Optimal Methodology for Synchronized Scheduling of Parallel
Station Assembly with Air Transportation
#
Viswanath Kumar Ganesan#, Li Kunpeng*, and Sivakumar Appa Iyer#
Innovation in Manufacturing Systems and Technology, Singapore MIT Alliance, Singapore 639798
*
School of Mechanical & Production Engineering, Nanyang Technological University, Singapore
Abstract— We present an optimal methodology for
synchronized scheduling of production assembly with air
transportation to achieve accurate delivery with minimized
cost in consumer electronics supply chain (CESC). This
problem was motivated by a major PC manufacturer in
consumer electronics industry, where it is required to
schedule the delivery requirements to meet the customer
needs in different parts of South East Asia. The overall
problem is decomposed into two sub-problems which consist
of an air transportation allocation problem and an assembly
scheduling problem. The air transportation allocation
problem is formulated as a Linear Programming Problem
with earliness tardiness penalties for job orders. For the
assembly scheduling problem, it is basically required to
sequence the job orders on the assembly stations to minimize
their waiting times before they are shipped by flights to their
destinations. Hence the second sub-problem is modelled as a
scheduling problem with earliness penalties. The earliness
penalties are assumed to be independent of the job orders.
Index Terms — Scheduling, Supply chain, Optimization, air
transportation
I. INTRODUCTION
T
HE synchronization problem of CESC studied in this
work is observed in a major PC assembly
manufacturing industry facing a challenge in its
performance of on time delivery. The company has its
major assembly plant in Singapore. The industry receives
their orders through many sources including email, World
Wide Web, fax and phone. Orders come randomly, and the
company commits the delivery time to the customers. Air
transportation is commonly used for the distribution of
high value MTO consumer electronics products to global
customers and in general, commercial cargo flights are
Manuscript received November 19, 2004. This work was supported in
part by the Singapore MIT Alliance and School of Mechanical and
Production Engineering, Nanyang Technological University.
V. K. Ganesan is a Research Fellow in Singapore-MIT Alliance, 50
Nanyang Avenue, Singapore 639798. Phone: 65 6790 6397; Fax: 65 6862
7215; E-mail: vkganesan@ntu.edu.sg.
K. P. Li is a Ph D Scholar in School of Mechanical and Production
Engineering, Nanyang Technological University, Singapore 639798. Email: PG01538891@ntu.edu.sg.
A. I. Sivakumar is with Singapore-MIT Alliance and School of
Mechanical and Production Engineering, Nanyang Technological
University, 50 Nanyang Avenue, Singapore 639798. E-mail:
msiva@ntu.edu.sg.
often used. The important dynamic factors, which dictate
the outbound logistics of consumer electronics supply
chain are (a) the number of available flights for the
distribution planning horizon, (b) the departure and arrival
time of the flights, (c) the designated capacity and the
corresponding transportation cost, and (d) the possible
special capacity in each flight with the corresponding
freight cost. Hence, the air transportation allocation
involves selection of capacities in flights for all orders to
minimize the cost of transportation. Costs or penalties are
incurred by delivering an order either earlier or later than
the customers’ due dates. The delivery earliness costs
could result from the need for storage and insurance. The
delivery tardiness cost includes customer dissatisfaction,
contract penalties, loss of sales, and potential loss of
reputation. Costs are also incurred when an order is
completed earlier than its scheduled transportation
departure time. The costs can be taken as either for storing
them at the production facility or waiting charges at the
airport. Unlike the basic assembly and transportation cost
of the products, these penalty costs can be minimized by
achieving better synchronization in CESC.
The Just-in-Time (JIT) production philosophy has lead
to a growing interest both in production and transportation
scheduling problems considering earliness and tardiness
penalties [1]-[3]. Research on Production scheduling, in
general, has addressed problems with out taking into
account the final delivery and transportation requirements
after processing the jobs on machines. Only a few
researchers have considered the joint optimization of
machine scheduling and job transporting. The first study
explicitly considering the transportation issue is by Maggu
and Das [4]. They studied a two-machine flow-shop
makespan problem in which they assumed that there are
unlimited buffers on both machines and a sufficient
number of transporters available to transport jobs from one
machine to the other with job-dependent transportation
times. The problem was solved by using a generalization of
Johnson's rule [5]. Potts [6] and Hall and Shmoys [7]
studied a single-machine problem with unequal job arrival
times and delivery times. In their model, they implicitly
assumed that a sufficient number of vehicles are available
in the system in order to deliver immediately a processed
job to the customer. They provided a heuristic and a worstcase analysis. Woeginger [8] studied the same problem in
the parallel-machine environment with equal job arrival
times and provided a heuristic with a worst-case analysis.
Machine scheduling problems with jobs delivered in
batches after processing is reported in Herrmann and Lee
[9], Chen [10], and Cheng et al. [11]. They did not
consider transportation times, i.e., it was assumed that
deliveries can be made instantaneously. Lee and Chen [12]
studied machine scheduling problems with explicit
transportation considerations. Two types of transportation
situations are considered in their models. The first type,
Type-1, involves intermediate transportation of jobs from
one machine to another for further processing. The second
type, Type-2, involves the transportation provided to
deliver finished jobs to their destinations. Jobs are
delivered in batches by transporter(s). They assumed that
all jobs require the same physical space on the transporter.
Both transportation capacity and transportation times are
considered in their models. This class of scheduling
problems is computationally difficult.
The problem under study consists of two tasks. The first
task is to allocate accepted orders to available flights’
capacities to minimize total transportation cost and delivery
(earliness & tardiness) penalties. The second task is to
determine a schedule for assembly production that
minimizes the waiting time before air transportation.
Transportation allocation is constrained by assembly, i.e.,
allocation should be balanced with assembly capacity. The
assembly schedule should ensure that each order does not
miss its scheduled flight and also minimize the average
waiting time between assembly and transportation. The
assembly schedule decides assembly release time for each
order. We assume that there are no delays or interruptions
in assembly line during the planning period and the
assembly capacity is sufficient enough to produce all the
orders on time to meet their transportation departure times.
II. ASSUMPTIONS
The air transportation problem and the assembly
problem are formulated in this work are based on the
following assumptions:
• Decisions of transportation allocation and the
assembly scheduling are for the orders accepted in the
previous planning periods.
• Order fulfillment is considered to be achieved when
the order reach the destination airport on time.
• Orders can be split and allocated to different flights
• All the packed products have same or similar
dimensions.
• There are multiple flights in the planning period with
different transportation specifications such as cost,
capacity, etc.
• Only one flight departs any given instant of time.
• Each flight has a normal capacity area with normal
transportation cost and a special capacity area with special
transportation cost for orders that exceed normal capacity.
Normal capacity can be considered as the forecasted
capacity, while special capacity can be taken as maximum
extension from the forecasted capacity in the planning
period.
• Business processing time and cost, together with
loading time and loading cost for each flight are included
in the transportation time and transportation cost.
• Local transportation time and cost are included in
assembly time and assembly cost. Local transportation
transfer products from the assembly plant to airport.
• Orders released into assembly flow shop for the
planning period are delivered within the same planning
period which means there are no assembly backlogs.
• The assembly flow line is a balanced one and hence it
is treated as a single machine.
• Setup times are included in the processing times of
assembly manufacturing.
• Total assembly manufacturing time of an order is
directly proportional to the order’s quantity.
• Waiting time penalties for jobs before transportation
are job independent i.e., they are not determined based on
any job characteristics.
• The starting time of the planning period is set equal to
zero.
III. METHODOLOGY
The synchronization problem is investigated in this
section by adopting a two stage approach which involves
decomposing the overall problem into two sub-problems,
consisting of an air transportation allocation problem and
an assembly scheduling problem. The air transportation
allocation problem is formulated as an Integer Linear
Programming (ILP) Problem with earliness tardiness
penalties for job orders. For the assembly scheduling
problem, it is basically required to sequence the job orders
on the parallel assembly stations to minimize the waiting
times of job orders before they are shipped by flights to
their destinations. Hence the second sub-problem is
modeled as a parallel machine scheduling problem with job
independent earliness penalties. The air transportation
problem is solved first to obtain the transportation
allocation schedule. Then the assembly scheduling problem
is solved optimally by formulating as an Integer Linear
programming problem to obtain the release time of each
order. The input data for the scheduling problem are the
flight allocation results of the air transportation allocation
problem.
IV. TRANSPORTATION ALLOCATION MODEL
The air transportation problem, the first stage of the
research problem, formulated as an Integer Linear
Programming (ILP) model allocates orders to the existing
transportation
capacity
with
minimum
cost.
Synchronization is incorporated into the ILP model by the
constraint that balances the production rate of the assembly
facility with the flight allocation.
The following notations are defined:
the order index, i=1, 2, …. N
the flight index, f=1,2,……F
the departure time of flight f at the local place
where the manufacturing plant is located
the arrival time of flight f at the destination;
Af
NCf
the transportation cost for per unit product which
allocated to normal capacity area of flight f;
SCf
the transportation cost for per unit product which
allocated to special capacity area of flight f;
the available normal capacity of flight f;
NCapf
SCapf
the available special capacity of flight f;
Qi
the quantity of order i;
αi
the delivery earliness penalty cost (/unit/hour) of
order i;
βi
the delivery tardiness penalty cost (/unit/hour) of
order i;
di
the due date of order i;
pi
the priority of order i;
WTi
the waiting time of order i between assembly
manufacturing and transportation;
PEif
the per unit delivery earliness penalty cost for
order i when it is transported by flight f;
PEif = Max(0,di-Af)* αi
(1)
PLif
the per unit delivery tardiness penalty cost for
order i when it is transported by flight f;
PLif = Max(0,Af-di)* βi
(2)
Zif
the quantity of order i allocated to flight f;
Xif
the quantity of the portion of order i allocated to
flight f’s normal capacity area;
Yif
the quantity of the portion of order i allocated to
flight f’s special capacity area
PR
the production rate of assembly manufacturing
Desi
the order i’s destination
Desf
the flight f’s destination
B
a large number
|B|
the absolute value of B
i
f
Df
The ILP model for the multi-destination transportation
allocation problem is expressed as follows:
for all i, f
(4)
X if + Yif = Z if
B * X if * | Des i − Des f |< 1
(5)
B * Yif * | Des i − Des f |< 1
(6)
∑X
if
≤ NCap f
for all f
(7)
≤ SCap f
for all f
(8)
i
∑Y
if
i
∑ (X
if
+Yif ) = Qi
for all i
(9)
f
f
∑∑
f =1 i
( X if +Yif ) ≤ D f PR for all f
(10)
(11)
X if + Yif = Z if
The decision variables are: Xif, Yif, and Zif. All decision
variables are non-negative integer variables. The objective
is to minimize overall total cost which consists of total
transportation cost for the orders allocated to the normal
flight capacity, total transportation cost for orders allocated
to the special flight capacity, total delivery earliness
penalty cost and total delivery tardiness penalty cost.
Constraint (4) ensures that the quantity of the proportion of
order i allocated into flight f consists of quantities of the
proportion of order i allocated into normal capacity area of
flight f and the proportion of order i allocated to special
capacity area of flight f. Constraints (5) and (6) ensures that
if order i and flight f have different destination, order i can
not be allocated to flight f. Constraint (7) ensures that the
normal capacity of flight f is not exceeded. Constraint (8)
ensures that the special capacity of flight f is not exceeded.
Constraint (9) ensures that order i is completely allocated.
Constraint (10) ensures that allocated orders do not exceed
production capacity. It ensures that allocated quantity can
be supplied based on assembly capacity.
Fig 1. Equality of air transportation problem (order split allowed) to the
unbalanced transportation problem.
A. Complexity analysis of the transportation allocation
problem
In this section, we establish the equality of the above air
transportation allocation problem with the unbalanced
transportation problem. In general, a transportation
problem is specified by three elements: a set of supply
points, a set of demand points, cost of shipping each unit
from a supply point to a demand point. If demand larger
than supply, or supply larger than demand, it is called an
unbalanced transportation problem. For the above problem,
each order can be taken as a supply point and each flight’s
capacity can be taken as a demand point inline with the
transportation problem. It is noted that the normal capacity
and special capacity of each flight are considered as two
demand points at different transportation cost. The unit
transportation cost from a supply point A to a demand
point B is the sum of the unit transportation cost and the
unit delivery earliness (or tardiness) penalty cost of order A
when transported by flight B. As total quantity of all orders
is less than total capacity of all flights, it is an unbalanced
transportation problem with supply less than demand. The
equality establishment is also denoted in Figure 1.
V. PARALLEL MACHINE ASSEMBLY SCHEDULING PROBLEM
This section models the assembly scheduling problem
as a parallel machine scheduling problem, and studies the
synchronization of the order completions with the
departure times of the flights to multiple destinations. We
use the term ‘job’ to refer customer order in the following
sections. To address a more general problem, the machines
are assumed to be unrelated machines. For n independent
jobs (or orders) J={J1, J2,…, Jn}to be scheduled on the m
unrelated machines, there is a processing time pjm
associated with each pair of job and machine. The
assembly due-dates for the jobs are the departure time of
the flights. Hence we consider to schedule the jobs grouped
based on their assembly due-date. In this work we assume
that only one flight departs on a given assembly due-date.
The parallel station scheduling problem is decomposed into
F parallel machine scheduling problems (as we have F
due-dates) such that a production schedule is associated
with each assembly due-date. Sequence for jobs in each
group on m parallel machines on a particular due-date is
obtained by solving an unrelated parallel machine problem.
We assume sufficient accumulated assembly production
capacity, and the objective is to complete each order on or
before its assembly due date, and further, the lateness for
each order is not suppose to happen. Therefore, the
objective is to minimize total weighted earliness.
The common due-date parallel machine problem is
addressed in literature [13], [14] is solvable optimally
when earliness penalties are job independent. To achieve
synchronization, one more parameter of unavailable time
Ulf is introduced for each machine on each stage.
Sometimes the jobs scheduled for flight f may start even
before the departure time of the f-1 the flight, when the
total processing time of all jobs scheduled for the flight f is
greater than the production time available between
departure times of fights f and f-1. Hence some jobs
scheduled for fight f will occupy a certain amount of
processing time on each machine in the assembly time
available for fight f-1 as shown in Figure 2.
Let us denote the scheduling of nf jobs (belonging group
f) on m machines for fight f as PMf. Let the r1l and r2l+1 is
the release time of the orders to be processed first on
machine l and l+1 in job-group f, which are ahead of the
departure time of flight f-1 denoted by df-1. The jobs in
machine l will be processed from time r1l till df. The jobs in
group f-1 which is prior to group f cannot be processed
between times r1l and df-1 on machine l. Thus, (df-1 - r1l) is
the unavailable time on machine l in for (f-1)th job group.
Similarly, (df-1 – r2l) is the unavailable time on machine
l+1. The, unavailable time of the machines for each job
group is determined by the release times of first scheduled
job on each machine in next job group. This necessitates us
to solve starting from the last job group with has the set of
jobs to be transported by the last flight in the planning
period.
Machines
M1
Ml
Ml+1
Ml+2
r1l
2l+1
df-1
df TimeÆ
Fig. 2. Demonstration of machine unavailable time concept
A. Parallel machine scheduling with job independent
weights
The unrelated parallel machine scheduling problem for
the objective of minimizing mean absolute deviation from
a common due-date with job independent weights can be
formulated as a transportation problem [14]. The parallel
machine scheduling problem presented in this section
mainly has two differences from the above problem. It
minimizes total earliness without tardiness. In addition, a
parameter of machine unavailable time is introduced in this
problem. Kubiak et al. [14] formulation is extended in this
research by considering the two differences in formulating
the presented problem as a transportation problem. A
transportation problem is specified by a set of supply
points, a set of demand points, and the shipping costs from
each supply point to each demand point. Before giving the
formulation of the present scheduling problem, the
following notations are defined:
xilk is equal to 1 if job i is processed on machine l and
finishes on or before the due-date, and there are k-1 jobs
processed after it;
pil
the processing time of job i on machine l;
df
the common due date for sub-problem f, i.e.,
departure time of flight f;
the unavailable time on machine l in stage f;
the penalty cost for job earliness.
Ulf
w
On each machine, a job can be completed on the duedate or before it. The waiting cost could be taken as the
finished goods inventory cost before transportation. In this
case, penalty cost is assumed to be identical for all orders.
It is noted that there are nfm positions for each job and each
one could be assigned to among them m parallel machines
for each sub-problem f. The parallel machine scheduling
problem with job independent weights can be viewed as a
transportation problem by assuming nf jobs equivalent to nf
supply points. nfm possible assignment positions for each
job are equivalent to nfm demand points. For a job i, when
xilk =1, the waiting time of job i after its completion is
k
∑ pil , where k-1 jobs are scheduled on machine l
+ Max(d − U lf x, 0)]wxilk
(11)
∀ i=1, 2, …, nf
(12)
The algorithm for scheduling N jobs on m parallel
machines for F flights is presented below:
Step 1: Sequence and group the jobs based on duedates using Earliest Due-Date (EDD) rule.
Let f = F, where F is the number of groups of
jobs determined based of the number of
flights considered in the planning period.
Step 2: Sequence nf jobs in the group f on m parallel
machines by solving the problem as a
transportation model.
Step 3: Schedule the last job in the group f such that
Ci,f = df. The last jobs on m machines are
scheduled to complete at df.
Step 4: If df > r1l,f+1 then Cil,f = r1l,f+1.
Step 5: Set ril,f = ril,f+1 –pj+1. i=i-1.
Step 4: Schedule the next last job such that Cil,f = r i+1l,f
–pi+1.
Step 4.1: Go to step 4 unless all jobs in the group are
scheduled.
Step 5: If (f = = 0) go to next step else f=f-1 go to step
2.
Step 6: Calculate each job’s waiting time using the
information on jobs’ completion time and
transportation departure time.
(13)
The above algorithm generates optimal schedule by solving
F transportation problems.
(14)
VI. CONCLUSION
i =1
between its completion and the departure time of the
assigned flight. It can be taken as the shipping cost from
supply point i to demand point l. A job can only be
assigned to one position, and it can be viewed that each
supply point can only supply one demand point. Other the
other side, each position only needs one job to be assigned
to, which denotes that the demand for each demand point is
1. Since nf is less than nfm, the formulation is a
transportation problem with total supply less than total
demand. The 0-1 Linear programming formulation which
gives the sequence for the set of jobs associated with each
assembly due-date is presented below:
nf
Min
m
nf
∑∑∑[(k − 1) p
i =1 l =1 k =1
il
Subject to
m
nf
∑∑ x
l =1 k =1
m
∑x
i =1
ilk
ilk
=1
≤1
xilk ≥ 0
to be scheduled to finish on or before the due date. So there
should be no inserted idle time between each two jobs on
each machine to minimize total earliness. And each job’s
release time is determined by the start time of the next job
on the same machine. The last job in f-j-1 job-group is
assigned on machine l, after solving stage f-j, should be
finished either at the common due-date (departure time of
its flight), if the due-date is earlier than the release time of
the first job in job-group f-j. Otherwise, the last job is to be
finished at the release time of the first job in job-group f-j
for the corresponding machine.
l=1, 2, …, m, k=1, 2, …, nf.
The decision variable is xilk. The objective is to minimize
total earliness, or to say in other words, to minimize total
waiting time. In addition, if each machine has an
unavailable time Ulf, an additional waiting time of Ulf
should be included in waiting time of order i. It can be
viewed as additional fixed transportation cost for
transporting from supply points to the particular demand
point l. Constraint (12) ensures that each job is assigned to
a position on a machine. Constraint (13) ensures that each
position on each machine has at most one job assigned to.
The job assignment to each machine and the sequence
on each machine are determined by solving f transportation
problems for each job-group. The sequence for jobs on fth
group is obtained first followed by f-1th and so on. All jobs
in each job group share a common due-date and they have
In this paper we present an optimal methodology for
synchronized scheduling of production assembly with air
transportation to achieve accurate delivery with minimized
cost in consumer electronics supply chain (CESC). The
overall problem is decomposed into two sub-problems
which consist of an air transportation allocation problem
and an assembly scheduling problem. Both the air
transportation problem and parallel machine assembly
scheduling problem are found to have the structure of a
general transportation problem. Formulations and exact
algorithms are presented to achieve synchronization of one
with another to achieve deliveries with minimized delivery
cost.
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Viswanath Kumar Ganesan is a Research Fellow in Singapore MIT
Alliance, Nanyang Technological University Singapore. He received is
Ph. D and Masters degree from Indian Institute of Technology Madras,
India and PSG College of Technology Coimbatore, India respectively. His
research interests include combinatorial optimization and heuristic search
methods with application to production scheduling problems.
Kunpeng Li is a Ph.D. student in the Center for Supply Chain
Management, School of Mechanical and Production Engineering,
Nanyang Technological University, Singapore. He received his Bachelors
of Engineering in 2001 from Huazhong University of Science and
Technology, P. R. China. His research interests are in the areas of
synchronized scheduling of production and transportation, and heuristic
searching methods.
Appa Iyer Sivakumar is an Associate Professor in the School of
Mechanical and Production Engineering at Nanyang Technological
University, Singapore and a Fellow of Singapore MIT Alliance. Prior to
this he was at Gintic Institute of Manufacturing Technology Singapore.
His research interests are in the area of simulation based optimization of
manufacturing performance and dynamic scheduling. He has held various
management positions for nine years in Lucas Systems and Engineering
and Lucas Automotive, UK. He received a Bachelors of Engineering and a
Ph. D from University of Bradford UK.