabrizio Marinelli - Cutting problems 132 Discretization of [KAN] model qk m nid z k q k k i q k kik q k kD           1integer,0 1)( min 1 1 1 *     p 0  X m q k k 1  redundant constraint at the end we obtain the [GG] model
133. 133. Fabrizio Marinelli - Cutting problems 133 CSP models relationship mjLyxl nidx yz j n i iji i m j ij m j jR          1 1 min 1 1 1 KAN [KAN] nkdx Ljf Lj jf xx fz k Alii lii Akj jk Aji ij R k k                 1 1,...,10 0 min ),( , ),(),( deC [deC] nidxp i N j jij  1 1   N j jR xz 1 GG min [GG] Discretization of knapsack (fractional) polyhedra KANGG RR zz  Discretization of flow conservation constraints (integer polyhedron) deCGG RR zz 
134. 134. Fabrizio Marinelli - Cutting problems 134 Solution methods: Lower bounds
135. 135. Fabrizio Marinelli - Cutting problems 135  Given an integer 0 <  < L / 2 let Martello & Toth combinatorial bound I1 = {i : li > L –  }  I2 = {i : L / 2 < li < L –  } I3 = {i :  < li < L / 2} L / 2                           L dl d dd z IIi ii Ii i Ii i Ii i LB )( 32 1 21 max)(  each part in I1 and I2 requires a distinct bin; parts in I3 are neglected continuous lower bound of parts in I2 and I3
136. 136. Fabrizio Marinelli - Cutting problems 136 Solution methods: heuristics
137. 137. Fabrizio Marinelli - Cutting problems 137 Primal heuristics low: constructive heuristics high: rounding of the fractional sol. of the continuous relaxation of [GG]  First Fit Decreasing (FFD)  Best Fit Decreasing (BFD)  Sequential heuristics Production volume (idi) The number of operations depends on the number of parts  The number of operations does not depend on the demand (is O(n2L) on average)  Thanks to the IRUP property, the optimal solution approximates the integer optimum as the demand gets larger
138. 138. Fabrizio Marinelli - Cutting problems 138 Low demand: on-line heuristics for BPP  Next Fit (NF): the next item is assigned to the current bin if it has a sufficient residual capacity, otherwise a new bin is open. zNF := 1 and S := 0 for i := 1 to n do if S + li > L then zNF := zNF + 1 and S := 0 pos(i) := zNF and S := S + li end for
139. 139. Fabrizio Marinelli - Cutting problems 139 Low demand: on-line heuristics for BPP  [Theorem] zNF < 2li / L  1 < 2z*  1   ]2,1[)(pos ii: i Ll1:   ]4,3[)(pos ii: i Ll2:    ],1[)(pos NFNF zzii: i LlzNF/2: li > L zNF /2 zNF < 2li / L zNF < 2li / L zNF < 2li / L  1 zNF < 2z*  1worst instance: {2, L  , 2, L  ,…}
140. 140. Fabrizio Marinelli - Cutting problems 140 Low demand: on-line heuristics for BPP  First Fit (FF): the next item is assigned to the bin with smallest index and enough residual capacity. If there is none, a new bin is open. for i := 1 to n do pos(i) := end for zFF := maxi=1,..,n pos(i)        jh ih j Lll )(posIN minarg
141. 141. Fabrizio Marinelli - Cutting problems 141 Low demand: on-line heuristics for BPP  Garey et al., 1976 zFF < 1.7 z*  Simchi-Levi, 1994 zFF < 1.75 z*  Asymptotic performance  Absolute performance  Boyar, 2012 zFF < 12/7 z*  1.7143 z*
142. 142. Fabrizio Marinelli - Cutting problems 142 Low demand: constructive heuristics  First Fit Decreasing (FFD): sort the items in non-increasing order and apply FF.  Best Fit Decreasing (BFD): sort the items in non-increasing order. Largest unplaced item is assigned to the bin with smallest residual capacity, but still sufficient to accommodate the item. If there is none, a new bin is open.  FFD and BFD have:  absolute performance zFFD < 3/2 z* (Simchi-Levi, 1994)  asymptotic performance zFFD < 11/9 z* + 4 (Johnson, 1974) zFFD < 11/9 z* + 3 (Baker, 1985) zFFD < 11/9 z* + 1 (Yue, 1991) zFFD < 11/9 z* + 2/3 (Dósa, 2007)  If Successive Divisibility property holds then FFD is optimal (Coffman, 1987)
143. 143. Fabrizio Marinelli - Cutting problems 143 FFD: absolute performance  [Theorem] (Simchi-Levi, 1994) zFFD < 3/2 z* j = 2/3 zFFD 1  1st case: the bin j contains an item k with lk > L/2 2 zFFD  Consider the bin j = 2/3 zFFD k  bins i < j has no space for k  Since items are sorted in non-increasing order, bins i < j contains items with length > L/2  At least j items has length > L/2  z* > j > 2/3zFFD
144. 144. Fabrizio Marinelli - Cutting problems 144 FFD: absolute performance  [Theorem] (Simchi-Levi, 1994) zFFD < 3/2 z* j = 2/3 zFFD 1 zFFD 2  Consider the bin j = 2/3 zFFD  bins j, j + 1, …, zFFD contain at least 2(zFFD j) + 1 items  2(zFFD j) + 1 > 2(zFFD 2/3 (zFFD +1)) + 1 = 2/3zFFD 1/3 > j  1  Since, none of such (at least) j  1 items fits in the first j  1 bins we have  li > L( j  1)  z* >  li / L > j > 2/3zFFD  2nd case: the bin j (and therefore the bins j +1,…, zFFD) contains no items with length > L/2 worst instance: L = 10, l1 = l2 = 4, l3 = l4 = l5 = l6 = 3
145. 145. Fabrizio Marinelli - Cutting problems 145 FFD: asymptotic performance L = 100 Stocks Production bill Q l1 = 51 l3 = 26 l2 = 27 d1 = 6 d2 = 6 d3 = 6 l4 = 23 d3 = 12 Optimal solution: 9 bins FFD solution: 11 bins
146. 146. Fabrizio Marinelli - Cutting problems 146 Low demand: constructive heuristics  Sequential heuristics: a solution is built up pattern after pattern by solving a sequence of integer knapsack problems. In general, sequential heuristics provide much better solutions than FFD or BFD while some parts are left do  Solve the problem max{yTa | lTa < L, a  Nn}  Cut mini:a > 0 d′i/ai stocks with the cutting pattern a  Update the demands d′ end while i SVC (BelovScheithauer, 2007)      i ii p i ii al lL kyky 21 Fill Bin (Vanderbeck, 1999) yi = li
147. 147. Fabrizio Marinelli - Cutting problems 147 High demand: [GG] rounding heuristics Solve the residual problem by FFD, BFD or by an exact algorithm for BPP x = solution of the continuous relaxation of [GG]  Column generation y = Rounding of x Residual demand = d  Ay  round-up, round-down or a combination of them  Sequential rounding of the most fractional variable (and fixing of integer variables)  Removing of oversupply Stadtler (1990), Waescher  Gau (1996), Belov  Scheithauer (2002)  Absolute performance: zH < z + n
148. 148. Fabrizio Marinelli - Cutting problems 148 [GG] rounding heuristic: performance  [Theorem] (de la Vega - Lueker, 1981) Let x be a basic solution (non necessarily optimal) of the continuous relaxation of [GG]. A solution xH of CSP with zH < z + (n  1)/2 cuts can be always obtained. [proof]  x  = round-down of x and x = round-up of x  R = residual instance and SUM(R) = iR li ► Solution of R 1) x  x is a solution of R with at most n cuts (x is a FBS) 2) Next Fit solves R with at most 2 SUM(R)/L  1 cuts
149. 149. Fabrizio Marinelli - Cutting problems 149 [GG] rounding heuristic: performance (cont’d) ► By combining 1) and 2), the number of required cuts is zR = min{n, 2 SUM(R)/L  1} < (n  1)/2 + SUM(R)/L min{a,b} < mean(a,b)   R N j jH zxz  1             N j n i iijjj lp L xx 1 1 1     N j jj xx 1               N j jj N j jH xx n xz 11 2 1          N j jx n 12 1  z n         2 1               L RSUMn )( 2 1
150. 150. Fabrizio Marinelli - Cutting problems 150 Solution of [GG]: column generation Several problems (cutting stock, crew scheduling, clustering,…) admit a natural formulation with exponentially many variables.  For such integer programs, even the continuous relaxation can be hard to solve. Indeed, it is already impractical to write down the program: the [GG] model has 50 constraints and much more than 250 (i.e., 1.125.899.906.842.624) variables for an instance with 50 part types.
151. 151. Fabrizio Marinelli - Cutting problems 151 Col Gen: the restricted master problem  [Idea] we solve a restricted version of [MP] (the Restricted Master Problem [RMP]) defined on a subset R of A and we dynamically generate only the columns of A that are part of the optimal solution or that guide the search toward the optimal solution. master problem [MP] z* R = min{cTx: Ax = b, x > 0} A(n  N), x  RN  [Observation] an optimal solution is an FBS (Feasible Base Solution): regardless the number N of variables, it has at most n << N non-zero components.
152. 152. Fabrizio Marinelli - Cutting problems 152 Col Gen: the algorithm while there exists a column Aj in A R with negative reduced cost do  Add the column Aj to the matrix R  Solve the new Restricted Master Problem end while The optimal solution of [RMP] is the optimal solution of [MP]  Explicit scan of A R is impractical.  If the mathematical structure of columns is known, a new optimization problem (pricing problem) that computes the most attractive column of A R, i.e., that with minimum reduced cost, can be defined.
153. 153. Fabrizio Marinelli - Cutting problems 153 Col Gen: pricing problem Restricted Master Problem zR = min{cRxR | RxR = b, xR > 0} Pricing Problem * = min{cj  TAj | Aj  AR} dual variables  attractive column Aj * < 0 * > 0 END (xR, 0N r) optimal solution  (xR, 0N r) is feasible for [MP]  If * > 0 then  is feasible for the dual of [MP].  zR = Tb and therefore (xR, 0N r) is optimal for [MP] (strong duality) The efficiency of column generation strictly depends on the complexity of the pricing problem
154. 154. Fabrizio Marinelli - Cutting problems 154 Column generation for [GG] model  The continuous relaxation of [GG] model is: Let yi be the number of parts of type i in a column (cols of A are cutting patterns) [GGR] z* R = min{1Tx: Ax > d, x > 0} A(n  N), x  RN 01 1   n i ii y The column is an attractive cutting pattern if Lyl n i ii 1  The column is a feasible cutting pattern if          integer,0,1min 11 * i n i ii n i ii yLylyPricing Problem:
155. 155. Fabrizio Marinelli - Cutting problems 155 Column generation for [GG] model  The pricing problem is a (bounded) integer knapsack  solved by:  dynamic programming in O(nL)  branch-and-bound: (Pisinger, MartelloToth, HorowitzSahni)          integer,0,1min 11 * i n i ii n i ii yLylyPricing Problem:          integer,0,max1 11 * i n i ii n i ii yLyly 
156. 156. Fabrizio Marinelli - Cutting problems 156 An example L = 8 Stocks Production bill Q l1 = 4 l3 = 2 l2 = 3 d1 = 5 d2 = 4 d3 = 8  A starting feasible solution (required to provide starting duals) can be easily obtained by considering a maximal cutting pattern for each part type:            0 0 2 1p            0 2 0 2p            4 0 0 3p
157. 157. Fabrizio Marinelli - Cutting problems 157 An example: 1st iteration zR = min x1 + x2 + x3 2x1  5 2x2  4 4x3  8 x1 x2 x3 > 0  First restricted master problem on patterns p1, p2 e p3 :  The optimal solution is xR = (2.5, 2.0, 2.0) and its value is zR = 6.5. The corresponding dual solution is  = (0.5, 0.5, 0.25)  The pricing problem is: * = 1  max{0.5 y1 + 0.5 y2 + 0.25 y3 | 4 y1 + 3 y2 + 2 y3 < 8, y1 y2 y3  N} The optimal solution is y = (0, 2, 1) with * = 0.25: the cutting pattern A = (0, 2, 1) has a negative reduced cost and hence is a candidate for entering in the basis of the restricted master problem.
158. 158. Fabrizio Marinelli - Cutting problems 158 An example: 2nd iteration zR = min x1 + x2 + x3 + x4 2x1  5 2x2 + 2x4  4 4x3 + x4  8 x1 x2 x3 > 0  The new restricted master problem is:  The optimal solution is xR = (2.5, 0.0, 1.5, 2.0) and its value is zR = 6.0. The corresponding dual solution is  = (0.5, 0.375, 0.25)  The pricing problem is: * = 1  max{0.5 y1 + 0.375 y2 + 0.25 y3 | 4 y1 + 3 y2 + 2 y3 < 8, y1 y2 y3  N} The optimal solution is y = (2, 0, 0) with * = 0.0: there are no cutting patterns with negative reduced cost, therefore xR = (2.5, 0.0, 1.5, 2.0) is an optimal solution of the continuous relaxation of [GG] model.
159. 159. Fabrizio Marinelli - Cutting problems 159 Solution methods: back to approximation results
160. 160. Fabrizio Marinelli - Cutting problems 160 Asymptotically exact algorithms  de la Vega – Lueker, 1981 zH < (1+)z* + 1/2 0 <  < 1/2  Jansen – Solis-Oba, 2011 zH < z* + 1 in O((log2 ∑di + log L)3 log5 ∑di) for fixed number of part-types  Karmarkar – Karp, 1982 zH < z* + O(log 2 n) AFPTAS
161. 161. Fabrizio Marinelli - Cutting problems 161 de la Vega  Lueker, 1981  0 <  < 1/2 Km1K0 K1 Km2 RF  = L /( + 1) L/2 |K0| < h1 |K1| = … …= |Km2| = |Km1| = |R| = h – 1 ym ym1 ym2 y2 y1
162. 162. Fabrizio Marinelli - Cutting problems 162 de la Vega  Lueker, 1981  Let  M = K0  K1  …  Km1  Q = {y1, …, y1, y2, …, y2,…, ym, …, ym} h1 h1 h1 CSP instance with m part-types  0 <  < 1/2  = L /( + 1) F K0 K1 Km2 Km1 R L/2 |K0| < h1 |K1| = … …= |Km2| = |Km1| = |R| = h – 1 ym ym1 ym2 y2 y1
163. 163. Fabrizio Marinelli - Cutting problems 163 de la Vega  Lueker, 1981  Clearly z* Q < z*  Compute by [GG] a packing of Q with zQ < z* Q + (m  1)/2. The number of cutting patterns is a constant N = O(mL/) therefore [GG] can be solved in polynomial time (Lenstra, 1983)  0 <  < 1/2  = L /( + 1) F K0 K1 Km2 Km1 R L/2 |K0| < h1 |K1| = … …= |Km2| = |Km1| = |R| = h – 1 ym ym1 ym2 y2 y1
164. 164. Fabrizio Marinelli - Cutting problems 164 de la Vega  Lueker, 1981  0 <  < 1/2  = L /( + 1) F K0 K1 Km2 Km1 R L/2 1) Since |R| = |Km1| = …= |K1| > |K0| and ym > l l  Km1 , … , y1 > l l  K0 a packing of M can be easily obtained from a packing of Q, hence zM = zQ 2) Moreover, parts in R can be packed in |R| bins, hence zR = h – 1 3) Pack F in the bins used for packing M and R (by any heuristic, e.g. FF) 4) Pack F´ (the residual of F after step 3.) by Next Fit
165. 165. Fabrizio Marinelli - Cutting problems 165 de la Vega  Lueker, 1981  Case 1: F´   since l < , l  F, each bin is filled for at least L   (except possibly the last) * 1 )1)(( LzlzL n i iH    1 1 *    Lz L zH  2 ** 1 )1(1)1(    zzzH  = L /( + 1)
166. 166. Fabrizio Marinelli - Cutting problems 166 de la Vega  Lueker, 1981         n i ilh 1          h Fn Lm ||  Case 2: F´ =  Let zH < 2 1**   m zz 2 1 )1( *   m z   2 11         L Fn Fn L zH < 2 2 * 2 1 )1(      z      n i il Fn L h Fn L 1  2 * 1 )1(    z zH = zM + zR   1 2 1*         h m z
167. 167. Fabrizio Marinelli - Cutting problems 167 Jansen – Solis-Oba, 2011  Split the instance into big and small parts L = 100  big parts B = {i | li > L}  small parts S = {i | li < L}12 1   n  L = 14.28 l1 = 35 l2 = 25 l3 = 12 l4 = 9  Split each pattern into a big and a small sub-pattern p pB pS  PB = set of big sub-patterns (including the empty subpattern)  PS = set of small sub-patterns
168. 168. Fabrizio Marinelli - Cutting problems 168 Jansen – Solis-Oba, 2011 N             ji i Pj jij i Pj jij Pj j B jj i i yx Sidxp Bidy zy Pjyx j B j B j j B i , : : : * : : p σ σ σp σ  [GGD]: activation level of big sub-patterns The total number of cuts must be z* big parts must be covered by big sub-patterns small parts must be covered  [Proposition] The solution x is optimal for [GG] if and only if (x, y) is feasible for [GGD]
169. 169. Fabrizio Marinelli - Cutting problems 169 Jansen – Solis-Oba, 2011  the formulation [GG] always admits an optimal solution x with at most 2n activated cutting patterns (EisenbrandShmonin, 2005). Relax the integrality constraints Choose < diz Choose a set of 2n big sub-patternsPB  B PzMILP , z PB PB PB xi > 0N             j i j jij i j jij j j jj i i y Sidxp Bidy y jyx B j j j j B i : : : : : p σ σ σp σ  PB
170. 170. Fabrizio Marinelli - Cutting problems 170 Jansen – Solis-Oba, 2011  [Remark] has a constant number (2n) of integer variables but an exponential number, |P| = O((di)n), of continuous variables.  B PzMILP ,  Solvable in polynomial time by the Lenstra’s algorithm and the ellipsoid method where the separation oracle is a knapsack with a constant number of variables.  Let (x*, y*) a solution of MILP(z*, PB*)  Integer variables y* describe an assignment of big parts to z* stocks  Continuous variables x* describe a fractional assignment of small parts  [Theorem] All the small parts can be assigned by using at most one additional stock
171. 171. Fabrizio Marinelli - Cutting problems 171 Jansen – Solis-Oba, 2011 any optimal solution x´ has at most Y + n positive variables * : ** : : :0 : min * BB jj i Pj jij B jj i i Pj j Pjx Sidxp Pjyx x BB j j B i j          p σ p σp p small parts assignment Y + n positive variables for Y constraints  at least Y  n are integer Y constraints (one for each positive yj * ) at most n constraints < 2n fractional variables
172. 172. Fabrizio Marinelli - Cutting problems 172 Jansen – Solis-Oba, 2011    12  nxx ii  the unpacked small parts require at most stocks  Apply Next Fit algorithm to 2n – 1 stocks of the solution (those without small parts) in order to assign such small parts. At the end of the procedure:  all the small parts have been assigned to stocks. We have done  some small parts still have to be assigned the residual of the 2n – 1 stocks is at most (2n – 1) L = (2n – 1)L /(2n – 1) the residual small parts require at most one additional stock
173. 173. Fabrizio Marinelli - Cutting problems 173 Set  = 1/(2n – 1) and z* = di Jansen – Solis-Oba, 2011 For each set of 2n big sub-patterns PB Find = min{1,2,…,z*} such that has a solution (x*, y*) B PzMILP , z Update z* Rounding of (x*, y*) with at most one additional stock O(log(di)) MILP (binary search)           n n 2 /1  times
174. 174. Fabrizio Marinelli - Cutting problems 174 Solution methods: exact algorithms Solution of [GG] by means of column generation + implicit enumeration
175. 175. Fabrizio Marinelli - Cutting problems 175 Literature  1996 – P. H. Vance, “Branch-and-price algorithms for the one-dimensional cutting stock problem”, Computational Optimization and Applications (n < 20)  1998 – J. M. de Carvalho, “Exact solution of cutting stock problems using column generation and branch-and-bound”, International Transactions in Operational Research  1999 – F. Vanderbeck, Computational study of a column generation algorithm for bin packing and cutting stock problems, Mathematical Programming (n < 50)  2006 – G. Belov and G. Scheithauer, “A branch-and-cut-and-price algorithm for one-dimensional stock cutting and two-dimensional two-stage cutting”, European Journal of Operational Research  2008 – C. Alves, J. M. de Carvalho, “A stabilized branch-and-price-and-cut algorithm for the multiple length cutting stock problem”, Computers & Operations Research (n < 100 and 4 stock sizes)
176. 176. Fabrizio Marinelli - Cutting problems 176 Branch-and-Price  Master problem initialization  Master problem solution  Node selection strategies  Branching
177. 177. Fabrizio Marinelli - Cutting problems 177 Master problem initialization Initialization for CSP  A maximal cutting pattern for each part-type (Vance, De Carvalho)  A unique super-pattern with bigM coeff. in the o.f. (Degraeve, Vanderbeck)  Primal solutions computed by FFD, BFD and Fill Bin heuristics (Vanderbeck)  Feasibility of the master problem must be guaranteed at each node of the enumeration tree.  Irrelevant or poorly chosen initial columns may cause the heading-in effect (Vanderbeck, 2005)
178. 178. Fabrizio Marinelli - Cutting problems 178 Master problem solution  Complexity: an LP model with an exponential number of variables is polynomially solvable by column generation if and only if the pricing problem is polynomial.  Algorithms: simplex, dual simplex, subgradient, hybrid methods simplex-subgradient (Barahona  Jensen, 1998 Degraeve  Peeters, 2003).  Running time: tailing-off effect (slow convergence and degeneracy)
179. 179. Fabrizio Marinelli - Cutting problems 179 Master problem solution: tailing-off iterations * Rz master problem solutions zR lower bound zL  The number of iterations can be reduced by:  interrupting the generation of columns (early termination)  increasing the convergence speed (acceleration techniques)
180. 180. Fabrizio Marinelli - Cutting problems 180 Early termination: lower bound’s lower bound  Let A* be the column with minimum reduced cost 1  TA* < 1  TAj for any column j TAj < TA* TAj / TA* < 1 ( / TA*)TAj < 1 dual feasible solution the corresponding value Tb / TA* is a lower bound to z* R optimal solution zR of the current master problem optimal solution * of the pricing problem
181. 181. Fabrizio Marinelli - Cutting problems 181 Early termination: lower bound’s lower bound * * R R L z z z          If zL = zR then early termination (because z* R = zR )  If zL > zU then pruning for bounding  [Remark] the bound zL is not monotone throughout the column generation  (Farley, 1990) a lower bound zL of the master problem optimal value z* R is: master problem current value current pricing optimal value
182. 182. Fabrizio Marinelli - Cutting problems 182 Acceleration techniques General methods:  Trust region (Madsen, 1975): box constraints on dual variables  Stabilization (Ben Amor 2002, Du Merle et al. 1999): perturbation of RHS by means of surplus and slack variables and penalization of their usage Problem specific methods:  Dual cuts (de Carvalho, 2000) adding of valid dual cuts to the master problem before starting column generation  Slow convergence is mainly due to the oscillation of dual variable values throughout the column generation.  The bounding of the dual space speeds-up the convergence
183. 183. Fabrizio Marinelli - Cutting problems 183 Acceleration techniques: stabilization 0x bAx xc   min T  Add surplus and slack variables y and y+  Variables y and y+ account for a perturbation of b by  [, +] helping to reduce degeneracy  The perturbation is penalized by  and + +  Ty + + Ty+  y+ + y y [0, ] y+ [0, +] An optimal solution of the perturbed problem is an optimal solution of the original problem when the best reduced cost is > 0 and y+ = y = 0  Select starting value of  to form a box containing an estimated dual solution  Solve the model.  If the new dual u is in the box [, +] reduce its size and increase the penalty , otherwise enlarge the box and decrease the penalty.
184. 184. Fabrizio Marinelli - Cutting problems 184 Acceleration techniques: stabilization 0ww wδuwδ cuA wεwεub        , max T TTT  Dual perspective  Dual variables u are restricted to the box [, +]  the deviation w or w+ of u from the box is penalized by  and + respectively.
185. 185. Fabrizio Marinelli - Cutting problems 185 Dual cuts  A dual cut is an inequality that cuts the dual space. From the primal point of view, a dual cut is a new variable u : extended master problem [MPE] 0ux bDuAx udx1    , min TT* REz  [Theorem] an optimal solution for [MP] can be obtained from any optimal solution of [MPE] If there exists a map between solutions of [MPE] and [MP] that preserves the value of the objective function. hence, in general, [MPE] is a relaxation of the master problem [MP], but restricting the dual  relaxing the primal  We are interested in dual cuts that preserve at least one optimal dual solution
186. 186. Fabrizio Marinelli - Cutting problems 186 Dual cuts for CSP For each part-type i and for each set S  Si = {s | ls < li} one has: i Ss s   0 Ss si  is a dual cut  [primal interpretation] a produced part of length li can be used for producing (at zero cost) a set of parts whose total length is no greater than li [Property] Let 1, 2 ,…,n be the dual variables of [GG]. If l1 < l2 <…< ln then 1 < 2 <…< n for any optimal solution. i Ss s ll  
187. 187. Fabrizio Marinelli - Cutting problems 187 An example [Instance] L = 9, n = 2, l = (2,3) d = (d1,d2) z* = min 1x1 + 1x2 + 1x3 + 1x4 4x1+ 3x2+ 1x3  d1 1x2+ 2x3+ 3x4  d2 x1 x2 x3 x4 > 0 [Primal] w* = max d11 + d22 4 1 < 1 31 + 12 < 1 11 + 22 < 1 32 < 1 1 , 2 > 0 [Dual] [Pricing] {y  N2 | 2y1+3y2 < 9} Dual space 1 2 41 = 1 11 + 22=1 31 + 2=1 32 = 1  2 + 1 < 0 Dual cut
188. 188. Fabrizio Marinelli - Cutting problems 188 From a solution of [MPE] to a solution of [MP] L = 20 l1 = 3 l2 = 4 l3 = 5 l4 = 7 l5 = 8 2 0 0 2 0 pattern p xp = 0.3 1 1 0 -1 0 + dual cut u u = 0.5 z = 0.3 = 1.1 0.5 0 0.1 0 2 0 0 2 0 pattern p xp = 0.05 4 2 0 0 0 + pattern q xq = 0.25 z = 0.3 = 1.1 0.5 0 0.1 0
189. 189. Fabrizio Marinelli - Cutting problems 189 From a solution of [MPE] to a solution of [MP] 0 3 6 7 13 20 0.5 0.5 0.5 0.05 0.05 0.05 0.05 0.3 2 0 0 2 0 p 0 3 6 9 1613 20 0.25 0.25 0.25 0.25 0.25 0.25  A dual cut is a cycle where the longest edge is traversed backward  cycles can be combined with paths in order to produce new paths 0.3 0.3 0.3 0.3 0.3 2 0 0 2 0 p 1 1 0 -1 0 u 4 2 0 0 0 q
190. 190. Fabrizio Marinelli - Cutting problems 190 Dual cuts for CSP  Dual cuts are exponentially many.  De Carvalho uses only those with | S | < 2  i + i + 1 < 0 i = 1,…, n – 1  i + j + k < 0 i, j, k : lj + lk < li Results:  Beasley, BPP instances: reduction in number of columns (43%) and in running time (20%).  Vance, CSP instances: reduction in number of columns (76%), in running time (78%) and in degenerate pivot (from 39.8% to 8.5%).  Belov, Multiple Lenght CSP instances: reduction in number of branch-and- bound nodes (70%) and in running time (50%)
191. 191. Fabrizio Marinelli - Cutting problems 191 Node selection strategies  The optimal value of IRUP instances is already known at the root node.  non-IRUP instances are uncommon.  The problem admits a lot of equivalent optimal solutions Depth-first search
192. 192. Fabrizio Marinelli - Cutting problems 192 Branching  A valid branching rule must  cut the current fractional solution,  ensure the finiteness of the algorithm.  A good branching rule should  keep the search space balanced,  preserve the structure of the pricing problem (robustness). branching rule for CSP  Dichotomy on variables of the pattern-based model (Belov, Degraeve)  Generalization of the Ryan-Foster rule (Vance)  General scheme (Vanderbeck)  Dichotomy on variables of the arc-flow model (de Carvalho)
193. 193. Fabrizio Marinelli - Cutting problems 193 Branching: dichotomy on pattern variables  * jj xx    1*  jj xx  Easy to implement (by setting bounds on variable) but produces an unbalanced enumeration tree (up-branching corresponds to a very small sub-tree)  up-branching does not destroy the structure of the pricing problem (the up- branching node corresponds to a residual CSP)  A variable with upper bound (set by down-branching) may correspond to the most attractive column generated by the pricing problem. Pricing problem turns out to be a k-knapsack problem. * jx down-branching up-branching
194. 194. Fabrizio Marinelli - Cutting problems 194 Ryan-Foster’s branching rule Branching rule for the set-partitioning ILP (and for BPP) min{cTx: Ax = 1, x {0,1}} 10 1:    skrk aak kx same column1 1:   skrk aak kx different columns0 1:   skrk aak kx  For any fractional solution x one can always identify two rows r and s such that:  Same column: rows r and s must be covered by the same column  Different columns: rows r and s must be covered by different columns
195. 195. Fabrizio Marinelli - Cutting problems 195 Ryan-Foster’s branching rule: example 0 1 0 1 0 1 1 1 0 0 0 0 1 1 0 0 1 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 0 1 0 0 0 1 1 = 1 = 1 = 1 = 1 = 1 0.3 0.3 0 0.7 0.3 0 0 0 0 0 0 0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 r s 0 1 1 1 0 1 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 0 0 0 1 1 1 0 0 0 1 x5 x7 x8 x9 x10 x11 1 1: 54   kk aak kx 0 1 0 1 1 1 0 0 0 1 1 0 0 0 0 1 0 0 1 0 1 1 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1 1 1 0 0 0 0 0 1 x1 x2 x3 x4 x6 x8 x9 x10 x12 0 1: 54   kk aak kx x = same rows: smaller problem disjoint rows: easier problem
196. 196. Fabrizio Marinelli - Cutting problems 196 Ryan-Foster’s branching rule: pricing 0 1 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 0 0 0 1 1 1 0 0 0 1 x5 x7 x8 x9 x10 x11 0 1 0 1 1 1 0 0 0 1 1 0 0 0 0 1 0 0 1 0 1 1 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1 1 1 0 0 0 0 0 1 x1 x2 x3 x4 x6 x8 x9 x10 x12 1 1: 54   kk aak kx 0 1: 54   kk aak kx r s duals 1 2 3 r s c  max{Ty | (y1 y2 y3 1 1)  feas. cols.) pricing c  max{Ty | (y1 y2 y3 yr ys)  feas. cols. and yr + ys < 1 } r s
197. 197. Fabrizio Marinelli - Cutting problems 197 Ryan-Foster’s branching rule: correctness ►[Theorem] Ryan-Foster (1981) For any fractional feasible basic solution x there always exists two rows r and s such that: 10 1:    skrk aak kx [proof] ►Suppose xi fractional and select row r such that ari = 1 (a such row always exists) ►Since ar Tx = 1 and xi fractional, there must exist some other basic column j such that 0 < xj < 1 and arj = 1. ►Since there are no duplicate columns in the basis, there must exist some row s such that asi = 1 or asj = 1 but not both.    1:1: 1 skrkrk aak k ak k xx xj 1 1 = 1……s 0… … …… = 1 xi r 1
198. 198. Fabrizio Marinelli - Cutting problems 198 Branching: generalization of Ryan-Foster  For any fractional solution x* one can always identify a sub-pattern S (i.e. a set of rows S (branching set) and a set of integers {i, i  S}) such that: S j j SSj x αa: * down-branching S j j SSj x αa: *  Summation boils down to one (fractional) variable x* k if the master problem consists of only maximal cutting patterns and one chooses S = ak up-branching  1 : *  S j j SSj x  αa fractional branching dichotomy on pattern variables
199. 199. Fabrizio Marinelli - Cutting problems 199 Branching: general scheme  matrix A of the master problem is converted into a matrix Ab whose columns are the binary encode of the columns of A.  For any fractional solution x* one can always identify a sub-pattern S (i.e., a set of rows S of Ab and a vector of bits {i, i  S}) such that:  If S = [1] the pricing problem remains a binary knapsack down-branching S j j SSj x  αab : * up-branching  1 b : *   S j j SSj x  αa S j j SSj x  αab : * fractional
200. 200. Fabrizio Marinelli - Cutting problems 200 Branching: dichotomy on arc-flow variables Our most valuable source of information are the original variables of the compact formulation; they must be integer, and they are what we branch and cut on (Lübbecke  Desrosiers, 2005) Solve [GG] model but branch on arc-flow variables of [deC] model
201. 201. Fabrizio Marinelli - Cutting problems 201 Branching: dichotomy on arc-flow variables  For any solution x* of [GG] one can always obtain a solution y* of [deC]. In particular, the flow yij on edge (i, j) is:   Pk kij xy ** P′ = set of cutting patterns with part-type of length j – i in position i Branching dichotomy on arc-flow variables * ijy  * ijy  1*  ijy Branching constraints on [GG] model * ijy  Pk kx*   1*   Pk kx
202. 202. Fabrizio Marinelli - Cutting problems 202 Branching: dichotomy on arc-flow variables robust branch-and-price  Branching constraints do not change the structure of the pricing problem (max path on acyclic graph)  j jR xz minGG [GG] constraintsnidxp ij jij  1 Branching constraints   UpBklx klklj j j   ),(1),(),(: p   DwBklx klklj j j   ),(),(),(: p Duals i : lk : lk :  In the pricing problem, the cost of edge (l, k) with k  l = i is    UpBkl lk DwBkl lkilkc ),(),( 
203. 203. Fabrizio Marinelli - Cutting problems 203 An example L = 8 Stocks Production bill Q l1 = 4 l3 = 2 l2 = 3 d1 = 5 d2 = 4 d3 = 8            0 0 2 1p            0 2 0 2p            4 0 0 3p zR = min x1 + x2 + x3 + x4 2x1  5 2x2 + 2x4  4 4x3 + x4  8 x1 x2 x3 > 0  The optimal solution is xR = (2.5, 0.0, 1.5, 2.0) and its value is zR = 6.0. The corresponding dual solution is  = (0.5, 0.375, 0.25)            1 2 0 4p
204. 204. Fabrizio Marinelli - Cutting problems 204 An example 0 2 31 4 5 6 87 2.5 2.5 1.5 1.5 1.5 1.5 2 2 2  Also y02 , y04 , y24 , y46 and y48 are candidate branching variables           0 0 2           0 2 0           4 0 0 2.5 0.0 1.5 2.0           1 2 0 y68 = 2 + 1.5 = 3.5 y68 > 4y68 < 3 Branching arc-flow variable x3 + x4 = 3.5 x3 + x4 > 4x3 + x4 < 3 Branching constraints on [GG] model
205. 205. Fabrizio Marinelli - Cutting problems 205 Research directions Theoretical issues  Is CSP polynomial for each given n? yes  Is MIRUP property true? Algorithmic and application issues  Solution of non-IRUP instances  Problem extension (multiple lengths, technological constraints, integration with scheduling issues)
206. 206. Fabrizio Marinelli - Cutting problems 206 Some other references  2002 – C. Arbib, F. Di Iorio, F. Marinelli, F. Rossi, “Cutting and Reusing: an Application from Automobile Component Manufacturing”, Operations Research  2005 – C. Arbib, F. Marinelli, “Integrating Process Optimization and Inventory Planning in Cutting-Stock with Skiving Option: an Optimization Model and its Application”, European Journal of Operational Research  2007 – C. Arbib, F. Marinelli, “An Optimization Model for Trim Loss Minimization in an Automotive Glass Plant”, European Journal of Operational Research  2009 – C. Arbib, F. Marinelli, “Exact and Asymptotically Exact Solutions for a Class of Assortment Problems”, INFORMS Journal on Computing  2011 – A. Aloisio, C. Arbib, F. Marinelli “On LP Relaxations for the Pattern Minimization Problem”, Networks  2011 – A. Aloisio, C. Arbib, F. Marinelli, “Cutting Stock with No Three Parts per Pattern: Work-in-process and Pattern Minimization”, Discrete Optimization  2012 – C. Arbib, F. Marinelli, F. Pezzella, “An LP-based tabu search for batch scheduling in a cutting process with finite buffers”, International Journal of Production Economics
207. 207. Fabrizio Marinelli - Cutting problems 207 models and algorithms for one-dimensional cutting problems Fabrizio Marinelli fabrizio.marinelli@univpm.it Università Tor Vergata Roma, November 2013 Università Politecnica delle Marche
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Dec. 15, 2019

a survey on complexity results, integer linear programs and algorithms for the one-dimensional cutting stock problem

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