1 Column Generation. 2 Outline trim loss problem different formulations column generation the trim...

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Column GenerationColumn Generation

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OutlineOutline

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

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Motivation Example Motivation Example for Column Generationfor 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?

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Motivation Example Motivation Example for Column Generationfor 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

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Formulation 1: A Naive OneFormulation 1: A Naive OneMotivation Example for Column GenerationMotivation Example for Column Generation

several formulations an upper bound of pipes used: 255

200 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

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Formulation 1: A Naive One Formulation 1: A Naive One Motivation Example for Column GenerationMotivation Example for Column Generation

segments: 1st type = 1.5 meters, 2nd type = 2 meters, and 3rd type = 4 meters

variables .

xij = # of type-j segments cut from pipe i

e.g., xi = (2, 2, 0) two 1.5-meter segments and two 2-meter segments from the ith pipe

1, if the th pipe is used, 1,..., 255

0, . .,ii

y io w

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formulation

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disadvantages of Formulation 1 many alternate optimal solutions due to the

symmetry

hard to solve

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Formulation 2: Variables on Cut PatternsFormulation 2: Variables on Cut PatternsMotivation Example for Column GenerationMotivation Example for Column Generation

defining variables on cut patterns parameters

aij = # of j type segments produced by the ith cut pattern

a1 = (a11, a12, a13)T = (0, 0, 1) trim loss t1 = 3

variables: xi = the # of steel pipes cut in the ith pattern

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cut patterns

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formulation:

question: Is it necessary to include inefficient cut patterns in this formulation?

yes

is there any formulation that uses only efficient cut patterns?

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Formulation 3: Formulation 3: Variables on Efficient Cut PatternsVariables on Efficient Cut Patterns

Motivation Example for Column GenerationMotivation Example for Column Generation

efficient cut patterns

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formulation

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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

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

min i xi,

s.t. i aijxi bj, j = 1, 2, 3,

xi 0, i = 1, 2, 3.

= (cB)TB-1 be the vector of dual variables

reduced cost of xi = ci - Tai = 1- Tai

minimal reduced cost from max Tai

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minimal reduced cost from solving the subproblem

max j jaj,

s.t. j ljaj L;

aj Z+{0}, i.

a knapsack problem, NP-hard, with pseudo-polynomial algorithm in L

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master problem: the problem with the columns selected providing

subproblem suggesting a new cut pattern, or optimal

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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

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Solving the Trim Loss Problem by Column Generation

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A Shortest-Path Subproblem A Shortest-Path Subproblem in Solving Network-Based Problem by Column Generationin 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: (cij, tij)

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A Time-Constrained A Time-Constrained Shortest-Path ProblemShortest-Path Problem

variables: xij = 1 if arc (i, j) is taken, else xij = 0

formulation constrained shortest-

path problem, NP-hard problem

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Path-Based Formulation of a Time-Path-Based Formulation of a Time-Constrained Shortest-Path ProblemConstrained Shortest-Path Problem P: set of all paths

parameters apij =

for path 1-2-4-6 to be the first path, a112 = 1, a124 = 1, a146 = 1, and other a1ij = 0.

possible to express xij in p, where p = the “probability” that path p is taken

1, if arc ( , ) is in path ,

0, . .

i j p

o w

By itself there is no guarantee p = 1 so that fractional xij

may be possible.

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set of all paths for the problem

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formulation

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relaxed version: master problem solved by column generation

1 and 0 be the dual

variable of the first and the second constraints, respectively

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reduced cost of variable p

1 0

( , ) ( , )

1 0( , )

=

p ij pij ij piji j i j

ij ij piji j

c c a t a

c t a

A A

A

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subproblem: shortest-path problem

1 0( , )

min ij ij iji j

c t x

A

1( , )

1,ji j

x

A

{ :( , ) } { :( , ) }0,ij ji

j i j j i jx x

A A

6( , )

1,ii j

x

A

xij {0, 1}, (i, j) A.

s.t.

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Solving the Time Constrained Shortest-Solving the Time Constrained Shortest-Path Problem by Column GenerationPath 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

1 Column Generation. 2 Outline trim loss problem different formulations column generation the trim...

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