Cutting Stock Problem

By Devendra Kumar MittalHemant DasKiran Kumar Nagar

Introduction

• It arises from many applications in industry. Imagine that you work in a paper mill and you have a number of rolls of paper of fixed width waiting to be cut, yet different customers want different numbers of rolls of various-sized widths. How are you going to cut the rolls so that you maximize the profit. To maximize profit you have to use minimum trim loss.

• It is a NP Hard problem.

Introduction(1 Dimensional)

The goal of the 1 Dimensional cutting stock problem:

is to find the "optimal" cutting patterns, where the total number of long steel bars(may be any other material) used is minimized, subject to the constraint that the desired shorter steel bars are cut in needed quantities.

..........

..........

Bar Length = 1200 cm

....................

Bar Length = 600 cm

Bar Length = 700 cmBar Length = 300 cm

12

35...

12

10...

12

15...

Trim Loss

300 cm 300 cm 300 cm 300 cm

300 cm 300 cm 600 cm

300 cm 700 cm

600 cm600 cm

Pattern 1:

Pattern 2:

Pattern 3:

Pattern 4:

2D Cutting Stock Problem

• In it cuts are made along both the directions (width and length).

• Two-dimensional cutting stock problems are more difficult to solve than one-dimensional problems because of the greater complexity of defining feasible cutting patterns.

Uses

• Industrial applications of cutting-stock problems for high production volumes arise especially when basic material is produced in large rolls that are further cut into smaller units. This is done e.g. in paper and plastic film industries but also in production of flat metals like steel or brass.

Formulation of 1 Dimensional Cutting Stock problem

• Suppose there is stock of rods of length • There are orders.• Each order requires rods of length where

Cont.

• Then problem isMin Subject to where And where

And no of patterns and no of times pattern is used.

And pattern and is item of pattern

Technique used for 1 Dimensional

• There are many techniques for solving 1 dimension cutting Stock Problem.

• The most useful technique is Column generation.

Column Generation

• There are many patterns for a problem but we use only few patterns to get the answer.

• These few patterns are basis patterns. • Denote Basis by B(list of all basis patterns).• Suppose the problem is to• Min • But here

• In column generation, Initial basis has some random patterns.

• Now chose a non basis pattern(which is not in basis) and if it has negative than it will enter the basis.

• In this(cutting stock problem) and where is dual variableSo

• So pattern with enters the basis.

Dual Formulation

• Dual of primal formulation of 1 D cutting stock problem is (with dual variable

Such that Where is unrestricted.

• Now our job is to find out pattern which enters the basis.

• Now problem is where are constants.• Now it becomes a linear programming

problem.• But it has only integer solution.• It can also be solved by bounded knapsack

algorithm.

Techniques to generate Pattern

• There are many techniques to generate pattern for 1 D cutting Stock problem.

• Some are1) Branch and Bound2) Cut Plane3) Branch and Cut– These all are Integer linear programming algorithms.

4) Genetic Algorithm

Branch and Bound• Branch and bound is widely used integer linear programming

algorithm.• Branch-and-bound is essentially a strategy of ‘‘divide and

conquer.’’ The idea is to partition the feasible region into more manageable subdivisions and then, if required, to further partition the subdivisions.

• Suppose there is an IP problem

• Find out the LP relaxation.

• If LP-relaxation has integral optimal solution x*, then x* is optimal for IP too.

• In our case, = (2.25, 3.75) is the optimal solution of the LP-relaxation.

• But, unfortunately, it is not integral.• The optimal value is 41.25 .• Choose a variable that is fractional in the

optimal solution to the LP-relaxation – say, .• Observe that every feasible IP point must have

either .

• With this in mind, branch on the variable x2 to create the following two sub problems:

• Another Example

• This chain will go till we find a optimal integer solution.

• Example of 1 D cutting stock problem

• Solution is

Cut Plane

• This technique is popularly used to find integer solutions to mixed integer linear programming (MILP) problems.

• The use of cutting planes to solve MILP was introduced by Ralph E. Gomory.

• In this also, First find out the LP relaxation of the problem.• When an optimal solution is obtained, and this solution

has a non-integer value for a variable• Then find linear constraints which are satisfied by all

feasible integer points but violated by the current fractional solution.

• If such an inequality is found, it is added to the linear program, such that resolving it will yield a different solution which is hopefully "less fractional".

• This process is repeated until an integer solution is found (which is then known to be optimal).

• In practice, the branch-and-bound procedures almost always outperform the cutting-plane algorithm.

• Nevertheless, the algorithm has been important to the evolution of integer programming.

• Historically, it was the first algorithm developed for integer programming that could be proved to converge in a finite number of steps

Example

Branch and Cut

• It is the mixer of “Branch and bound” and “Cut plane” method.

• First solve the problem using “Cut Plane”.• If no more furthers plane can be added and

the optimal solution has at least one non-integer variable then apply “Branch and Bound”.

Example

Fitness Function

• To reduce the probability of taking chromosomes that are violating constrain equation, fitness function must be design effectively.