Production Scheduling
Useful tools to assist in production
scheduling




Assignment problem (which job to which work
center?)
Sequencing rules (which job to process next?)
Next Best rule (minimize setup costs)
Sequencing through two successive work centers
(minimize the completion time for the last job
through the process)
Assignment problem




Objective is to minimize total process time
Number of jobs to be loaded = number of work
centers or machines
Each machine is assigned one and only one job
Linear programming problem
– Objective function (maximize or minimize) subject to
constraint conditions
– Manual solution procedure
– Excel Solver
Example of assignment problem

The metalworking work center of a job shop has been
assigned the task of processing four orders. Four machines in
the work center are all capable of processing any of the
orders. The foreman must decide which job to allocate to
each machine.
machine 1 machine 2 machine 3 machine 4
job 1
13 min
16
21
14
job 2
17
13
19
14
job 3
12
16
20
16
job 4
20
12
17
11
Manual solution procedure to
assignment problem


(1) Column reduction: Subtract the lowest cost in
each column from every cost element in that
column. Do this for every column. This new matrix
of opportunity costs is now used in the next step.
(2) Row reduction: Subtract the lowest cost in each
row from every cost element in that row. Do this
for every row. This new matrix of opportunity costs
is now used in the next step.
Manual solution procedure to
assignment problem (cont.)

(3) Cover the zeroes: Cover all the zero elements in
the matrix with horizontal and/or vertical lines. Find
the minimum number of lines necessary to cover all
zeroes. If the number of lines required is equal to
the number of machines available, an optimal
solution has been reached. The optimal
assignments are found by examining the zero
elements in the matrix. If the number of lines is
fewer than the number of machines, proceed to
step (4).
Manual solution procedure to
assignment problem (cont.)

(4) Create new zeroes: Begin with the matrix from
step (3). Find the smallest uncovered cost element
(not covered by a line) and subtract it from all
uncovered cost elements, including itself; add it to
all cost elements at the line intersections. All other
cost elements remain unchanged. Now erase all
horizontal and vertical lines and return to step (3).
Excel solution to assignment
problem







Load Excel Solver (click on “File” tab and go to
“Options,” choose “Add-ins” from the left menu,
select “Solver Add-in” and then click the “go”
button, check “Solver Add-in” in the checkbox)
Solver is now available under the “Data” tab
Define changing cells
Calculate total cost (or profit)
Set up resource usage
Set up Solver Parameters (Target Cell, Max/Min,
Changing Cells, Constraints, and Options)
For metalworking center example
Another assignment problem
example

Merchants Bank has four new tellers (A-D) with varying skills who
are to be assigned to the main bank or one of the branches
(locations 1-4). The criterion for assigning tellers to locations is
minimized customer waiting time. An index of customer waiting time
is shown in the table below for each of the four locations and varying
teller skills. Make teller assignments such that the overall waiting
time will be minimized. What is the total waiting time for the optimal
assignment?
teller A
teller B
teller C teller D
location 1
40 min
60
50
30
location 2
60
30
60
50
location 3
60
30
40
60
location 4
50
60
30
60
Other applications of linear
programming

Transportation problem

Objective function subject to resource constraints

Reports (Answer, Sensitivity, or Limits) can be
generated by clicking on any of the choices in the
window
Sequencing


Which job to process next?
Methodology to follow
– Shortest processing time: select the job having the least
processing time
– Earliest due date: select the job that is due the soonest
– First-come, first served: select the job that has been
waiting the longest for this workstation
– First-in-system, first-served: select the job that has been
in the shop the longest
– Slack per remaining operations: select the job with the
smallest ratio of slack to operations remaining to be
performed
Performance evaluation criteria

Choice of sequencing methodology to choose is
dependent on the performance evaluation criteria
to be applied
–
–
–
–
–
–
–
–
Total job completion time
Avg job completion time
Avg job waiting time
Avg job lateness
Avg number of jobs in the system
Avg number of jobs waiting
Set-up costs
In-process inventory costs
Sequencing example

Five jobs waiting, one workstation/machine
job (in order of arrival)
job time (days)
due date
A
4
6
B
17
20
C
14
18
D
9
12
E
11
12
Sequencing example evaluation

Using shortest processing time
–
–
–
–
–
–
–
–
Sequence is A-D-E-C-B
Construct timeline
Total job completion time: 55 days
Avg job completion time:
(4+13+24+38+55)/5 = 26.8 days/job
Avg job waiting time: (0+4+13+24+38)/5 = 15.8 days/job
Avg job lateness: (0+1+12+20+35)/5 = 13.6 days/job
Avg number of jobs in the system:
(4x5+9x4+11x3+14x2+17x1)/55 = 2.44 jobs
Avg number of jobs waiting:
(4x4+9x3+11x2+14x1)/55 = 1.44 jobs
Sequencing example evaluation
(cont.)


Using earliest due date: could be same sequence as
shortest processing time, A-D-E-C-B
Using first-come, first served
– Total job completion time: 55 days
– Avg job completion time:
(4+21+35+44+55)/5 = 31.8 days/job
– Avg job waiting time: (0+4+21+35+44)/5 = 20.8 days/job
– Avg job lateness: (0+1+17+32+43)/5 = 18.6 days/job
– Avg number of jobs in the system:
(4x5+17x4+14x3+9x2+11x1)/55 = 2.89 jobs
– Avg number of jobs waiting:
(4x4+17x3+14x2+9x1)/55 = 1.89 jobs
Comparison of SPT, EDD, and FCFS
evaluation criteria
SPT
EDD
FCFS
total job completion time (days)
55
55
avg job completion time (days/job)
26.8
same
as SPT
avg job waiting time (days/job)
15.8
20.8
avg job lateness (days/job)
13.6
18.6
avg number of jobs in system (jobs)
2.44
2.89
avg number of jobs waiting (jobs)
1.44
1.89

31.8
“For jobs to be processed in one work center, it (shortest
processing time rule) is consistently superior to other rules; it
is optimal for minimizing avg. (job) completion time, avg.
number of jobs in the system, and avg. job lateness.” (Adam
and Ebert, page 493)
Setup cost minimization

Next Best approach – given that a job (job i) is
being processed, select next the unassigned job
(job j) for which the setup cost is minimized
Next Best rule example


Assuming A is being processed, the NB approach asks “What
next?”
Setup costs ($):
Predecessor
Job i
Follower Job j
A
B
C
D
E
A
-
29
20
18
24
B
-
-
14
19
15
C
-
35
-
37
26
D
-
15
10
-
10
E
-
18
16
40
-
The least costly sequence?

Two sequences possible using NB approach:
– A-D-C-E-B ($18+$10+$26+$18=$72)
– A-D-E-C-B ($18+$10+$16+$35=$79)



Enumeration of all 24 possible sequences yields an optimal
solution of A-D-E-B-C with a cost of $18+$10+$18+$14=$60
NB approach is a heuristic – a rule or a set of rules that
generally leads to the optimal solution, but does not
guarantee the optimal solution
Assuming the NB approach and say, SPT, sequences do not
coincide then one must consider relative importance placed on
costs of machine setup (NB) versus the value of gaining
overall shop effectiveness and customer satisfaction (SPT)
Sequencing through two work
centers



All jobs have to be processed through two
successive work centers
Jobs must be processed in the same order on both
work centers
The following is an optimal procedure to minimize
the completion time for the last job through the
process
Sequencing through two work
centers example

Processing times (days):
Job
Work Center 1
(sheet metal)
Work Center 2
(paint)
A
4
5
B
17
7
C
14
12
D
9
2
E
11
6
Sequencing through two work
centers procedure





(1) Determine the minimum of all processing times
(2) If the minimum is associated with work center
1, place the corresponding job in the earliest
available position in the sequence
(3) If the minimum is associated with work center
2, place the corresponding job in the latest
remaining position in the sequence
(4) Cross out the times of the job just assigned
across all work centers to omit that job from further
consideration
(5) Now return to step (1) and repeat the
procedure by identifying the minimum of all
remaining process times
A second example of sequencing
through two work centers

Data Systems, Inc., processes all incoming jobs through two
successive work centers, A and B (each job goes first through
A, then through B). Five jobs await processing (see below).
Assign a job sequence that minimizes the completion time of
the last job processed. What is the total flow time for this
sequence?
Job
S
T
U
V
W
Processing time in A (minutes)
14
3
22
31
7
Processing time in B (minutes)
8
9
4
17
12
Summary of sequencing methods

“… an abundance of sequencing methods is
available and they can affect shop performance in
different ways. … in choosing among these
methods, you should carefully evaluate them in
terms of the criteria that are of greatest importance
for your organization’s competitive posture.”