TheoryScheduling ProblemsThese are expressed generally as n job
m machine problemsThere are a few rules that are used - These are
known as priority rules- FIFO : First In First Out- LIFO : Last In
First Out- EDD : Earliest Delivery Date first- SPT : Shortest
Processing Time First- Most Profitable job- Least remaining
operation timeA few TermsFlow Time - The time it takes for the job
to flow out of the factory. It is the difference between the job
was available for processing till the time it is completedTardiness
- It is the difference between the actual completion date/time of
the job and the required date/time of the job for the jobs that
were delivered late. For early and on time jobs, the lateness is
zeroEarliness - It is the difference between the required
completion date/time of the job and the actual date/time of the job
for the jobs that were delivered early. For on time and late jobs,
the earliness is zeroLateness - It is the difference between the
required completion date/time of the job and the actual date/time
of the job. This may be positive, negative or zero1n job 1
machineRegardless of sequence, the total makespan remains same.SPT
rule minimises mean flow time and mean lateness2n job 2 machine -
Same SequenceJohnson's Algorithma) Find the least operation time
for the jobs that are to be scheduled. If it is for the first
machine, schedule it first, else schedule it last. Break ties
arbitrarily.b) Tick off the scheduled job.c) If all jobs are not
scheduled, go to step a3n job 2 machine - Different
SequenceJackson's Algorithma) Group the jobs in 4 groups, viz A, B,
AB and BAb) For groups AB and BA, internal sequence of the group to
be decided using Johnson's algorithm. For groups A and B, internal
sequence of the group is immaterial, choose any sequence.c) For
machine A, the jobs are to be processed as group AB, group A and
group BA with the internal sequence of the group as per step bd)
For machine B, the jobs are to be processed as group BA, group B
and group AB with the internal sequence of the group as per step
b4n job 3 machines - Same sequenceJohnson's 3 Machine rulea) Find
the minimum on machine A - min(A), maximum on machine B - max(B)
and minimum on machine C - min(C)b) If min(A)>=max(B) then
machine B is said to be dominated by machine Ac) If
min(C)>=max(B) then machine B is said to be dominated by machine
Cd) If either of the conditions b, c is satisfied, then convert the
given 3 machine problem into equivalent 2 machine problem ase)
Duration on mac-1 = duration on machine A + duration on machine Bf)
Duration on mac-2 = duration on machine B + duration on machine Cg)
Solve this problem using Johnson's 2 machine algorithm. The
resulting sequence is optimal for the original 3 machine problem52
Job m machinesGraphical methoda) On a graph, plot the two jobs on
two axisb) For each job, Assuming that you are having only one job,
mark the axis with the machine for the duration of operationc) For
each machine, mark the areas of interferrence by moving along both
axis for the machine under consideration.d) Starting from origin,
try to move at 45 degrees till either the border of graph is
encountered or an interferrence area is encountered.e) Try to get
around the interferrence area by travelling horizontally or
vertically as appropriate. When a choice is available, select the
shortest path as travel along interference area indicates idle
time.Once interference is overcome, move again at 45 degreesf) If
the border of the graph is reached, then travel along the border to
reach the top right cornerg) Idle time for each job is travel along
interferrence area in the direction of other axis
n Job 1 MachineProblem 1JobDurationDelivery DateOn Time delivery
ProfitLate delivery Penalty (Rs/Day)Early delivery bonus
(Rs/Day)FIFOLIFOEDDSPT112787219110212775881152910537326862710386643192341674774514075505564861109712665977515400146748984263402088312952337518892331077630115101510Find
the profit/loss based on the following rulesFIFOLIFOEDDSPTAlso find
a sequence that will maximise the profit.Excel
ToolJobDurationStartEndReq Delivery DateNormal ProfitLate delivery
Penalty (Rs/Day)Early delivery bonus (Rs/Day)Net
Profit11012787023432151124075019026561231097042139433619234091325846102634001284687510151540000400951520233750243992720277588220036837273432686540632107344176303740256Total4135126487093573LDPEDBNormal
ProfitTotalEDD87335122639FIFO90351635123125SPT64870935123573
n Job 2 MachineJohnson's 2 Machine AlgorithmProblem
1Job12345678Job12687543A4581065106A4556106108B810245766B810766542SequenceABABABABABABABABSchedule12345678Find
the sequence, Makespan and idle time for machine B using the
following methods1) FIFO2) LIFO3) Johnson's algorithmExcel
ToolJob87655321TotalDur (A)61056685450Dur(B)66755210849Start on
A06162127334146End on A616212733414650Start on B616222934414656End
on B122229343943566464Idle Time6400023015Jackson's AlgorithmProblem
4Job1234567891011121314151617181920TotalA45840510663071147204885B8108457607104529050346103SequenceBABAABBABBAABAABBABABBAABAABABABBAFind
Schedule using Jackson's method, total makespan and idle time on
each machineExcel
ToolJob1234567891011121314151617181920TotalA45840510663071147204885B8108457607104529050346103SequenceBABAABBABBAABAABBABABBAABAABABABBASeq
On A37912141619815171246101320TotalDur
(A)810671746424545318000Start on
A08182431323943495355596468737677858585End on
A81824313239434953555964687376778585858585Idle
Time000000000000000000000Seq On
B12461013205111837912141619Dur(B)8104710265438675954000Start on
B08182229394147525659677380859499103103103End on
B8182229394147525659677380859499103103103103103Idle
Time000000000000000000000
2 job m machineJob1Job2MCDurStartEndIdle TimeMCDurStartEndIdle
TimeInterferrenceMC23030MC330300JOB223MC2MC55380MC44370022MC458130MC558131021MC1313160MC1613190Inter20MC3216180MC251924001918MC1TOTAL1818023241171615141312MC51110987MC46543MC321MC2MC5MC4MC1MC3123456789101112131415161718JOB1
