Spring 2000IE 5141 Topic 36 Reservation Systems without Slack.

Spring 2000 IE 514 1

Topic 36

Reservation Systems without Slack


Overview of Service Industry Models

(Lot sizing)

Reservation systems

Timetabling

Workforce scheduling


Reservation without Slack

Reservation system with m machines, n jobs

Release date rj, due date dj, weight wj

No slack

If processed job must start at time rj

Should we process the job? Maximize number of jobs processed

rjj rdp


Example: A Car Rental

Four types of cars: subcompact, midsize, full size, and sport utility

Fixed number of eachMachine = type of carJob = customer requesting a carMay request certain machine(s)

Job can be processed on a subset of machines


Problem Formulation

Integer Programming ProblemsFixed time periodsAssume H periodsLet xij be 1 if job j assigned to ith

machine (and zero otherwize)

Hlmix

njx

lJjij

m

iij

,...,1 ,,...,1 1

,...,1 ,11

Job requiringprocessing inperiod l


Feasibility Problem

Can we process every job?

Can we assign jobs to machines such that job j is assigned to a set Mj of machines

Relatively easy to solve


Optimization Problem

Maximize the total profit

In general NP-hardSpecial solvable cases

All processing times = 1 Decomposition into separate time units

m

i

n

jijijxw

1 1


Identical Weights and Machines

Assume wj = 1 and Mj = {1,2,…,m}Do not have a time decompositionSimple algorithm maximizes number

of jobs that are processed

Order jobs in increasing release date order

nrrr ...21


Algorithm

Step 1Set J= and j=1

Step 2If a machine available at time rj assign job j to the machine,

include it in J and go to Step 4

Step 3Select j* such that

If do not include j in J and go to Step 4Else delete job j* from J, assign j to freed machine & include in

J

Step 4If j=N STOP; otherwise set j=j+1 and go back to Step 2

*jjjj CprC kk

Jkk

JkjprCC

maxmax*

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Unlimited Machines

No slack, arbitrary processing times, equal weights, identical machines

Infinitely many machines in parallelMinimize the number of machines

used

Easily solvedOrder jobs as before

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Algorithm

Assign job 1 to machine 1Suppose first j-1 jobs have been

processedTry to assign jth job to a machine in useIf not possible assign to a new machine

Graph theory: node coloring problem

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Node Coloring Problem

Consider a graph with n nodes If an arc (j,k) connects nodes j and k

they cannot be colored with the same color

How many colors do we need to color the graph?

Number of colors needed = Number of machines needed

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Topic 37

Reservation Systems with Slack

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Reservation with Slack

Now allow slack

Trivial case all processing times = 1, identical

weights, identical machines decomposition in time

rjj rdp

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General Equal Processing Times

Now assume processing times are equal to some p 2

Interaction between time unitsBarriers algorithm

wait for critical jobs to be released start the job with the earliest deadline

Slot number l = lth job to startS(l) starting time of lth slot

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Barrier Algorithm

Barrier ordered pair (l,r) constraint

A k-partial scheduleStarting with a k-partial schedule

construct a (k+1)-partial schedule add a new barrier to the barrier list Lb

and start over from scratch

Slotnumber Release

time

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Earliest Deadline with Barriers

Selects machine h for job in (k+1)st slot

Compute the starting time

otherwisemod)1(

of multiple 1

mk

mkmh

4321 ,,,max tttt

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Computing the Starting Time

Let be the earliest time the job can start 4321 ,,,max tttt

)(),...,1(max1 kSSt Minimumrelease dateof jobs left

otherwise)1(

103 pmkS

mkt

bLrkrt ),1(:max4

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Selecting a Job

Select for (k+1)th slot job j’ with the earliest deadline

If creation of (k+1) partial schedule was successful

Otherwise a ‘crisis’ occurred and job j’ is a ‘crisis job’

Crisis routine

'jd

pdj

'

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Crisis Routine

Backtracks to find job with the latest starting time such that the deadline is greater than the crisis job

If no such job, optimal schedule does not include crisis job

Otherwise this job is a pull jobAdd new barrier with the minimum

release time r* of all the jobs Jr after the pull job

Start over with the barrier list

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Example: 2 machines, 4 jobs

Jobs 1 2 3 4

rj 0 2 5 5

dj 20 30 19 30

Processing time p = 10

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Barriers Algorithm

0 10 20 30

Machine 1

Machine 2

1

2

3

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Barriers Algorithm

0 10 20 30

Machine 1

Machine 2

1

2

3

193 d

Crisis job

Pull job

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Barriers Algorithm

Found a crisis job and a pull job exists:

Setup a barrier Lb={(2,5)}

Setup restricted set Jr={3}

Start from scratch

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Barriers Algorithm

0 10 20 30

Machine 1

Machine 2

1

3

2

4

Optimal Schedule

Second iteration:

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Generalizations

Non-identical processing times NP-hard No efficient algorithm exists Need heuristics

Composite dispatching rule Preprocessing: flexibility of jobs and

machines Dispatch least flexible job first on the least

flexible machine, etc

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Preprocessing

Let jk be the number of jobs that can be processed on machine j in the slot [k-1,k]

Let Mj be the number of machines job j can be processed on

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Priority Indices

Priority index for jobs

Priority index for machines

|)|,/( jjjj MpwfI

j

j

j

pkikiki

j

p

llki

pkikiki pg

,2,1,

1,

,2,1,

,...,,max

),...,,(

Low values =high priority

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AlgorithmStep 0

Calculate both priority indicesOrder jobs according to job priority index (Ij)

Step 1Set j=1

Step 2For job j select the machine and time slot with lowest rankDiscard job j if it cannot be processed at all

Step 3If j=n STOP; otherwise set j=j+1 and go back to Step 2

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Topic 38

Timetabling

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Timetabling

Infinite identical machines in parallelAll of n jobs must be processedTooling constraints

Many tools Need one or more tools for each job

Resource constraints Single resource of quantity R Need certain amount for each job

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Tooling Constraints

One of each toolObjective: minimize makespan

Special case of resource-constrained project scheduling problem with one of each resource, that is, Ri = 1

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Problem Complexity

NP-hard

Assume all processing times = 1

Decomposition?

Equivalent to node coloring problem

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Node Coloring Problem

Job = nodeJobs need same tool = arc between

nodes

Feasibility problem: Can the graph be colored with H colors?

Optimization problem: What is the lowest number of colors

needed? Chromatic number of the graph

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Relation to Reservation Models

Closely related to reservation problem with zero slack and arbitrary processing times

Special case of timetabling problem tools in common = overlapping time slots tools in common nodes connected colors = machines minimizing colors = minimizing machines adjacent time slots vs tools need not be adjacent

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Heuristics

Many heuristics exist for graph coloring degree of node = number of arcs connected to

node saturation level = number of colors connected to

node

Intuition: Color high degree nodes first Color high saturation level nodes first

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Algorithm

Step 1Order nodes in decreasing order of degree

Step 2Use color 1 for first node

Step 3Choose uncolored node with maximum saturation

level, breaking ties according to degreeStep 4

Color using the lowest possible number colorStep 5

If all nodes colored, STOP; otherwise go back to Step 3

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Topic 39

Example: Scheduling Meetings

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Example

Schedule 5 meetings with 4 people meetings = jobs and people = tools all meetings take one hour

1 2 3 4 5

Joe 1 1 0 1 1

Lisa 1 1 1 0 0

Jane 1 0 1 0 0

Larry 0 1 0 1 1

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Corresponding Graph

1

2

5 4

3

Degree =4

Degree =2Degree =4

Degree =3Degree =3

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Select First Job

Can select either job 1 or 2 firstSay, select 1 and color with the first

color

1

2

5 4

3Saturation level = 1for every node

Select 2 becausehighest degree

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Color Second Job

Color Job 2 with the next color

1

5 4

3

2 Saturation level = 2for every node

Select 4 becausehighest degree

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Color Third Job


1

5 4

3

2 Saturation level = 2 for node 3 3 for node 5

Select 5 next

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Color Fourth Job


1

5 4

3

2

Select 3 next

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Color Final Job

Color Job 3 with same color as Job 4

1

5 4

3

2

Had to use 4 colors Makespan = 4

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Example

Suppose now we have a reservation system:

Job 1 2 3

jp 2 3 1

jr 0 2 3

jd 2 5 4

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Equivalent Timetabling Problem

Job 1 2 3

Tool 1 1 0 0

Tool 2 1 0 0

Tool 3 0 1 0

Tool 4 0 1 1

Tool 5 0 1 0Toool 5

Job 1 mustbe processedin time [0,2]

Job 2 mustbe processedin time [2,5]

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Topic 40

Timetabling with Resource Constraints

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Resource Constraints

One type of tool but R units of it (resource)

Job j needs Rj units of this resource

Clearly, if Rj + Rk > R then job j and k cannot be processed at the same time, etc

Applications scheduling a construction project (R=crew

size) exam scheduling (R=number of seats)

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Bin-Packing

Assume all processing times = 1Unlimited number of machinesMinimize makespanEquivalent to bin-packing problem

each bin has capacity R item of size Rj

Pack into the minimum number of bins

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Solving the Bin-Packing Problem

Known to be NP-hardMany heuristics developedFirst fit (FF) heuristic

Always but an item in the first bin it fits into

Know that2)OPT(

10

17)FF( maxmax CC

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Example

Assume 18 items and R=2100 Jobs 1-6 require 301 resources Jobs 7-12 require 701 resources Jobs 13-18 require 1051 resources

FF heuristic: We assign the first 6 jobs to one interval

(3016=1806) We then assign jobs two at a time to next 3 intervals

(7012=1402) We then assign just one of the remaining jobs to

each interval

Very poor performance due to order of jobs

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First Fit Decreasing (FFD)

An improvement of FF:Order jobs in decreasing order firstKnow that

FF and FFD can be extended to different release dates

4)OPT(9

11)FFD( maxmax CC

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Discussion

Only considered very simple models

In practice: Dynamic reservation systems Price considerations (yield

management)

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Topic 41

Workforce Scheduling

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Workforce Scheduling

Overview: Days-Off Scheduling Shift Scheduling Cyclic Staff Problem Crew Scheduling Simultaneous Worker and Job

Scheduling

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Days-Off Scheduling

Number of workers assigned to each day

Fixed size of workforce

Problem: find minimum number of employees to cover a weeks operation

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Constraints

Demand per day nj, j=1,2,…,7

k1 out of every k2 weekends (day 1&7) off

Work 5 out of 7 daysWork no more than 6 consecutive

days

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Optimal Schedule

Algorithm for one week

Repeat for next week

Cyclic schedule when repeat

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Lower Bounds on Minimum Workforce W

Weekend constraint

Total demand constraint

Maximum daily demand constraint

1

12

712 ,maxB

kk

nnkW

2

7

15

1BnW

jj

3721 ,...,,max BnnnW

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Optimal Schedule

Define

Have

7,1

6,...,2

,max

,,max

71

321

jnn

jnWu

nnn

BBBW

j

jj

nuj

j 27

1

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Algorithm

Schedule weekends off

Determine additional off days

Categorize employees

Assign off-day pairs

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Optimality

Can be shown that schedule is feasible

Schedule is optimal

Why?

A cyclic optimal schedule exists

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Topic 42

Shift Scheduling

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Shift Scheduling

Fixed cycle of length m periodsHave bi people assigned to ith periodHave n shift patterns:

Cost cj of assigning a person to shift j

Decision variable: xj = people assigned to j

),...,,( 21 mjjj aaa

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Integer Program

0

...

...

...

2211

22222121

11212111

j

mnmnmm

nn

nn

x

bxaxaxa

bxaxaxa

bxaxaxa

nnxcxcxc ...2211Minimize

Subject to

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Solution

NP-hard in general

Special structure in shift pattern matrix

Solve LP relaxation Solution always integer when each column

contains a contiguous set of ones

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Topic 43

Cyclic Staffing

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Cyclic Staffing Problem

An m-period cyclic scheduleMinimize costConstraint bi for ith periodEach worker works for k consecutive

periods and is free for the next m-kExample: (5,7)-cyclic staffing

problem

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Integer Program Formulation

Shift patterns:

1111100

0111110

0011111

1001111

1100111

1110011

1111001

A

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Solution

Solution to LP relaxation ‘almost right’STEP 1: Solve LP relaxation to get

if integer STOP; otherwise continueSTEP 2: Formulate two new LPs with

The best integer solution is optimal

''' ,...,,21 nxxx

'''

21

'''21

......

......

21

21

nn

nn

xxxxxx

xxxxxx

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Example

(3,5)-cyclic staffing problem

7

4

6

4

3

11100

01110

00111

10011

11001

bA

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Solution

Solve the LP relaxation

cxx

min

0x

bAxS.t.

)5.2,0,5.4,0,5.1(' xNon-integer

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Solution

Add together:

Add constraint

No feasible solutionAdd constraint

Solution:

5.85.205.405.1

854321 xxxxx

854321 xxxxx

)2,1,4,0,2(x Optimal

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Days-Off Scheduling

We can represent our days-off scheduling problem as a cyclic staffing problem as long as we can determine all the shift patterns

Difficulty: unknown cycle length

Difficulty: many patterns larger problem

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Example

Two days off in a week + no more than 6 consecutive workdays

1100

0011

0111

1111

1111

1010

1101

0111

0111

1111

1111

1111

1000

1000

A

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Cyclic Staffing with Overtime24-hour operation8-hour shifts with up to 8 hour

overtime

1U0

01U

U01

A

100000000

110000000

111000000

111100000

111110000

111111000

111111100

111111110

U

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Penalties for Under/Overstaffing

Demand not fixed Linear penalty for understaffing Linear penalty for overstaffing Let denote the level of understaffing

IP

'ic''ic

'ix

0, '

'

xx

bxx IA

'''''min xxbcxcxc A

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Topic 44

Crew Scheduling

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Crew Scheduling

Have m jobs, say flight legsHave n feasible combination of jobs

a crew is permitted to do

1 2

3 45

6

6m

Set partitioningproblem

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Notation

Cost cj of round trip jDefine

otherwise0

tripround ofpart is leg if1 jiaij

otherwise0

selected is tripround if1 jx j

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Integer Program

1,0

1...

1...

1...

2211

2222121

1212111

j

nmnmm

nn

nn

x

xaxaxa

xaxaxa

xaxaxa

nnxcxcxc ...2211Minimize

Subject to

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Set Partitioning

Constraints called partitioning equations

The positive variables in a feasible solution called a partition l

NP-HardWell studied like TSP, graph-coloring,

bin-packing, etc.

}1:{ lj

l xjJ

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Row Prices

Say that

is a set of feasible row prices if for

Cost of covering a job

),...,,( 21lm

lll

m

ijij

li ca

1

lJj

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Change Partition

Let Z1 (Z1) denote the objective value of partition 1 (2)

Then

Potential savings of including column j is

If all negative then optimal

2 1

112

Jj

m

ijiji caZZ

m

ijijij ca

1

1

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Heuristic

Start with some partitionConstruct a new partition as follows:

Find the column with highest potential savings

Include this column in new partition If all jobs covered stop; otherwise

repeat

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ConnectionsHere

Production requirements per shift fixed Optimize number of people per shift

Timetabling with resource constraints Number of people fixed Optimize job schedule Determines production requirements per

shiftOptimize simultaneously!

Spring 2000IE 5141 Topic 36 Reservation Systems without Slack.

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