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(Solved) I need help solving problem number 4 in AMPL. Can anyone help?


I need help solving problem number 4 in AMPL. Can anyone help?
Problem Set 7

 

IE406 Introduction to Mathematical Programming

 

Dr. Ralphs

 

Due October 31, 2007

 

1. Bertsimas 5.12

 

2. Bertsimas 5.13

 

3. Consider the following linear programming problem and its optimal ?nal tableau shown below.

 

min ?2x1 ? x2 + x3

 


 

x1 + 2x2 + x3 ? 12

 


 

is

 

ar stu

 

ed d

 

vi y re

 

aC s

 

ou

 

ou rc

 

rs e

 

eH w

 

er as

 

o.

 

co

 

m

 


 

s.t.

 


 

?x1 + x2 ? 2x3 ? 3

 

x1 , x2 , x3 ? 0

 


 

Final tableau:

 


 

x1 x2 x3 x4 x5

 

0 3

 

3

 

2 0 24

 

1 2

 

1

 

1 0 12

 

0 3 ?1 1 1 15

 


 

(a) Determine the optimal dual solution by examining the tableau.

 


 

(b) Determine the range of values of the right hand side of the ?rst constraint for which the

 

basis shown above remains optimal.

 

(c) Suppose that after obtaining the optimal solution depicted in the ?nal tableau above, it

 

was revealed that the following set of constraints were left out and must also be satis?ed:

 

2x1 + 3x2 ? 20

 


 

x1 ? x2 + x3 ? 11

 


 

Th

 


 

2x1 ? 3x3 ? 23

 


 

Use constraint generation to obtain an optimal solution after augmenting the original

 

LP with these three new constraints. (Hint: This only requires a few calculations.)

 


 

sh

 


 

4. The output of a paper mill consists of standard rolls 110 inches (110?) wide, which are cut

 

into smaller rolls to meet orders. This week, there are orders for smaller rolls of the following

 

widths:

 


 

1

 


 

https://www.coursehero.com/file/221289/ps7/

 


 

Width

 

20?

 

45?

 

50?

 

55?

 

75?

 


 

Orders

 

48

 

35

 

24

 

10

 

8

 


 

The owner of the mill wants to know what cutting patterns to apply so as to ?ll the orders

 

using the smallest number of 110? rolls.

 


 

is

 

ar stu

 

ed d

 

vi y re

 

aC s

 

ou

 

ou rc

 

rs e

 

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m

 


 

A cutting pattern consists of a certain number of smaller rolls of each width that can be cut

 

from one larger roll, such as two of 45? and one of 20?, or one of 50? and one of 55? (and 5?

 

of waste). Notice that the sum of the widths of the smaller rolls in the pattern must be less

 

than 110?. For example, we could consider the following six patterns:

 

Width

 

20?

 

45?

 

50?

 

55?

 

75?

 


 

1

 

3

 

0

 

1

 

0

 

0

 


 

2

 

1

 

2

 

0

 

0

 

0

 


 

3

 

0

 

0

 

1

 

1

 

0

 


 

4

 

2

 

0

 

0

 

1

 

0

 


 

5

 

1

 

0

 

0

 

0

 

1

 


 

6

 

3

 

1

 

0

 

0

 

0

 


 

The pattern in the ?rst column represents 3 20? rolls and one 50? roll.

 


 

(a) Develop an AMPL model, allowing for any set of order widths and any set of patterns,

 

that minimizes the total number of 110? rolls used, assuming that the number of smaller

 

rolls produced need only be greater than or equal to the number ordered (ignore the fact

 

that the solutions may be fractional).

 

(b) Using the given data, how many rolls should be cut according to which pattern to

 

minimize the total number of 110? rolls used in this example?

 

(c) Find another pattern that, when added to those above, improves the optimal solution.

 


 

sh

 


 

Th

 


 

(d) As noted, all of the solutions above use fractional numbers of rolls. Can you ?nd solutions

 

that satisfy the constraints, but also uses an integer number of rolls in each pattern?

 

How much does your integer solution cause the value of the objective function value to

 

go up?

 


 

2

 


 

https://www.coursehero.com/file/221289/ps7/

 


 

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