Notes 6IE3121 Knapsack Model Intuitive idea: what is the most valuable collection of items that can be fit into a backpack?

Notes 6 IE312 1

Knapsack Model Intuitive idea: what is the most

valuable collection of items that can be fit into a backpack?

Notes 6 IE312 2

Race Car Features

1 2 3 4 5 6Cost (thousand) 10.2$ 6.0$ 23.0$ 11.1$ 9.8$ 31.6$ Speed increase (mph) 8 3 15 7 10 12

Proposed Feature (j )

Budget of $35,000

Which features should be added?

Notes 6 IE312 3

Decision variables

ILP

Formulation

otherwise0

added is feature if1 jx j

1 2 3 4 5 6

1 2 3 4 5 6

max 8 3 15 7 10 12

10.2 6.0 23.0 11.1 9.8 31.6 35s.t

0 or 1j

x x x x x x

x x x x x x

x

Notes 6 IE312 4

LINGO FormulationMODEL:

SETS:

FEATURES /F1,F2,F3,F4,F5,F6/: INCLUDE,SPEED_INC,COST;

ENDSETS

DATA:

SPEED_INC = 8 3 15 7 10 12;

COST = 10.2 6.0 23.0 11.1 9.8 31.6;

BUDGET = 35;

ENDDATA

MAX = @SUM( FEATURES: SPEED_INC * INCLUDE);

@SUM( FEATURES: COST * INCLUDE) <= BUDGET;

@FOR( FEATURES: @BIN( INCLUDE));

END

Specifyindex sets

All theconstants

Objective

Constraints

Variables indexed bythis set

Decision variablesare binary

Note ; to end command: to begin an environment

Notes 6 IE312 5

Solve using Branch & Bound

5 0x 5 1x

1 2 3 4 6

1 2 3 4 6

max 8 3 15 7 12

10.2 6.0 23.0 11.1 31.6 35s.t

0 or 1j

x x x x x

x x x x x

x

1 2 3 4 6

1 2 3 4 6

max 8 3 15 7 12

10.2 6.0 23.0 11.1 31.6 35s.t

[0,1]j

x x x x x

x x x x x

x

Solution?

Candidate Problem

Relaxed Problem

Notes 6 IE312 6

What is the Relative Worth?

1

2

3

4

5

6

8: 0.7810.23: 0.5615: 0.65237: 0.6311.110: 1.029.812: 0.3831.6

x

x

x

x

x

x

Want to add thisfeature first

Want to add thisfeature second

Notes 6 IE312 7

Solve Relaxed Problem

5 0x 5 1x

1 2 3 4 6

1 2 3 4 6

max 8 3 15 7 12

10.2 6.0 23.0 11.1 31.6 35s.t

[0,1]j

x x x x x

x x x x x

x

Solution:

Relaxed Problem

1

3

4

1 Still have $35 $10.2 $24.8

1 Still have $24.8 $23.0 $1.8

1.8 0.2577

x

x

x

Objective 24.8

8 1 15 1 7 0.257

Notes 6 IE312 8

Now the other node…

5 0x 5 1x

1 2 3 4 6

1 2 3 4 6

max 8 3 15 7 12 10

10.2 6.0 23.0 11.1 31.6 35 9.8s.t

[0,1]j

x x x x x

x x x x x

x

Relaxed Problem

Solution:

1

3

1 Still have $25.2 $10.2 $15

15 0.65223

x

x

Objective 27.8

Notes 6 IE312 9

Next Step?

5 0x 5 1x

Objective 24.8 Objective 27.8

Notes 6 IE312 10

Rule of Thumb: Better Value

5 0x 5 1x

Obj 24.8

1 0x 1 1x

2 3 4 6

2 3 4 6

max 3 15 7 12 18

6.0 23.0 11.1 31.6 15s.t

[0,1]j

x x x x

x x x x

x

Relaxed Problem

Solution:

315 0.65223x

2 3 4 6

2 3 4 6

max 3 15 7 12 10

6.0 23.0 11.1 31.6 25.2s.t

[0,1]j

x x x x

x x x x

x

Relaxed Problem

3

4

1 Still have $25.2 $23 $2.2

2.2 0.19811.1

x

x

Obj. 26.4 Obj. 27.8

Solution:

Notes 6 IE312 11

Next Level

5 0x 5 1x

Obj 24.8

1 0x 1 1x

Obj. 26.4

3 0x 3 1x

Infeasible

2 4 6

2 4 6

max 3 7 12 18

6.0 11.1 31.6 15s.t

0,1j

x x x

x x x

x

Candidate Problem

4 2 61, 0x x x Solution:

Obj. = 25

(This turns out to be true.)

Now What?

Notes 6 IE312 12

Next Steps …

5 0x 5 1x

Obj 24.8

1 0x 1 1x

Obj. 26.43 0x 3 1x

InfeasibleObj. = 25

Obj 25

Still need to continue branching here.

Finally we will have

accounted for every solution!

Notes 6 IE312 13

Capital Budgeting Multidimensional knapsack

problems are often called capital budgeting problems

Idea: select collection of projects, investments, etc, so that the value is maximized (subject to some resource constraints)

Notes 6 IE312 14

NASA Capital Budgeting2000- 2005- 2010- 2015- 2020- Not Depends

j Mission 2004 2009 2014 2019 2024 Value With On1 Communication satellite 6 2002 Orbital microwave 2 3 33 Io lander 3 5 204 Uranus orbiter 2020 10 50 5 35 Uranus orbiter 2010 5 8 70 4 36 Mercury probe 1 8 4 20 37 Saturn probe 1 8 5 38 Infrared imaging 5 10 119 Ground-based SETI 4 5 200 14

10 Large orbital structures 8 4 15011 Color imaging 2 7 18 8 212 Medical technology 5 7 813 Polar orbital platform 1 4 1 1 30014 Geosynchronous SETI 4 5 3 3 185 9

Budget 10 12 14 14 14

Budget Requirements

Notes 6 IE312 15

Formulation Decision variables

Budget constraints

otherwise0

chosen is mission if1 jx j

1 2 3 7 9 126 2 3 1 4 5 10 (year 1)x x x x x x

Notes 6 IE312 16

Formulation Mutually exclusive choices

Dependencies

1

1

1

149

118

54

xx

xx

xx

34

211

xx

xx

Notes 6 IE312 17

Set Packing, Covering, and Partitioning

Notes 6 IE312 18

Select Locations

Notes 6 IE312 19

Ways of Splitting the Set Set covering constraints

Set packing constraints

Set partitioning constraints

Jj

jx 1

Jj

jx 1

Jj

jx 1

Notes 6 IE312 20

Example: Choosing OR Software

Algorithm 1 2 3 4LP Yes Yes Yes YesIP Yes YesNLP Yes YesCost 3 4 6 14

Software Package (j )

Formulate a set covering problem to acquire the minimum costsoftware with LP, IP, and NLP capabilities.

Formulate set partitioning and set packing problems. What goalsdo they meet?

Notes 6 IE312 21

Maximum Coverage Perhaps the budget only allows $9000 What can we then do

Maximum coverage How do we now formulate the

problem? Need new variables

otherwise0

providednot ALG if1ALGy

Notes 6 IE312 22

Travelling Salesman Problem (TSP)

Ames

Fort Dodge

BooneCarrollMarshalltown

West Des Moines

Waterloo

What is the shortest route,starting in Ames, that visitseach city exactly ones?

Notes 6 IE312 23

TSP Solution

Ames

Fort Dodge

BooneCarrollMarshalltown

West Des Moines

Waterloo

Notes 6 IE312 24

Not a TSP Solution

Ames

Fort Dodge

BooneCarrollMarshalltown

West Des Moines

Waterloo

Notes 6 IE312 25

Applications

Routing of vehicles (planes, trucks,

etc.)

Routing of postal workers

Drilling holes on printed circuit boards

Routing robots through a warehouse,

etc.

Notes 6 IE312 26

Formulating TSP A TSP is symmetric if you can go

both ways on every arc

otherwise0

and between leg includes route theif1,

jix ji

i ij

jiji xc ,,min

2, ij

jix

Notes 6 IE312 27

Example1

5

2

6

3 4

10

10

10

11

1

1

1 1

Formulate a TSP

Notes 6 IE312 28

Subtours It is not sufficient to have two arcs

connected to each node Why?

Must eliminate all subtours Every subset of points must be

exited

Notes 6 IE312 29

How do we eliminate subtours?

1

5

2

6

3 4

10

10

10

11

1

1

1 1

Notes 6 IE312 30

Asymmetric TSP Now we have decision variables

Constraints

otherwise0

to from goes route theif1,

jix ji

)(enter 1

) (leave 1

,

,

ix

ix

ijij

ijji

Notes 6 IE312 31

Asymmetric TSP (cont.) Each tour must enter and leave

every subset of points

Along with all variables being 0 or 1, this is a complete formulation

1, Si Sj

jix

Notes 6 IE312 32

Example1

5

2

6

3 4

10

10

10

11

1

1

1 1

Assume a two unit penalty for passing from a high to lowernumbered node.

This is now an asymmetric TSP. Why?

Notes 6 IE312 33

Subtour Elimination Making sure there are no subtours

involves a very large number of constraints

Can obtain simpler constraints if we go with a nonlinear objective function

otherwise0

is tedpoint visith theif1,

iky ik

Notes 6 IE312 34

Quadratic Assignment Formulation

1,0

1

1

min

,

,

,

,1,,

ik

kik

iik

i j kjkikji

y

y

y

yyd

Notes 6 IE312 35

Example: reformulate1

5

2

6

3 4

10

10

10

11

1

1

1 1