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ENM 503 Block 3Lesson 12 – Discrete Optimization Models

Combinatorial Problems and their Solutions

An Introduction to Discrete Optimization

Narrator: Charles EbelingUniversity of Dayton

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The Lineup

The Assignment Problem The Knapsack Problem The Traveling Salesman Problem The Postman problem

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Combinatorial Optimization Problems

Optimization problem having a finite number of discrete solutions.

One solution approach is to explicitly generate and evaluate all possible solutions

Explicit enumeration

Consider a problem with 100 variables where xj = 0, 1, 2, …, 50; j = 1,2, …, 100

number of possible solutions = 51100

Explicit enumeration may not be possible Implicit enumeration – attempt to account for all

possible solution without enumerating all of them Problem dependent

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The Assignment Problem

Assign n workers to n tasksA Problem in Permutations

I heard that this is really good!

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A problem

Three workers are to be assigned to one of 4 distinct tasks. Each worker can perform each task in a different time. The objective is to minimize the total time to complete the assigned tasks.

worker task times

We need to add a dummy worker.

time task1 task2 task3 task4w1 12 13 10 11w2 10 12 14 10w3 14 11 15 12

Complete Enumeration

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time task1 task2 task3 task4w1 12 13 10 11w2 10 12 14 10w3 14 11 15 12

(4)(3)(2) = 24 possible solutions

  w1 w2 w3 time   w1 w2 w3 time

1 task1 task2 task3 35 13 task1 task4 task3 37

2 task1 task3 task2 37 14 task1 task3 task4 38

3 task2 task1 task3 38 15 task4 task1 task3 36

4 task2 task3 task1 41 16 task4 task3 task1 39

5 task3 task1 task2 31 17 task3 task1 task4 32

6 task3 task2 task1 36 18 task3 task4 task1 34

7 task4 task2 task3 38 19 task1 task2 task4 36

8 task4 task3 task2 36 20 task1 task4 task2 33

9 task2 task4 task3 38 21 task2 task1 task4 35

10 task2 task3 task4 39 22 task2 task4 task1 37

11 task3 task4 task2 35 23 task4 task1 task2 32

12 task3 task2 task4 34 24 task4 task2 task1 37

A Bigger Problem

What if there were 20 workers and 25 tasks to be completed?

25P20

= 29,260,083,694,425,000,000,000 What if it takes 1 second to evaluate

each assignment? 29,260,083,694,425,000,000,000 sec/ (60 sec/min X 60 min/hr x 24 hr/day x 365

days/yr = 4,098,810,365,754,210 years

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Some Applications

workers to tasks jobs to machines facilities to locations Truck drivers to customer pick-up points Umpire crews to baseball games Judges to court dockets State inspectors to construction sites Weapons to targets

These are some terrific applications.

The Knapsack Problem

A total of m items whose weights are w1, w2, …, wm are available for packing a knapsack. The total weight to be packed cannot exceed Wtotal. The objective is to pack as many items as possible. 9

The Knapsack Problem - a binary problem

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m

ii=1

1

0 if itemi is not to be packedLet

1 if itemi is to be packed

Maximize

subject to:

i

m

i i totali

x

p x

w x W

New & Improved Knapsack Problem

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1

1

max

subject to:

n

i ii

n

i ii

p v x

w x Weight

It is comforting to

know my load is

optimal!

A thief robbing a store can carry a maximum weight of w. There are n items and ith  item weighs wi and is worth vi dollars. What items should thief take?

A Knapsack Problem to Solve

Johnny has five friends who would like to go on a fishing trip with him to the remote island of Chichagof located 70 miles northwest of Sitka, Alaska.

However, Johnny can only carry an additional 319 pounds of weight in his seaplane – his only means of transportation.

Johnny has assigned a “pleasure” index to each friend – the larger the value the more Johnny would enjoy the friend’s company.

Which friend(s) should Johnny take with him?

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The Data

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Friend Jane Joan Judy Joy Jim

Pleasure Index 8 6 9 10 3

Weight (lb.) 130 180 140 125 185

25 = 32 alternatives

TheResults

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Friend Jane Joan Judy Joy Jim      Pleasure Index 8 6 9 10 3      Weight (lb.) 130 180 140 125 185 value weight    1 1 1 1 1 36 760 1  1 1 1 1 0 33 575 2  1 1 1 0 1 26 635 3  1 1 0 1 1 27 620 4  1 0 1 1 1 30 580 5  0 1 1 1 1 28 630 6  1 1 1 0 0 23 450 7  1 1 0 0 1 17 495 8  1 0 0 1 1 21 440 9  0 0 1 1 1 22 450 10  1 1 0 1 0 24 435 11  1 0 1 0 1 20 455 12  0 1 0 1 1 19 490 13  0 1 1 1 0 25 445 14  0 1 1 0 1 18 505 15  1 0 1 1 0 27 395 16

feasible 1 1 0 0 0 14 310 17feasible 1 0 0 0 1 11 315 18feasible 0 0 0 1 1 13 310 19feasible 1 0 1 0 0 17 270 20

  0 1 0 0 1 9 365 21feasible 0 1 0 1 0 16 305 22

  0 1 1 0 0 15 320 23feasible 0 0 1 1 0 19 265 24feasible 1 0 0 1 0 18 255 25

  0 0 1 0 1 12 325 26feasible 1 0 0 0 0 8 130 27feasible 0 1 0 0 0 6 180 28feasible 0 0 1 0 0 9 140 29feasible 0 0 0 1 0 10 125 30feasible 0 0 0 0 1 3 185 31feasible 0 0 0 0 0 0 0 32

319 lb.

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Let’s count the number of solutions…

Dad, how many solutions are there for the knapsack problem if I have 30 items

to consider?

You should know that Johnny. It is 230 or

1,073,741,824 solutions if you include the infeasible

ones as well.

Other Binary Selection Problems

Menu selection Select items from a menu to maximize protein Calorie and carbohydrate or cost constraints

Cargo loading on trucks or aircraft Which crates or pallets to load – maximize value Volume and weight constraints

Project selection – n potential engineering projects to fund

Maximize expected profit Cost or resource constraint

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The Traveling Salesman Problem

There once was a farmer’s daughter…

You got to know the territory!

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Traveling Salesman Problem

A salesman must visit each of n cities once and only once returning to his starting city. What route should be followed so that the total distance (cost or time) traveled is minimized?

(n-1)! possible routes

ItineraryDaytonCincinnatiDenverNew YorkAtlantaChicagoBostonSan FranciscoWapakoneta

My secretary always finds for

me, the minimum

distance itinerary.

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A From-to MatrixFrom / To

City A

City B

City C

City D

City A

25 30 12

City B

25 17 23

City C

30 17 37

City D

12 23 37distances in miles

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Traveling on a network

2

5

5

3 4

6

7

8

11A

BC

D

E

F

G

6! = 720 routesA solution isa permutation!

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A Traveling Salesman Problem

P. Rose, currently unemployed, has hit upon the followingscheme for making some money. He will guide a group ofpeople on a tour of all National League baseball parks. Thetour will start and end in Cincinnati. What route should hefollow in order to minimize total distances (costs)?

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Distances between parks in miles

ATL CHI CIN HOU LA MON NY PHI PIT STL SD SFATL 702 454 842 2396 1196 864 772 714 554 2363 2679CHI 702 324 1093 2136 764 845 764 459 294 2184 2187CIN 454 324 1137 2180 798 664 572 284 338 2228 2463HOU 842 1093 1137 1616 1857 1706 1614 1421 799 1521 2021LA 2396 2136 2180 1616 2900 2844 2752 2464 1842 95 405MON 1196 764 798 1857 2900 396 424 514 1058 2948 2951NYK 864 845 664 1706 2844 396 92 386 1002 2892 3032 PHI 772 764 572 1614 2752 424 92 305 910 2800 2951PIT 714 459 284 1421 2464 514 386 305 622 2512 2646STL 554 294 338 799 1842 1058 1002 910 622 1890 2125SD 2363 2184 2228 1521 95 2948 2892 2800 2512 1890 500SF 2679 2187 2463 2021 405 2951 3032 2951 2646 2125 500

11! = 39,916,800 alternatives

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A Heuristics AlgorithmI need a goodheuristic tosolve this problem.

Joe EngineerP. Rose

Heuristic: A procedure for solving problems by an intuitiveapproach in which the structure of the problem can be interpretedand exploited intelligently to obtain a reasonable solution.

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Nearest Neighbor Heuristic – a greedy heuristic algorithm

ATL CHI CIN HOU LA MON NY PHI PIT STL SD SFATL 702 454 842 2396 1196 864 772 714 554 2363 2679CHI 702 324 1093 2136 764 845 764 459 294 2184 2187CIN 454 324 1137 2180 798 664 572 284 338 2228 2463HOU 842 1093 1137 1616 1857 1706 1614 1421 799 1521 2021LA 2396 2136 2180 1616 2900 2844 2752 2464 1842 95 405MON 1196 764 798 1857 2900 396 424 514 1058 2948 2951NYK 864 845 664 1706 2844 396 92 386 1002 2892 3032 PHI 772 764 572 1614 2752 424 92 305 910 2800 2951PIT 714 459 284 1421 2464 514 386 305 622 2512 2646STL 554 294 338 799 1842 1058 1002 910 622 1890 2125SD 2363 2184 2228 1521 95 2948 2892 2800 2512 1890 500SF 2679 2187 2463 2021 405 2951 3032 2951 2646 2125 500

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LAX

SNF

CIN

MON

SND

ATL

HOU

CHI PITSTL

PHI

NYK

DISTANCE = 8015 MILES

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LAX

SNF

CIN

MON

SND

ATL

HOU

CHI PITSTL

PHI

NYK

DISTANCE = 8015 MILES

X

X

OPTIMUM= 7577 MILES

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Other Applications

manufacture of circuit boards assembly line reconfiguration mixing paints colors in vats searching for stars

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The Chinese Postman Problem

A study in the optimal delivery of the postal mail

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The problem defined The Chinese postman problem concerns

a postman who has to deliver mail to houses along each of the streets in a particular housing district and wants to minimize the distance that has to be walked.

The problem was first considered by the Chinese mathematician Mei-ko Kwan in the 1960s.

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More of the Chinese Postman Problem

Given a network with distances assigned to each arc,find the minimum distance walk that walks each arcat least once and returns to the starting node.

2

5

5

3 4

6

7

8

11A

BC

D

E

F

G

I get tired walking

this route. It is too

long.

Solve by Trial and Error

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A – B – C – D – F – G – E – D –E –B –E – G – A - 67 miles

Start and end At AMinimum possible distance = 51 miles

2

5

5

3 4

6

7

8

11A

BC

D

E

F

G

A – B – C – D – F – G – E – D – E – B – A – G – A - 63 miles

What if I need to walk every street once in each direction?

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The Postman Problem Model

Given a network with distances assigned to each arc, find the minimum distance walk that walks each arc at least once and returns to the starting node.

1

1 1

1 1

,

min

1 or 1 1

1,2,3,...

ij

n n

ij ijj i

i j

ij ji ij ji

n n

ik kji ji j j i

let x nbr times arc i j walked

z d x

st

x x x and x

x x for k n

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Other Applications

trash pickup newspaper delivery and door-to-door

soliciting railroad track inspections and maintenance snow plowing, salting streets, and street

cleaning meter reading school bus routes

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Play ball! Now it’s your turn.

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