CUTTING STOCK PROBLEM

COMBINATORIAL OPTIMIZATION

MADE BY :

LUCKY AGARWAL Y08UC076

RADHIKA LOOMBA Y08UC099

VENKATESH VISWANATHAN Y08UC138

UNDER GUIDANCE OF:

Dr. ANJULIKA GUPTA

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INTRODUCTION

• The cutting-stock problem is an optimization problem,

or more specifically, an integer linear programming problem

• It arises from many applications in industry , eg. Paper

mills, sheet metal cutting etc.

• Solving this problem to optimality can be economically significant: a difference of

1% for a modern paper machine can be

worth more than one million USD per year.

• Can be classified in several ways:1 D: cutting wires/cables/pipes etc

2 D: furniture/glass production etc3 D: Shipping container packing.

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TYPES OF 1D CUTTING STOCK PROBLEM

There are 2 types of 1DCSPs.

• Single stage 1DCSP

• Multistage 1DCSP

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SINGLE STAGE 1DCSP

• The finished product is obtained in

one single step

example:

or

STOCK BAR

F1 F2 F4F3

LEFT

STOCK BAR

F1 F2

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MULTI STAGE 1DCSP

• In multistage 1DCSP the cutting

process is distributed over several successive stages

First stage

Second stage

Third stage

S1

F1 I1 S2

F2 F3 F4 I2

F5 F6

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SINGLE STAGE

1 DIMENSIONAL CUTTING

STOCK PROBLEM

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FORMULATION

• The standard formulation starts with a

list of m orders, each requiring qj, j = 1,...,m pieces

• We then construct a list of all possible

combinations of cuts (often called "patterns")

• A positive integer variable xi denotes

how many times each pattern is to be used

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Example:

The paper rolls all have width of 100

inches. There are orders for 4 different

order widths as follows:

Then there are 37 ways of having a

pattern.

Quantity Ordered Order Width

(inches)

97 45

610 36

395 31

211 14

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37 Patterns•45

•45 45

•45 36

•45 36 14

•45 31

•45 31 14

•45 14

•45 14 14

•45 14 14 14

•36

•36 36

•36 36 14

•36 36 14 14

•36 31

•36 31 31

•36 31 14

•36 31 14 14

•36 14

•36 14 14

•36 14 14 14

•36 14 14 14 14

•31

•31 31

•31 31 31

•31 31 14

•31 31 14 14

•31 14

•31 14 14

•31 14 14 14

•31 14 14 14 14

•14

•14 14

•14 14 14

•14 14 14 14

•14 14 14 14 14

•14 14 14 14 14 14

•14 14 14 14 14 14 14

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SOLUTION APPROACH

• The integer linear program can be

stated as:

Minimize

s.t.

For all j=1……m

and >=0 integer

Where is the number of times order j

appears in pattern i

is the cost (often the waste) of pattern i

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• The above formulation's quantity constraints are

minimum constraints (at least the given amount of each order must be produced,but possibly more)

• When = 1 the objective minimises the number of

utilised master items

• The most general formulation has two-sided constraints

(and in this case a minimum-waste solution may

consume more than the minimum number of master

items):

• In general, the number of possible patterns

grows exponentially as a function of m.

As the number of orders increases, it maytherefore become impractical to

enumerate the possible cutting patterns

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1 D CUTTING STOCK

PROBLEM WITH USABLE

LEFTOVERS

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DESCRIPTION

• We will consider the1DCSPUL, focusing on minimizing

the pseudo bar cost in respect of the following three aspects:

1. The bar cost. It is the actual cost of the bars required

to fulfill the item demand.

2. The profit obtained from the leftovers.

The residual length of a cutting pattern is

taken as a leftover if it is longer than athreshold; as trim loss otherwise. The

leftovers are returned to inventory forfuture use.

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DESCRIPTION (contd…)

3. The profit obtained from shorter

stocks reduction. Shorter stocks may

represent the leftovers generated

from previous stock cutting processes, which in general are more

difficult to use than longer bars.

The proposed algorithm consists of a linear programming (LP) procedure.

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FORMULATION OF 1DCSPUL

• The 1DCSPUL is characterized by the following data:

1. The demand of items:

m :number of item types

(l1,…, lm) :item lengths

(b1,…, bm) :item demands

2. The leftover:

K :number of leftover types

(g1,…, gK) :leftover lengths with g1<…<gK

(u1,…, uK) :upper bound on no. of leftovers

(w1,…, wK) :leftover values

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3. The stock bar usage:

M :number of bar types

(L1,…, LM) :bar lengths

(B1,…, BM) :bar supply

(p1,…, pM) :bar prices

Now, let n=m+K & N=m+K+M

The notations for the linear program are:

z : total cost of the cutting plan

q : number of cutting patterns, q ≥ N

cj : the price of the stock bar used bypattern j

xj : the frequency of pattern j

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: when 1 ≤ i ≤ m , is the number of items

type i appearing in pattern j

when m < i ≤ n , is the number of

leftovers type i-m appearing in pattern j

when n < i ≤ N , =1 if pattern j uses bar

type i-n ; =0 otherwise

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• The 1DCSPUL can be formulated as the following integer

linear programming problem:

• Min

• St

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• Where in the objective function

represents the total value of the leftovers in pattern j.

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1 D CUTTING STOCK

PROBLEM WITH TWO

STAGES

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PROBLEM AT HAND:List of roll sizes available

List of stock sizes.

List of intermediate sizes.

List of finished sizes. Stock

Final Product

Intermediatea two-stage CSP with one width of stock roll that is to be cut at the first stage into several intermediate rolls

Constraints for intermediate rolls are strict: For every size the total number of consumed rolls cannot exceed the produced amount

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assumption: the width of an intermediate roll coming out of the first

stage and going to the second one satisfies minimum–maximum

restrictions

x1 and x2 are pattern activities for the first and the second

stages

b is a vector of customer demands on finished rolls.

Cutting patterns of the first stage and the second stage: A11

and A22 matrices

A12 showing the relations between the two:

If column j represents a pattern then the value is

non-zero (-1) only for the position corresponding to the

intermediate roll i that should be cut

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DUAL PROBLEM

vectors u1 and u2 are vectors of dual variables

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I) Generating a cutting pattern for the first stage

This is identical to solving the given Knapsack problem. We

basically stress on maximum usage of patterns

(Knapsack I)

u1 is a vector of values that are of the dual

variables of the master problem

y is a vector of intermediate roll widths

w0 is the stock roll width

vector a is a vector of variables.

If the objective function value exceeds 1.0, a new column a for the first stage is generated.

The solution vector of u1T ≤ A11 1Tenters as a column into matrix

A11. 24

II) Generating a cutting pattern for the final rolls

Here also we can design it as the given Knapsack problem.

(Knapsack II)

u2 is a vector of values that are the dual variables of

the master problem;

w is a vector of finished sizes;

emin is a mandatory minimal edge, always ≥0

yj is width of intermediate roll j

vector a is a vector of variables.

If the objective function value exceeds u1j, a new column a for the second stage is generated.

The solution vector of u2TA22 ≤−u1

TA12enters as a column into matrix A22.

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If we do not have uncertainty in the

intermediate rolls, two types of

knapsacks shown above – Knapsack I

and Knapsack II – are sufficient to

solve the problem optimally

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REFERENCES

• http://en.wikipedia.org/wiki/Integer_li

near_programming.

• www.sciencedirect.com.

• Research Paper: Modeling multistage

cutting stock problems -Eugene J. Zak

• Research Paper: J.M.V. Carvalho and

A.J.G. Rodrigues , An LP-based

approach to a two-stage cutting stock problem. European Journal of

Operational Research 84 3 (1995), pp.

580–589.

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THANK

YOU

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