Abstract—Raw material yards at dry bulk terminals act as temporary buffers for inbound and outbound raw materials. Both discharging and charging processes are supported by raw material yards in most cases. Stacker-reclaimers are dedicated equipments in yards for raw material handling. The efficiency of yard operation depends to a great extent on the productivity of stacker-reclaimers. Stacker-reclaimer scheduling problem is discussed in this paper. Given a set of handling operations in a yard, the objective is to find an optimal operation sequence on each stacker-reclaimer so as to minimize the makespan. A mixed integer programming model is formulated. Since the considered problem is NP-hard in nature, a parthenogenetic algorithm is developed to obtain near optimal solutions. Computational experiments are conducted to examine the presented model and the solution algorithm. The computational results show that the proposed parthenogenetic algorithm is effective.

I. INTRODUCTION ITH the continuous growth in steel production, steel producers in China consume tremendous raw materials annually, leaving a shortfall in raw materials to be met

through importing international supplies. The import of raw materials grows to 627.78 million tons in 2009. The raw materials are brought in by vessels to dry bulk terminals, and then transshipped by inland carriers. A dry bulk terminal plays an important role in raw material transportation by serving as a multi-modal interface. A typical layout a dry bulk terminal is illustrated in Fig. 1.

Vessel

Inland carrier

Unloader

Stacker-reclaimer

Quay side

Raw material yards

Inbound raw materials

Outbound raw materials

Fig. 1. A typical layout of a dry bulk terminal

Manuscript received March 31, 2010. This work was supported in part by Science and Technology Commission of Shanghai Municipality under the Industry-Academia Research Project.

Dayong Hu is with the Institute of Manufacturing Technology & Process Automation, School of Mechanical Engineering, Shanghai Jiao Tong University, 800 Dong Chuan Road, Shanghai 200240, China (phone: 86-021-34206583; fax: 86-021-34206583; e-mail: dyhu@sjtu.edu.cn).

Zhenqiang Yao is with the Institute of Manufacturing Technology & Process Automation, School of Mechanical Engineering, Shanghai Jiao Tong University, 800 Dong Chuan Road, Shanghai 200240, China (e-mail: zqyao@sjtu.edu.cn).

The raw materials are discharged from a vessel, and then transported to raw material yards through belt-conveyor system. The discharged raw materials are stacked at raw material yards by stacker-reclaimers (S-Rs). When demanded by inland carriers, the stored raw materials are reclaimed by S-Rs. In view of the differences in arrival times of vessels and inland carriers, raw material yards act as temporary buffers for inbound and outbound raw materials. It is clear that both discharging and charging processes are usually supported by raw material yards. As the rapid increase in raw material throughput, improvement of yard operation becomes more urgent than ever. S-Rs are dedicated equipments in yards, which perform stacking and reclaiming raw materials. The efficiency of yard operation depends to a great extent on the productivity of S-Rs. Therefore, it is necessary to discuss the problem of S-R scheduling for efficient yard operation.

These research problems that arise in container terminals have been extensively studied in the past few years. That is mainly composed of allocating and scheduling berths, storage space and handling equipments, which show main production processes in container terminals. Comprehensive surveys of literature on container terminal operation were given by Vis and Koster [1], Steenken, Voß, and Stahlbock [2], Stahlbock and Voß [3]. However, little research has been conducted on dry bulk terminals.

This study focuses on a dry bulk terminal to deploy S-Rs for raw material yard operation. The rest of the paper is organized as follows: Section 2 describes the problem and presents a mixed integer programming model. As there is no polynomial time algorithm for the exact solution of the S-R scheduling problem, Section 3 develops a parthenogenetic algorithm to obtain its near optimal solutions. Computational experiments are conducted, and the results are reported in Section 4. Finally, conclusions are mentioned in Section 5.

II. PROBLEM FORMULATION As shown in Fig. 2, a yard consists of multiple storage

areas. For the raw materials stored in the same yard, different raw materials need to be separated by at least the given clearance distance to prevent blending. Top view of a yard is given in Fig. 3. The yard is served by two S-Rs that run on respective rails. Through running, luffing and slewing movements, S-R1 and S-R2 can stack raw materials in storage areas or reclaim raw materials from storage areas.

Fig. 2. Side view of a yard

Stacker-reclaimer Scheduling for Raw Material Yard Operation Dayong Hu, Zhenqiang Yao

W

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Third International Workshop on Advanced Computational Intelligence August 25-27, 2010 - Suzhou, Jiangsu, China

978-1-4244-6337-4/10/$26.00 @2010 IEEE

Fig. 3. Top view of a yard

The raw material handling in a storage area is defined as a task. It is assumed that all tasks are all available at the beginning of the schedule. S-R1 and S-R2 are identical, and only one S-R can work on a task at a time until it completes the task. The times required for each S-R to perform running, luffing and slewing movements are generally regarded as setup times. S-Rs perform preventive maintenance after the schedule is completed, and S-Rs are also setup for the beginning of next schedule. Therefore, no setup time is needed for each S-R at the beginning of a schedule.

In order to formulate the problem of S-R scheduling, the following parameters and decision variables are introduced: Parameters: n The number of tasks in a yard, labeled with n,...,1

ip The time required by one S-R to process task i

ijs The setup time required by one S-R to process

task j immediately after processing task i M A sufficiently large positive constant number Decision variables:

iC The completion time of task i kiX 1, if task i is processed by SR k ; 0, otherwise k

ijY 1, if SR k process task i before task j ; 0, otherwise

With these assumptions and definitions, the problem of S-R scheduling described above can be formulated as follows: Minimize:

iiCmax (1)

Subject to:

∑=

=2

11

k

kiX ni ≤≤∀1 (2)

ii pC ≥ ni ≤≤∀1 (3)

( )kij

kj

kiiijjj YXXMCspC −−−−≥−− 3

21, and,,1 ≤≤∀≠≤≤∀ kjinji (4) kj

ki

kji

kij XXYY =+

21, and,,1 ≤≤∀≠≤≤∀ kjinji (5)

10,, =kij

ki YX

21, and,,1 ≤≤∀≠≤≤∀ kjinji (6) The objective function (1) is to minimize the maximum completion time criterion called makespan at which all tasks are completed. Constraints (2) denote that each task must be processed only by one S-R. Constraints (3) define the relationship between the completion time of a task and its

processing time. Constraints (4) give the relationship between the completion time of a task and that of its predecessor. Constraints (5) ensure the order relationship between two tasks processed by the same S-R. Constraints (6) are simple binary constraints.

III. A PARTHENOGENETIC ALGORITHM The S-R scheduling problem can be classified into parallel

machines scheduling problem to minimize the makespan [4], [5]. Even for two parallel machines, it has turned out that parallel machines scheduling for minimizing the makespan is NP-hard [6], [7]. Thus there is no polynomial time algorithm for the exact solution of the S-R scheduling problem.

A parthenogenetic algorithm [8]-[10] is then developed to obtain near optimal solutions. A parthenogenetic algorithm simulates the asexual reproductive mode in biology through single parent to generate offspring. Compared with standard genetic algorithm, a parthenogenetic algorithm acts on single chromosome, and crossover and mutation are replaced by swap, reverse, and insert that are commonly used in it. The procedure of the proposed parthenogenetic algorithm is illustrated in Fig. 4, and the details are presented as follows.

Chromosome representation

Generate initial population

Evaluation each chromosome

Selection

Swap

Reverse

Insert

Elitism

Maximum generation?

Bit mutation

Start

End

Yes

No

Fig. 4. The flowchart of the proposed parthenogenetic algorithm

A. Chromosome Representation A feasible solution is represented as a chromosome that is a

form of character string. In order to achieve good performance, a suitable representation scheme for a possible solution is a primary task. S-R assignment and task sequencing are two issues to be considered for the chromosome representation of a feasible solution. The two-string structure is employed to represent a feasible

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solution [11], [12]. This structure requires two strings with the same length n . The first string provides a permutation of tasks, and the second assigns a S-R to each of the tasks in the corresponding position of the first string. The chromosome representation can be illustrated in Fig. 5. Tasks 7, 2, 6, 4, 1 and 3 (in that order) are processed by S-R1, and tasks 10, 11, 8, 9 and 5 (in that order) are processed by S-R2.

Fig. 5. An illustration of the chromosome representation

B. Initial Population Generation The algorithm starts searching with a population of

solutions represented by chromosomes. The initial population is generated using the following procedure, and the population size ( popsize ) is defined in advance.

Step1: Set 1i = ;

Step2: Generate a random permutation of tasks, and assign

the permutation to the first string of chromosome i ; Step3: Each gene value for the second string is generated as a

discrete uniform random number between 1and 2; Step4: Set 1i i= + , if i popsize> STOP, else go to Step 2.

C. Fitness Evaluation and Selection A fitness function is used to evaluate the quality of a chromosome. The objective of the S-R scheduling problem is to minimize the makespan that is always positive. Hence the fitness value of a chromosome is set to be the reciprocal of its objective function value.

iiC

Fitnessmax

1= (7)

Based on the idea of survival of the fittest, the selection component is used to guide the evolution of chromosomes towards the optimum region in search space. In this paper, the roulette wheel approach is adopted as the selection procedure [13]. It belongs to the fitness-proportional selection, and then the chromosome with a higher fitness value has greater probability of being selected.

D. Genetic operators Offspring is produced from each selected chromosome

through a series of genetic operations. In this paper, the genetic procedure is done by using four genetic operators: except for basic swap, reverse, insert, bit mutation is also introduced.

Swap: as shown in Fig. 6, a pair of points in a chromosome is selected randomly. Then the contents of the two points are swapped to generate a new chromosome with a given swap probability.

Fig. 6. An illustration of the swap

Reverse: as shown in Fig. 7, a segment of a chromosome is selected randomly. Then the contents of the segment are reversed to generate a new chromosome with a given reverse probability.

Fig. 7. An illustration of the reverse

Insert: as shown in Fig. 8, a segment of a chromosome is selected randomly. Then the last gene in the segment is transferred to the first position of the segment, and all the other genes in the segment are moved backwards to generate a new chromosome with a given insert probability.

Fig. 8. An illustration of the insert

Bit mutation: as shown in Fig. 9, for the two-string structure, it is clear that the basic swap, reverse, and insert operator can only act on task sequencing but have no effect on S-R assignment. Therefore, bit mutation operator is introduced to maintain the diversity of the population. When a bit of the second string in a chromosome is chosen for bit mutation, its value is changed to the other value with a given mutation probability.

Fig. 9. An illustration of the bit mutation

E. Elitism strategy Elitism is often employed in standard genetic algorithm

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where the chromosome with the best fitness is copied to the next generation. The next generation is composed through selection and genetic operations from the current generation. The next generation replaces the current generation only after the new population is completely created. Elitism is also applied in the parthenogenetic algorithm to replace the worst chromosome of next population with the best chromosome of current population.

IV. COMPUTATIONAL EXPERIMENTS Computational experiments are conducted to evaluate the

performance of the proposed model and parthenogenetic algorithm. The parthenogenetic algorithm described in the previous section is coded in C program and executed in a Pentium IV 2.68GHz GHz PC with 512MB RAM. The population size, the swap probability, the reverse probability, the insert probability, the probability of bit mutation and the limit of generations are set as 150, 0.6, 0.8, 0.6, 0.3 and 350 respectively.

Considering the common planning horizon of which the length is set to be 8 hours, the sizes of tasks in a realistic production schedule are usually no more than 15. To solve the presented model, a number of test instances of different sizes are randomly generated. According to the practical data collected from a dry bulk terminal of Shanghai Port, the processing times and setup times are drawn from the uniform distribution [30, 60] and [5, 15] respectively. Each instance performs 15 independent runs, and average objective value and mean computational time are reported. As a comparison, ILOG CPLEX 10.0 (a commercial MIP solver) and standard genetic algorithm are introduced.

CPLEX is firstly employed to exactly solve these instances with small sizes and executed in a Core Duo 2.53GHz PC with 4GB RAM. As shown in Table I, the computational time of CPLEX grows exponentially as the instance size increases. However, the proposed parthenogenetic algorithm can obtain near optimal solutions in short time. The percentage deviation from corresponding optimal solution that is defined as the gap is less than 3%, and the average gap is only about 1.95%.

TABLE I

COMPARISON WITH CPLEX Size CPLEX P-GA Gap (%) Value CPU (s) Ave Mean CPU (s) 6 154 1 156 0.128 1.30 7 194 20 197 0.156 1.55 8 205 87 209 0.165 1.95 9 214 853 219 0.171 2.34 10 229 9640 235 0.175 2.62

The rest of the instances can not be solved by CPLEX due

to lack of memory. Standard genetic algorithm is then applied to solve these instances, which is also coded in C program and executed in the same PC as the proposed parthenogenetic algorithm. The standard genetic algorithm utilizes the same chromosome representation, roulette wheel selection and elitism strategy. Ordered crossover is employed in crossover operation, and mutation operations consist of swap and bit mutation. The population size, the crossover probability, the swap probability, the probability of bit mutation and the limit

of generations are 150, 0.35, 0.05, 0.06 and 350 respectively. As shown in Table II, the proposed parthenogenetic algorithm shows better performance in the results of the instances than those of standard genetic algorithm.

TABLE II COMPARISON WITH STANDARD GENETIC ALGORITHM

Size Standard GA P-GA Ave Mean CPU (s) Ave Mean CPU (s) 11 268 0.156 265 0.187 12 285 0.187 284 0.219 13 352 0.218 348 0.218 14 366 0.203 365 0.218 15 383 0.223 383 0.281 20 523 0.258 523 0.265 25 666 0.296 664 0.281 30 782 0.328 781 0.343 35 898 0.359 898 0.390 40 1084 0.421 1080 0.421 45 1205 0.468 1200 0.437 50 1347 0.489 1345 0.515

V. CONCLUSIONS This paper focuses on raw material yard operation in dry

bulk terminals. The S-R scheduling problem is discussed and formulated as a mixed integer programming model. That deals with two S-Rs to complete a given set of tasks in a yard, which considers sequence-dependent setup times. As the problem is NP-hard in nature, a parthenogenetic algorithm is developed to solve it. The performance of the proposed parthenogenetic algorithm is evaluated by a number of test instances. The computational results show that the proposed parthenogenetic algorithm has been effective in solving the S-R scheduling problem.

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