INTRODUCTION
A supply chain consists of all parties involved, directly or indirectly,
in fulfilling a customer request. The supply chain includes suppliers,
manufacturers, distributors, warehouses, retailers and even customers
themselves. The key problem in a supply chain is a coordinated management
and control of these activities.
Traditional scheduling problems assume that there are always infinitely
many resources for delivering finished jobs to their destinations and
no time is needed for their transportation, so that finished products
can be transported to customer without delay. In accordance with this
view there is a need for a synchronized procedure for generating more
realistic production and distribution scheduling to be used in the supply
chain. In this study, supply chain is shown in fig. 1.
Within this chain, components are stored in inventory. On the receipt
of an order from the customer, components and materials required for production
are transferred to production line and then finished products are transferred
to customers using air transportation to meet their due dates. Synchronization
of production and air transportation is important, as the cost of missing
a shipment in a scheduled flight is quite heavy and therefore, the missed
shipment should be transported by special flights or commercial flights.

fig. 1: 
Supply chain stages synchronization 
Therefore, in this study, the extra cost corresponding to commercial flights that we need to bear is called departure
time tardiness. The departure time earliness costs could result from the
need for storing the order at the production facility or waiting charges
at the airport. Delivery penalties are incurred by delivering an order
either earlier or later than the committed due date to customers. The
delivery tardiness cost includes customer dissatisfaction, contract penalties,
loss of sales and potential loss of reputation for manufacturer and retailers.
If arrival time of allocated orders in air transportation model is earlier
than its due date, retailers encounter delivery earliness. Therefore,
delivery earliness cost considered as storing cost of orders by retailers.
We study the problem under two policies and they are as such: first policy
considers delivery tardiness and the second one assumes that no delivery
tardiness is authorized. The overall problem is decomposed into two coordinated
tasks in each policy. The first task is to allocate accepted orders to
available flights` capacities to minimize the total transportation cost
and delivery earliness tardiness penalties according to the related situation
and policy. The allocation is constrained by production such that allocation
should be balanced with production capacity in the same situation and
policy.
There seems to be little research on production scheduling considering
air transportation. Li et al. (2004) studied the synchronization
of single machine scheduling and air transportation with single destination.
The overall problem is decomposed into air transportation problem and
single machine scheduling problem. They formulated two problems and then
presented a backward heuristic algorithm for single machine scheduling.
Li et al. (2005) extended their previous work to consider multiple
destinations in air transportation problem. Li et al. (2006a) showed
the air transportation allocation have the structure of regular transportation
problem, while the single machine scheduling problem is NPhard. They
also proposed a forward heuristic and a backward heuristic for single
machine (Li et al., 2006b). Li et al. (2008) extended their
work by considering parallel machines in production. The problem was formulated
as a parallel machine with departure time earliness penalties. They also
showed the parallel machine scheduling problem is NPComplete and a simulated
annealing based heuristic algorithm was presented to solve the parallel
machine problem. They compared their simulated annealing algorithm with
an operation method of a factory in Singapore (Li et al., 2007).
There also have been some discussions on synchronization of production
and road transportation with emphasis on vehicle routing scheduling problem
(Blumenfeld et al., 1991; Fumero and Vercellis, 1999; Chen, 2000;
Lee and Chen, 2001; Chang and Lee, 2004; Chen and Vairaktarakis, 2005;
Li et al., 2005; Soukhal et al., 2005; Li and Ou, 2005;
Wang and Lee, 2005; Wang and Cheng, 2006; Zhong et al., 2007; Yuan
et al., 2007; Chen and Lee, 2008). In addition, considerable research
has been conducted in productiondistribution integration. There are reviews
on integrated analysis of productiondistribution systems for more details
see Vidal and Goetschalckx (1997), Erenguc et al. (1999), Sarmiento
and Nagi (1999) and Goetschalckx et al. (2002).
GENERAL ASSUMPTIONS
The problem is formulated based on the following assumptions:
• 
The plant treated as a single machine 
• 
Decisions of air transportation allocation and production scheduling
are for the orders accepted in the previous planning period 
• 
There are multiple flights in the planning period with different
transportation specifications such as cost, capacity, etc. 
• 
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 transfers products from the plant to the airport.
Local transportation time is assumed to be included in transportation
time 
• 
Local transportation can transfer an order to the airport when the
order is produced completely 
• 
Orders released into plant for the planning period are delivered
within the same planning period, which means there are no production
backlogs 
DELIVERY TARDINESS
The air transportation allocation problem: The air transportation model
allocates orders to the existing transportation capacities that minimizes the
total transportation cost and weighted delivery earliness tardiness penalties.
We first corrected and illustrated the model proposed by Li et al. (2006)
and then extended the model with considering due window and scheduling policies
which are completely explained below. Synchronization is incorporated into the
model by including the constraint that balances the production rate of the plant
with the flight allocation.
The notations used here are defined as follows:
i,i`,j 
: 
The order or job index, i, i`, j = 1,2,...,N 
f,f 
: 
The flight index, f, f` = 1,2,...,F 
k 
: 
The destination index, k = 1,2,...,K 
D_{F} 
: 
The departure time of flight f at the local airport 
A_{f} 
: 
The arrival time of flight f at the destination 
NC_{f} 
: 
The transportation cost for per unit product when allocated to normal
capacity area of flight f 
SC_{f} 
: 
The transportation cost for per unit product when allocated to special
capacity area of flight f 
Ncap_{f} 
: 
The available normal capacity of flight f 
Scap_{f} 
: 
The available special capacity of flight f 
Q_{i} 
: 
The quantity of order i 
α_{i} 
: 
The delivery earliness penalty cost (/unit/h) of order i 
β_{i} 
: 
The delivery tardiness penalty cost (/unit/h) of order i 
d_{i} 
: 
The due date of order i 
X_{if} 
: 
The quantity of the portion of order i allocated to flight f`s normal
capacity area 
Y_{if} 
: 
The quantity of the portion of order i allocated to flight f`s special
capacity area 
Des_{i} 
: 
The order i`s destination 
des_{f} 
: 
The flight f`s destination 
LN 
: 
A large positive number 
p_{i} 
: 
The processing time of order i 
e_{i},l_{i} 
: 
The due window of order i, where e_{i} is the earliest due
date and l_{i} is the latest due date 
Without loss of generality, it is assumed that D_{1}≤D_{2}≤...≤D_{F}.
The mathematical programming formulation of the model is shown as follow:
Subject to:
The objective is to minimize overall total cost which consists of total
transportation cost for the orders allocated to normal flight capacity,
total transportation cost for the orders allocated to special flight capacity,
total delivery earliness tardiness penalties cost. Constraint sets 2 and
3 ensure that if order i and flight f have different destinations, order
i cannot be allocated to flight f. Constraint sets 4 and 5 ensure that
the normal and special capacity of flight f is not exceeded. Constraint
set 6 ensures that order i is completely allocated. Constraint set 7 ensures
that allocated orders do not exceed production capacity. It ensures that
total orders related to allocated quantities can be produced by sufficient
production capacity.
We can also use constraint set 9 or constraint sets 10 and 11 or constraint
set 12 instead of constraint sets 2 and 3.
For the air transportation problem, each order can be taken as a supply
point and each flight`s capacity can be taken as a demand point. It is
noted that the normal capacity and special capacity of each flight are
considered as two demand point with different transportation costs.
Due window: Typically the customers accept small deviation from
delivery date, as they tolerate a small degree of uncertainty on the supplier`s
side. This uncertainty might come about as a result of production problems
such as defect in raw material, machine malfunctioning or problems with
delivery itself such as flight`s delay, traffic jam, etc. It is generally
agreed to accept small deviations from a delivery date and thus a delivery
window (or due window) is arranged as shown in fig. 2
(Biskup and Feldmann, 2005).
The earliness time of order i is equal to max (0, e_{i}A_{f})
and the tardiness time of order i is equal to max (0, A_{f}l_{i}).
Hence the objective function is transformed as follows:

fig. 2: 
The penalty function around e_{i} and l_{i} 
The models with due window are generalized case of models with delivery
date, because when both e_{i} and l_{i} be equal
to d_{i} the problem is transformed to models with delivery date.
An illustration: In order to validate and verify the proposed
models, a common small problem is solved by the Lingo 8 software in all
models. Consider a case of two orders (N = 2) with distinct destination
1 and 2 (Des_{1} = 1 and Des_{2} = 2) with quantities
30 and 40 (Q_{1} = 30 and Q_{2} = 40) such that each order
can be transported by two flights with different departure times (F =
4, des_{1} = 2, des_{2} = 1, des_{3} = 1 and des_{4}
= 2). The other parameters values for this example are as follows:
p_{1} = 4, p_{2} = 7, e_{1} = 12, l_{1}
= 14, e_{2} = 15, l_{2} = 17, α_{1}
= 4, α_{2} = 3, β_{1} = 7, β_{2} = 5, Des_{1} = 1,
Des_{2} = 2, des_{1} = 2, des_{2} = 1, des_{3}
= 1, des_{4} = 2, D_{1} = 8, D_{2} = 11, D_{3}
= 16, D_{4} = 18, A_{1} = 9, A_{2} = 13, A_{3}
= 17, A_{4} = 20, NCap_{1} = 20, NCap_{2} = 20, NCap_{3} = 20, NCap_{4} = 25, SCap_{1} = 20,
SCap_{2} = 20, SCap_{3} = 10, SCap_{4} = 15, NC_{1}
= 20, NC_{2} = 30, NC_{3} = 15, NC_{4} = 10, SC_{1} = 30, SC_{2} = 50, SC_{3}
= 20, SC_{4} = 15.
The results obtained from solving this example are as follows:
X_{11} = 0, X_{12} = 20, X_{13} = 10, X_{14}
= 0, X_{21} = 0, X_{22} = 0, X_{23} = 0, X_{24}
= 25, Y_{11} = 0, Y_{12} = 0, Y_{13} = 0, Y_{14}
= 0, Y_{21} = 0, Y_{22} = 0, Y_{23} = 0, Y_{24}
= 15.
The production scheduling problem: The next task of the solution
process is to determine the sequence and completion time for the allocated
orders in production. This requires solving a production scheduling problem
to ensure that allocated orders catch their flights so that total departure
time earliness cost and plant cost is minimized. Transportation allocation
results are the inputs for the production problem which include the order`s
quantities allocated to flights. The required notation to present the
model is as follows:
c_{i} 
: 
The completion time of order or job i 
α`_{i} 
: 
The per hour earliness penalty of order or job i for production 
p 
: 
The position or sequence of order i p = 1,2,...,N 
u_{ip} 
: 
1 if order i be in position p, 0 otherwise 
λ 
: 
The per hour plant costs (including machine cost, operator wages
and other production variable costs which is completely related to
the length of working hours) 
I_{i} 
: 
The idle time before order i in the schedule 
_{Cmax}: 
: 
The maximum completion time of orders that is equal to shut down
time of shop 
Subject to:
The decision variables are c_{i}, I_{i}, u_{ip}
and C_{max}. The objective function is to minimize the total weighted
earliness penalties of jobs and plant cost. Constraint sets 15 and 16
state that each job has to be assigned to a position and each position
has to be covered by a job. Constraint set 17 calculates completion time
of jobs, considering inserted idle times among jobs. Constraint set 18
ensures that order i catches all of its departure times or the completion
time of order is less than or equal to minimum of its related departure
times. It means that all jobs must catch their all related scheduled flights.
Constraint set 19 calculates C_{max} and can be replaced by the
constraint set 22.
The total cost of overall problem is the sum of objective function of
air transportation and production scheduling models. Thus the total cost
is as follows:
An illustration: The other required parameters are as follows:
The solutions are as follows:
NO DELIVERY TARDINESS
The air transportation allocation problem: Since no tardiness
is authorized, the objective function does not include the delivery tardiness
costs and minimizes the total transportation costs and weighted delivery
earliness penalties. Therefore, constraint set 25 ensures that the arrival
time of all flights allocated to the order i is less than or equal to
its delivery due date. The problem under study can be formulated as follows:
Subject to:
The other constraints of the model are the same as constraint sets 48
and 12.
Due window: The objective function 24 and constraint set 25 are
changed as follows:
Subject to:
An illustration: The solutions are as follows:
X_{11} = 0, X_{12} = 20, X_{13} = 0, X_{14}
= 0, X_{21} = 20, X_{22} = 0, X_{23} = 0, X_{24}
= 0, Y_{11} = 0, Y_{12} = 10, Y_{13} = 0, Y_{14}
= 0, Y_{21} = 20, Y_{22} = 0, Y_{23} = 0, Y_{24}
= 0.
The production scheduling problem: Similar to the previous presented
model for production scheduling the objective function is to minimize
the weighted earliness penalties and plant cost and all jobs must catch
their scheduled flights. So the objective function and constraints of
the model are the same as that model. Total cost of the overall problem
is the sum of objective function of two subproblem of this section and
is as follows:
An illustration: The solutions are as follows:
CONCLUSION
In this research, we studied supply chain synchronization problem. We
have presented mathematical models with considering due window and scheduling
policies. Numerical examples were performed to validate and verify the
proposed models. Since there are a few researches about this subject,
many researches can develop this paper. Further research can be conducted
to consider other production configuration such as, parallel machine,
flow shop, job shop, etc. Meta heuristics can also be applied to solve
the proposed models. Future research can also be conducted by all assumption
that studied in production scheduling and transportation scheduling research
such as, set up time, ready time, stochastic processing time, nonsplit
in transportation allocation, etc.