3.10 Column generation method

Transcription

1 3.10 Column generation method Many relevant decision-making (discrete optimization) problems can be formulated as ILP problems with a very large (exponential) number of variables. Examples: cutting stock, crew scheduling, vehicle routing, combinatorial auctions, multicommodity flows,... General principle: enumerate all partially feasible solutions and represent any additional constraints in a set covering, set packing or set partitionning type of formulation. How can we tackle such very large formulations without explicitly considering all variables? Edoardo Amaldi (PoliMI) Optimization Academic year / 11

2 Example: cutting stock problem A paper company produces large rolls of width W. Customer demand: b i small rolls of width w i (assume w i W ), with 1 i m. Smaller rolls are obtained by cutting large rolls according to certain patterns. E.g., if W = 15, w 1 = 6 and w 2 = 2, a feasible cutting pattern consists of two small rolls with w 1 and one with w 2, with a waste of 1. Given large rolls of width W, customer demands for b i small rolls of width w i, with i = 1,..., m, decide how to cut the large rolls into small rolls so as to minimize the number of large rolls used, while satisfying the customer demand. Illustration: NP-hard problem Edoardo Amaldi (PoliMI) Optimization Academic year / 11

3 ILP formulation (Gilmore and Gomory) Variables: x j = number of large rolls cut according to j-th pattern, with 1 j n z ILP = min s.t. n j=1 x j n j=1 a ijx j b i i I = {1,..., m} x j Z + j J = {1,..., n} where a ij is the number of small rolls of width w i in the j-th cutting pattern. The number n of variables (patterns) grows exponentially with the number m of rows (types of small rolls). A generalization of the set covering problem. Note: For large values of the integers b i, rounding an optimal solution of the LP relaxation leads to satisfactory integer solutions. Edoardo Amaldi (PoliMI) Optimization Academic year / 11

4 Column generation scheme Idea: no need to include all variables a priori, new variables are generated on demand. Main steps: 1) consider the LP relaxation of the original ILP problem, choose an initial subset of variables J 0 J, and set k = 0, 2) solve the LP Restricted Master problem (LPRM) with subset of variables J k, 3) solve a pricing subproblem for LPRM with J k to search for an improving non basic variable x l (with negative reduced cost if min problem) and the associated column, 4) if such x l, update J k+1 := J k {l}, set k := k + 1 and goto (2); otherwise LPRM optimal solution is also optimal for the LP relaxation of the original ILP problem. Observation: Column Generation yields an optimal solution of the LP relaxation of the ILP formulation and hence a bound on the value of an optimal ILP solution. Strong impact in practice due to great flexibility to model complicated restrictions. Edoardo Amaldi (PoliMI) Optimization Academic year / 11

5 Example cont.: cutting stock problem LP relaxation of Master problem (LPM): z LPM = min s.t. n j=1 x j n j=1 a ijx j b i i I = {1,..., m} x j 0 j J = {1,..., n}. When solving with the Simplex method: Since c N = c N c t B B 1 N, the reduced cost of the non basic variable x j is c j = 1 m i=1 a ijy i, where y = c t B 1 is the (complementary) dual solution. The current basic feasible solution is optimal if c j 0 for all non basic variables. Dual of LPM problem: max s.t. m i=1 b iy i m i=1 a ijy i 1 j J = {1,..., n} y i 0 i I = {1,..., m}. Edoardo Amaldi (PoliMI) Optimization Academic year / 11

6 Start with a LP Restricted Master problem (LPRM) with a subset J 0 J = {1,..., n} of patterns, guaranteeing a feasible solution. LPRM problem with subset J 0: z LPRM = min s.t. n j=1 x j j J 0 a ij x j b i i I = {1,..., m} x j 0 j J 0. The reduced cost of the non basic variable x j is still c j = 1 m i=1 a ijy i. Dual of LPRM problem with subset J 0: max m i=1 b iy i s.t. m i=1 a ijy i 1 j J 0 y i 0 i I = {1,..., m}. Let x and y be optimal solutions of LPRM and its dual, respectively. Edoardo Amaldi (PoliMI) Optimization Academic year / 11

7 Systematic way to search for new improving variables (columns/patterns)? Look for a non basic variable with smallest reduced cost and the corresponding pattern α {0, 1} m by solving the pricing subproblem: min s.t. c = 1 m i=1 y i α i m i=1 w iα i W (1) α i Z + i I = {1,..., m} Integer Knapsack problem that can be solved in O(mW ) using Dynamic Programming. Two cases: if c 0 then the optimal solution of current LPRM is also and optimal solution of the LP relaxation of the original ILP, if c < 0 then adding to current LPRM the non basic variable associated to cutting pattern α {0, 1} m improves (decreases) the objective function value. Edoardo Amaldi (PoliMI) Optimization Academic year / 11

8 Numerical example: Cutting stock instance with W = 3.9 m, w = and b = Initial patterns: A 1 = 1 waste of 0.05, A 2 = 1 waste of 0.5, A 3 = 0 waste of 0.6, A 4 = 3 waste of From J. Lundgren, M. Rönnqvist, P. Värbrand, Optimization, Studentlitteratur AB, Lund, Sweden, LP Restriced Master problem: min z = 4 j=1 x j s.t. 1 x x x x x j 0 j J 0 = {1, 2, 3, 4} Edoardo Amaldi (PoliMI) Optimization Academic year / 11

9 Optimal solution of LPRM: x = (35, 21, 0, 38.33) t with value z = Optimal dual solution: y = ( 2 9, 1 3, 2 9 )t Pricing subproblem: min c = 1 ( 2 9 α α2 + 2 α3) 9 s.t. 1.25α α α (2) α 1, α 2, α 3 0 integer Optimal solution (integer knapsack): α = (0, 3, 1) t with value c = 2 9. Since c < 0, adding new pattern A 5 = (0, 3, 1) t will improve (decrease) the objective function value. Optimal solution of LPRM with J 1 = {1, 2, 3, 4, 5}: x = (35, 6.625, 0, 0, ) t with value z = Optimal dual solution: y = ( 1 4, 1 4, 1 4 )t Edoardo Amaldi (PoliMI) Optimization Academic year / 11

10 Pricing subproblem: min c = 1 ( 1 4 α α2 + 1 α3) 4 s.t. 1.25α α α (3) α 1, α 2, α 3 0 integer Optimal solution (integer knapsack): α = (0, 3, 1) t (as before!) with c = 0. Thus x = (35, 6.625, 0, 0, ) t is an optimal solution of LP relaxation of the original ILP formulation. Rounding up we have the integer solution x = (35, 7, 0, 0, 44) t with z = 86. Since z LPM = the lower bound is 85. Optimal solution ILP: x ILP = (36, 6, 0, 0, 43)t with z ILP = 85 Edoardo Amaldi (PoliMI) Optimization Academic year / 11

11 General remarks For LPs with m rows and n columns, basic feasible solutions have at most m nonzero variables the basic variables. Since m n, only a very small part of the variables (columns) are needed for optimal solution. Initial set of columns (indexed by J 0) has a strong impact: rich enough to guarantee an initial feasible solution but not too large to reduce the computational load. Use heuristics for the pricing subproblem as long as an improving variable (column) is found. Exact method only to certify that LPRM solution is also optimal for LPM. To find an optimal solution of the original ILP formulation, Column Generation can be embedded in a Branch-and-Bound framework Branch-and-Price method. Third computer laboratory devoted to a Column Generation approach to the airline crew pairing problem. Edoardo Amaldi (PoliMI) Optimization Academic year / 11