1 Column Generation

2 Outline trim loss problem different formulations column generation the trim loss problem master problem and subproblem in column generation the time constrained shortest-path problem shortest-path subproblems for network-based problems

3 Motivation Example for Column Generation trim-loss problem, 7-meter steel pipes to cut 150 pieces of 1.5-meter (pipe) segments 250 pieces of 2-meter segments, and 200 pieces of 4-meter segments loss: pipes cut and segments left after satisfying the demands how to cut the steel pipes so as to minimize the loss?

4 Motivation Example for Column Generation X: the total number of steel pipes used trim loss = 7X – (150)(1.5) – (250)(2) – (200)(4) = 7X-1525 minimizing trim loss = minimizing X, the number of steel pipes used

5 Formulation 1: A Naive One Motivation Example for Column Generation several formulations an upper bound of pipes used: pipes cut by the (0, 1, 1) pattern 38 pipes cut by the (4, 0, 0) pattern 17 pieces cut by the (0, 3, 0) pattern

6 Formulation 1: A Naive One Motivation Example for Column Generation segments: 1 st type = 1.5 meters, 2 nd type = 2 meters, and 3 rd type = 4 meters variables. x ij = # of type-j segments cut from pipe i e.g., x i = (2, 2, 0) two 1.5-meter segments and two 2-meter segments from the ith pipe

7 Formulation 1: A Naive One Motivation Example for Column Generation formulation

8 Formulation 1: A Naive One Motivation Example for Column Generation disadvantages of Formulation 1 many alternate optimal solutions due to the symmetry hard to solve

9 Formulation 2: Variables on Cut Patterns Motivation Example for Column Generation defining variables on cut patterns parameters a ij = # of j type segments produced by the ith cut pattern a 1 = (a 11, a 12, a 13 ) T = (0, 0, 1) trim loss t 1 = 3 variables: x i = the # of steel pipes cut in the ith pattern

10 Formulation 2: Variables on Cut Patterns Motivation Example for Column Generation cut patterns

11 Formulation 2: Variables on Cut Patterns Motivation Example for Column Generation formulation: question: Is it necessary to include inefficient cut patterns in this formulation? yes is there any formulation that uses only efficient cut patterns?

12 Formulation 3: Variables on Efficient Cut Patterns Motivation Example for Column Generation efficient cut patterns

13 Formulation 3: Variables on Efficient Cut Patterns Motivation Example for Column Generation formulation

14 Formulation 3: Variables on Efficient Cut Patterns Motivation Example for Column Generation challenges faced by the formulation hard problem, an integer program explosive number of variables for long pipes and multiple demands solved by column generation ignoring the integral constraints

15 Solving the Trim Loss Problem by Column Generation min i x i, s.t. i a ij x i b j, j = 1, 2, 3, x i 0, i = 1, 2, 3. = (c B ) T B -1 be the vector of dual variables reduced cost of x i = c i - T a i = 1- T a i minimal reduced cost from max T a i

16 Solving the Trim Loss Problem by Column Generation minimal reduced cost from solving the subproblem max j j a j, s.t. j l j a j L; a j Z + {0}, i. a knapsack problem, NP-hard, with pseudo- polynomial algorithm in L

17 Solving the Trim Loss Problem by Column Generation master problem: the problem with the columns selected providing subproblem suggesting a new cut pattern, or optimal

18 Solving the Trim Loss Problem by Column Generation Generic Column Generation Algorithm for the trim loss problem: 1 Select columns for the initial basis. 2 Solve the master problem to get 3 Solve the knapsack subproblem for new cut pattern; stop if optimal, else go to 2 with the new cut pattern added to the master problem

19 Solving the Trim Loss Problem by Column Generation Solving the Trim Loss Problem by Column Generation

20 A Shortest-Path Subproblem in Solving Network-Based Problem by Column Generation a time-constrained shortest-path problem minimum cost path from node 1 to node 6 such that the total time 14 label of arcs: (c ij, t ij )

21 A Time-Constrained Shortest-Path Problem variables: x ij = 1 if arc (i, j) is taken, else x ij = 0 formulation constrained shortest- path problem, NP- hard problem

22 Path-Based Formulation of a Time- Constrained Shortest-Path Problem P: set of all paths parameters a pij = for path to be the first path, a 112 = 1, a 124 = 1, a 146 = 1, and other a 1ij = 0. possible to express x ij in p, where p = the probability that path p is taken By itself there is no guarantee p = 1 so that fractional x ij may be possible.

23 A Time-Constrained Shortest-Path Problem set of all paths for the problem

24 A Time-Constrained Shortest-Path Problem formulation

25 A Time-Constrained Shortest-Path Problem relaxed version: master problem solved by column generation 1 and 0 be the dual variable of the first and the second constraints, respectively

26 A Time-Constrained Shortest-Path Problem reduced cost of variable p

27 A Time-Constrained Shortest-Path Problem subproblem: shortest-path problem x ij {0, 1}, (i, j) A. s.t.s.t.

28 Solving the Time Constrained Shortest- Path Problem by Column Generation Generic Column Generation Algorithm for the time constrained shortest-path problem: 1 Start arbitrarily with a feasible path 2 Solve the master problem to get 3 Solve the shortest-path subproblem for new path; stop if optimal, else go to 2 with the new path added to the master problem