A multi-stage dynamic soft scheduling algorithm for the uncertain steelmaking-continuous casting scheduling problem

Accepted Manuscript Title: A multi-stage dynamic soft scheduling algorithm for the uncertain steelmaking-continuous casting scheduling problem Authors: Sheng-long Jiang, Zhong Zheng, Min Liu PII: DOI: Reference:

S1568-4946(17)30427-1 http://dx.doi.org/doi:10.1016/j.asoc.2017.07.016 ASOC 4344

To appear in:

Applied Soft Computing

Received date: Revised date: Accepted date:

17-1-2017 11-6-2017 9-7-2017

Please cite this article as: Sheng-long Jiang, Zhong Zheng, Min Liu, A multi-stage dynamic soft scheduling algorithm for the uncertain steelmaking-continuous casting scheduling problem, Applied Soft Computing Journalhttp://dx.doi.org/10.1016/j.asoc.2017.07.016 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

A Multi-Stage Dynamic Soft Scheduling Algorithm for the Uncertain Steelmaking-Continuous Casting Scheduling Problem Sheng-long Jiang1,*, Zhong Zheng1, Min Liu2 1.

School of Materials Science and Engineering, Chongqing University, Chongqing 400044, PR China 2.

Department of Automation, Tsinghua University, Beijing 100084, PR China

* Corresponding author. Tel.: 086 13520412520. E-mail address: [email protected] (SH. L Jiang).

Highlights



The uncertain SCCSP is decomposed into the global and local scheduling problems.



To solve the uncertain SCCSP, a multi-stage dynamic soft scheduling (MDSS) algorithm is proposed.



To solve the global scheduling problem, a dynamic multi-objective differential evolutionary based on decomposition is proposed.



To solve the local scheduling problem, a knowledge-based differential evolutionary based on interval TOPSIS is proposed.



Computational results demonstrate that the MDSS algorithm outperforms previously described algorithms

Abstract: The steelmaking-continuous casting (SCC) manufacturing system is usually regarded as a cornerstone as well as a bottleneck in a modern integrated steel company. In this study; we consider an uncertain scheduling problem that arises from the SCC manufacturing system where the processing times and arrival times are in intervals. To solve this problem; we propose a multi-stage dynamic soft scheduling (MDSS) algorithm based on an improved differential evolution. In the proposed algorithm; the uncertain SCC scheduling problem is decomposed into global and local scheduling problems. The global scheduling problem comprising cast units is solved by a dynamic multi-objective differential evolutionary algorithm based on decomposition where each solution is evaluated in the worst-case 1

scenario. The local scheduling problem comprising charge units is solved by the knowledge-based differential evolutionary algorithm where all the solutions are sorted by the interval TOPSIS method. A modified critical ratio-based rule is also developed for real-time dispatching. Finally; computational results demonstrate that the MDSS algorithm outperforms previously described algorithms. Keywords: multi-stage optimization; scheduling; steelmaking; uncertainty modeling

1 Introduction The steelmaking-continuous casting (SCC) manufacturing system is usually regarded as a cornerstone as well as a bottleneck in a modern integrated steel company. Effective and robust scheduling approaches are crucial for SCC manufacturing systems to improve their automatic and smart performance. As shown in Fig. 1, liquid iron resources are converted into solid slabs by the SCC manufacturing system in the following stages.

 The steelmaking stage, in which liquid iron, scrap, and slagging material are mixed and transformed into liquid steel in a basic oxygen furnace (BOF). Carbon, sulfur, silicon, and other impurities in liquid steel are reduced in the BOF.  The refining stage, in which impurities are further eliminated from the liquid steel and the requisite alloy ingredients are also added to obtain the desired grade of liquid steel. More than one refining stage is required to obtain high-grade steel, such as a ladle furnace (LF) and a Ruhrstahl Hearers (RH) refining machine. 

The casting stage, in which liquid steel is cast continuously into solid slabs by a continuous caster machine (CCM) without any stoppages.

In the SCC manufacturing process, the basic production unit is called a charge, which refers to liquid steel in the same ladle. In the casting stage, the basic production unit in the casting stage is called a cast, which refers to charges set in the same tundish. The SCC scheduling problem (SCCSP) may be considered as a special version of the hybrid flow shop (HFS) scheduling problem because each charge sequentially visits all of the stages in the same direction, each of which has several identical parallel machines placed, and the following additional constraints are considered in the casting stage.

 All charges are grouped into several casts according to the life of the tundish, which is a critical component of the CCM.  All charges in the same cast must be processed consecutively on the same CCM, but the precedence of charges in each cast is given.  The sequence-independent setup time must be considered before the first charge is 2

started in each cast. Besides these computational complexities caused scheduling algorithms not easy to obtain an optimal solution of the SCCSP, the varying processing times, dynamic arrival times, unforeseen machine breakdowns and uncertain factors existed in practical SCC manufacturing systems make the optimal solution often suffers from performance deterioration even infeasibility. To overcome these challenges, we propose a multi-stage dynamic soft scheduling (MDSS) algorithm in this study which is guided by the insight of the SCCSP [1,2]. Differential evolution (DE) is one of the most powerful metaheuristic algorithm proposed by Storn and Price [3]. It has been applied to solve optimization problems from science to industry fields, and have been published in top-tier journals and conferences [4,5]. The variants of standard DE algorithm have led other similar algorithm in the competitions organized by the IEEE Congress on Evolutionary Computation (CEC) (for details please visit http://www.ntu.edu.sg/home/epnsugan/index_files/cecbenchmarking.htm). When DE is applied in solving discrete scheduling problems, most researchers have modified its crossover and mutation operators [6,7,8,9]. Unlike these fashions, we propose a DE algorithm with multi-stage optimization strategies in this study to make the MDSS achieves a better performance. The remainder of this study is organized as follows. In Section 2, we review previous studies related to the uncertain SCCSP. In Section 3, the uncertain SCCSP with complex constraints is formulated. In Section 4, based on the characteristics of the SCCSP, a soft-form schedule and a multi-stage optimization paradigm are introduced. In Section 5, the multi-stage dynamic soft scheduling (MDSS) algorithm based on DE is proposed for the uncertain SCCSP. In Section 6, we present and analyze our experimental results and comparisons. Finally, in Section 7, we provide our conclusions and suggestion for further research.

2 Literature review HFS scheduling problem is a thoroughly investigated decision-maaking problem arising from steelmaking, semiconductor, bio-pharmacy and other industries [10,11]. Because uncertainty is a common feature exists in practical production environments, the uncertain HFS scheduling problem has become a hot research topic, such as uncertain due dates [12], unforeseen breakdowns [13], stochastic and interval processing times [14,15]. As a common scheduling problem, the scheduling strategies to address uncertain HFS scheduling problem are also categorized into three types: reactive scheduling, proactive scheduling, and predictive-reactive scheduling [16].



“Reactive scheduling”, which dispatches jobs by priority rules rather than the initial 3

schedule. It is able to make a quick decision under practical environments, but unable to obtain better performance since only local information is observed and used. 

“Proactive scheduling”, which generates an initial schedule by predicting some random disturbances. It is able to keep production stability in a certain scope of uncertainty, but it always is vulnerable and conservative, because specifying all uncertain factors is impossible, and real-time information is not utilized while the jobs are being dispatched.



“Predictive-reactive scheduling”, which is the most popular approach used for handling uncertain scheduling problem in previous studies. It generates a predictive schedule in the initial stage and the initial schedule is revised during execution.

In practical SCC manufacturing systems, random disturbances and unforeseen events occur frequently, and make the initial schedule suffers from low-optimality and infeasibility. In these case, most previously scheduling algorithms investigated on the uncertain SCCSP generate an initial schedule with deterministic parameters, and rescheduling after any uncertain factor realized. To identify these uncertainties, Roy et al. developed a comprehensive knowledge model to manage some common types of disturbance and support decision-making in SCC shop [14], Hou and Li [18] comprehensively analyzed typical disturbances and their effects on the SCC shop floor. Ouelhadj et al. [19] and Cowling et al. [20] developed a multi-agent scheduling architecture for solving the dynamic steel production scheduling problem, where the initial job sequence is optimized by a Tabu search algorithm. When unforeseen events occur, all of the agents cooperate with each other based on the contract net protocol in order to find a globally near-optimal schedule. Yu and Pan [21] analyzed how an operational time delay affects the feasibility of the initial schedule and proposed a heuristic rescheduling policy, which includes batch splitting, forward scheduling, and backward scheduling. To address the similar problem, Yu et al. [22] proposed a heuristic rescheduling algorithm to quickly react uncertain factors, which is able to make the schedule feasible and optimal. Tang et al. [23] proposed an improved differential evolution algorithm with an incremental mechanism, which re-optimizes the job sequence, machine assignment, and timetable of the practical SCCSP in dynamic environments. To address the uncertain SCCSP considering machine breakdowns and processing time variations, Mao et al. [24] formulated a timeindex rescheduling problem, which is solved by an effective Lagrangian relaxation approach combined with a sub-gradient and a dynamic programming algorithm; Li et al. [25] proposed a hybrid fruit fly optimization algorithm embedded with iterated greedy (IG) local search. Moreover, other scheduling strategies implement proactive scheduling to generate initial schedules in uncertain environments. Machine breakdowns can occur each day, so Worapradya and 4

Thanakijkasem proposed a robust predictive scheduling algorithm using a minimax genetic algorithm [26]. To solve the uncertain SCCSP with unpredictable breakdowns, Worapradya and Thanakijkasem also proposed a proactive scheduling method based on a genetic algorithm, where the performance is estimated by a decomposed artificial neural network [27]. To address the SCCSP with uncertain demand, Ye et al. [28] proposed a robust optimization approach and a scenario-based stochastic programming to obtain a preventive schedule that is more robust than the nominal schedule. The SCCSP is more complex than the typical HFS scheduling problem and strong uncertainty exists, so the traditional scheduling methods mentioned above cannot directly obtain a better solution within a short time. To overcome this challenge, we introduce a soft scheduling algorithm, which exploits the rapid solving capacity of dispatch rules [29], such as the longest processing time (LPT) and the short processing time (SPT), as described in Section 4. We have published research results based on this concept, i.e., Hao et al. [30] proposed a two-layer soft scheduling approach. In the offline layer, the soft decision variable called the cast workload is optimized by a particle swarm optimization (PSO) algorithm. In the online layer, the starting time and the processing machine are dispatched by a heuristic method. Jiang et al. [31] proposed a two-phase soft optimization approach where the cast buffer ratio is treated as a characteristic index to solve the uncertain SCCSP with stochastic processing times.

3 Problem Description Because the SCC system can be identified a special hybrid flow shop, its process flow is illustrated using block diagram (as shown in Fig. 2). The liquid iron goes through the multiple stages in the SCC system, and is transformed to the final product. In this process flow, the waiting time caused temperature drop should be minimized as far as possible. According to above characteristics, we formulate the mathematical model of the SCCSP in this section.

3.1 Notations 

Set of casts, = 1,

, l, ,   .

J

Set of charges, J = 1,

Q

Set of iron resources, Q  1,

l

Charge set of cast l , where l , r is the

G

Set of stages, G  1,

Mi

Set of machines in stage i , M i  mi ,1 ,

, j, , J  .

i,

, q, , Q  .

r th charge and

, G.



, mi ,k

5



mi , M i .

l

l  J .

mri ,k

Release time of machine mi ,k .

atq

Arrival time of the q th liquid iron.

rt j

Release time of charge

Oi , j

Operation of charge

pi , j

Processing time of operation Oi , j .

tri1 ,i2

Transfer time between stage i1 and i2 .

sul

Setup time of cast l .

J i ,k

Charge set allocated on machine mi ,k .

j

Stage set of charge

up  i, j 

Upstream stage of operation Oi , j .

dw  i, j 

Downstream stage of operation Oi , j .

Lk

Cast set allocated on the machine m G ,k .

j.

j in the i th stage.

j,

 j , r is the

r th stage of charge j .

In the parameters above, we assume that both the processing time pi , j and arrival time atq are   uncertain variables, where pi , j   pi , j , pi , j  , pi , j is the standard processing time, and a is the standard arrival interval.

3.2 Decisions To obtain a executable solution for the SCCSP, the following decision variables must be determined. xi , j

Allocated machine for operation Oi , j .

yj

Liquid iron assigned for charge j .

 l1 , l2 

Two adjacent casts allocated to machine m G ,k .

si , j

Starting time for operation Oi , j .

3.3 Constraints (1) For two adjacent operations in the same charge, the next operation can be started only after the previous operation has been completed and transferred. 6

si , j  pi , j  tri ,dwi , j   sdwi , j , j , j  J , i  j \ G 

(1)

(2) A machine only process one charge at a time at most. si , j1  pi , j1  si , j2  si , j1  pi , j1  si , j2 , j1 , j2  J i ,k , j1  j2

(2)

(3) A machine is available only after its release time. si , j  mri ,k , j  J i ,k

(3)

(4) In the casting stage, two adjacent charges in the same cast must be processed continuously.

s G , j  p G , j  s G , j  p G , j ,   j, j  1 l

(4)

(5) For two casts assigned to the same CCM, the next can be started after the setup is finished only when the previous cast is finished.

s G ,

l1 , l 1

 p G ,

l1 , l 1

 sul1  s G ,

l2 , l 2

,  l1 , l2   Lk

(5)

(6) The release time of each charge is the arrival time of the assigned liquid iron resource.

rt j  at y j , j  J

(6)

(7) For each charge, the first operation can be started only after the charge is released.

s1, j  rt y j , j  J

(7)

3.4 Objectives Considering the constraints given above, the overall objective of the SCCSP considered in this study can be formulated as follows:

min f   w  f w   b  fb    w   s G , j  rj   pi , j  tri ,dwi , j  jJ  i j \ G  





l 1     b   s G , ,r 1  s G , ,r  p G , ,r l l l  l r 1 





(8)

where f w represents the waiting time cost (WTC) and f b represents the cast-break penalty (CBP). After all decision variable are determined to satisfy all constraints and minimize the objective function, the optimal solution of SCCSP can be obtained and represented by using a “Gantt chart”, as shown in Fig. 3.

7

4 Multi-stage soft scheduling In practical environments, scheduling algorithms for the SCCSP must make rapid decisions. However, the uncertain SCCSP considered in this study contains a large number of decision variables and uncertain factors. Therefore, traditional optimization algorithms have the following disadvantages.



Simulation-based optimization. An optimization algorithm must perform many repeated simulations to evaluate the expected objectives for candidate solutions, which is time consuming and difficult to apply in practical environments.



Robust optimization (RO). To obtain a robust solution with interval variables, the worst scenario where all the uncertain parameters take their worst values is optimized. This is a conservative approach for achieving better performance in practical environments.

According to these disadvantages, we propose a novel soft-form scheduling approach based on a multi-stage optimization mechanism.

4.1 Soft-form schedule In the traditional SCC solution method, the charge sequence or starting times are fixed in each stage. However, it is difficult to protect this “rigid” form of schedule against uncertain factors. Thus, we propose a novel schedule with a “soft” form, which can be applied in practical SCC systems. To reduce the problem’s complexity, the SCC system can be decomposed into two subsystems (as shown in Fig. 4): upstream stages and the last stage. After this decomposition, the starting time for each charge in the last stage can be estimated by the due-date assignment method [32] and formulated as follows:

s G , j  rj 





i  j \ G



p

i, j

where aw j is the average waiting time for charge

 



 tri ,dwi , j    j  1  aw j , j  J

(9)

j.

According to queuing theory, a CCM in the last stage can be treated as a server, and the charges in each cast processed in the CCM can be treated as a no-break server process. To reduce the CBP in the casting stage, it is necessary to keep a certain number of waiting charges to protect the cast server from a “starving” status. According to Little’s rule [33],

L  W , and thus the waiting time of each charge

helps to avoiding a break in casting because the waiting time are proportional to the queuing length.

8

According to the characteristics defined above, we propose a soft-form schedule that includes critical decisions and characteristic indices to solve the uncertain SCCSP. Instead of determining all the decision variables, the soft schedule only fixes starting times for charges in the casting stage, which are treated as critical decisions, and the average waiting times are used, which are treated as characteristic indices for determining the starting times in upstream stages. For example, the starting time of charge

j in the r th stage can be estimated as follows: G 1

s j ,r , j  s G , j 

 p

i  j ,r

i, j



 tri ,dwi , j    G  r   aw j , j  J

(10)

4.2 Multi-stage optimization According to the arrival sequence of all the casts, the uncertain SCCSP can be treated as a multistage optimization process. In particular, for the casts that have arrived, the processing time of each charge in the casting stage is known [34]. According to the RO paradigm [35], the decisions for arrived casts are “here and now” whereas the decisions for unarrived casts are “wait and see”. The conservativeness of the soft schedule obtained by the interval optimization approach can be improved using multi-stage strategies. According to assumptions given above, the uncertain SCCSP can be divided into the following two types of sub-problems based on the arrival status of the cast. The multi-stage optimization method is implemented via the decomposition way illustrated in Fig. 5).



Global scheduling problem ( P glb ): Many uncertain factors exist in an actual SCC production system, so it is difficult for the algorithm to comprehensively consider all the uncertainties that might affect the feasibility and objective of the solution. According to the “wait and see” rule in RO, we assume that the processing time in the casting stage is fixed at a standard value and we propose a multi-objective optimization



algorithm to optimize P glb by considering the WTC and CBP in the worst scenario. Local scheduling problem ( Ploc ): For arrived casts, the uncertain processing time is known or adjustable in the casting stage. According to the “here and now” rule in the RO, we construct a subproblem that only involves the charges in arrived casts, and we propose an interval-based evolution computation algorithm for optimizing Ploc by considering the interval values of the WTC and CBT.

9

5 Proposed algorithm 5.1 Main framework According to the soft-form solution and the multi-stage optimization mechanism proposed in Section 4, the MDSS algorithm (as shown in Fig. 6) based on DE is divided into two stages: the global and local optimization stage. In the first stage, a dynamic multi-objective differential evolutionary algorithm based on decomposition (MODE/D) is proposed for determining a non-dominated solution set (NDS) for all casts, where the soft schedule is formulated as the cast priorities and average waiting times for all the casts. In the local optimization stage, we propose a knowledge-based differential evolution (KBDE) algorithm to optimize the average waiting times for the charges in arrived casts. When dispatching charges online, a modified critical ratio (MCR)-based heuristic method is applied. If unforeseen events occur during the execution of the soft schedule, the MDSS algorithm reinitializes the population with the NDS and re-optimizes the soft schedule.

5.2 Global optimization stage The MODE/D algorithm is a new version of the multi-objective evolutionary algorithm [34], which decomposes a multi-objective optimization problem (MOP) into a number of single-objective optimization problems (SOPs). This decomposition reduces the algorithm’s complexity and increases the speed of convergence [36, 37]. In the present study, the dynamic MODE/D is proposed for solving the global scheduling problem Input： N glb population size, vector,



P glb and its procedure is described as follows.

T

the number of the weight vectors in the neighborhood of each weight

the probability that parent solutions are selected from the neighborhood， nb the maximal

number of solutions replaced by each child solution,  glb differential scaling, and  glb the crossover probability. Step 1：Initialization (1.1) For each id  1, 2,

, N glb randomly generate a solution.

(1.2) For each id  1, 2,

, N glb construct a neighborhood Bid .

(1.3) Initialize reference point z   z1 ,

, zd , , zD  , where zd  min1id , N

(1.4) Initialize external NDS  . Step 2：Update

10

glb

 F  X  . d

id

For

id  1 to N glb , do (2.1) Construct the differential set

E , with following equation  B  id   E glb  1, 2, , N

rand  

(11)

otherwise

(2.2) Differential evolution Set r1  id , randomly select indices r2 and r3 from E , and generate a new solution X new with the differential operator described in Subsection 5.2.3. (2.3) Evaluation: evaluate the objective F  X new  using the method in Subsection 5.2.4. (2.4) Update reference point: For d  1,

, D , if zd  Fd  X new  , then zd  Fd  X new  .

(2.5) Update solutions in the neighborhood: set

cnt  0 , and do the following steps.

(a) If cnt  nb or E  0 , go to Step 4; otherwise, randomly select an index

id from

E.

id id (b) If   X new  , z     X id  , z  , then set

cnt  cnt  1 , where the single object function 





X id  X new ,



F  X id   F  X new  ,

is defined as follows:

 X id  id , z  max  id  Fd  X id   zd 

(c) Delete

id from

E

(12)

1 d  D

and go to (a).

(2.6) Update  : delete solutions dominated by F  X new  in F  X new  . If no solution is

glb dominated by F  X new  , then put X new into  . If   N 5 , remove

 N

glb

5 solutions

using the fuzzy clustering method described in Subsection 5.2.5. Step 3：Local Search If  is not updated, to improve the depth search ability of the dynamic MODE/D, randomly select the top

5%

of the solutions from  to perform the local search procedure in Subsection 5.2.6.

Step 4：Stopping Criterion If the stop criterion is satisfied, then output  and exit; otherwise, go to step 2. Output: the non-dominated global soft schedule set  .

5.2.1 Encoding and decoding A feasible global soft schedule X can be represented by the following two-part vector: 11

X   pr1 , pr2 , , pr aw1 , aw2 , , aw   

(13)

where the first part represents the cast priorities and the second part represents the average waiting times for the casts. The priorities are real-value variables and the average waiting times are integer variables, so P glb represented by the vector above is a mixed-variable optimization problem. After determining all the priorities and average waiting times, we can construct the starting time for each cast in the soft schedule according to the following procedure. Input: soft schedule X . Step 1: Casting stage scheduling (1.1) For the finished and processing casts, keep the processing machines and start times unchanged. (1.2) For the unstarted casts, sort then in descending order based on their priorities, and select the processing machine according to the earliest available machine rule. (1.3) According to the continuity constraint on each cast, calculate the starting times for all the unstarted charges in the casting stage. Step 2: Refining stage scheduling (2.1) According to the starting time s G , j , calculate the latest completion times for all the unstarted operations in the refining stage.

ciL, j  sdw(i , j ), j  tri ,dw(i , j )  awl , j l ,1  i  G

(14)

L

(2.2) Sort all the operations in the refining stage in descending order based on ci , j , and then select the processing machine and determine the starting times according to the latest available machine (LAM) rule. Step 3: Steelmaking stage scheduling (3.1) According to the starting times in the refining stage, calculate the latest completion times for all the unstarted charges in the steelmaking stage.

c1,L j  sdw1, j , j  tr1,dwi1, j   awl , j l

12

(15)

L

(3.2) Sort all the charges in the steelmaking stage in descending order based on c1, j , and then select the processing machines in reverse order and the calculate starting times using the LAM rule. (3.3) According to the starting time for each operation and the release time for each machine, calculate the value of the overlap time:

  T  max  max  mr1,k  min  si , j  , 0   jJ1,k kM1   

(16)

Step 4: Right shift If T  0 , right shift all the unstarted operations on the CCM with T time units. Output: Starting time for each cast Sl .

5.2.2 Population Initialization To initialize the N glb -size population, perform the following steps. Step 1: Initialize cast priorities (1.1) For each completed cast: prl  2.0  1.0 Sˆl , where Sˆl is the actual starting time for cast l . (1.2) For each processing cast: prl  1.0  1.0 Sˆl . (1.3) For each unstarted cast: randomly generate a uniform distribution U  0.0,1.0  . Step 2: Initialize the average waiting times (2.1) For each completed cast: awl is set to an actual value. (2.2) For each unstarted or processing cast: randomly generate an average waiting time awl U  awmin , awmax  , where awmin and

awmax represent the minimum and maximum values, respectively.

5.2.3 Differential operator In step 2.2 of the dynamic MODE/D, the new solution X new is generated using the DE/rand/1/bin method, where



glb   X r ,k    X r2 ,k  X r3 ,k X new,k   1   X r1 ,k

13



rand   glb rand   glb

, k  1, 2  l 

(17)

The starting times of the completed and processing casts are real, so their priories remain unchanged in the differential procedure given above and average waiting times of the completed casts are fixed. The size of the soft solution X is fixed in the dynamic process, so the values of  glb and

 glb remain unchanged in MODE/D.

5.2.4 Evaluation using the worst scenarios After determining the soft schedule, if the processing times of all the operations are the minimum, w,  b, then the WTC of X id is the maximum (denoted as Fid ) and CBP is the minimum (denoted as Fid ); if w,  all the processing times are the maximum, then the WTC of X id is the minimum (denoted as Fid ) and b, CBP is the maximum (denoted as Fid ). According to these characteristics, the objective of X id includes

the worst cases for the WTC and CBP,

Fid   Fidw, , Fidb, 

(18)

5.2.5 Maintaining diversity In the standard MOEA/D, the diversity of the NDS is controlled by uniform weights instead of a maintenance policy. This method can decompose a MOP into several SOPs with a uniform Pareto frontier, but it cannot be applied to the MOP with a non-uniform frontier. To overcome this disadvantage, we employ fuzzy clustering [38] to rearrange the NDS  and improve diversity, which is implemented according to the following steps.

    1    , where 1 and  2 are the minimum and  2 

Step 1: Set the cluster number Csize  1   maximum size of each cluster, respectively.

Step 2: Initialize the fuzzy coefficient   2 ，convergence precision   0.05 , and membership matrix

 

U    uid ,ic , where 0

0



Csize ic 1

uid 0,ic  1 ; set it  0 . u   F      u   

Step 3: Calculate clustering center set V

 it 

 vic  , where vic

id 1



id 1

Step 4: Update membership matrix U

it 1

  , where u

 uid ,ic  it 1

 it 1 id ,ic

it id ,ic



it id ,ic

id

.

  C  Fid  vic    iksize 1   F v ik  id  

1

2   1    .   

 it   it 1   , then go to step 6; otherwise, it  it  1, repeat steps 3 and 4. Step 5: If V  V

Step 6: Remove the redundant elements from the cluster with the maximum size, update  , and exit.

14

5.2.6 Local search To enhance the exploitation ability of the dynamic MODE/D, we perform a local search procedure on the soft schedule X id , as follows. Step 1: Set X new  X id , and generate a random number rand   0,1 . If rand  0.5 , then go to step 2; else, go to step 3. Step 2: Randomly select two elements X id ,k1 and X id ,k2 ( 1  k1 , k2   ) from solution X id , and generate new solution X new by swapping: X new,k1  X id ,k2 , X new,k2  X id ,k1 . Go to Step 4. Step 3: Randomly select an element X id , k (   1  k  2  ) from the solution X id , and generate a new solution X new as follows:





If the cast-break penalty of cast    k  is greater than 0, X id ,k  rand X id ,k , awmax ;





otherwise, X id ,k  rand awmin , X id ,k . Step 4: Evaluate X new , and update NDS  .

5.3 Local optimization stage In this study, we assume that the CBP, which reflects the violation degree of feasibility, is more important than the WTC, which reflects the cost level. The best global soft schedule is selected from  based on the following hierarchical decision rule:

 

X best  arg min Fidw, , min  Fidb,  X id 



(19)

A feasible local soft schedule x can be represented by the following vector:



x  aw1 , aw2 ,

, aw j ,

, aw J c



where each element represents the average waiting times of casts and J c represents the set of charges that has arrived in cast set c .

5.3.1 Differential evolution According to their processing speeds, all of the casts can be divided into the following two categories.



High-speed casts with higher processing speeds than the standard value.



Low-speed casts with lower processing speeds than the standard value.

15

(20)

In each category, the average waiting time for each cast can be calculated based on the maximum H H L L and the minimum values:  awmin , awmax  and  awmin , awmax  .

After the global soft schedule has been determined, all the cast priorities remain unchanged and the average waiting times for charges in the arrived casts are optimized using the KBDE, as follows. Input: N loc population size,  loc differential scaling,  loc crossover probability. Step 1: Initialize iteration counter. Step 2: Based on the best soft solution X best , initialize N loc solutions for the arrived casts using KBDE.









(2.1)

H H If the cast is high-speed, then aw j  rand awmin , awmax .

(2.2)

L L If the cast is low-speed, then aw j  rand awmin , awmax .

Step 3: DE/best/1/z differential operator. (3.1)

Select the best solution xbest .

(3.2)

For id  1, 2,

xnew,k

, N loc ，generate a new solution xnew :



 xbest ,k   loc xr  xr  2,k 3,k   xr1 ,k



rand   loc rand   loc

, k  1, xbest 

(21)

Evaluate xnew and place it in the population.

(3.3)

Step 4: Sort all the solutions in the population and only retain N loc 2 promising solutions. Step 5: If the stop criterion is satisfied, then exit; otherwise, iter  iter  1 , and go to Step 3.

5.3.2 Evaluation and sorting In the local scheduling problem, the processing times are uncertain in the casting stage, so the upper and the lower limits of the WTC and CBP can be estimated for the solution xid , which are denoted as

f

，fidw,  and  fidb, , fidb,  , respectively. In different scenarios, these objective values

w,  id

cannot be simply tackled using the weighted sum, and the optimization algorithm for solving

Ploc must

respond rapidly; thus, we use the interval-based technique for order preference by similarity to ideal solution (TOPSIS) [39, 40] to sort all the candidate local soft schedules. The main idea is that all of the solutions are sorted according to their distance from the best and worst solutions. The sorting procedure is described as follows. w,  w,  b, b, Step 1: Calculate the normalized decision matrix  id ,  id  ,  id ,  id  .

16

 idw,  fidw,



 f w, 2   f w, 2  , id  1, id id 1   id 

, N loc

(22)

 idw,  fidw,



 f w, 2   f w, 2  , id  1, id id 1    id

, N loc

(23)

 idb,  fidb,



 f b, 2   f b, 2  , id  1, id  id 

, N loc

(24)

 idb,  fidb,



 f b, 2   f b, 2  , id  1, id  id 

, N loc

(25)

N loc

N loc

N loc id 1

N loc id 1

w,  w,  b, b, Step 2: Calculate the weighted normalized decision matrix  id , vid  , vid , vid  .  idw,   w   idw, , id  1, , N loc

(26)

 idw,   w   idw, , id  1,

, N loc

(27)

 idb,   b   idb, , id  1,

, N loc

(28)

 idb,   b   idb, , id  1,

, N loc

(29)

Step 3: Determine the positive ideal solution A and the negative ideal solution A .

A   w, , b,  



A   w, , b,  





min vidw, ,

id 1, , N loc



  max v 

max loc vidw, ,

id 1, , N

min vidb,

id 1, , N loc

id 1, , N

loc

b, id

(30)

(31)

Step 4: Calculate the separation measures using the n-dimensional Euclidean distance. 2 2 did   idw,  w,    idb,  b,   , id  1, , N loc  

(32)

2 2 did   idw,  w,    idb,0  b,   , id  1, , N loc  

(33)

12

12

Step 5: Calculate the relative closeness Rid to the ideal solution. Rid  did

d

 id

 did  , id  1,

Step 6: Rank all of the solutions based on their relative closeness Rid .

17

, N loc

(34)

5.4 Real-time dispatching After determining the best local soft schedule, all of the unstarted charges can be dispatched by the MCR rule, as follows. Input: The starting time for each charge in the casting stage, s G , j ; and the average waiting time for each charge, aw j . Step 1: Steelmaking stage scheduling At time t , when machine m1,k is released, construct the unstarted operation set





OS  Oi , j ati , j  t .

(1.1) Check if available iron resources exist. If resources exist, then go to step (1.2); else, wait. (1.2) Calculate the critical ratio (CR) of for each charge in the set OS ,

sG , j  t

CR j 





i  j \ G



 pˆ

1, j

 tri ,dwi , j 



(35)

Set the due date of O1, j , ODD1, j  CRj  pˆ i , j , where pˆ i , j is the realized processing time.





t Calculate the priorities of the charges based on CR+SPT rule,  j  max ODD1, j , pˆ i , j .

* Select Oi , j for processing on m1,k , Oi , j  arg Omin OS

*

i, j

  . t j

Set rt j  atq . Step 2: Refining stage scheduling At time t , when machine mi ,k is released, select the charge using the SPT rule. Step 3: Casting stage scheduling At time t , when machine m G , k is released, select charge j * , which is the first to arrive. (3.1) If j * is null, and no cast is processed on m G , k , then keep it idle. * (3.2) If j * is the first charge in cast l * , then the processing cast on m G , k is l * , where j l* ; and

set its starting time for OG , j* using s G , j*  etk  sul* , where etk is the earliest available time of m G , k .

(3.3) If j * is not the first charge in cast l * , then set the starting time for OG , j* using s G , j*  t ; if

j * is not the first charge in cast l * , then the processing cast on m G ,k is null. 18

6 Computational Results 6.1 Instance generation Two steelmaking manufacturing systems with three stages and four stages were considered to generate the test instances. The parameters employed for generating the instances are shown in Table 1. As shown in Table 2, the processing times in each stage were extracted from the statistical results in a manufacturing execution system database. The noise levels of charges in the same cast and the same stage were equal. In stage i , the interval processing time for Oi , j is defined as follows:  pi, j , pi, j   1 i ,l  pi , j , 1  i ,l  pi , j  ,

j l

(36)

In the following experiments, the realized processing times were generated using a

truncated normal distribution N  pi , j ,1 [41], according to the algorithmic code provided by Dollé and Maze (http://miv.u-strasbg.fr/mazet/rtnorm/). For the 3-stage and 4-stage systems, the number of arrived iron resources was set to three and their intervals were atq ~ Exp  20  and

atq ~ Exp  30  ; thus, nine and six liquid iron resources arrived at each system every hour,

respectively.

6.2 Parameter Tuning A metaheuristic algorithm with well-tailored parameter values can achieve better performance than on with poorly inappropriate values. In this section, we select appropriate parameter levels in proposed algorithm with recommendation from literature and design of experiment (DOE). In practical environment, the scheduling decision should be made a quick response. In the global optimization stage, the time limit of dynamic MODE/D (Tm) is set to be 120s. In the local optimization stage, the time limit of KBDE is set to be 20s. According to the recommendation by Li and Zhang [42], the control parameters of MODE/D are set as follows: 

Neighborhood size: T  20 .



Selection probability:   0.9 .



Maximum value for the updated sub-problem: nb  2 .

To tune appropriate parameter values for DE in global and local stage, we have applied the Taguchi design method to choose a set of recommended values. In following experiments, we consider 6 parameters (population size, differential scale, crossover probability). each of them 19

has 3 value level. Then the orthogonal array L27(36), which means an instance with 27 parameter combinations has to be tested. A 8×10 instance is adopted for the tested in the 3-stage and 4stage manufacturing system is adopted for the test, and the result data (S/N ratios, i.e. signal-tonoise ratios) based on 10 independent runs are shown in Fig. 7 (output by the Minitab® software). As the figure suggests, the initial parameter settings for MODE/D an KBDE are as follows: 

Differential step:  glb  0.5



Crossover probability:  glb  0.6



Population size: N glb  200



Mating probability:  loc  0.5



Crossover probability:  loc  0.5



Population size: N loc  60

6.3 Basic components test We tested the basic components of the MDSS, such as the effectiveness of object evaluation, convergence of the dynamic MODE/D, and exploitation of the KBDE. In the global optimization stage, the MDSS optimizes the worst scenario in an uncertain scheduling problem, thereby estimating the upper limits of the WTC and CBP by processing the time intervals. To validate the upper estimate, we calculated the hit ratio (HR) as follows:

HR 

CN  f wg  f wg ,

fbg  fbg , 

(37)

CN

where CN are total number of simulation replicates and CN 

 is a replicate that satisfied some

constraints.

For the three-stage and four-stage systems, we selected a 6×10 instance and randomly generated 10 different soft schedules. For uncertain environments with noise levels of 0.05, 0.10, and 0.15, each soft schedule was simulated 500 times and we calculated its mean HR using Equation (9). As shown in Table 3, the mean HR for most instances was approximately 1.0, and thus the estimated upper limit of MDSS can be considered effective. According to the results in Table 3, we may also conclude that the mean HR decreased as the noise level increased.

20

To validate the convergence performance of the dynamic MODE/D, we calculated the crowding distance metric proposed by Deb et al. [43]. The sigma-based multi-objective PSO (MOPSO) algorithm [44] and the Pareto-based multi-objective differential evolutionary algorithm (MODE) [45] were compared with the dynamic MODE/D. A 6×10 instance was tested using both the three-stage and fourstage systems. We also fixed the noise level at i ,l  0.1 and ran each algorithm 20 times. According to the comparison of the results shown in Fig. 8, the dynamic MODE/D exhibited better convergence performance than MOPSO and MODE. In the local optimization stage, KBDE uses a knowledge-based population initialization method, where the maximum and the minimum values of the average waiting time are calculated based on historical production data. In general, the population in the differential evolution stage is often initialized randomly from its solution space. To analyze the exploitation performance of the knowledgebased initialization method, we compared the proposed KBDE with random initialization-based differential evolution (RBDE). We assumed that the cast priorities were fixed and set all the noise levels to 0.1. Each algorithm was run for 60 iterations and the convergence curves for the total objective in the worst case are shown in Fig. 8, which demonstrate that the exploitation performance of KBDE was better than that of RBDE.

6.4 Comparison of the results In the following experiments, 20 instances were generated for both the three-stage and four-stage systems, where numbers of the casts were 6,7,8,9,10 , the number of charges in each cast was

generated from a uniform distribution U  8,15 , and each cast level was generated for four instances.

6.4.1 Comparison with Cooperative Co-evolutionary Artificial Bee Colony (CCABC) Pan [46] decomposed the deterministic counterpart to the uncertain SCCSP into two subproblems, i.e., a cast scheduling problem with parallel machines and a charge scheduling problem with a hybrid flow shop, and proposed a CCABC algorithm. Numerical computations based on both synthetic and realworld instances showed that the CCABC is effective for solving the deterministic SCCSP. In the static CCABC algorithm, all of the uncertain parameters are unknown and they are assumed to have a standard value, so the final solution is relatively poor. If all the uncertain parameters are known, the final solution obtained by the CCABC is relatively optimal. To test the performance of the proposed MDSS, we calculated the objective values optimized using the CCABC with standard parameters and actual parameters, i.e., CCABC-S and CCABC-R, respectively, and the results are given in Tables 4 and 5.

21

RPD 

f  f min 100% f min

(38)

In Tables 4 and 5, f w , f b , and f represent the mean values for the WTC, CBP, and the total objective (the best result is marked in bold), respectively, which were calculated by running each algorithm 10 times. The statistical results (averages and confidence intervals) in terms of the relative percentage deviation (RPD) index are shown at the bottom of each table. Box plots for the RPD are shown in Fig. 8 where the CBP values are very large because CCABC-S does not consider the uncertainty of the arrival times and processing times. By contrast, the CCABC-R knows all these uncertainties, so no CBP is generated. Our comparison of these two algorithms shows that the MDSS improves the CBP at the expense of a larger WTC, and thus the total objective value is improved. Table 5 Comparison of the results obtained using the CCABC (four-stage system)

6.4.2 Comparison with similar algorithms No previous studies have focused on the problem considered, so we constructed a two-stage PSO (2-PSO) algorithm and a two-stage differential evolution (2-DE) algorithm, both of which are based on a multi-stage optimization policy.

 2-PSO where the global scheduling problem is solved by the sigma-based MOPSO and the local scheduling problem is solved by PSO based on the optimal computation budget allocation (OCBA) method [47]. 

2-DE where the global scheduling problem is solved by the Pareto-based MODE and the local scheduling problem is solved by differential evolution based on the upper confidence bound (UCB1) method [48].

In Tables 6 and 7, f w , f b , and f represent the mean values of the WTC, CBP, and the total objective (the best result is marked in bold), respectively, which were calculated by running each algorithm 10 times. The statistical results for the RPD (averages and confidence intervals) are shown at the bottom of each table. Box plots of the RPD are shown in Fig.10, which indicate that the MDSS obtained better results in terms of both the WTC and the CBP. The results were better in the following respects: (1) the multi-objective convergence performance of the dynamic MODE/D was better than that of MOPSO and MODE; and (2) compared with the OCBA-based PSO and UCB1-based DE, the TOPSISbased KBDE exhibited faster convergence.

22

Table 7 Comparison with the results obtained by similar algorithms (4-stage system)

7 Conclusion and further research In this paper, we studied the uncertain SCCSP with interval processing times and interval arrival times. Based on the problem-specific characteristics, we constructed a soft-form schedule where cast priorities are treated as critical decisions and average waiting times are treated as characteristic indices, and we proposed the MDSS algorithm based on a multi-stage DE. To solve the global scheduling problem, we developed a worst scenario-based dynamic MODE/D algorithm. To solve the local scheduling problem, we presented an interval TOPSIS-based KDBE algorithm. In the real-time dispatching process, we use the MCR rule to determine other non-critical decisions. Finally, our computational results demonstrate that the MDSS algorithm is effective and it outperforms other similar algorithms. In the proposed algorithm, the standard DE algorithm is modified by reflecting the structural characteristics of the soft-form schedule and running characteristic of the multi-stage optimization. To improve the performance of the MDSS, future research should consider the following issues. 1) Only the interval uncertainty set is selected in the MDSS and the worst scenario is a minor probability event, so other types of uncertainty set should be considered, such as ellipsoidal, polyhedral, and norm sets. 2) When a soft schedule becomes infeasible or degraded, the MDSS algorithm can dynamically optimize the uncertain SCCSP. However, it certainly lags behind the changes in the practical environment. Therefore, predicting environmental change is another challenging problem.

Acknowledgements We thank the anonymous reviewers and the editors for their constructive and pertinent comments. This study was supported by the Fundamental Research Funds for the Central Universities (No. 106112017CDJXY), the National Natural Science Foundation of China (No. 51474044).

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[21] Yu S, Chai T, Tang Y. An Effective Heuristic Rescheduling Method for Steelmaking and Continuous Casting Production Process With Multirefining Modes[J]. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2016, 46(12): 1675-1688. [22] Tang, L., Zhao, Y., & Liu, J. (2014). An improved differential evolution algorithm for practical dynamic scheduling in steelmaking-continuous casting production. IEEE Transactions on Evolutionary Computation, 18(2), 209-225. [23] Mao, K., Pan, Q. K., Pang, X., & Chai, T. (2014). An effective Lagrangian relaxation approach for rescheduling a steelmaking-continuous casting process. Control Engineering Practice, 30, 67-77. [24] Li, J. Q., Pan, Q. K., & Mao, K. (2016). A hybrid fruit fly optimization algorithm for the realistic hybrid flowshop rescheduling problem in steelmaking systems. IEEE Transactions on Automation Science & Engineering, 13(2), 932-949. [25] Hou, D. L., & Li, T. K. (2012). Analysis of random disturbances on shop floor in modern steel production dynamic environment. Procedia Engineering, 29, 663-667. [26] Worapradya, K., & Thanakijkasem, P. (2010, December). Worst case performance scheduling facing uncertain disruption in a continuous casting process. 2010 IEEE International Conference on Industrial Engineering and Engineering Management (IEEM), (pp. 291-295). [27] Worapradya, K., & Thanakijkasem, P. (2015). Proactive Scheduling for Steelmaking-Continuous Casting Plant with Uncertain Machine Breakdown Using Distribution-Based Robustness and Decomposed Artificial Neural Network. Asia-Pacific Journal of Operational Research, 32(02), 1550010. [28] Ye, Y., Li, J., Li, Z., Tang, Q., Xiao, X., & Floudas, C. A. (2014). Robust optimization and stochastic programming approaches for medium-term production scheduling of a large-scale steelmaking continuous casting process under demand uncertainty. Computers & Chemical Engineering, 66, 165-185. [29] Cowling, P., & Johansson, M. (2002). Using real time information for effective dynamic scheduling. European journal of operational research, 139(2), 230-244. [30] Hao, J., Liu, M., Jiang, S., & Wu, C. (2015). A soft-decision based two-layered scheduling approach for uncertain steelmaking-continuous casting process. European Journal of Operational Research, 244(3), 966-979. [31] Jiang, S., Liu, M., & Hao, J. (2017). A two-phase soft optimization method for the uncertain scheduling problem in the steelmaking industry. IEEE Transactions on Systems Man & Cybernetics: Systems, 47(3), 416-431. [32] Enns, S. T. (1995). A dynamic forecasting model for job shop flowtime prediction and tardiness control. The international journal of production research, 33(5), 1295-1312. [33] Little, J. D. (1961). A proof for the queuing formula: L= λ W. Operations research, 9(3), 383-387. [34] Zhang, Y., & Dudzic, M. S. (2006). Industrial application of multivariate SPC to continuous caster startup operations for breakout prevention. Control Engineering Practice, 14(11), 1357-1375. [35] Gorissen, B. L., Yanıkoğlu, İ., & den Hertog, D. (2015). A practical guide to robust optimization. Omega, 53, 124-137. [36] Zhang, Q., & Li, H. (2007). MOEA/D: A multiobjective evolutionary algorithm based on decomposition. IEEE Transactions on evolutionary computation, 11(6), 712-731. [37] Li Y.R., Zhou Y.R., Zhan Z.H., Zhang J. (2016). A primary theoretical study on decomposition-based multiobjective evolutionary algorithm, 20(4), 563 - 576

25

[38] Bezdek, J. C. (2013). Pattern recognition with fuzzy objective function algorithms. Springer Science & Business Media. [39] Behzadian, M., Otaghsara, S. K., Yazdani, M., & Ignatius, J. (2012). A state-of the-art survey of TOPSIS applications. Expert Systems with Applications, 39(17), 13051-13069. [40] Jahanshahloo, G. R., Lotfi, F. H., & Izadikhah, M. (2006). An algorithmic method to extend TOPSIS for decision-making problems with interval data. Applied mathematics and computation, 175(2), 1375-1384. [41] Chopin, N. (2011). Fast simulation of truncated Gaussian distributions. Statistics and Computing, 21(2), 275-288. [42] Li, H., & Zhang, Q. (2009). Multiobjective optimization problems with complicated pareto sets, MOEA/D and NSGA-II. IEEE Transactions on Evolutionary Computation, 13(2), 284-302. [43] Deb, K., Pratap, A., Agarwal, S., & Meyarivan, T. A. M. T. (2002). A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE transactions on evolutionary computation, 6(2), 182-197. [44] Moslehi, G., & Mahnam, M. (2011). A Pareto approach to multi-objective flexible job-shop scheduling problem using particle swarm optimization and local search. International Journal of Production Economics, 129(1), 14-22. [45] Qian, B., Wang, L., Hu, R., Wang, W. L., Huang, D. X., & Wang, X. (2008). A hybrid differential evolution method for permutation flow-shop scheduling. The International Journal of Advanced Manufacturing Technology, 38(7-8), 757-777. [46] Pan, Q. K. (2016). An effective co-evolutionary artificial bee colony algorithm for steelmakingcontinuous casting scheduling. European Journal of Operational Research, 250(3), 702-714. [47] Zhang, R., Song, S., & Wu, C. (2012). A two-stage hybrid particle swarm optimization algorithm for the stochastic job shop scheduling problem. Knowledge-Based Systems, 27, 393-406. [48] Zhang, R., Song, S., & Wu, C. (2013). A hybrid differential evolution algorithm for job shop scheduling problems with expected total tardiness criterion. Applied Soft Computing, 13(3), 1448-1458.

26

SCC Manufacturing System

Tundish Slab

BOF

……

RH

LF

Refining

Steelmaking

Billet

CCM

Casting

Fig. 1. A typical process flow for an SCC manufacturing system

Liquid iron resources

Steelmaking

Refining 1

1# BOF

1# LF)

Refining 2

Casting 1# CC

1# RH new

1

…

n

2# BOF

2# LF

2# CC 2# RH

3# BOF

3# LF

3# CC

Fig. 2. The process diagram of SCC as a HFS

M11 M21 M22 0 M31

O(1,1)

O(1,3)

O(1,2)

O(2,1)

O(1,4)

O(1,5)

O(2,2)

O(2,3)

O(1,6)

Pu

O(2,5)

O(2,4)

O(3,1)

O(2,6)

O(3,2)

setup Pl

M32

O(3,3)

O(3,4)

O(3,5)

Fig. 3. Gantt chart representation of the SCC solution

27

O(3,6)

pi,j

rj

Q1

O1,1

O2,1

Q2

O1,2

O2,2

Q3

O1,3

O2,3

p*g,j

O3,1

O3,2

O3,3

last stage

wi,j

upstream stages

Fig. 4. The decomposed SCC system

New cast is arriving

Realized processing times in all stages

Standard processing times in the casting stage

Standard processing times in the casting stage

Finished cast set

Arrived cast set

Unarrived cast set

Local scheduling problem

Soft-form schedule

Global scheduling problem

Interval values of WTC and CBP The worst values of WTC and CBP

Fig. 5. The paradigm for multi-stage optimization

28

Global Op tim iz at ion

Initialization dynamic multi-objective differential evolution based on decomposition (MODE/D)

Loc al Opt imiz at io n

Non-dominant solution set (NDS)

Reinitialization

Min-max global optimal solution

Yes

knowledge-based differential evolution（KBDE）

No

Is event happens?

Real-time dispatching with the MCR rule

Dynamic events Steelmaking

Refining

Casting

Fig. 6. Overview of the MDSS algorithm or new cast arrived

(a) 3-stage system

(b) 4-stage system

Fig. 7. Average S/N ratio for each level of the parameters.

29

8.5

8.5

7.5

8

7

7.5

distance metric

distance metric

8

6.5 6

7 6.5

5.5

6

5

5.5 4.5

5 MODE/D

MOPSO

(a)

MODE

MODE/D

MOPSO

(b)

3-stage system

MODE

4-stage system

Fig. 8. Comparison of the performance using each algorithm

1400

1800

RBDE KBDE

RBDE KBDE

1750

1350

1700 1650

objective

objective

1300 1600 1550

1250 1500 1450

1200

1400

1150

0

10

20

(a)

30 iteration

40

50

1350

60

0

10

20

(b)

3-stage system

30 iteration

40

50

60

4-stage system

Fig. 7. Convergence curves for KBDE and RBDE

MDSS No. 1#

CCABC-S

fw

fb

2535

0

f

2535

fw

fb

2266

800

30

CCABC-R f

3066

fw

fb

2550

0

f

2550

2#

2102

0

2102

2029

1700

3729

2159

0

2159

3#

2447

0

2447

2443

500

2943

2565

0

2565

4#

2243

0

2243

2223

700

2923

2794

0

2794

5#

4158

0

4158

3622

1000

4622

4317

0

4317

6#

3241

0

3241

2952

900

3852

3510

0

3510

7#

2451

600

3051

2737

1800

4537

2798

0

2798

8#

3440

300

3740

3320

1200

4520

4040

0

4040

9#

3411

0

3411

3498

3100

6598

3396

0

3396

10#

3634

0

3634

2948

3300

6248

3816

0

3816

11#

4050

700

4750

3230

3100

6330

4833

0

4833

12#

3548

0

3548

3461

1700

5161

4614

0

4614

13#

5152

0

5152

4831

2100

6931

5202

0

5202

14#

5069

0

5069

4935

1300

6235

5761

0

5761

15#

4669

0

4669

4105

2000

6105

4960

0

4960

16#

5927

0

5927

5314

2800

8114

6172

0

6172

17#

6387

0

6387

5734

3300

9034

7333

0

7333

18#

5926

0

5926

5417

3000

8417

7059

0

7059

19#

5591

400

5991

5188

3900

9088

6537

0

6537

20#

6254

0

6254

6057

2000

8057

7570

0

7570

AVG.

4112

100

4212

3816

2010

5826

4599

0

4599

AVGRPD

8.14

-

0.47

0.73

-

39.80

20.33

-

8.94

95% lower

4.72

-

0.00

0.00

-

29.56

14.97

-

4.82

95% upper

11.56

-

1.42

1.98

-

50.05

25.68

-

13.05

45 100

40 90 80

30

70

25

60

RPD

RPD

35

20

50 40

15 30

10

20 10

5

0

0 MDSS

(a)

CCABC-S

 

MDSS

CCABC-R

(b)

RPD f w in the three-stage stage

31

CCABC-S

 

CCABC-R

RPD f in the three-stage stage

50 90

45 80

40 70

35

60

RPD

RPD

30 25

50 40

20 15

30

10

20

5

10

0

0

MDSS

(c)

CCABC-S

 

CCABC-R

MDSS

RPD f w in the four-stage stage

(d)

CCABC-S

 

RPD f in the four-stage stage

Fig. 8. RPD box plots for the results obtained using the CCABC

32

CCABC-R

25

15

10

RPD

RPD

20

15 10

5 5

0

0

MDSS

(a)

 

2-DE

2-PSO

MDSS

RPD f w in the three-stage system

(b)

 

2-DE

2-PSO

RPD f in the three-stage system

40

25

35 30 25

15

RPD

RPD

20

10

20 15 10

5

5

0

0

MDSS

(c)

 

2-DE

MDSS

2-PSO

(d)

RPD f w in the four-stage system

 

2-DE

RPD f in the four-stage system

Fig. 10. RPD box plots of the results obtained by similar algorithms

33

2-PSO

Table 1 Parameters for generating instances Parameter

3-stage

4-stage

Stage number

3

4

Parallel machine number in each stage

{3,4,3}

{3,4,2,3}

Release time of each machine

0

0

Skip probability of the LF or RH stage

[LF=0.3]

[RH= 0.6]

Set up time of each cast

40 min

60 min

Transfer time between two adjacent stages

5 min

8 min

Coefficients of objectives (  w ,  b )

(1.0, 100.0)

(1.0, 100.0)

Table 2 Parameters for generating the processing times 3-stage system

Stage

4-stage system

Standard

Noise level

Standard

Noise level

Steelmaking

U(20, 25) min

U(0.0, 0.15)

U(30,35) min

U(0.0, 0.15)

LF refining

U(20, 40) min

U(0.1, 0.3)

U(25,50) min

U(0.1, 0.3)

RH refining

-

-

U(25,35) min

U(0.1, 0.3)

Casting

U(25, 35) min

U(0.0, 0.2)

U(35,45) min

U(0.0, 0.2)

Table 3 Parametric levels for DEs Parameter



Levels

glb

0.4 0.5 0.6

 glb

0.4 0.5 0.6

N glb

100 200 300

 loc

0.4 0.5 0.6

 loc

0.4 0.5 0.6

N loc

40 50 60

34

Table 3 Mean hit ratios for the test instances 3-stage system

4-stage system

No.

  0.05

  0.10

  0.15

  0.05

  0.10

  0.15

1# 2# 3# 4# 5# 6# 7# 8# 9# 10# 11# 12#

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 0.9891 1.0000 1.0000 1.0000 0.9931 1.0000 1.0000 1.0000 1.0000 1.0000

0.9766 1.0000 1.0000 0.9924 0.9667 1.0000 1.0000 0.9890 1.0000 0.9885 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 0.9718 1.0000 0.9891 1.0000 1.0000 0.9789 1.0000 0.9682 1.0000 1.0000 0.9676

1.0000 0.9609 1.0000 0.9535 0.9774 1.0000 0.9864 0.9634 1.0000 1.0000 0.9630 0.9562

13#

1.0000

1.0000

0.9608

1.0000

1.0000

1.0000

14# 15# 16# 17# 18# 19# 20# AVG.

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.9724 0.9872 1.0000 1.0000 1.0000 1.0000 1.0000 0.9971

1.0000 0.9630 1.0000 0.9518 1.0000 1.0000 1.0000 0.9894

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.9748 1.0000 1.0000 1.0000 1.0000 0.9622 1.0000 0.9906

0.9802 1.0000 1.0000 1.0000 0.9783 0.9575 1.0000 0.9838

Table 4 Comparison of the results obtained using the CCABC (three-stage system) MDSS No.

fw

fb

1# 2#

1788 1970

3#

2280

0 0 500

4#

1987

5#

CCABC-S f

CCABC-R f

fw

fb

1788 1970

1575 1801

1000 1400

2575 3201

2780

2116

2600

4716

1634 2336

0

1987

1481

1000

2481

2113

2448

0

2448

2361

900

3261

6#

2477

0

2477

2275

1900

7#

1663

0

1663

1659

8#

2583

2583

9# 10#

2971 2772

0 400

3371 2772

0

f

fw

fb

2106

0 0

2106

0 0

2336 2113

2454

0

2454

4175

2988

0

2988

1600

3259

2377

0

2377

2532

1500

4032

2765

0

2765

2633 2697

2300 1600

4933 4297

3410 2988

0 0

3410 2988

35

1634

11#

3711

0

3711

3396

1500

4896

4139

0

4139

12#

3697

0

3697

3387

1600

4987

3884

0

3884

13#

3917

0

3917

3734

2400

6134

4421

0

4421

14#

3730

0

3730

3473

2300

5773

3946

0

3946

15#

4935

0

4935

3772

2700

6472

5320

0

5320

16# 17#

2973 3863

0

2973

2469

3863

3182 4339

0

3182 4339

18#

4257

4557

3692 4283

3969 6392

0

0 300

1500 2700 2200

6483

0

4246

19#

4374

0

4374

4266

2700

6966

4246 4759

0

4759

20#

3569

3569

3275

2300

5575

3841

0

3841

AVG.

3098

0 60

3158

2843

1885

4728

3362

0

3362

AVGRPD

10.24

-

2.34

0.55

-

54.92

19.98

-

9.14

95% lower

5.78

-

0.00

0.00

-

44.49

13.64

-

4.55

95% upper

14.71

-

5.24

1.62

-

65.33

26.33

-

13.72

Table 4 Comparison of the results obtained using the CCABC (three-stage system) MDSS No.

fw

fb

1#

1788

0

2#

1970

3#

2280

0 500

4#

1987

5#

CCABC-S f

fw

fb

1788 1970

1575 1801

1000

2780

0

2448

6#

CCABC-R f

f

fw

fb

2575

2106

0

2106

1400

3201

0

1634

2116

2600

4716

1634 2336

0

1987

1481

1000

2481

2113

0

2336 2113

0

2448

2361

900

3261

2454

0

2454

2477

0

2477

2275

1900

4175

2988

0

2988

7# 8#

1663 2583

1663 2583

1659 2532

1600 1500

3259 4032

2377 2765

0 0

2377 2765

9#

2971

0 0 400

3371

2633

2300

4933

3410

0

3410

10#

2772

0

2772

2697

1600

4297

2988

0

2988

11#

3711

0

3711

3396

1500

4896

4139

0

4139

12#

3697

0

3697

3387

1600

4987

3884

0

3884

13#

3917

0

3917

3734

2400

6134

4421

0

4421

14#

3730

0

3730

3473

2300

5773

3946

0

3946

15#

4935

0

4935

3772

2700

6472

5320

0

5320

16#

2973

0

2973

2469

1500

3969

3182

0

3182

17#

3863

3863

6392

4339

0

4339

4257

4557

3692 4283

2700

18#

0 300

2200

6483

0

4246

19# 20#

4374 3569

4374 3569

4266 3275

2700 2300

6966 5575

0 0

4759 3841

AVG.

3098

0 0 60

4246 4759 3841

3158

2843

1885

4728

3362

0

3362

AVGRPD

10.24

-

2.34

0.55

-

54.92

19.98

-

9.14

36

95% lower

5.78

-

0.00

0.00

-

44.49

13.64

-

4.55

95% upper

14.71

-

5.24

1.62

-

65.33

26.33

-

13.72

Table 6 Comparison with the results obtained using similar algorithms (three-stage system) MDSS No.

2-PSO

2-DE

fw

fb

f

fw

fb

f

fw

fb

f

1# 2#

1760

0

1760

300 0

0

1979 2068

2509

0

2509

2217

4#

2184 2111

2040 2484

1979 2068

3#

0 300

1967 2394

0

2040

1667 2394

0

2111

0

2179

2365 2393

0 400

2365 2793

2258

0 600

2032 2466 2858

2217 2479

5# 6#

2032 2466

0 300

2516 2334

0 700

2516 3034

7#

1631

0

1631

0

1808

1850

0

1850

8#

2708

0

2708

1808 3077

0

3077

2982

0

2982

9# 10#

2993 2682

0 0

2993 2682

3306

0 800

3306 3394

3237 2663

3237 2663

11#

3844

0

0

4175

3793

12#

3756

3885

0

3885

3721

4443

4198

0

4198

14#

3743 3892

3721 4089

0

13#

0 700

3844 3756

0 0 600

0

3892

0

3838

5080

0

5080

0

3726 5924

4089 4638

15#

3726 5924

0 800

5182

16#

2994

0

2994

3239

0

3239

2998

0 500

17# 18#

0 0

0 0

4079 4681

4595 4431

4402

3657

0

3657

3139

70

3209

130

3428

3269

0 190

3966

AVG.

3558 3298

4402 4458

4737

20#

0 900

0 0 900

4595 4431

0

4016 4300 4630

4079 4681

19#

4016 4300 4630

AVGRPD

1.68

-

1.79

6.67

-

9.20

6.19

-

9.70

95% lower

0.63

-

0.21

3.71

-

5.62

3.67

-

5.71

95% upper

2.72

-

3.38

9.62

-

12.79

8.72

-

13.70

2594 4175

37

5182

3966

4393

3498

5637 3459

Table 7 Comparison with the results obtained by similar algorithms (4-stage system) MDSS No.

2-PSO

2-DE

fw

fb

f

fw

fb

f

fw

fb

f

1#

2382

0

2579

2495

0

2495

2097 2432

2382 2497

2579

2#

0 400

2396

0

2396

2326

0

2432

0

0 300

2222 4518

2617 2838

2366 2577

0

2326 3466 2577

5#

2222 4218

2617 2838

0 1100

4426

4426

6#

3341

0

3341

3091

0 900

3991

3874 3341

0 100

3874 3441

7#

2467

0

2467

600

2996

2672

0

2672

8#

3338

0

3338

2396 3624

0

3624

3608

0

3608

0 0

3893

4130

800

4930

3700

0

0

13#

5174

0

5174

3464 5491

3464 5491

3862 3584

0 0 500

3893

12#

3519 4070 3700

4043

11#

3519 4070

0 500

3493

10#

3493 3543

0 700

3233

9#

3233 3706

14# 15#

5314 4914

0 0

5314 4914

5266

5266 4847

4926 4945

0

4956 5726 4945

16#

5729 6578

0

6573

6557

0

6557

0

5729 6578

4847 5873

0 0 700

0 800

6126

0

6126

6203

0

6203

18# 19#

6045 5553

0 600

6045 6153

5919 5700

0 0

6468 6002

6386

0

6386

6423

5919 5700 7623

6468 6002

20#

0 0 1200

6985

6985

AVG.

4154

65

4219

4212

235

4447

4249

0 160

AVGRPD

3.23

-

3.34

5.91

-

10.06

5.80

-

8.98

95% lower

1.75

-

1.34

2.47

-

5.56

3.30

-

3.99

95% upper

4.72

-

5.35

9.36

-

14.67

8.29

-

13.97

3# 4#

17#

38

0

0

4956

4406 3862 4084

4409