The Online Target Date Assignment Problem · The Online Target Date Assignment Problem S. Heinz1 S.O. Krumke2 N. Megow3 J. Rambau4 A. Tuchscherer1 T. Vredeveld5 DFG Research Center

The Online Target Date Assignment Problem

S. Heinz1 S. O. Krumke2 N. Megow3 J. Rambau4

A. Tuchscherer1 T. Vredeveld5

DFG Research Center MATHEON

1Zuse Institute Berlin

2Technical University Kaiserslautern

3Technical University Berlin

4University Bayreuth

5Maastricht University

10th Aussois Workshop on Combinatorial OptimizationJanuary 12, 2006

Andreas Tuchscherer The Online Target Date Assignment Problem

Technical Customer Service

MoMo TuTu

We Th Fr

repair order

repair order

customer places a repair orderimmediately assign “good” targetdate (online target date assignment)complete information about tasks fornext day at closing timeoptimize technician dispatch for nextday (offline vehicle dispatching)

Goalonline target date assignment ⇒ total costs as small as possible



Mo

Mo

Tu

Tu

We Th Fr

repair order

repair order

customer places a repair orderimmediately assign “good” targetdate (online target date assignment)

complete information about tasks fornext day at closing timeoptimize technician dispatch for nextday (offline vehicle dispatching)




Mo

Mo

Tu

Tu

We Th Fr

repair order

repair order






Mo

Mo

Tu

Tu

We Th Fr

repair order

repair order






Mo

Mo

Tu

Tu

We Th Fr

repair order

repair order






Mo

Mo

Tu

Tu

We Th Fr

repair order

repair order

customer places a repair orderimmediately assign “good” targetdate (online target date assignment)complete information about tasks fornext day at closing time

optimize technician dispatch for nextday (offline vehicle dispatching)




Mo

Mo

Tu

Tu We Th Fr

repair order

repair order





Mo

Mo

Tu

Tu We Th Fr

repair order

repair order




Two Stage Problem Structure

1 Assign a given request to some target dateOnline

2 Optimize the processing of all requests assigned to the sametarget date

Offline (current day is not allowed)Online (current day is also allowed)

AssumptionThe second stage problem can be solved offline to optimality.















Definition (ONLINETDAP)1 An instance of the ONLINETDAP consists of

deferral time δ ∈ N,downstream minimization problem Π,request sequence σ = r1, r2, . . . with release dates t(ri) ∈ N0.

2 Feasible target dates of ri are T (ri) := {t(ri) + 1, . . . , t(ri) + δ}.3 σd := subset of requests assigned to date d .4 Downstream cost downcost(σd) := optimal cost of Π on σd .5 Online cost is a function of the incurred downstream costs, e. g.,

the sum or the maximum.6 Task: assign online: requests → feasible target dates

such that the online cost is as small as possible.





2 Feasible target dates of ri are T (ri) := {t(ri) + 1, . . . , t(ri) + δ}.

3 σd := subset of requests assigned to date d .4 Downstream cost downcost(σd) := optimal cost of Π on σd .5 Online cost is a function of the incurred downstream costs, e. g.,







2 Feasible target dates of ri are T (ri) := {t(ri) + 1, . . . , t(ri) + δ}.3 σd := subset of requests assigned to date d .

4 Downstream cost downcost(σd) := optimal cost of Π on σd .5 Online cost is a function of the incurred downstream costs, e. g.,







2 Feasible target dates of ri are T (ri) := {t(ri) + 1, . . . , t(ri) + δ}.3 σd := subset of requests assigned to date d .4 Downstream cost downcost(σd) := optimal cost of Π on σd .

5 Online cost is a function of the incurred downstream costs, e. g.,the sum or the maximum.

6 Task: assign online: requests → feasible target datessuch that the online cost is as small as possible.






the sum or the maximum.

6 Task: assign online: requests → feasible target datessuch that the online cost is as small as possible.









Considered Downstream Problems

Bin-Packing

bins of unit capacityminimize #required bins to pack given itemsrequest r has size 0 < s(r) ≤ 1

(Nonpreemptive) Parallel-Machine Schedulingm identical parallel machinesminimize the makespan for given jobsrequest r has processing time p(r) > 0

Traveling Salesman Problem

metric space (X , d) with an origin ominimize tour length for a set of pointsrequest r has point x(r) ∈ X

o



Bin-Packing





o



Bin-Packing





o



Bin-Packing





o



Bin-Packing





o



Bin-Packing





o



Bin-Packing





o



Bin-Packing





o



Bin-Packing





o


Competitive Analysis

Definition (c-competitive, competitive ratio)1 A deterministic online algorithm ALG is c-competitive if for each

sequence of requests σ we have:

ALG(σ) ≤ c · OPT(σ).

2 The competitive ratio of ALG is the infimum of all c ≥ 1 such thatALG is c-competitive.


Result Overview

1 Minimize total downstream cost:∑

d downcost(σd)

downstream problem lower bound upper boundbin-packing 3/2 2scheduling

√2 2

traveling salesman√

2 2Table: Bounds on competitive ratio.

2 Minimize maximum downstream cost: maxd downcost(σd)

downstream problem lower bound upper boundbin-packing 2 min{4, δ}scheduling 3/2 3− 1/δtraveling salesman 2 2δ − 1

Table: Bounds on competitive ratio.


Result Overview


d downcost(σd)


√2 2







Min-Total ONLINETDAPs

min∑

d

downcost(σd)


The Algorithm PACKTOGETHERORDELAY (PTD)

Algorithm PTD

Assign a request r to any feasible target date which is already used.If no used target date is feasible, assign r to t(r) + δ.



Algorithm PTD


Example (PTD for Downstream Bin-Packing)Let δ = 3.

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time t1 2 4

1 2

3 5 6

4 5 63

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Algorithm PTD



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4 5 63

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Algorithm PTD



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time t

1 2 4

1 2 3 5 64 5

63

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Algorithm PTD



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1 2 3 5 64 5

63

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Algorithm PTD



↑ ↑

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3 5 6

4 5 63

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Algorithm PTD



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3 5 6

4 5 63

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Algorithm PTD


TheoremConsider the min-total ONLINETDAP w. r. t. downstream problem Π.Assume that each instance of Π is feasible and that the followingproperties hold for any subinstance σ̄ of each σ:

1 OPT(σ̄) ≤ OPT(σ).

2 For each disjoint partition σ(1), . . . , σ(k) of σ̄ we havedowncost(σ̄) ≤

∑ki=1 downcost(σ(i)).

Then, PTD is 2-competitive.


Analysis of PTD

Corollary

PTD is 2-competitive for the min-total ONLINETDAP for thedownstream problems bin-packing, scheduling, and the TSP.

Assumptions1 OPT(σ̄) ≤ OPT(σ).2 For each disjoint partition σ(1), . . . , σ(k) of σ̄ we have

downcost(σ̄) ≤∑k

i=1 downcost(σ(i)).

Proof (Theorem).d1 < d2 < . . . < dk used target datesσodd (σeven) subsequence of requests assigned to date di with iodd (even)

di+1 − di ≥ δ

time td1. . . d2

. . . d3. . . d4

. . . . . . . . . . . .

PTD(σodd) = OPT(σodd) and PTD(σeven) = OPT(σeven)

PTD(σ) = PTD(σodd) + PTD(σeven)

= OPT(σodd) + OPT(σeven) ≤ 2 · OPT(σ)


Analysis of PTD

Corollary





Proof (Theorem).d1 < d2 < . . . < dk used target datesσodd (σeven) subsequence of requests assigned to date di with iodd (even)

di+1 − di ≥ δ

time td1. . . d2

. . . d3. . . d4

. . . . . . . . . . . .





Analysis of PTD

Corollary





Proof (Theorem).d1 < d2 < . . . < dk used target datesσodd (σeven) subsequence of requests assigned to date di with iodd (even)di+1 − di ≥ δ

time td1. . . d2

. . . d3. . . d4

. . . . . . . . . . . .





Analysis of PTD

Corollary






time td1. . . d2

. . . d3. . . d4

. . . . . . . . . . . .





Analysis of PTD

Corollary






time td1. . . d2

. . . d3. . . d4

. . . . . . . . . . . .





A Lower Bound for Downstream Bin-Packing

TheoremNo deterministic online algorithm for the min-total ONLINETDAP withdownstream bin-packing is better than 3/2-competitive.

Proof (δ = 2).

ALG(σ) = 0

ALG(σ) = 2ALG(σ) = 1ALG(σ) = 2ALG(σ) = 3

↑↓

↑↓↑↓

↑↓

time t

2 3

41

22 433

. . .

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. . . . . .. . .. . . . . .. . .

. . .

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OPT(σ) = 0

OPT(σ) = 1OPT(σ) = 2

time t

2 3

41

22 433

. . .

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. . . . . .. . .. . . . . .. . .

. . .

. . .




Proof (δ = 2).

ALG(σ) = 0


↑↓

↑↓↑↓

↑↓

time t2 3 41

22 433

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. . .. . . . . .. . .

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OPT(σ) = 0


time t2 3 41

22 433

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Proof (δ = 2).

ALG(σ) = 0


↑↓

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↑↓

time t

2 3

41 2

2 4

3

3

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OPT(σ) = 0


time t

2 3

41 2

2 4

3

3

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Proof (δ = 2).

ALG(σ) = 0ALG(σ) = 2

ALG(σ) = 1


↑↓

↑↓↑↓

↑↓

time t

2 3

41 2

2 4

3

3

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OPT(σ) = 0

OPT(σ) = 1

OPT(σ) = 2

time t

2 3

41 2

2 4

3

3

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Proof (δ = 2).


ALG(σ) = 1


↑↓

↑↓

↑↓

↑↓

time t

2 3

41

2

2 43

3

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OPT(σ) = 0

OPT(σ) = 1

OPT(σ) = 2

time t

2 3

41

2

2 43

3

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. . .. . .




Proof (δ = 2).

ALG(σ) = 0

ALG(σ) = 2

ALG(σ) = 1ALG(σ) = 2ALG(σ) = 3

↑↓

↑↓

↑↓

↑↓

time t

2 3

41

2

2 43

3

. . .

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OPT(σ) = 0

OPT(σ) = 1

OPT(σ) = 2

time t

2 3

41

2

2 43

3

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Proof (δ = 2).


ALG(σ) = 1


↑↓

↑↓

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↑↓

time t

2 3

41

2

2 43

3

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OPT(σ) = 0

OPT(σ) = 1

OPT(σ) = 2

time t

2 3

41

2

2 43

3

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Proof (δ = 2).


ALG(σ) = 1


↑↓

↑↓↑↓

↑↓

time t

2 3 4

1

2

2 43

3

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OPT(σ) = 0

OPT(σ) = 1

OPT(σ) = 2

time t

2 3 4

1

2

2 43

3

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Proof (δ = 2).


ALG(σ) = 2

ALG(σ) = 3

↑↓

↑↓↑↓

↑↓

time t

2 3 4

1

2

2 43

3

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OPT(σ) = 2time t

2 3 4

1

2

2 43

3

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. . .. . .




Proof (δ = 2).


ALG(σ) = 2

ALG(σ) = 3

↑↓

↑↓↑↓

↑↓

time t

2 3 4

1

2

2 4

3

3 . . .

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OPT(σ) = 2time t

2 3 4

1

2

2 4

3

3 . . .

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Proof (δ = 2).


ALG(σ) = 3

↑↓

↑↓↑↓

↑↓

time t

2 3 4

1

2

2 4

3

3 . . .

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OPT(σ) = 2time t

2 3 4

1

2

2 4

3

3 . . .

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. . . . . .. . .

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. . . . . .. . .




Proof (δ = 2).


ALG(σ) = 3

↑↓

↑↓↑↓

↑↓

time t

2 3 4

1

2

2 4

3

3 . . .

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. . . . . .. . .

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OPT(σ) = 2time t

2 3 4

1

2

2 4

3

3 . . .

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Min-Max ONLINETDAPs

min maxd

downcost(σd)


The Algorithm BALANCE (BAL)

BALAssign a request to the earliest feasible target date such that theincrease in the objective (maximum downstream cost) is minimal.

LemmaAfter BAL assigned a given request r define

σ̄ as the subsequence of requests assigned within T (r)B as the total number of required bins in T (r)

Then if B > 1, we have B/2 ≤∑

r∈σ̄ s(r).




Example (BAL for Downstream Bin-Packing)Let δ = 2.

↑

↑ ↑

time t21 2

1

3

3

4 5 6

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r∈σ̄ s(r).





↑↑ ↑

time t21 2

1

3

3

4 5 6

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r∈σ̄ s(r).





↑

↑

↑

time t21 2

1

3

3

4 5 6

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r∈σ̄ s(r).





↑↑ ↑

time t21 2

1

3

3

4 5 6

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r∈σ̄ s(r).





↑↑

↑time t2

1 2

1

3

3 4 5 6

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r∈σ̄ s(r).





↑↑ ↑

time t2

1 2

1

3

3 4 5 6

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r∈σ̄ s(r).




TheoremBAL is 4-competitive for the min-max ONLINETDAP w. r. t.downstream bin-packing.




r∈σ̄ s(r).




TheoremBAL is 4-competitive for the min-max ONLINETDAP w. r. t.downstream bin-packing.




r∈σ̄ s(r).


Analysis of BAL for Downstream Bin-Packing

Proof (Theorem).

rk : first request in σ such that BAL(σ) = BAL(r1, . . . , rk )

σ̄: subsequence of requests assigned to T (rk ) up to rk

time t

. . .

. . .t(rk )− δ + 2

. . .

. . .t(rk ) t(rk ) + 1 t(rk ) + δ

rk↑

. . . . . .

The optimal cost is at least:

OPT(σ) ≥ 12δ − 1

∑r∈σ̄

s(r) >12δ

∑rk∈σ̄

s(r)

time t. . .

. . .t(rk ) t(rk ) + 1 t(rk ) + δ

rk↑

. . . . . .

BAL assigns rk to date t(rk ) + 1 and B = δ(BAL(σ)− 1) + 1Lemma implies B/2 ≤

∑r∈σ̄ s(r) which gives:

BAL(σ) ≤ 2δ

∑r∈σ̄

s(r) + 1− 1δ

< 4OPT(σ) + 1− 1δ

Integrality of BAL(σ) and OPT(σ) gives BAL(σ) ≤ 4OPT(σ)



Proof (Theorem).



time t

. . .

. . .t(rk )− δ + 2

. . .

. . .t(rk ) t(rk ) + 1 t(rk ) + δ

rk↑

. . . . . .


OPT(σ) ≥ 12δ − 1

∑r∈σ̄

s(r) >12δ

∑rk∈σ̄

s(r)

time t. . .

. . .t(rk ) t(rk ) + 1 t(rk ) + δ

rk↑

. . . . . .



BAL(σ) ≤ 2δ

∑r∈σ̄

s(r) + 1− 1δ

< 4OPT(σ) + 1− 1δ




Proof (Theorem).



time t. . .

. . .t(rk )− δ + 2

. . .

. . .t(rk ) t(rk ) + 1 t(rk ) + δ

rk↑

. . . . . .


OPT(σ) ≥ 12δ − 1

∑r∈σ̄

s(r) >12δ

∑rk∈σ̄

s(r)

time t. . .

. . .t(rk ) t(rk ) + 1 t(rk ) + δ

rk↑

. . . . . .



BAL(σ) ≤ 2δ

∑r∈σ̄

s(r) + 1− 1δ

< 4OPT(σ) + 1− 1δ




Proof (Theorem).



time t. . .

. . .t(rk )− δ + 2

. . .

. . .t(rk ) t(rk ) + 1 t(rk ) + δ

rk↑

. . . . . .


OPT(σ) ≥ 12δ − 1

∑r∈σ̄

s(r) >12δ

∑rk∈σ̄

s(r)

time t. . .

. . .t(rk ) t(rk ) + 1 t(rk ) + δ

rk↑

. . . . . .



BAL(σ) ≤ 2δ

∑r∈σ̄

s(r) + 1− 1δ

< 4OPT(σ) + 1− 1δ




Proof (Theorem).



time t. . .

. . .t(rk )− δ + 2

. . .

. . .t(rk ) t(rk ) + 1 t(rk ) + δ

rk↑

. . . . . .


OPT(σ) ≥ 12δ − 1

∑r∈σ̄

s(r) >12δ

∑rk∈σ̄

s(r)

time t. . .

. . .t(rk ) t(rk ) + 1 t(rk ) + δ

rk↑

. . . . . .BAL assigns rk to date t(rk ) + 1 and B = δ(BAL(σ)− 1) + 1Lemma implies B/2 ≤


BAL(σ) ≤ 2δ

∑r∈σ̄

s(r) + 1− 1δ

< 4OPT(σ) + 1− 1δ




Proof (Theorem).



time t. . .

. . .t(rk )− δ + 2

. . .

. . .t(rk ) t(rk ) + 1 t(rk ) + δ

rk↑

. . . . . .


OPT(σ) ≥ 12δ − 1

∑r∈σ̄

s(r) >12δ

∑rk∈σ̄

s(r)

time t. . .

. . .t(rk ) t(rk ) + 1 t(rk ) + δ

rk↑

. . . . . .



BAL(σ) ≤ 2δ

∑r∈σ̄

s(r) + 1− 1δ

< 4OPT(σ) + 1− 1δ




Proof (Theorem).



time t. . .

. . .t(rk )− δ + 2

. . .

. . .t(rk ) t(rk ) + 1 t(rk ) + δ

rk↑

. . . . . .


OPT(σ) ≥ 12δ − 1

∑r∈σ̄

s(r) >12δ

∑rk∈σ̄

s(r)

time t. . .

. . .t(rk ) t(rk ) + 1 t(rk ) + δ

rk↑

. . . . . .

BAL assigns rk to date t(rk ) + 1 and B = δ(BAL(σ)− 1) + 1

Lemma implies B/2 ≤∑

r∈σ̄ s(r) which gives:

BAL(σ) ≤ 2δ

∑r∈σ̄

s(r) + 1− 1δ

< 4OPT(σ) + 1− 1δ




Proof (Theorem).



time t. . .

. . .t(rk )− δ + 2

. . .

. . .t(rk ) t(rk ) + 1 t(rk ) + δ

rk↑

. . . . . .


OPT(σ) ≥ 12δ − 1

∑r∈σ̄

s(r) >12δ

∑rk∈σ̄

s(r)

time t. . .

. . .t(rk ) t(rk ) + 1 t(rk ) + δ

rk↑

. . . . . .



BAL(σ) ≤ 2δ

∑r∈σ̄

s(r) + 1− 1δ

< 4OPT(σ) + 1− 1δ




Proof (Theorem).



time t. . .

. . .t(rk )− δ + 2

. . .

. . .t(rk ) t(rk ) + 1 t(rk ) + δ

rk↑

. . . . . .


OPT(σ) ≥ 12δ − 1

∑r∈σ̄

s(r) >12δ

∑rk∈σ̄

s(r)

time t. . .

. . .t(rk ) t(rk ) + 1 t(rk ) + δ

rk↑

. . . . . .



BAL(σ) ≤ 2δ

∑r∈σ̄

s(r) + 1− 1δ

< 4OPT(σ) + 1− 1δ




Proof (Theorem).



time t. . .

. . .t(rk )− δ + 2

. . .

. . .t(rk ) t(rk ) + 1 t(rk ) + δ

rk↑

. . . . . .


OPT(σ) ≥ 12δ − 1

∑r∈σ̄

s(r) >12δ

∑rk∈σ̄

s(r)

time t. . .

. . .t(rk ) t(rk ) + 1 t(rk ) + δ

rk↑

. . . . . .



BAL(σ) ≤ 2δ

∑r∈σ̄

s(r) + 1− 1δ

< 4OPT(σ) + 1− 1δ



Analysis of BAL for Downstream Scheduling

TheoremBAL is (3− 1/δ)-competitive for the min-max ONLINETDAP w. r. t.downstream scheduling.

Proof.

r : first request in σ causing the maximum makespanw : load of least loaded machine in T (r) when r is released

time t. . .

. . .t(r) t(r) + 1 t(r) + δ

r↑

w

Total load in T (r) is at least wmδ + p(r) yielding:

OPT(σ) ≥ wmδ + p(r)(2δ − 1)m

>wδ

2δ − 1

⇒ w <

(2− 1

δ

)OPT(σ)

For BAL we have:




Proof.r : first request in σ causing the maximum makespan

w : load of least loaded machine in T (r) when r is released

time t. . .

. . .t(r) t(r) + 1 t(r) + δ

r↑

w


OPT(σ) ≥ wmδ + p(r)(2δ − 1)m

>wδ

2δ − 1

⇒ w <

(2− 1

δ

)OPT(σ)

For BAL we have:




Proof.r : first request in σ causing the maximum makespanw : load of least loaded machine in T (r) when r is released

time t. . .

. . .t(r) t(r) + 1 t(r) + δ

r↑

wTotal load in T (r) is at least wmδ + p(r) yielding:

OPT(σ) ≥ wmδ + p(r)(2δ − 1)m

>wδ

2δ − 1

⇒ w <

(2− 1

δ

)OPT(σ)

For BAL we have:





time t. . .

. . .t(r) t(r) + 1 t(r) + δ

r↑

w


OPT(σ) ≥ wmδ + p(r)(2δ − 1)m

>wδ

2δ − 1

⇒ w <

(2− 1

δ

)OPT(σ)

For BAL we have:





time t. . .

. . .t(r) t(r) + 1 t(r) + δ

r↑

w


OPT(σ) ≥ wmδ + p(r)(2δ − 1)m

>wδ

2δ − 1

⇒ w <

(2− 1

δ

)OPT(σ)

For BAL we have:





time t. . .

. . .t(r) t(r) + 1 t(r) + δ

r↑

w


OPT(σ) ≥ wmδ + p(r)(2δ − 1)m

>wδ

2δ − 1⇒ w <

(2− 1

δ

)OPT(σ)

For BAL we have:





time t. . .

. . .t(r) t(r) + 1 t(r) + δ

r↑

w


OPT(σ) ≥ wmδ + p(r)(2δ − 1)m

>wδ

2δ − 1⇒ w <

(2− 1

δ

)OPT(σ)

For BAL we have:

BAL(σ) ≤ w + p(r) <

(2− 1

δ

)OPT(σ) + OPT(σ)





time t. . .

. . .t(r) t(r) + 1 t(r) + δ

r↑

w


OPT(σ) ≥ wmδ + p(r)(2δ − 1)m

>wδ

2δ − 1⇒ w <

(2− 1

δ

)OPT(σ)

For BAL we have:

BAL(σ) ≤ w + p(r) <

(3− 1

δ

)OPT(σ)


Summary


d downcost(σd)


√2 2







Reference

S. Heinz, S. O. Krumke, N. Megow, J. Rambau, A. Tuchscherer,and T. VredeveldThe Online Target Date Assignment Problem.Approximation and Online Algorithms, LNCS, 2005

Thank You!


Reference

S. Heinz, S. O. Krumke, N. Megow, J. Rambau, A. Tuchscherer,and T. VredeveldThe Online Target Date Assignment Problem.Approximation and Online Algorithms, LNCS, 2005

Thank You!


Documents

The Online Target Date Assignment Problem · The Online Target Date Assignment Problem S. Heinz1 S.O. Krumke2 N. Megow3 J. Rambau4 A. Tuchscherer1 T. Vredeveld5 DFG Research Center