1. Lecture 5 Set Packing Problems Set Partitioning Problems1

2. Outline
set packing problems
set partitioning problems 2

3. Set Packing Problems 3

4. Context
a set S = {1, 2…, m}
a collection of subsets of S, , such that each subset carry a value
problem: to maximize the total value of subsets selected such that no element is selected more than once 4

5. Set Packing Problems S = {1, 2, 3, 4, 5, 6}
 = {{1, 2, 5}, {1, 3}, {2, 4}, {3, 6}, {2, 3, 6}}
examples
{1, 2, 5}, {3, 6}: a pack
{1, 3}, {2, 4}: a pack
{1, 2, 5}, {2, 3, 6}: not a pack
{2, 3, 6}: a pack
assumption: every subset of value = 1 5

6. Set Packing Problems S = {1, 2, 3, 4, 5, 6}
 = {{1, 2, 5}, {1, 3}, {2, 4}, {3, 6}, {2, 3, 6}}
Property 9.4: maximization with all  constraints
i  {0, 1}
: element 1
: element 2
: element 3
: element 4
: element 5
: element 6
Property 9.5: all RHS coefficients = 1
Property 9.6: all matrix coefficients = 0 or 1
set 1:
set 2:
set 3:
set 4:
set 5: 6

7. Primal-Dual Pair
“An interesting observation is that the LP problem associated with a set packing problem with objective coefficients of 1 is the dual of the LP problem associated with a set covering problem with objective coefficients of 1.” (pp 192 of [7]) 7

8. Primal-Dual Pair special properties between the primal-dual pair
Dual (Primal)
special properties between the primal-dual pair
obj function of min  obj function of max
unbounded primal  infeasible dual
optimal primal  optimal dual
same objective function value
easy to deduce the optimal of one from the other
possible to have infeasible primal and infeasible dual 8

9. Comments on Set Packing Problems
set covering problem with objective coefficients = 1
dual of each other
i  0
set packing problem with objective coefficients = 1
The two LPs are dual of each other, though the S and  in one problem are different from those of the other problem. 9

10. Comments on Set Packing Problems
similar generalization as in set covering problems
weighted set packing problems: RHS  positive integers > 1
generalized set packing problems: matrix coefficients = 0 or  1 10

11. Matching Problem: A Special Type of Set Packing Problem
matching: select the maximum numbers of arcs such that there is no overlapping of nodes involved
(1, 2), (3, 6), (4, 5) 2 3 1 5 4 6 11

12. Exercise
formulate the matching problem of the RHS network as a set packing problem 2 3 1 5 4 6 12

13. Set Partitioning Problems13

14. Set Partitioning Problems
to cover all the members of S by elements of  without overlapping
S = {1, 2, 3, 4, 5}
 = {{1, 2}, {1, 3, 5}, {2, 4, 5}, {3}, {1}, {4, 5}}
e.g., {1, 2}, {3}, and {4, 5} form a partition 14

15. Set Partitioning Problems
 = {{1, 2}, {1, 3, 5}, {2, 4, 5}, {3}, {1}, {4, 5}}
either maximization or minimization is all right
i  {0, 1} 15

16. Equivalence Between Set Packing Problem and Set Partitioning Problem
i  {0, 1}, 1 to 11
Set Partitioning Problem 
i  {0, 1}
Set Packing Problem 16

17. Equivalence Between Set Packing Problem and Set Partitioning Problem
illustrate the transformation of the first equality constraint into an -constraint
introduce a dummy variable  to the first constraint
i  {0, 1}
Set Partitioning Problem
max 1 + 2 + 3 +  5 + 6, 17

18. Equivalence Between Set Packing Problem and Set Partitioning Problem
max 1 + 2 + 3 +  5 + 6,
i,  {0, 1}
max 1 + 2 + 3 +  5 + 6  M, 
i,  {0, 1}
max (M+1)1 + (M+1)2 + 3 + 4
+ (M+1) 5 + 6  M, 
i,  {0, 1}
max (M+1)1 + (M+1)2 + 3 + 4
+ (M+1) 5 + 6  M,  18

19. Further Comments
set covering problems different from set packing problems and set partitioning problems
possible to transform a set packing problem into a set covering problem, but in general not the other way around
set covering problem more difficult to solve than the other two problems 19

20. A Simplified Air Crew Scheduling Problem20

21. A Simplified Air Crew Scheduling Problem
six flights every day, for cities A, B, and C, where A is the base 8 10 12 18 16 14 22 20 A B C
leg 5
leg 3
leg 1
leg 6
leg 4
leg 2 21

22. A Simplified Air Crew Scheduling Problem
pairings of legs for air crew: rules from regulation bodies and union
simplified rules in the example
at most eight hours flying time in a pairing
flying time in a pairing = sum of flying times in all legs of the pairing
at most two duties in a pairing
at least nine hours for overnight rest (OR) between duties 22

23. A Simplified Air Crew Scheduling Problem
cost of a pairing = time away from the base  flying time of the pairing
cost of pairing 1 = 3610 = 26 23

24. A Simplified Air Crew Scheduling Problem
suppose only considering covering the 6 legs in a day
let xj = 1 if the jth pairing is used, and xj = 0 otherwise 24

25. A Simplified Air Crew Scheduling Problem
a set partitioning problem
min 26x1 + 20x2 + 2x3 + 26x4 + 20x5 + 26x6,
s.t.
x1 + x2 = 1,
x x6 = 1,
x3 + x4 + x5 + x6 = 1,
x2 + x3 + x4 + x5 = 1,
x x5 = 1,
x5 + x6 = 1,
xi  {0, 1} 25

26. A Simplified Air Crew Scheduling Problem
The dual of the partitioning problem
min 1 + 2 + 3 + 4 + 5 + 266,
s.t.
1 +  ≤ 26
1 + 4 +  ≤ 20
3 +  ≤ 2
3 +  ≤ 26
3 + 4 + 5 + 6 ≤ 20
2 +  6 ≤ 26 26

27. To Construct a Simplified Air Crew Scheduling Problem27

28. Six Cities Tokyo Seoul 2 hr 1 2 8 am 012 0 flight
2 hr 20 min
3 hr 10 min
Taipei
1 hr 30 min
3 hr 40min
Hong Kong 3 5
Bangkok
3 hr 45 min
2 hr 15 min 4
Singapore 28

29. Assumptions for Planes
It takes 90 minutes to load supply and passengers before departure (including unloading passengers for the previous flight leg, if applicable).
All flights are on exact time.
It takes 30 minutes for a plane to reach the terminal after arrival.
It takes 30 minutes to unload passengers and clean up a plane at the end of its service. 29

30. Itinerary of the Plane for 0-1-2-0 Tour
Time
Activities
7:30
Loading
9:00
Leaving Taipei
11:20
Arriving Seoul
11:50
Parked at Terminal; unloading and re-loading passengers; loading suppliers
13:20
Leaving Seoul
15:20
Arriving Tokyo
15:50
17:20
Leaving Tokyo
20:30
Arriving Taipei
21:00
Parked at Terminal; unloading passengers
21:30
Parking overnight 30

31. Exercise give the itinerary of the plane for 0-3- 4-5-0 tour
based on the itineraries of the planes for the and tours, construct the requirements for air stewards and for pilots
put down all assumptions made 31