ie 400 principles of engineering management solving
TRANSCRIPT
![Page 1: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/1.jpg)
IE 400 Principles of Engineering Management
Solving Integer Programming Models
Branch And Bound
![Page 2: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/2.jpg)
Geometry of IP
2
max x2
s.t.
-x1 + x2 ≤ 1/2
x1 + x2 ≤ 7/2
x1, x2 ≥ 0, integer
![Page 3: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/3.jpg)
• By removing the integrality conditions, we
get an LP problem.
• This is called the LP-relaxation and is often
a good way starting IP problems.
• If we solve the LP of the example:
3
![Page 4: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/4.jpg)
Infeasible
x1 = 3/2
x2 = 2
z = 2
NOT INTEGER
x1 =1, or 2
x2 = 2
z = 2
ROUNDING
![Page 5: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/5.jpg)
INTEGER PROGRAMMING
Proposition 1: For an integer programming
problem, the optimal value of the LP-
relaxation is at least as good as the optimal
value of the IP
Proof: The LP relaxation has a larger feasible
region and so more alternatives
Proposition 2: If the optimal solution of the
LP relaxation is integer valued, then that
solution is also optimal for the IP.
(proof of optimality) 5
![Page 6: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/6.jpg)
6
Knapsack Problem
![Page 7: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/7.jpg)
7
Solving Knapsack LP
Order the variables in non-increasing of per unit size utility values and
fill the knapsack with respect to this order!
![Page 8: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/8.jpg)
8
Solving IPs
![Page 9: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/9.jpg)
9
![Page 10: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/10.jpg)
10
![Page 11: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/11.jpg)
11
![Page 12: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/12.jpg)
12
![Page 13: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/13.jpg)
13
![Page 14: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/14.jpg)
14
Solving IPs (Branch and Bound)
![Page 15: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/15.jpg)
15
![Page 16: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/16.jpg)
16
The Essence of Branch and Bound
Select nodes of the “enumeration tree” one at a timebut do not branch from a node if none of itsdescendents can be a better solution than that of theincumbent
Eliminate subtree 2 if(for Maximization)
Incumbent ZI
Bound zLP(2)zLP(2) ≤ zI
![Page 17: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/17.jpg)
17
![Page 18: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/18.jpg)
18
![Page 19: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/19.jpg)
19
![Page 20: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/20.jpg)
20or even and incumbent of 44 would work!
![Page 21: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/21.jpg)
21
![Page 22: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/22.jpg)
22
Fathom by Infeasibility
![Page 23: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/23.jpg)
23
Fathom by Bound
![Page 24: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/24.jpg)
24
Fathom by Integrality
![Page 25: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/25.jpg)
25
Branch Further
![Page 26: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/26.jpg)
26
![Page 27: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/27.jpg)
27
![Page 28: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/28.jpg)
28
![Page 29: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/29.jpg)
29
![Page 30: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/30.jpg)
30
![Page 31: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/31.jpg)
31
![Page 32: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/32.jpg)
32
![Page 33: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/33.jpg)
33
![Page 34: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/34.jpg)
34
![Page 35: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/35.jpg)
35
![Page 36: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/36.jpg)
36
![Page 37: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/37.jpg)
37
![Page 38: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/38.jpg)
38
![Page 39: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/39.jpg)
39
![Page 40: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/40.jpg)
40
![Page 41: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/41.jpg)
41
![Page 42: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/42.jpg)
42
![Page 43: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/43.jpg)
43
![Page 44: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/44.jpg)
44
![Page 45: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/45.jpg)
45
![Page 46: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/46.jpg)
46
![Page 47: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/47.jpg)
47
![Page 48: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/48.jpg)
48
![Page 49: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/49.jpg)
49
![Page 50: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/50.jpg)
50
![Page 51: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/51.jpg)
51
![Page 52: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/52.jpg)
52
![Page 53: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/53.jpg)
53
![Page 54: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/54.jpg)
54
![Page 55: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/55.jpg)
55
![Page 56: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/56.jpg)
56
![Page 57: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/57.jpg)
57
![Page 58: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/58.jpg)
58
![Page 59: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/59.jpg)
59
![Page 60: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/60.jpg)
60
![Page 61: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/61.jpg)
61
![Page 62: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/62.jpg)
62
![Page 63: IE 400 Principles of Engineering Management Solving](https://reader031.vdocuments.us/reader031/viewer/2022012420/6174da2d8523d4273d6b4993/html5/thumbnails/63.jpg)
63