Scheduling+concerns+opBmal+allocaBon+or+assignment+of+resources,+over+Bme,+to+a+set+of+tasks+or+acBviBes.+++Resourses:+machines,+people,+space+Tasks:+ + ++producBon,+jobs,+classes,+flights+

+–  schedule+may+be+represented+by+GanS+charts+– m+machines+M1,...,Mm++++and+n+jobs+J1,...,Jn++++

+

Henry0Gan30(VS)01861::19190

Examples++

Gate+assignment+ Containers+

Processor+scheduling+Graph+drawing+

General+NotaBon+•  m+machines+1,...,m+•  n+jobs.+++J++is+set+of+all+jobs:+++J={1,...,n}+.+++Data+for+jobs:+•  pj++:+processing+Bme+of+job+j+•  pij+:+processing+Bme+of+job+j+on+machine+i++

+ +(if+processing+Bme+of+job+j+depends+on+the+machine)+•  rj+:+release+date+of+job+j +(earliest+starBng+Bme)++•  dj+:+due+date+of+job+j+ + +(commiSed+compleBon+Bme)+•  wj+:+weight+of+job+j+ + +(importance)+++

ClassificaBon+of+Scheduling+Problems++

+

(Many)+Scheduling+problems+can+be+described+by+a+three+field+

notaBon+α|β|γ,+where++

+α+describes+the+machine+environment+

+β+describes+the+job+characterisBcs,+and+

+γ+describes+the+objecBve+criterion+to+be+minimized++

+

Remark:++A+field+may+contain+more+than+one+entry+but+may+also+be+empty+

+

EXAMPLE:++

+++1+|+rj+|++Σ

j+C

j+++++++++++Single+machine.++

+ + + +Jobs+have+release+Bmes.++

+ + + +ObjecBve+is+minimizing+the+sum+of+the+compleBon+Bmes.+++

+ α|β|γ,+

ClassificaBon+d+Machine+environment++(α+)+++

•  Single+machine+(α+=+1)+•  IdenBcal+parallel+machines+(α+=+P+or+Pm)++

–  m+idenBcal+machines;+if+α+=+P,+the+value+m+is+part+of+the+input+and+if+α+=+Pm,+the+value+is+considered+as+a+constant+(complexity+theory)+

–  each+job+consist+of+a+single+operaBon+and+this+may+be+processed+by+any+of+the+machines+for+pj+Bme+units++

•  Unrelated+parallel+machines+(α+=+R+or+Rm)++–  m+different+machines+in+parallel+

pij+is+the+process+Bme+of+job+j+if+scheduled+completely+on+machine+i+

+

ClassificaBon+d+Job+characterisBcs+(β+)+++

•  release+dates+(rj+is+contained+in+β+field)+–  if+rj+is+not+in+β+field,+jobs+may+start+at+any+Bme+–  +if+rj+is+in+β+field,+jobs+may+not+start+processing+before+their+release+

date++

•  preempBon+(pmtn+is+contained+in+β+field)++–  processing+of+a+job+on+a+machine+may+be+interrupted+and+resumed+at+a+

later+Bme+even+on+a+different+machine+

•  unit+processing+Bmes+(pj+=+1+or+pij+=+1+in+β+field)+–  +each+job+(operaBon)+has+unit+processing+Bmes++

•  other+’obvious’+specificaBons+(i.e.+dj+=+d)+++

ClassificaBon+d+Job+characterisBcs+(β+)+++

•  precedence+constraints+(prec+in+β+field)+–  between+jobs+precedence+relaBons+are+given:+a+job+may+not+start+its+

processing+before+another+job+has+been+finished+–  may+be+represented+by+an+acyclic+graph+(verBces+=+jobs,+arcs+=+

precedence+relaBons)++

+

ClassificaBon+d+ObjecBve+criterion+(γ++)+++

NotaBons+–  Cj+:+compleBon+Bme+of+job+j++

–  Lj+=+Cj+−+dj+:+lateness+of+job+j+

ObjecBves:+

–  Makespan+ + + + +(γ+=+Cmax+)+++ +Cmax+=+max+{C1,...,Cn}++

–  Maximum+lateness+ + +(γ+=+Lmax+)+ +Lmax+=+max{L1,...,Ln}++

–  Total+compleBon+Bme+ + +(+γ+=++Σj+Cj+)+

–  Total+weighted+compleBon+Bme+(+γ+=++Σj+wjCj+)+

+

Many+more+models+in+literarture+!+

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