New results on single-machine two-agent scheduling problems

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New results on single-machine two-agent
scheduling problems

• Alessandro Agnetis, Università di Siena
• joint work with
• Gianluca De Pascale, Università di Siena
• Dario Pacciarelli, Università di Roma Tre
• Marco Pranzo, Università di Siena

CIRM, Marseille 12/5/2008
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Multi-agent problems

• A set of m agents, each owning a set of jobs
• Job j requires processing time pj
• There is a single machine
• Each agent k wants to minimize a cost function f
  k(s) which only depends on the schedule of
  his/her jobs

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• Trains competing for railroad resource usage
  Brewer and Plott 1996

• Quay cranes at container terminals competing for
  movers Lau et al 2007

• Allocation of airport time slots to incoming
  aircraft Ball et al. 2000

• Different data packets competing for radio
  resources Meiners and Torng 2007

• Resource allocation in industrial districts
• Albino, Carbonara and Giannoccaro 2006

• Schedule adjustment upon arrival of new jobs
• Leung, Pinedo and Wan 2007

4

• Protocols
• Auction mechanisms (Wellman et al 2002)
• Combinatorial auctions (Kutanoglu and Wu 1999,
  Lau et al 2007)
• Automated protocols (Fink 2006)
• Cooperative/Noncooperative games
• Sequencing games (Curiel et al 1989)
• Decentralization cost, mechanism design (Hain and
  Mitra 2006, Bukchin and Hanany 2007)

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• Bargaining models, multi-agent scheduling
• Nash (1950), Mariotti (1998), Peha (1995), A. et
  al (2004, 2007), Baker and Smith (2003), Arbib et
  al (2005), Cheng, Ng, Yuan (2006, 2007), Leung,
  Pinedo and Wan (2007), Meiners and Torng (2006)

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Two-agent problems bargaining

• The set of PO (Pareto-optimal) schedules may be
  viewed as the bargaining set over which the
  agents will negotiate
• The number of PO schedules and the complexity of
  their computation depends on the agents cost
  functions

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e-constrained problem

• The problem is to compute the schedule which
  minimizes the cost for agent A such that the cost
  for B does not exceed Q
• By varying Q, one can generate all PO schedules

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f B
f A
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Agents utility

• The bargaining set also contains a point (dA,dB)
  representing the agents utility if negotiation
  fails (disagreement point)
• We consider the agents utilities
• uA(s)d A f A(s)
• uB(s)d B f B(s)

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Bargaining solutions

• Among PO schedules, there are some satisfying
  particular axioms in terms of equity and
  stability
• The Nash Bargaining Solution (NBS) is the one
  maximizing the Nash value
• N(s)(d A f A(s))(d B f B(s))

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Individual cost functions

• We consider two scenarios
• f A(s) Si wiACiA
• f B(s) Si wiBCiB
• f A(s) Si wiACiA
• f B(s) LmaxB max CiB-diB

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1 Si wiBCiB ? Q Si wiACiA
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Complexity

• The e-constrained problem is NP-hard, even if all
  jobs have equal weights
• The number of PO schedules is pseudopolynomial
• Finding the Nash solution is also NP-hard (A., de
  Pascale and Pranzo 2007)

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• min Si wiACiA (s)
• Si wiBCiB (s) ? Q
• s ? S
• If we relax the constraint, we get the Lagrangian
  problem
• L(l) min Si wiACiA (s) l (Si wiBCiB (s) - Q)
• s ? S

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Lagrangian relaxation

• The Lagrangian problem is simply solved by
  ranking the jobs in nondecreasing order of dk
  where
• dkA wkA/ pkA if k ? A
• dkB l wkB / pkB if k ? B

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Optimal schedules for decreasing l
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PO schedules and NBS

• The Lagrangian bound is used in a branch and
  bound algorithm to generate PO schedules
• To find the NBS, we adopt the approach
• Generate all extreme schedules
• Locate the triangle containing the NBS
• Enumerate PO solutions in the triangle

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Locating the Nash triangle

• The Nash triangle can be found evaluating the
  angle between the convex hull of extreme
  schedules and the gradient of the Nash function

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Computational experiments

• The approach has been run on several instances of
  various sizes
• JA10, 20, 30, 40
• JB10, 20, 30, 40
• All weights and processing times uniformly
  distributed in 1,25

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nA
nB
T1
TPO
TNASH
E
PO
10 10 60 603 0.01
4 0.06
10 20 119 4595 0.05
217 1.72
10 30 178 15383 0.14
2185 12.30
10 40 232 74771 0.32
24061 102.27
20 10 118 4698 0.05
227 1.86

1. 20 239 15601 0.14
   2110 8.96

20 30 345 74547 0.33
24912 69.21
20 40 451 220000 0.65
146400 322.87
30 10 178 15413 0.14
2120 12.48
30 20 346 75225 0.31
22883 65.24
30 30 510 219000 0.64
140700 282.58
30 40 662 950000 1.12
1056000 1571.23
40 10 233 73952 0.33
24427 110.23
40 20 452 220000 0.63
138000 311.06
40 30 653 947000 1.09
1062000 1551.55
40 40 862 2397000 1.81
4218000 4974.83
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1 LmaxB? Q Si wiACiA
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1LmaxB? Q Si wiACiA

• min Si wiACiA (s)
• C1B(s) ? Q1B
• C2B(s) ? Q2B
• CnBB(s) ? QnBB
• s ? S

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Complexity

• The e-constrained problem is strongly NP-hard (A.
  et al 2004, Cheng,Ng and Yuan 2006)
• The number of PO schedules is pseudopolynomial
• Even finding extreme schedules is NP-hard (Baker
  and Smith 2003, Hoogeveen 2002)

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Lower bounds

• The problem is a special case of
• 1dj SjwjCj
• Pan (2003) solves instances with up to 100 jobs,
  based on a bounding approach by Posner (1995)

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Example
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Optimal solution
6
X
Z
Y
5
4
4
3
10
7
5 3 4 7 4 10 83
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Posners preemptive bound
6
Z
4
5 1/3
4
5 2/3
weights
2
10
4
7
4 2 10/3 4 5/3 7 4 10 73
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Lagrangian relaxation

• Relaxing all the constraints, one has
• L(l)
• min Si wiACiA (s) Sj lj(CjB(s) - QjB)
• s ? S

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Lagrangian relaxation

• The Lagrangian problem is solved by ranking the
  jobs in nondecreasing order of dj where
• diA wiA/ piA if i ? A
• dkB lk / pkB if k ? B

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Lagrangian dual

• Theorem In an optimal solution to the Lagrangian
  dual, for each B-job k there exists an A-job i
  such that
• dkB diA

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Structure of an optimal solution of the
Lagrangian dual
Note the ordering within each cluster is
immaterial
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Solving the Lagrangian dual

• To solve the Lagrangian dual, we only need to
  find the partition of the B-jobs
• Let Rj QjB - pjB
• The window of a B-job is the time span Rj , Qj

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Windows
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Windows
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2.5
Windows
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Lagrangian bound

• Theorem The bound provided by the optimal
  solution to the Lagrangian dual dominates
  Posners bound

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Lagrangean bound
6
Z
4
5
4
2 5/3
weights
2
10
4
7
4 2 5 7 4 10 2(5/3)(-2) 76.333
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Conclusions

• The Lagrangian approach appears viable for
  efficiently deriving good lower bounds for these
  classes of two-agent scheduling problems
• The number of PO schedules may grow rapidly, but
  the NBS can still be computed in reasonable time
• The best extreme schedule approximates the NBS
  very well

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and future research

• Extension to other cost functions
• Simulations to compare bargaining vs.
  auction-type approaches
• Other decentralized protocols