Coordination of production and transportation in supply chain scheduling

Article (PDF Available)inJournal of Industrial and Management Optimization 11(2):399-419 · April 2015with 151 Reads
DOI: 10.3934/jimo.2015.11.399
Abstract
This paper investigates a three-stage supply chain scheduling problem in the application area of aluminium production. Particularly, the first and the third stages involve two factories, i.e., the extrusion factory of the supplier and the aging factory of the manufacturer, where serial batching machine and parallel batching machine respectively process jobs in different ways. In the second stage, a single vehicle transports jobs between the two factories. In our research, both setup time and capacity constraints are explicitly considered. For the problem of minimizing the makespan, we formalize it as a mixed integer programming model and prove it to be strongly NP-hard. Considering the computational complexity, we develop two heuristic algorithms applied in two different cases of this problem. Accordingly, two lower bounds are derived, based on which the worst case performance is analyzed. Finally, different scales of random instances are generated to test the performance of the proposed algorithms. The computational results show the effectiveness of the proposed algorithms, especially for large-scale instances.
JOURNAL OF INDUSTRIAL AND doi:10.3934/jimo.2015.11.399
MANAGEMENT OPTIMIZATION
Volume 11, Number 2, April 2015 pp. 399–419
COORDINATION OF PRODUCTION AND TRANSPORTATION
IN SUPPLY CHAIN SCHEDULING
Jun Pei,1,2, Panos M. Pardalos2,3, Xinbao Liu1,4
Wenjuan Fan1,5and Shanlin Yang1,4
1School of Management, Hefei University of Technology
Hefei 230009, China
2Center for Applied Optimization, Department of Industrial and Systems Engineering
University of Florida, Gainesville, FL 32611, USA
3Laboratory of Algorithms and Technologies for Networks Analysis, National Research
University Higher School of Economics, Niznhy Novgorod 603093, Russia
4Key Laboratory of Process Optimization
and Intelligent Decision-making of Ministry of Education
Hefei 230009, China
5Department of Computer Science, North Carolina State University
Raleigh 27695, USA
Ling Wang
Shanghai Key Laboratory of Power Station Automation Technology
School of Mechatronics and Automation
Shanghai University, Shanghai 200444, China
(Communicated by Eugene Levner)
Abstract. This paper investigates a three-stage supply chain scheduling prob-
lem in the application area of aluminium production. Particularly, the first and
the third stages involve two factories, i.e., the extrusion factory of the supplier
and the aging factory of the manufacturer, where serial batching machine and
parallel batching machine respectively process jobs in different ways. In the
second stage, a single vehicle transports jobs between the two factories. In our
research, both setup time and capacity constraints are explicitly considered.
For the problem of minimizing the makespan, we formalize it as a mixed in-
teger programming model and prove it to be strongly NP-hard. Considering
the computational complexity, we develop two heuristic algorithms applied in
two different cases of this problem. Accordingly, two lower bounds are derived,
based on which the worst case performance is analyzed. Finally, different scales
of random instances are generated to test the performance of the proposed al-
gorithms. The computational results show the effectiveness of the proposed
algorithms, especially for large-scale instances.
2010 Mathematics Subject Classification. Primary: 90B35, 90C27; Secondary: 90C59.
Key words and phrases. Supply chain scheduling, batching, transportation, heuristic algorithm.
This work is supported by the National Natural Science Foundation of China (Nos. 71231004,
71171071, 71131002). Prof. Panos M. Pardalos is partially supported by LATNA laboratory, NRU
HSE, RF government grant, ag. 11.G34.31.0057.
Corresponding author. Email: peijun@ufl.edu.
399
400 J. PEI, P. M. PARDALOS, X. LIU, W. FAN, S. YANG AND L. WANG
1. Introduction. In recent years, effective supply chain management has become
of greater and greater importance with the integration of global cooperation. A
supply chain includes all stages that have added value to a product, and all inter-
actions among suppliers, manufacturers, distributors, and customers [9]. Most of
the literature about supply chain focuses on stochastic models to analyze inventory
control issues at a strategic level, and there is a small amount of related research
combining scheduling problems with supply chains until Hall and Potts’s paper [15].
In their work, Hall and Potts considered a variety of scheduling, batching, and de-
livery problems that arose in an arborescent supply chain, where a supplier made
deliveries to several manufacturers and a manufacturer also made deliveries to cus-
tomers. They also derived efficient dynamic programming algorithms for a variety
of problems, and identified incentives and mechanisms for cooperation. Afterwards,
the problem of supply chain scheduling has attracted more attention of researchers.
In this paper, we study a three-stage supply scheduling problem, which arises
from the real aluminium production supply chain. Based on the orders of a kind
of special industrial aluminum profiles, the raw materials of aluminium ingots need
to be processed through extrusion and aging by the factories located in this supply
chain. We conclude the whole process as three stages, 1) in the upstream of the
supply chain, the supplier (extrusion factory) produces the jobs of the orders from
the manufacturer, 2) in the midstream, the vehicle carries the jobs from the sup-
plier to the manufacturer after their processing is completed in the supplier, and
3) in the downstream, the manufacturer (aging factory) produces the jobs trans-
ported from the supplier. Based on shared economic interests, the supply chain
participants try to cooperate more efficiently to increase the overall productivity by
making better cooperative production decisions together. This is the motivation to
conduct the supply chain scheduling problem on the integration of production and
transportation considered in this paper.
In the first study of supply chain scheduling by Hall and Potts [15], they de-
scribed the concept of supply chain scheduling as follows, “Whereas much of the
supply chain management literature focuses on inventory control or lot-sizing issues,
this paper considers a number of issues that are important to scheduling in supply
chains. Central to this literature is the idea of coordination between different parts
of a supply chain. Where decision makers at different stages of a supply chain make
decisions that are poorly coordinated, substantial inefficiencies can result. This pa-
per considers the coordination of scheduling, batching, and delivery decisions, both
at a single stage and between different stages of a supply chain, to eliminate those
inefficiencies. The ob jective is to minimize the overall scheduling and delivery cost.
This is achieved by forming batches of orders, each of which is delivered from a
supplier to a manufacturer or from a manufacturer to a customer, in a single ship-
ment.” Since this problem is frequently encountered in many real-life supply chains,
it has attracted the attention of many researchers over the last ten years. Basically,
there are mainly three types of algorithms for solving supply chain scheduling prob-
lems: exact algorithms, heuristic algorithms, and metaheuristic approaches. Here
we conduct a brief review of them, respectively.
Some exact algorithms, which are mainly dynamic programming and branch and
bound algorithms, were used to solve supply chain scheduling problems. How op-
timization techniques were applied to scheduling problems, in particular dynamic
programming and branch and bound, was described in [28]. A number of remarks
with regard to areas of applications were also presented. Gordon and Strusevich [12]
COORDINATION OF PRODUCTION AND TRANSPORTATION 401
studied the supply chain scheduling problem derived between the manufacturer and
customer. In their work, the job processing time depended on its position in the
processing sequence, and the objective was to minimize the sum of the cost of chang-
ing the due dates and the total cost of discarded jobs. They developed dynamic
programming algorithms with CON (constant flow allowance) and SLK (slack time)
due date assignment methods. Grunder [14] addressed a production and transporta-
tion problem in a single-stage supply chain. The goal was to minimize the sum of
production, transportation, and holding costs. They proposed the algorithm to
solve the problem based on the efficient dynamic programming scheme. Cheng and
Wang [10] considered the machine scheduling problems where the jobs of different
classes needed to be processed and then delivered to customers. When the first
job was processed on a machine or the following job belonged to another class, a
setup time was required. Their goal was to minimize the weighted sum of the last
arrive time of jobs and transportation cost. They proposed a dynamic program-
ming algorithm to solve it optimally. Yeung et al. [35] studied a two-echelon supply
chain scheduling problem, where the upstream suppliers first processed materials
and delivered the semi-finished jobs to the manufacturer due to its time window and
then the manufacturer processed these semi-finished jobs and delivered the finished
jobs to the downstream retailers within its own production and delivery time win-
dows. Some dominance properties were derived, and based on these properties fast
pseudo-polynomial dynamic programming algorithms were developed to solve the
problem optimally. Chretienne et al. [11] considered the integrated batch sizing and
scheduling problem in a supply chain. Each customer order could be only handled
by a single machine at a time. Their goal was to minimize the sum cost of the
tardiness and the setup. They proposed dynamic programming algorithms based
on some structural properties for the problem.
In addition to the mentioned dynamic programming algorithms, there were some
studies which developed branch and bound algorithms for the proposed problem.
One of them could refer to Mazdeh et al. [22] which studied the scheduling prob-
lem where the jobs were processed on a single machine and then transported to
customers in batches for further processing. Their goal was to minimize the sum
cost of flow time and delivery. They used a branch and bound algorithm to solve it
and proved the proposed algorithm to be far more efficient than the only existing
algorithm for this problem. Bard and Nananukul [4] investigated the scheduling
problem including a single production facility, a set of customers, and a fleet of
vehicles. Their goal was to minimize the total cost of production, inventory, and
delivery. An approach combining heuristic algorithms and the branch-and-price
algorithm was proposed, where a novel column generation heuristic algorithm and
a rounding heuristic algorithm were developed to improve the algorithm. Rasti-
Barzoki et.al [31] studied the integrated production and delivery scheduling problem
in a supply chain, where a manufacturer first processed the orders from a customer
on one or two machines and then the finished jobs were delivered to the customer.
Their objective was to minimize the sum cost of the total weighted number of tardy
jobs and the delivery costs. They designed a new branch and bound algorithm
for solving both the single machine scheduling problem and the two-machine flow-
shop problem. Rasti-Barzoki and Hejazi [30] proposed a supply chain scheduling
problem with the due date assignment and the capacity-constrained deliveries for
multiple customers, and the objective was to minimize the weighted number of
tardy jobs in the single machine environment. The problem was formulated as an
402 J. PEI, P. M. PARDALOS, X. LIU, W. FAN, S. YANG AND L. WANG
integer programming model, and a branch and bound algorithm was presented for
solving it. Chang et al. [8] considered a supply chain scheduling problem integrat-
ing production and distribution, where the production and distribution stages are
modelled as the parallel machines scheduling problem and the capacitated vehicle
routing problem, respectively. They proposed column generation techniques in con-
junction with a branch and bound approach to solve this problem. Despite the
fact that these exact algorithms were proposed to solve various small-scale supply
chain scheduling problems, they cannot obtain solutions for large-scale scheduling
problems effectively.
Heuristic algorithms were widely used in supply chain scheduling problems.
Chang and Lee [9] considered the machine scheduling and finishing product de-
livery, where jobs were processed on a single machine and delivered by a single
vehicle to one custom area. They provided a proof of NP-hardness and a heuris-
tic algorithm with worst-case analysis. The worst-case performance ratio for their
heuristic algorithm was proven to be 5/3, and the bound was tight. Jetlund and
Karimi [17] discussed the maximum-profit scheduling of the logistic chemicals man-
ufacturing and delivery. They proposed a mixed-integer linear programming formu-
lation and a heuristic decomposition algorithm. Their approach was illustrated on a
real industrial case study and shown an increase of 37.2% in profit compared to the
original plan. Selvarajah and Steiner [32] studied the batch scheduling problem in
a two-level supply chain from the view of the supplier. In this problem, the supplier
produced multiple products and delivered them in batches, and the objective was
to minimize the sum of the total inventory holding cost and batch delivery cost of
the supplier. They proposed an algorithm which exhibited polynomial complexity
time to solve it. Agnetis et al. [1] considered the scheduling problem of finding an
optimal supplier’s schedule, an optimal manufacturer’s schedule, and optimal sched-
ules for both in a two-stage supply chain. They developed a polynomial algorithm
to minimize total interchange cost and buffer storage cost. Averbakh and Xue [3]
examined the on-line supply chain scheduling problem with preemption, where a
manufacturer processed jobs and delivered them to the customers. The jobs were
processed in batches and delivered to the customers as single shipments. Their ob-
jective was to minimize the sum cost of the total flow time and total delivery. They
presented an on-line two-competitive algorithm for the single customer and consid-
ered an extension of the algorithm for the case of multiple customers. Lin et al. [20]
studied a two-stage supply chain scheduling problem, including two machines in
the first stage and one machine in the second stage. Before each batch was formed
on the second-stage machine, a constant setup time was required. They proposed
several heuristic algorithms to minimize the makespan. Lee et al. [19] addressed a
scheduling problem with multi-attribute setup times, the objective of which was to
minimize the total setup time on a single machine. They developed a constructive
heuristic algorithm based on several theorems to solve the problem. Although many
heuristic algorithms were applied in the various supply chain scheduling problems,
there are no particular heuristic algorithms for our studied problem in this paper.
In recent years, the technological advancements in computer science have en-
couraged metaheuristic approaches such as genetic algorithm (GA) to be applied
in supply chain scheduling problems. Chan et al. [7] studied distributed schedul-
ing problems in a multi-factory and multi-product environment. They proposed an
adaptive genetic algorithm, using a new crossover mechanism named as dominated
gene crossover. They also carried out a number of experiments, in which significant
COORDINATION OF PRODUCTION AND TRANSPORTATION 403
improvement was obtained by the proposed algorithm. Torabi et al. [33] considered
a supply chain scheduling problem where a single supplier produced components
and delivered them to an assembly facility. They developed a mixed integer non-
linear program and proposed a hybrid genetic algorithm to solve it. Naso et al. [27]
focused on the supply chain scheduling problem about the distribution of ready-
mixed concrete industry in a real-world case. After developing a detailed model for
the problem, they proposed a hybrid genetic algorithm combined with constructive
heuristic algorithms to solve it. Zegordi et al. [37] investigated a two-stage sup-
ply chain scheduling problem. In the first stage, there were msuppliers producing
jobs at different speeds. In the second stage, each of lvehicles had a different
speed and different transport capacity. They developed a mixed integer program-
ming model and proposed a gendered genetic algorithm to solve it. Yimer and
Demirli [36] studied a build-to-order (BTO) supply chain scheduling problem for
material procurement, components fabrication, product assembly, and distribution.
They decomposed the problem into two subsystems and evaluated them sequen-
tially. They also proposed a genetic algorithm based solution procedure to solve it.
Cakici et al. [6] focused on a multi-objective supply chain scheduling problem which
integrated production and distribution, and the objective was to minimize the total
weighted tardiness and total distribution costs. They developed different heuristic
algorithms based on a genetic algorithm for a Pareto-optimal set of solutions.
Some other metaheuristic approaches were also applied in supply chain scheduling
problems. Moon et al. [24] considered the integration of planning and scheduling
in a supply chain, which took into account sequence and precedence constraints.
They developed a new evolutionary search approach based on a topological sort to
minimize the makespan within a reasonable computing time. Pardalos et al. [29]
proposed a new metaheuristic for the job shop scheduling problem. They used the
backbone and “big valley” properties of the job shop scheduling problem and ob-
tained new upper bounds for many problems. Zhang et al. [38] discussed the proce-
dures of order planning in details and constructed a nonlinear integer programming
model for the order planning problem. They designed a hybrid Particle Swarm Op-
timization (PSO) and Tabu Search (TS) algorithm, in which new heuristic rules to
repair infeasible solutions were proposed. Mehravaran and Logendran [23] focused
on a non-permutation flowshop scheduling problem with sequence-dependent setup
times in a supply chain. Their goal was to minimize the inventory for the producer
and maximize the service level for the customers. They developed a metasearch
heuristic algorithm employing a new concept known as TS to solve it. Liu and
Chen [21] considered the inventory, routing, and scheduling problem in a supply
chain. After developing an integrated model, they proposed a metaheuristic of
variable neighborhood search algorithm for the problem.
Another important research perspective, flowshop scheduling problems have been
well studied during the last 60 years. Intuitively, the models of flowshop schedul-
ing and the supply chain scheduling problem considered in this paper have some
similarities. However, there are still some significant differences between them. In
traditional flowshop scheduling problems, it is a common assumption that there is
no transportation time between machines. In other words, the jobs completed on
the previous machine can be processed on the new machine immediately. However,
it might be unrealistic in the supply chains of the real-world industries. In recent
years, there were several papers concerning flowshop scheduling problems consid-
ering the transportation. Hurink and Knust [16] addressed a flowshop scheduling
404 J. PEI, P. M. PARDALOS, X. LIU, W. FAN, S. YANG AND L. WANG
problem, where if a job was transferred from one machine to another, then the
job was delivered by a single robot and the transportation time was job-dependent
or job-independent. Their main contribution is that they derived new complexity
results for special cases with constant processing or transportation times. Allaoui
and Artiba [2] investigated a hybrid flowshop scheduling problem subject to the
machine maintenance, where setup, cleaning, and transportation times were con-
sidered simultaneously. They developed the approach by using three dispatching
rules, the simulated annealing heuristic, and a flexible simulation model. Naderi
et al. [26] pointed out that in the literature there was almost no work considering
the features of sequence-dependent setup and transportation times simultaneously
in hybrid flowshop scheduling problems, and the related studies with the minimiza-
tion objectives based on due date were far less. Then, they investigated the hybrid
scheduling problem with the above two features, and their objective was to minimize
both total completion time and total tardiness. A simulated annealing was devel-
oped to solve this problem, where the optimum parameters with the experiments’
least possible number were selected by using Taguchi method. Naderi et al. [25] in-
vestigated the permutation flowshop problem, where the transportation time of jobs
from one machine to the next machine was considered. Both multi-transporter and
single-transporter systems were considered, and six different mixed integer linear
programs were formulated for them. Some well-known heuristics were provided, and
the metaheuristics were developed based on artificial immune systems, combining
an effective local search heuristic and simulated annealing. Khalili and Tavakoli-
Moghadam [18] studied a bi-objective flowshop scheduling problem which was to
minimize the makespan and total weighted tardiness, where jobs’ transportation
times between machines were considered. Based on the attraction-repulsion mech-
anism of electromagnetic theories, a new multi-objective electromagnetism algo-
rithm was developed to solve the problem. Although there was some research work
studying on flowshop scheduling problems with transportation times, specific job
processing patterns were seldom considered in previous research, and most of them
considered the situation that a single machine processes one job at one time. In our
considered supply chain scheduling problem, two specific job processing patterns
are considered in different factories which are located in the same supply chain.
In summary, there are more and more algorithms proposed to solve supply chain
scheduling problems. Most literature focused on the problem of coordination of
production and transportation among the participants in supply chain. However,
few authors considered the concrete production modes and constraints in the real
factories, which is inadequate when confronted with practical situations. Also,
most previous approaches cannot be directly applied in the large-scale scheduling
problem, while this is particularly the reality in nowadays supply chain schedul-
ing problems. Thus, this is our motivation to conduct such research. The main
contributions of this paper can be summarized as follows:
(1) New research is pursued by considering the features of setup times and two
typical processing ways of serial batching and parallel batching in a supply chain
scheduling problem. To the best of our knowledge, this paper is the first time
to consider the practical modes of production both in extrusion factory and aging
factory and some particular production constraints simultaneously in a supply chain
scheduling problem. The studied problem is also proved to be strongly NP-hard.
(2) The proposed problem is agilely analyzed in two different cases based on the
relationship between the round-trip transportation time and the job processing time
COORDINATION OF PRODUCTION AND TRANSPORTATION 405
in the manufacturer. Two novel heuristic algorithms are developed for the prob-
lem, which can obtain high quality solutions for large-scale supply chain scheduling
problems within a reasonable time.
(3) We derive the lower bounds of two different cases, based on which the worst
case performance of two proposed heuristic algorithms is also analyzed.
The reminder of this paper is organized as follows: we start section 2 with the
problem description. A mathematical model of the studied supply chain scheduling
problem is proposed and the problem is also proved to be strongly NP-hard in
section 3. In section 4 two heuristic algorithms are designed to solve the problem,
and afterwards the performance of the heuristic algorithms is also analyzed. The
computational experiments are conducted and discussed in section 5. We conclude
the paper with a summary and give future research directions in section 6.
2. Problem description. The practical production system is complicated in the
aluminium product supply chain, which is consisted of multiple participating en-
terprises. In this paper, we focus on the supply chain scheduling problem from the
supplier to the manufacturer, i.e., extrusion factory and aging factory. Due to the
large-scale feature of the problem, it is difficult for these enterprises to make an
effective schedule for the whole supply chain. Until now, few enterprises can make
the production plan considering upstream and downstream enterprises, and most
enterprises even still rely on human experience to make manufacturing schedules.
The poor performance on schedules reduces the competitiveness of the supply chain.
Therefore, an effective solution for the scheduling problem is imperative.
Supplier(extrusion factory) Transportatio n Manufacturer(aging factory)
Figure 1. The structure of the supply chain scheduling between
the supplier and the manufacturer
The structure of the studied scheduling problem is shown in Figure 1. A set
{J1, J2, ..., Jn}of nindependent and non-preemptive jobs is available to be processed
at time zero in the supplier. All jobs are processed in batches, and once a batch is
initiated, no jobs in the batch can be released until the whole batch is completely
processed. The machines and vehicle have the identical capacity of size, which is
denoted as c, i.e., each machine and vehicle can handle no more than cjobs in a
batch simultaneously. Our problem can be divided into 3 stages, i.e., the production
stage on the supplier’s machine, the transportation stage from the supplier to the
manufacturer, and the production stage on the manufacturer’s machine, which is
described as follows:
(1) In the first stage, jobs are processed on the supplier’s serial batching machine.
The setup time is required before each batch is processed, which is denoted as s.
Let piand Pkdenote the processing time of job Jiand batch bkon the supplier’s
machine, respectively. In the serial batch production, jobs are processed one after
406 J. PEI, P. M. PARDALOS, X. LIU, W. FAN, S. YANG AND L. WANG
another [34], so the processing time of batch bkis Pk=PJibkpi(i= 1,2, ..., n).
There is no buffer for the processed batches, therefore the setup for a new batch
cannot be started on the serial batching machine until the previous batch on the
machine has been transported.
(2) In the second stage, all batches which have been processed are transported
by only one vehicle. A round-trip transportation time of the vehicle is assumed to
be a constant T, and one-way time is T/2. The vehicle can carry any one batch
from the supplier in one-time shipment. We assume Tpi+s(i= 1,2, ..., n) for
emphasizing the importance of transportation.
(3) In the third stage, jobs are processed on the manufacturer’s parallel batching
machine, where all jobs in a batch are processed simultaneously [34]. The process-
ing time of a batch on the parallel batching machine is determined by the batch
processing time independent of jobs, which is supposed to be a constant P. It is also
assumed that Ppi+s(i= 1,2, ..., n) according to actual production condition.
Our objective is to minimize the makespan. Using the standard three-field notion
α|β|γintroduced by Graham et al. [13], the problem can be denoted as SM, 1|b=
c, tk=T|Cmax. In this notation, the symbols Sand Mstand for the supplier and
the manufacturer respectively, and the symbol “1” represents that the number of
machines in both the supplier and the manufacturer is one. The conditions b=cand
tk=Tindicate that the capacity of the batching machines and vehicle is c, and the
round-trip transportation time of each batch between supplier and manufacturer
is equal to T, respectively. The symbol Cmax denotes that the objective of the
scheduling problem is to minimize the completion time of the last job.
setup
transportation
Supplier
Manufacturer
Schedule
b1={J3,J4}
b2={J1,J2}
1 5 6 9 10 14 19
Figure 2. An example of the scheduling problem SM , 1|b=
c, tk=T|Cmax
The following assumptions are considered for the problem formulation:
All the facilities (i.e., machines and vehicle) are available at time zero in the
usage time.
The setup time on the supplier’s machine is independent of the jobs sequence
and batching.
Pre-emption is prohibited, i.e., once a batch is initiated, no jobs can be released
from the batch until the whole batch is completely processed.
To illustrate this problem, we give an example in Figure 2, where a set of four
jobs with parameters c= 2, s= 1, T= 8, P= 5, p1= 2, p2= 3, and p4= 1
is considered. Figure 2 shows that the schedule πcontains two batches, which are
b1={J3, J4}and b2={J1, J2}. The completion times for b1in the supplier and
the manufacturer are 5 and 14, and for b2in the supplier and the manufacturer are
10 and 19, respectively. Then, the makespan is 19.
COORDINATION OF PRODUCTION AND TRANSPORTATION 407
3. Model formulation and complexity analysis.
3.1. Model formalization. In this section, a mixed integer programming model
is formulated. The parameters and variables are described and the model is given
below.
Parameters
n: total number of jobs;
i: job index, i= 1,2, ..., n;
l: total number of batches, dn
ce ≤ ln;
k, f : batch index, k, f = 1,2, ..., l;
c: the capacity of the batching machines and corresponding vehicle;
pi: processing time of job Jion the supplier’s machine;
Pk: processing time of batch bkon the supplier’s machine;
P: processing time of each batch on the manufacturer’s machine;
s: setup time on the supplier’s machine;
T: round-trip transportation time of the vehicle between the supplier and
the manufacturer;
M: a large enough positive constant.
Decision variables
xik : 1,if job Jiis in batch bk; 0, otherwise;
ykf : 1,if batch bkis processed before batch bfon the supplier’s machine; 0,
otherwise;
S1k: start time of batch bkon the supplier’s machine;
C1k: completion time of batch bkon the supplier’s machine;
Dk: departure time of batch bkfrom the supplier to the manufacturer;
S2k: start time of batch bkon the manufacturer’s machine;
C2k: completion time of batch bkon the manufacturer’s machine;
Cmax : the makespan;
Mixed integer programming model
Minimize Cmax (1)
Subject to
l
X
k=1
xik = 1, i = 1,2, ..., n (2)
n
X
i=1
xik c, k = 1,2, ..., l (3)
l
X
k=1
n
X
i=1
xik =n(4)
C1k=S1k+X
Jibk
pi, k = 1,2, ..., l (5)
408 J. PEI, P. M. PARDALOS, X. LIU, W. FAN, S. YANG AND L. WANG
DkC1k, k = 1,2, ..., l (6)
S1(k+1) =Dk+s, k = 1,2, ..., l 1 (7)
S2k=Dk+T
2, k = 1,2, ..., l (8)
C2k=S2k+P, k = 1,2, ..., l (9)
C1kC1f+s+Pf(1 ykf )M0, k, f = 1,2, ..., l, k 6=f(10)
C2kC2f+Pf(1 ykf )M0, k, f = 1,2, ..., l, k 6=f(11)
Cmax C2k, k = 1,2, ..., l (12)
xik 0,1,i, k (13)
xkf 0,1,k, f (14)
The objective function (1) is to minimize the makespan. Constraint (2) ensures
that any job should be contained in only one batch. Constraint (3) guarantees that
the number of jobs in a batch cannot exceed the capacity of the batching machines
and corresponding vehicle. Constraint (4) indicates that the total number of jobs in
all batches should be equal to the total number of all jobs. By setting constraint (5),
the completion time of each batch on the supplier’s machine is defined. Constraint
(6) specifies that any batch can be only transported from the supplier to the man-
ufacturer after it is completely processed on the supplier’s machine. Constraint (7)
restricts that a constant setup time is required before each batch is processed on the
supplier’s machine, and the setup of a new batch cannot be started on the supplier’s
machine until the previously processed batch has been transported. Constraint (8)
ensures that any batch cannot start to be processed on the manufacturer’s machine
until it reaches the manufacturer. Constraint (9) indicates the completion time of
each job on the manufacturer’s machine. Constraints (10) and (11) guarantee that
there is no overlapping situation between any two batches on the supplier’s and
manufacturer’s machines, respectively. Constraint (12) defines the property of the
maximum completion time. Finally, the ranges of the decision variables are defined
by constraints (13) and (14).
3.2. Complexity analysis. In the following content, we present the strongly NP-
hard proof for the problem SM, 1|b=c, tk=T|Cmax.
Theorem 1. The Problem SM, 1|b=c, tk=T|Cmax is strongly NP-hard.
Proof. We construct an instance to perform the reduction with respect to the fol-
lowing 3-PARTITION problem, which is known to be strongly NP-hard [37].
3-PARTITION: Given an integer Mand a set Aof 3hpositive integers {x1, x2, ...,
x3h},M
4< xi<M
2,1i3h, such as P3h
i=1 xi=hM, does there exist a partition
A1,A2,...,Ahof the set Asuch that |Al|= 3 and PxiAlxi=M, 1lh?
We construct the following instance of problem SM, 1|b=c, tk=T|Cmax .
Number of jobs: n= 3h+ 3.
Capacity of the batching machines and vehicle: c= 3.
Setup time: s=M.
Processing time of the jobs on the extrusion factory’s serial batching machine:
pi=xi,i= 1,2, ..., 3h.pi= 0, i= 3h+ 1,3h+ 2,3h+ 3.
Processing time of each job on the aging factory’s parallel batching machine:
P= 2M.
Round-trip transportation time: T= 2M.
COORDINATION OF PRODUCTION AND TRANSPORTATION 409
Threshold value: y= (2h+ 4)M.
We claim that there is a solution to 3-PARTITION problem if and only if there
exists an optimal schedule for the instance of problem SM, 1|b=c, tk=T|Cmax
with makespan no greater than (2h+4)M.
() We can construct A1, A2, . . . , Ahto be a partition for the set Ain this 3-
PARTITION problem, and the schedule is shown in Figure 3. In the constructed
schedule, the first batch A1contains three jobs {J3h+1, J3h+2, J3h+3 }and departs
from the supplier at the time M. It is easy to derive that the makespan is (2h+
4)M=y.
Conversely, suppose that there exists an optimal schedule with makespan no
greater than yfor the problem SM, 1|b=c, tk=T|Cmax . The minimum number
of possible batches is hdue to the capacity of the batching machines and vehicle.
Thus, the minimum sum of the setup time and processing time of all batches on
the extrusion factory’s serial batching machine is (h+ 1)M+hM. The sum of the
processing time of one batch on the aging factory’s parallel batching machine and
its one-way transportation time is 2M+T
2. We have 2M+T
2=y[(h+1)M+hm].
Besides, the earliest possible departure time of the first batch from the supplier is
M, the minimum sum of transportation time is (2h+ 1)M, and the processing time
of one batch on the aging factory’s parallel batching machine is 2M. Then, we
have M+ 2M=y(2h+ 1)M. Therefore, for y= (2h+ 3)M, (a) there is no
idle time for the batches setup operation and processing on the extrusion factory’s
serial batching machine in the interval [0,(2h+ 1)M], and (b) there is no idle time
for transportation in the interval [M, (2h+ 2)M].
2
A
2
A
1h
A
3
A
1
2
A
2
2
A
3
2
A
3
3
A
2
3
A
1
3
A
3
1h
A
2
1h
A
1
1h
A
3
A
1h
A
s
etup
s
etup
s
etup
ggg
ggg
4
M
6
M
(2 2)h M
8
M
(2 4)h M
s
etup
1
A
2
M
1
A
Figure 3. The optimal schedule in Theorem 1
We can prove this by contradiction. Suppose that there is a batch Ak, where
PJiAkpi6=M. There are two situations as follows:
If PJiAkpi< M, it is easy to infer from Figure 3 that there is idle time
during the setup operation and processing on the extrusion factory’s serial batching
machine, and it contradicts (a).
If PJiAkpi> M, we can also see from Figure 3 that there is idle time during the
transportation between the extrusion factory and aging factory, which contradicts
(b).
Therefore, we can get a solution for 3-PARTITION problem.
Combining the “if” part and the “only if” part, we have proved the proposed
theorem.
4. Heuristic algorithms and performance analysis. In this section, we present
two heuristic algorithms for two different cases. Also, two lower bounds for the
studied problem and the worst case performance ratios of the proposed algorithms
are established.
410 J. PEI, P. M. PARDALOS, X. LIU, W. FAN, S. YANG AND L. WANG
4.1. Heuristic algorithms. In this section, we will discuss two different cases.
4.11. Case 1. TP
Heuristic Algorithm H1
Step 1. Create l=dn
cebatches based on the capacity c, and lshould be the smallest
number of possible batches. Index the jobs {J1, J2, . . . , Jn}in non-increasing order
of the processing time pi, i.e., p1p2. . . pn;
Step 2. Set i= 0 and Pf= 0(f= 1,2, . . . , l), where Pfdenotes the total processing
time of jobs which have been assigned to batch bf.
Step 3. i=i+ 1. Assign job Jito batch bk, where k=argminf=1,2,...,l {Pf
|bf|< c, s +PJibfpiT
}. If there are multiple batches with the same least
processing time, then the smallest indexed batch is selected. If there exists no such
batch, then we create a new batch and assign Jito it. Set l=l+ 1 and Pl=pi;
Step 4. If i<n, then return to step 3. Otherwise, turn to step 5;
Step 5. Re-index all batches in SPT order (the smallest processing time first order),
i.e., P1P2. . . Pl. Schedule the batches on the machines and the vehicle
according to the sequence of the batches.
Theorem 2. Given TP, we have Cmax(H1) = s+P1+ (l1)T+T
2+P.
Proof. According to the rule of H1, we have s+P1s+P2. . . s+Pl. There
are three situations with respect to P,T, and s+Pk, k = 1,2, . . . , l.
(1) Ps+P1s+P2. . . s+PlT;
(2) s+P1. . . P. . . s+PlT.
(3) s+P1s+P2. . . s+PlPT.
In the first situation, for the first batch b1,d1=s+P1. Then, C21 =d1+T
2+
P=s+P1+T
2+P. For the second batch b2,d2=d1+T=s+P1+T, and
C22 =d2+T
2+P=s+P1+T+T
2+P. Intuitively, the completion time of the
kth batch bkis Ck=s+P1+ (k1)T+T
2+P(k= 1,2, . . . , l). Therefore, we can
get the makespan of the schedule Cmax(H1) = s+P1+ (l1)T+T
2+P.
The derivation of the second and third situations is similar to the first one, so
we can get the makespan of the schedule Cmax(H1) = s+P1+ (l1)T+T
2+P.
The proof is completed.
4.12. Case 2. T < P
Heuristic Algorithm H2:
Step 1. Create l=dn
cebatches based on the capacity c, and lshould be the smallest
number of possible batches. Index the jobs {J1, J2, . . . , Jn}in non-increasing order
of the processing time pi, i.e.,p1p2. . . pn;
Step 2. Set i= 0 and Pf= 0(f= 1,2, . . . , l), where Pfdenotes the total processing
time of jobs which have been assigned to batch bf;
Step 3. i=i+ 1. Assign job Jito batch bk, where k=argminf=1,2,...,l {Pf
|bf|< c, s +PJibfpiP
}. If there are multiple batches with the same least
processing time, then the smallest indexed batch is selected. If there exists no such
batch, then we create a new batch and assign Jito it. Set l=l+ 1 and Pl=pi;
Step 4. If i<n, then return to step 3. Otherwise, turn to step 5;
Step 5. Re-index all batches in SPT order, i.e., P1P2. . . Pl. Schedule the
batches on the machines and the vehicle according to the sequence of the batches.
COORDINATION OF PRODUCTION AND TRANSPORTATION 411
Theorem 3. Given T < P , we have Cmac(H2) = s+P1+T
2+lP .
Proof. According to the rule of H2, we have s+P1s+P2. . . s+Pl. There
are three situations with respect to P,T, and s+Pk, k = 1,2, . . . , l.
(1) s+P1. . . s+Pθ1Ts+Pθ. . . s+PlP(2 θl);
(2) Ts+P1. . . P. . . s+PlP;
(3) s+P1s+P2. . . s+PlT < P .
In the first situation, for the first batch b1, we have d1=s+P1. Then, C21 =
d1+T
2+P=s+P1+T
2+P. For the (θ1)th batch bθ1, we can deduce that
dθ1=s+P1+ (θ2)T, and C2(θ1) =s+P1+T
2+ (θ1)P. The θth batch
bθdeparts from the supplier at the time dθ=s+P1+ (θ2)T+s+Pθ, and it
reaches the manufacturer at the time dθ+T
2=s+P1+ (θ2)T+s+Pθ+T
2.
The earliest time when the manufacturer’s machine can process the θth batch bθis
s+P1+T
2+(θ1)P, where s+P1+T
2+(θ1)Ps+P1+ (θ2)T+s+Pθ+T
2,
so we have C2θ=s+P1+T
2+ (θ1)P+P=s+P1+T
2+θP . Intuitively, the
completion time of the last batch blis C2l=s+P1+T
2+lP . Therefore, the
makespan of the schedule Cmax(H2) = s+P1+T
2+lP when T < P .
The derivation of the second and third situations is similar to the first one, so
we can get the makespan of the schedule Cmax(H2) = s+P1+T
2+lP .
The proof is completed.
4.2. Lower bounds. In order to evaluate the performance of the proposed heuris-
tic algorithms, we derive two lower bounds of the makespan under two situations.
When TP, we assume that there is no additional idle time on the vehicle. The
lower bound in the first case can be derived as the sum of the completion time of the
first batch on the supplier’s serial batching machine, the round-trip transportation
time of (l1) batches, the one-way transportation time of the last batch, and the
processing time of the last batch on the manufacturer’s parallel batching machine,
i.e. s+P1+ (l1)T+T
2+P, where ldenotes the total number of the batches.
Since l≥ dn
ceand P1mini=1,2,...,n{pi}, we have the lower bound for the first
case as LB1=mini=1,2,...,n{pi}+s+ (dn
ce − 1)T+T
2+Pwhen TP.
When T < P , we assume that there is no additional idle time on the manu-
facturer’s parallel batching machine. The lower bound in the second case can be
derived as the sum of the completion time of the first batch on the supplier’s se-
rial batching machine, the one-way transportation time of the first batch, and the
total processing time of all batches on the manufacturer’s parallel batching ma-
chine, i.e. s+P1+T
2+lP , where ldenotes the total number of the batches.
Since l≥ dn
ceand P1mini=1,2,...,n{pi}, the lower bound for the second case is
LB2=mini=1,2,...,n{pi}+s+T
2+dn
cePwhen T < P .
4.3. Worst case analysis. Let the symbol Cdenote the optimal makespan, and
we have the makespan Cmax(H1) and Cmax(H2) when TPand T < P according
to the Theorems 2 and 3, respectively. The following theorems are proposed to
analyze heuristic algorithms H1and H2.
Theorem 4. Given TP, the worst case ratio of H1is no more than 3c.
Proof. Let the lower bound of the first case in the supply chain scheduling problem
be CLB1. We obtain Cmax(H1) and CLB1as follows:
Cmax(H1) = s+P1+ (l1)T+T
2+P,
CLB1=mini=1,2,...,n{pi}+s+ (dn
ce − 1)T+T
2+P.
412 J. PEI, P. M. PARDALOS, X. LIU, W. FAN, S. YANG AND L. WANG
Hence,
Cmax(H1)
CCmax(H1)
CLB1s+P1+(l1)T+T
2+P
mini=1,2,...,n{pi}+s+(dn
ce−1)T+T
2+P
lT +1
2T
(dn
ce− 1
2)Tn+1
2
n
c1
2
(1+ 1
2n)c
1c
2n2(1 + 1
2n)c3c.
Theorem 5. Given T < P , the worst case ratio of H2is no more than 2c.
Proof. Let the lower bound of the second case in the supply chain scheduling prob-
lem be CLB2. We obtain Cmax(H2) and CLB2as follows:
Cmax(H2) = s+P1+T
2+lP ,
CLB2=mini=1,2,...,n{pi}+s+T
2+dn
ceP.
Hence,
Cmac(H2)
CCmax(H2)
CLB2s+P1+T
2+lP
mini=1,2,...,n{pi}+s+T
2+dn
ceP
P+lP
dn
cePn+1
dn
ce(1 + 1
n)c2c.
5. Computational experiments. In this section, we conduct computational ex-
periments to evaluate the performance of H1when TPand H2when T < P ,
respectively. We designed two types of computational tests for both situations with
a number of random instances. One type of computational tests involved small-
scale random instances, and the other one was performed with large-scale random
instances. The parameters for the test problems are randomly generated based on
the real aluminium production as follows:
(a) Number of jobs n:
For small-scale random instances, we take n∈ {50,60,70,80,90,100};
For large-scale random instances, we take n∈ {500,600,700,800,900,1000}.
(b) Capacity of the batching machines and vehicle c, job processing time on the
supplier’s serial batching machine pi, and setup time on the supplier’s serial batching
machine sare generated from the continuous uniform distributions U[6,8], U[1,15],
and U[2,4], respectively.
(c) Transportation time between the supplier and manufacturer Tis generated
from the continuous uniform distributions U[50,60), and U[60,70]:
For TP, we take TU[60,70];
For T < P , we take TU[50,60).
(d) Job processing time on the manufacturer’s parallel batching machine Pis
generated from the continuous uniform distributions U[50,60] and U[60,70]:
For TP, we take PU[50,60];
For T < P , we take PU[60,70].
To evaluate the performance of the proposed heuristic algorithms, the solutions
reported by H1and H2were also compared with two other approaches applied in
previous literature, namely algorithm LOE (Last-Only-Empty) and LP T (longest
processing time), and algorithm F OE (First-Only-Empty) and S P T (shortest pro-
cessing time) [5]. These different approaches are compared by measuring the relative
gap between the makespan reported by each approach and the lower bounds derived
in section 4.2. The relative gap percentage (GapH) between the approach Hand
the lower bound (LB) is calculated as in Eq. 15.
GapH=CH
max LB
LB ×100% (15)
A factorial experiment was designed to determine the impact of the factors on
the performance of the obtained solution. There are two factors in the factorial
COORDINATION OF PRODUCTION AND TRANSPORTATION 413
experiment, which characterize the number of jobs (i.e., n), and the capacity of
machines and vehicles (i.e., c). Each treatment for the combination (n, c) was
replicated fifty times, and both average gap and maximum gap were analyzed.
All the algorithms were coded in PowerBuilder 9.0 language and their code was
run on a Pentium(R)-4, 300 MHz PC with 2GB of RAM.
Case 1. TP
(1) Analysis of results for small-scale random instances
Figure 4 summarizes the computational results of the average relative gaps for
small-scale random instances when TP. As seen in Figure 4, it is clear that the
average gap percentage of H1decreased with the number of jobs increasing except
when n= 90, and it ranged from approximately 4.45% to 7.77%. H1outperformed
the other two algorithms in this case, and the algorithm F OE and S P T performed
better than the algorithm LOE and LP T . The algorithm LOE and LP T reported
an average gap of 9.98% in the 100-job problem instance. However, H1reported an
average gap of only 4.45%.
Figure 5 summarizes the computational results of the maximum relative gaps
for small-scale random instances when TP. From Figure 5, we observe that H1
worked better than the other two algorithms, and the maximum gaps ranged from
approximately 5.51% to 16.59%.
(2) Analysis of results for large-scale random instances
In Figure 6, we compare the impact of problem scale on the average relative gaps
for large-scale random instances when TP. From the results, we observe that
the effectiveness of H1increased with the number of jobs increasing, and it ranged
from approximately 0.48% to 1.09%. Compared to H1, the other two algorithms
performed poorly. The algorithm F OE and S P T also performed better than the
algorithm LOE and LP T . The algorithm LOE and LP T reported an average gap
of 10.29% in the 1000-job problem instance. However, H1reported an average gap
of only 0.48%.
Figure 7 shows the maximum relative gaps for large-scale random instances when
TP. The maximum gaps appeared in a decreasing trend as the number of jobs
increased except when n= 900, and it ranged from approximately 0.61% to 7.47%.
The proposed heuristic algorithm also worked better than the other two algorithms.
Figure 4. Computational results of the average gap percentage for
cU[6,8],piU[1,15],sU[2,4],TU[60,70],PU[50,60],n
{50,60,70,80,90,100}
414 J. PEI, P. M. PARDALOS, X. LIU, W. FAN, S. YANG AND L. WANG
Figure 5. Computational results of the maximum gap percent-
age for cU[6,8],piU[1,15],sU[2,4],TU[60,70],P
U[50,60],n∈ {50,60,70,80,90,100}
Figure 6. Computational results of the average gap percentage for
cU[6,8],piU[1,15],sU[2,4],TU[60,70],PU[50,60],n
{500,600,700,800,900,1000}
Case 2. T < P
(1) Analysis of results for small-scale random instances
Figure 8 presents the comparison of the different approaches on the average gap
percentage for small-scale random instances when T < P . It can be seen that the
average gap percentage of H2decreased with the number of jobs increasing except
when n= 70, and it ranged from approximately 4.67% to 9.61%. H2obtained
better solutions than the other two algorithms, and the algorithm F OE and S P T
worked better than the algorithm LOE and LP T . The algorithm F OE and SP T
reported an average gap of 10.81% in the 100-job problem instance. However, H2
reported an average gap of only 4.67%.
Figure 9 summarizes the computational results of the maximum relative gaps for
small-scale random instances when T < P . From Figure 9, we observed that H2
worked better than the other two algorithms, and the maximum gap ranged from
approximately 6.67% to 25.22%.
(2) Analysis of results for large-scale random instances
COORDINATION OF PRODUCTION AND TRANSPORTATION 415
Figure 7. Computational results of the maximum gap percent-
age for cU[6,8],piU[1,15],sU[2,4],TU[60,70],P
U[50,60],n∈ {500,600,700,800,900,1000}
Figure 10 displays the performance comparison of these three approaches in terms
of the average gap percentage for large-scale random instances when T < P . The
results indicated that the average gap percentage of H2also decreased with the
number of jobs increasing except when n= 900, and it ranged from approximately
0.80% to 1.67%. It is clear that H2performed better than the other two algorithms,
and the algorithm F OE and S P T also performed better than the algorithm LOE
and LP T . The algorithm F O E and S P T reported an average gap of 12.36% in the
1000-job instance. However, H2reported an average gap of only 0.80%.
Figure 11 shows the maximum relative gaps in large-scale random instances when
T < P . We observed that H2performed better than the other two algorithms, and
its maximum gaps ranged from approximately 6.57% to 13.33%.
In summary, H1and H2outperformed the other two algorithms on either small-
scale or large-scale problems. Even when n500, for both situations of TP
and T < P , the average gaps between the proposed heuristic algorithms and the
lower bounds were below 1.70%, indicating that H1and H2are quite effective
for large-scale random instances. It is also worthwhile to note that the proposed
heuristic algorithms can obtain solutions within 6 seconds for each problem instance,
even for the instance up to 1000 jobs. Thus, the computational burden of the
procedure is low enough to run the proposed heuristic algorithms for large-scale
random instances.
6. Conclusions. In this paper, we focus on a scheduling problem in an aluminum
production supply chain. The strongly NP-hard problem can be characterized into
the features such as different batching operations, setup time, capacity constraints,
and numerical relationship between processing time and transportation time. In
order to address the problem, two different heuristic algorithms were proposed re-
spectively for the cases that TPand T < P . Meanwhile, we developed two lower
bounds to analyze their worst case performance and verify the effectiveness of the
proposed heuristic algorithms. The experimental results showed that the proposed
heuristic algorithms outperformed the other two algorithms in the previous litera-
ture, and they can solve both small-scale and large-scale problems effectively and
efficiently.
416 J. PEI, P. M. PARDALOS, X. LIU, W. FAN, S. YANG AND L. WANG
Figure 8. Computational results of the average gap percentage for
cU[6,8],piU[1,15],sU[2,4],TU[50,60),PU[60,70],n
{50,60,70,80,90,100}
Figure 9. Computational results of the maximum gap percent-
age for cU[6,8],piU[1,15],sU[2,4],TU[50,60),P
U[60,70],n∈ {50,60,70,80,90,100}
Figure 10. Computational results of the average gap percent-
age for cU[6,8],piU[1,15],sU[2,4],TU[50,60),P
U[60,70],n∈ {500,600,700,800,900,1000}
COORDINATION OF PRODUCTION AND TRANSPORTATION 417
Figure 11. Computational results of the maximum gap percent-
age for cU[6,8],piU[1,15],sU[2,4],TU[50,60),P
U[60,70],n∈ {500,600,700,800,900,1000}
Despite this completed work, there are several interesting future directions based
on this research. The first interesting one is to combine other objective functions,
such as minimizing the sum of completion time, minimizing maximum lateness, and
minimizing the number of tardy jobs. The second future exploration may study the
case with different job release times in the first stage. Last but not least, the other
future research may consider multiple batching machines and vehicles in each stage
to extend the practice of the model.
Acknowledgments. We would like to thank the editors and referees for their
valuable comments and suggestions.
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Received June 2013; 1st revision January 2014; final revision March 2014.
E-mail address:peijun@ufl.edu
E-mail address:pardalos@ufl.edu
E-mail address:lxinbao@126.com
E-mail address:wfan3@ncsu.edu
E-mail address:hgdysl@gmail.com
E-mail address:wangling.sh@gmail.com
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