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=PJi∈bkpi(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 T≥pi+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 P≥pi+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 S→M, 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 S→M , 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 ≤ l≤n;

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

Ji∈bk

pi, k = 1,2, ..., l (5)

408 J. PEI, P. M. PARDALOS, X. LIU, W. FAN, S. YANG AND L. WANG

Dk≥C1k, 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)

C1k−C1f+s+Pf−(1 −ykf )M≤0, k, f = 1,2, ..., l, k 6=f(10)

C2k−C2f+Pf−(1 −ykf )M≤0, 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 S→M, 1|b=c, tk=T|Cmax.

Theorem 1. The Problem S→M, 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

2,1≤i≤3h, such as P3h

i=1 xi=hM, does there exist a partition

A1,A2,...,Ahof the set Asuch that |Al|= 3 and Pxi∈Alxi=M, 1≤l≤h?

We construct the following instance of problem S→M, 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 S→M, 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 S→M, 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

PJi∈Akpi6=M. There are two situations as follows:

If PJi∈Akpi< 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 PJi∈Akpi> 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. T≥P

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., p1≥p2≥. . . ≥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 +PJi∈bfpi≤T

}. 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., P1≤P2≤. . . ≤Pl. Schedule the batches on the machines and the vehicle

according to the sequence of the batches.

Theorem 2. Given T≥P, we have Cmax(H1) = s+P1+ (l−1)T+T

2+P.

Proof. According to the rule of H1, we have s+P1≤s+P2≤. . . ≤s+Pl. There

are three situations with respect to P,T, and s+Pk, k = 1,2, . . . , l.

(1) P≤s+P1≤s+P2≤. . . ≤s+Pl≤T;

(2) s+P1≤. . . ≤P≤. . . ≤s+Pl≤T.

(3) s+P1≤s+P2≤. . . ≤s+Pl≤P≤T.

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+ (k−1)T+T

2+P(k= 1,2, . . . , l). Therefore, we can

get the makespan of the schedule Cmax(H1) = s+P1+ (l−1)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+ (l−1)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.,p1≥p2≥. . . ≥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 +PJi∈bfpi≤P

}. 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., P1≤P2≤. . . ≤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+P1≤s+P2≤. . . ≤s+Pl. There

are three situations with respect to P,T, and s+Pk, k = 1,2, . . . , l.

(1) s+P1≤. . . ≤s+Pθ−1≤T≤s+Pθ≤. . . ≤s+Pl≤P(2 ≤θ≤l);

(2) T≤s+P1≤. . . ≤P≤. . . ≤s+Pl≤P;

(3) s+P1≤s+P2≤. . . ≤s+Pl≤T < 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 (θ−1)th batch bθ−1, we can deduce that

dθ−1=s+P1+ (θ−2)T, and C2(θ−1) =s+P1+T

2+ (θ−1)P. The θth batch

bθdeparts from the supplier at the time dθ=s+P1+ (θ−2)T+s+Pθ, and it

reaches the manufacturer at the time dθ+T

2=s+P1+ (θ−2)T+s+Pθ+T

2.

The earliest time when the manufacturer’s machine can process the θth batch bθis

s+P1+T

2+(θ−1)P, where s+P1+T

2+(θ−1)P≥s+P1+ (θ−2)T+s+Pθ+T

2,

so we have C2θ=s+P1+T

2+ (θ−1)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 T≥P, 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 (l−1) 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+ (l−1)T+T

2+P, where ldenotes the total number of the batches.

Since l≥ dn

ceand P1≥mini=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 T≥P.

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 P1≥mini=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 C∗denote the optimal makespan, and

we have the makespan Cmax(H1) and Cmax(H2) when T≥Pand T < P according

to the Theorems 2 and 3, respectively. The following theorems are proposed to

analyze heuristic algorithms H1and H2.

Theorem 4. Given T≥P, 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+ (l−1)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)

C∗≤Cmax(H1)

CLB1≤s+P1+(l−1)T+T

2+P

mini=1,2,...,n{pi}+s+(dn

ce−1)T+T

2+P

≤lT +1

2T

(dn

ce− 1

2)T≤n+1

2

n

c−1

2

≤(1+ 1

2n)c

1−c

2n≤2(1 + 1

2n)c≤3c.

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)

C∗≤Cmax(H2)

CLB2≤s+P1+T

2+lP

mini=1,2,...,n{pi}+s+T

2+dn

ceP

≤P+lP

dn

ceP≤n+1

dn

ce≤(1 + 1

n)c≤2c.

5. Computational experiments. In this section, we conduct computational ex-

periments to evaluate the performance of H1when T≥Pand 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 T≥P, we take T∈U[60,70];

For T < P , we take T∈U[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 T≥P, we take P∈U[50,60];

For T < P , we take P∈U[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. T≥P

(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 T≥P. 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 T≥P. 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 T≥P. 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

T≥P. 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

c∈U[6,8],pi∈U[1,15],s∈U[2,4],T∈U[60,70],P∈U[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 c∈U[6,8],pi∈U[1,15],s∈U[2,4],T∈U[60,70],P∈

U[50,60],n∈ {50,60,70,80,90,100}

Figure 6. Computational results of the average gap percentage for

c∈U[6,8],pi∈U[1,15],s∈U[2,4],T∈U[60,70],P∈U[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 c∈U[6,8],pi∈U[1,15],s∈U[2,4],T∈U[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 n≥500, for both situations of T≥P

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 T≥Pand 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

c∈U[6,8],pi∈U[1,15],s∈U[2,4],T∈U[50,60),P∈U[60,70],n∈

{50,60,70,80,90,100}

Figure 9. Computational results of the maximum gap percent-

age for c∈U[6,8],pi∈U[1,15],s∈U[2,4],T∈U[50,60),P∈

U[60,70],n∈ {50,60,70,80,90,100}

Figure 10. Computational results of the average gap percent-

age for c∈U[6,8],pi∈U[1,15],s∈U[2,4],T∈U[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 c∈U[6,8],pi∈U[1,15],s∈U[2,4],T∈U[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.

REFERENCES

[1] A. Agnetis, N. G. Hall and D. Pacciarellir, Supply chain scheduling: Sequence coordination,

Discrete Applied Mathematics,154 (2006), 2044–2063.

[2] H. Allaoui and A. Artiba, Integrating simulation and optimization to schedule a hybrid ﬂow

shop with maintenance constraints,Computers &Industrial Engineering,47 (2004), 431–450.

[3] I. Averbakh and Z. Xue, On-line supply chain scheduling problems with pre-emption, Euro-

pean Journal of Operational Research,181 (2007), 500–504.

[4] J. F. Bard and N. Nananukul, A branch-and-price algorithm for an integrated production and

inventory routing problem,Computers &Operations Research,37 (2010), 2202–2217.

[5] J. Behnamian, S. M. T. Fatemi Ghomi, F. Jolai and O. Amirtaheri, Realistic two-stage

ﬂowshop batch scheduling problems with transportation capacity and times,Applied Mathe-

matical Modelling ,36 (2012), 723–735.

[6] E. Cakici, S. J. Mason and M. E. Kurz, Multi-ob jective analysis of an integrated supply chain

scheduling problem,International Journal of Production Research,50 (2012), 2624–2638.

[7] F. T. S. Chan, S. H. Chung and P. L. Y. Chan, An adaptive genetic algorithm with dominated

genes for distributed scheduling problems,Expert Systems with Applications,29 (2005), 364–

371.

[8] Y. C. Chang, K. H. Chang and T. K. Chang, Applied column generation-based approach to

solve supply chain scheduling problems,International Journal of Production Research,51

(2013), 4070–4086.

[9] Y. C. Chang and Y. C. Lee, Machine scheduling with job delivery coordination,European

Journal of Operation Research,158 (2004), 470–487.

[10] T. C. E. Cheng and X. Wang, Machine scheduling with job class setup and delivery consid-

erations,Computers &Operations Research,37 (2010), 1123–1128.

[11] P. Chretienne, O. Hazir and S. Kedad-Sidhoum, Integrated batch sizing and scheduling on a

single machine,Journal of Scheduling,14 (2011), 541–555.

418 J. PEI, P. M. PARDALOS, X. LIU, W. FAN, S. YANG AND L. WANG

[12] V. S. Gordon and V. A. Strusevich, Single machine scheduling and due date assignment

with positionally dependent processing times,European Journal of Operation Research,198

(2009), 57–62.

[13] R. L. Graham, E. L. Lawler, J. K. Lenstra and A. H. G. Rinnooy Kan, Optimization and ap-

proximation in deterministic machine scheduling: A survey,Annals of Discrete Mathematics,

5(1979), 287–326.

[14] O. Grunder, Lot sizing, delivery and scheduling of identical jobs in a single-stage supply

chain, International Journal of Innovative Computing Information and Control,6(2010),

3657–3668.

[15] N. G. Hall and C. N. Potts, Supply chain scheduling: Batching and delivery,Operations

Research,51 (2003), 566–584.

[16] J. Hurink and S. Knust, Makespan minimization ﬂow-shop problems with transportation

times and a single robot,Discrete Applied Mathematics,112 (2001), 199–216.

[17] A. S. Jetlund and I. A. Karimi, Improving the logistics of multi-compartment chemical tankers,

Computers &Chemical Engineering,28 (2004), 1267–1283.

[18] M. Khalili and R. Tavakoli-Moghadam, A multi-objective electromagnetism algorithm for

a bi-objective ﬂowshop scheduling problem,Journal of Manufacturing Systems,31 (2012),

232–239.

[19] C. H. Lee, C. J. Liao and C. W. Chao, Scheduling with multi-attribute setup times,Computers

&Industrial Engineering,63 (2012), 494–502.

[20] B. M. T. Lin, T. C. E. Cheng and A. S. C. Chou, Scheduling in an assembly-type production

chain with batch transfer,Omega-International Journal of Management Science,35 (2007),

143–151.

[21] S. C. Liu and A. Z. Chen, Variable neighborhood search for the inventory routing and sched-

uling problem in a supply chain,Expert Systems with Applications,39 (2012), 4149–4159.

[22] M. M. Mazdeh, M. Sarhadi and K. S. Hindi, A branch-and-bound algorithm for single-machine

scheduling with batch delivery and job release times,Computers &Operations Research,35

(2008), 1099–1111.

[23] Y. Mehravaran and R. Logendran, Non-permutation ﬂowshop scheduling in a supply chain

with sequence-dependent setup times,International Journal of Production Economics,135

(2012), 953–963.

[24] C. Moon, Y. H. Lee, C. S. Jeong and Y. Yun, Integrated process planning and scheduling in

a supply chain,Computers &Industrial Engineering,54 (2008), 1048–1061.

[25] B. Naderi, A. Ahmadi Javid and F. Jolai, Permutation ﬂowshops with transportation times:

Mathematical models and solution methods,International Journal of Advanced Manufactur-

ing Technology,46 (2010), 631–647.

[26] B. Naderi, M. Zandieh, A. Khaleghi Ghoshe Balagh and V. Roshanaei, An improved simulated

annealing for hybrid ﬂowshops with sequence-dependent setup and transportation times to

minimize total completion time and total tardiness,Expert Systems with Applications,36

(2009), 9625–9633.

[27] D. Naso, M. Surico, B. Turchiano and U. Kaymak, Genetic algorithms for supply-chain sched-

uling: A case study in the distribution of ready-mixed concrete,European Journal of Opera-

tional Research,177 (2007), 2069–2099.

[28] P. M. Pardalos and G. C. R. Mauricio (Eds), Handbook of Applied Optimization, Oxford

University Press, 2002.

[29] P. M. Pardalos, O. V. Shylo and A. Vazacopoulos, Solving job shop scheduling problems

utilizing the properties of backbone and “big valley”,Computational Optimization and Ap-

plications,47 (2010), 61–76.

[30] M. Rasti-Barzoki and S. R. Hejazi, Minimizing the weighted number of tardy jobs with due

date assignment and capacity-constrained deliveries for multiple customers in supply chains,

European Journal of Operational Research,228 (2013), 345–357.

[31] M. Rasti-Barzoki, S. R. Hejazi and M. M. Mazdeh, A branch and bound algorithm to minimize

the total weighed number of tardy jobs and delivery costs,Applied Mathematical Modelling,

37 (2013), 4924–4937.

[32] E. Selvarajah and G. Steiner, Batch scheduling in a two-level supply chain - A focus on the

supplier,European Journal of Operational Research,173 (2006), 226–240.

[33] S. A. Torabi, S. M. T. Fatemi Ghomi and B. Karimi, A hybrid genetic algorithm for the

ﬁnite horizon economic lot and delivery scheduling in supply chains,European Journal of

Operational Research,173 (2006), 173–189.

COORDINATION OF PRODUCTION AND TRANSPORTATION 419

[34] H. Xuan and L. Tang, Scheduling a hybrid ﬂowshop with batch production at the last stage,

Computers &Operations Research,34 (2007), 2718–2733.

[35] W. K. Yeung, T. M. Choi and T. C. E. Cheng, Supply chain scheduling and coordination

with dual delivery modes and inventory storage cost,International Journal of Production

Economics,132 (2011), 223–229.

[36] A. D. Yimer and K. Demirli, A genetic approach to two-phase optimization of dynamic supply

chain scheduling,Computers &Industrial Engineering,58 (2010), 411–422.

[37] S. H. Zegordi, I. N. Kamal Abadi and M. A. Beheshti Nia, A novel genetic algorithm for

solving production and transportation scheduling in a two-stage supply chain,Computers &

Industrial Engineering,58 (2010), 373–381.

[38] T. Zhang, Y. J. Zhang, Q. P. Zheng and P. M. Pardalos, A hybrid particle swarm optimization

and tabu search algorithm for order planning problems of steel factories based on the make-

to-stock and make-to-order management architecture,Journal of Industrial and Management

Optimization,7(2011), 31–51.

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