## (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

eH w

er as

o.

co

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/

Solution details:
STATUS
QUALITY
Approved

This question was answered on: Oct 15, 2019

Solution~000.zip (25.37 KB)

STATUS

QUALITY

Approved

Oct 15, 2019

EXPERT

Tutor

#### YES, THIS IS LEGAL

We have top-notch tutors who can do your essay/homework for you at a reasonable cost and then you can simply use that essay as a template to build your own arguments.

You can also use these solutions:

• As a reference for in-depth understanding of the subject.
• As a source of ideas / reasoning for your own research (if properly referenced)
• For editing and paraphrasing (check your institution's definition of plagiarism and recommended paraphrase).
This we believe is a better way of understanding a problem and makes use of the efficiency of time of the student.

### Order New Solution. Quick Turnaround

Click on the button below in order to Order for a New, Original and High-Quality Essay Solutions. New orders are original solutions and precise to your writing instruction requirements. Place a New Order using the button below.

WE GUARANTEE, THAT YOUR PAPER WILL BE WRITTEN FROM SCRATCH AND WITHIN A DEADLINE.