Free Essay

Stock Cutting

In: Computers and Technology

Submitted By greg1990
Words 595
Pages 3
Stock Cutting problems

This problem is used both to illustrate the need for good modelling, when the problem size increases the model does not increase exponentially in size, and the use of a Knapsack formulation in modelling a problem from a different area.

The roll of paper shown in diagram 1 is to be cut into smaller rolls to satisfy customer demand.

[pic]

For example:

The width of a roll is given by w = 20

and there is a demand for rolls of the size:-

6. rolls of width 3 9. rolls of width 7, and 11 rolls of width 9

The first model to represent this problem, starts by listing, as variables, all the possible cut patterns.

|Number of 9 widths |Number of 7 widths|Number of 3 widths |Waste |Number of rolls cut |
|cut |cut |cut | |to pattern |
|2(9 | | |2 |U |
|1(9 |1(7 |1(3 |0 |V |
|1(9 | |3(3 |2 |W |
| |2(7 |2(3 |0 |X |
| |1(7 |4(3 |1 |Y |
| | |6(3 |2 |Z |

waste is seen as being an important factor and has been calculated for each cot pattern.
The model now follows as:-

[pic]

with the objective “to minimise the total waste” when unused paper is considered to be waste.

Waste = amount cut - amount used, this is given by

[pic]

this can be simplified to

[pic]

that is “minimise the number of rolls cut”.

This is a simple model, which does give the correct solution, but the determination of the “cut-patterns” is not a trivial task, for example in similar problem where

• width of a roll w = 25

• and there is a demand for rolls of width 3, 7, and 9

the cut-patterns are

|Number of 9 widths cut |Number of 7 widths cut |Number of 3 widths cut |Waste |Number of rolls cut |
| | | | |to pattern |
|2(9 |1(7 | |1 |U |
|2(9 | |2(3 |0 |V |
|1(9 |2(7 | |2 |W |
|1(9 |1(7 |3(3 |0 |X |
|1(9 | |5(3 |1 |Y |
| |3(7 |1(3 |1 |Z |
| |2(7 |3(3 |2 |A |
| |1(7 |6(3 |0 |B |
| | |8(3 |1 |C |

Alternatively this problem can be represented by a Knapsack model, here let [pic] indicate that K rolls of width i are cut from roll j,

treating the roll as if it were a container.

Constraints

[pic]

Objective Several objectives are again possible, for example:

Minimise rolls cut [pic]

A better formulation.

A set of items are to be transported using a set of containers where container j has a capacity of Cj .

Here the objective may be to

Maximise the value of the packed items given a fixed number of containers

or

Minimise the number of containers needed to pack all the items

or

Minimise the cost of containers needed to pack all the items

Therefore let

Xij = 1 indicate that item i is packed in container j

and Xij = 0 indicate that item i is not packed in container j

then it follows that

( Xij = 1, item i is packed (all i)

( Wi Xij ( Cj contain j is packed (all j)

cost= ( ( Vi Xij to be maximised

or if all are to be packed in a minimum number of containers then

( Xij = 1

( Wi Xij ( (j Cj

containers = ( (j to be minimised (number of containers)

----------------------- (

Diagram 1

Number of 9 widths cut
Number of 7 widths cut
Number of 3 widths cut

Width restriction for roll i

Cut enough to satisfy the demand

Indicate whether roll j has been cut

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