A preference-inspired multi-objective soft scheduling algorithm for the practical steelmaking-continuous casting production

Accepted Manuscript A Preference-Inspired Multi-Objective Soft Scheduling Algorithm for the Practical Steelmaking-Continuous Casting Production Sheng-Long Jiang, Zhong Zheng, Min Liu PII: DOI: Reference:

S0360-8352(17)30515-6 https://doi.org/10.1016/j.cie.2017.10.028 CAIE 4965

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Computers & Industrial Engineering

Received Date: Accepted Date:

25 January 2017 28 October 2017

Please cite this article as: Jiang, S-L., Zheng, Z., Liu, M., A Preference-Inspired Multi-Objective Soft Scheduling Algorithm for the Practical Steelmaking-Continuous Casting Production, Computers & Industrial Engineering (2017), doi: https://doi.org/10.1016/j.cie.2017.10.028

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A Preference-Inspired Multi-Objective Soft Scheduling Algorithm for the Practical Steelmaking-Continuous Casting Production Sheng-Long Jianga,1, Zhong Zhenga, Min Liub, a. College of Material Science and Engineering, Chongqing University, Chongqing 400044, PR China b. Department of Automation, Tsinghua University, Beijing 100084, PR China

1

Corresponding authors. Tel.: 086 18223235220. E-mail address: [email protected], [email protected] (SH.L Jiang). 1

A Preference-Inspired Multi-Objective Soft Scheduling Algorithm for the Practical Steelmaking-Continuous Casting Production

Abstract Uncertainty is the most challenging problem for implementing scheduling algorithms under practical environments, since the schedule released into a shop floor with optimal objectives often deteriorates or even become infeasible during its execution period. This paper focuses on the uncertain scheduling problem arising from the steelmaking-continuous casting (SCC) manufacturing system and propose a multi-objective soft scheduling (MOSS) to overcome this challenge. In this study, a soft-form schedule including critical decisions and characteristic indicators is introduced to provide more flexibility against random disturbances. In the MOSS algorithm, we proposed a preference-inspired chemical reaction optimization (PICRO) algorithm to solve the uncertain SCC scheduling problem with soft-form solutions, in which the objectives of waiting time, cast-break and over-waiting are tackled by the preference-inspired method. In the PICRO, a simulation-

Preprint submitted to Computers & Industrial Engineering

October 7, 2017

based T-test method is use to evaluate solutions, and a knowledge-based local search (KLS) is embedded to enhance the convergence of PICRO. Following this, a clean-up procedure is proposed for ranking and selecting the best solutions in the final population output by the PICRO. Computational experiments for the randomly generated SCC scheduling instances demonstrate that the proposed MOSS algorithm can result in significantly better solutions compared to other algorithms under practical environments. Keywords: steelmaking, uncertain scheduling, preference, CRO

1

1. Introduction

2

Scheduling is an important decision-making process for most manufac-

3

turing systems, particularly in resource and energy-intensive industries such

4

as iron & steel, chemical process, non-ferrous metal, and electric power. Ef-

5

fective and efficient scheduling can significantly improve the manufacturing

6

cycle, rate of timely delivery, energy consumption, and other key performance

7

indicators (KPIs). The iron & steel industry is a typical resource and energy-

8

intensive industry, which determines the growth of automobile, construction,

9

transportation, military and other industries in the world.

10

This study focuses on a challenging scheduling problem arising from the

11

practical steelmaking-continuous casting (SCC) manufacturing system in the 2

12

iron & steel industry. The typical SCC manufacturing system primarily in-

13

cludes three stages of steelmaking, refining and casting, as shown in Figure

14

1. First, in the steelmaking stage, the incoming hot iron, which contains ex-

15

cessive amounts of carbon, silicon, phosphorus and other impurity elements,

16

is smelted through a basic oxygen furnace (BOF), and then the liquid steel

17

is produced and poured into an empty ladle. Next, the smelted liquid steel

18

in the ladle is transported to the refining stage by a crane. In this stage, the

19

liquid steel is further smelted to produce a specific steel grade by adjusting its

20

chemical compositions and temperature. For producing more high-precision

21

steel grades, the liquid steel should visits multiple machines, such as ladle

22

furnace (LF) , Ruhrstahl-Hausen vacuum (RH) and other refining furnaces.

23

Finally, the refined liquid steel is transported to the casting stage. In this

24

stage, the liquid steel is casted into slabs or billets through a continuous

25

caster (CC) where a series of liquid steel are continuously processed without

26

any break. In the above production process, the liquid steel once produced

27

by a BOF is called a charge which is the minimum scheduling unit, and the

28

charge series continuously processed on a CC is defined as a cast.

29

During the SCC manufacturing process, the liquid steel is processed under

30

a high temperature condition that almost above 1500◦ C, and a large number

3

31

of physical and chemical reactions also take place. Due to its strictures

32

on temperatures and chemical compositions, the scheduling process of the

33

SCC manufacturing system has to take into account numerous technological

34

constraints. For example

35

36

• Avoiding the large temperature drop, a waiting time between two adjacent stages is limited [1].

37

• According to the usage life of the tundish, an important component

38

installed in the CC, all charges in the same cast must be continuously

39

processed on the same CC.

40

41

• Before a new cast is processed on a CC, a sequence-independent setup time must be provided to install a tundish in the CC machine.

42

Because random disturbances and unforeseen events frequently happen in

43

the practical SCC system, there exist some uncertainties in the SCC schedul-

44

ing problems (SCCSPs), such as processing time variation, machine break-

45

down and route change. Therefore, how to achieve high scheduling perfor-

46

mance under low risk of constraint violations is a concerning problem while

47

executing a released schedule.

4

48

1.1. Literature Review

49

The SCCSP is always identified as a hybrid flow shop scheduling problem

50

(HFSSP) with complex constraints [2]. Because of their importance for the

51

iron & steel industry, SCCSPs have been thoroughly investigated by consid-

52

erable researchers from both academia and industry. The SCCSPs reported

53

in the literature are categorized into three types: static , dynamic and un-

54

certain scheduling problem.

55

In the static scheduling problem, all parameters are assumed to be de-

56

terministic and uninterrupted when all charges are released into the SCC

57

manufacturing system. This problem is built for a perfect environment, in

58

which there exist no uncertain factor. The algorithms for solving this type of

59

problem have attracted extensive attention in earlier studies. For example,

60

Atighehchian et. al [3] proposed a hybrid ant colony optimization (ACO) al-

61

gorithm to solve three-stage SCCSPs. Job sequence and machine assignment

62

were optimized by the ACO algorithm in the first phase, while the timing

63

of jobs was determined via a non-linear programming method in the second

64

phase. Based on problem-specific characteristics, Pan et. al [4] suggested

65

an artificial bee colony (ABC) algorithm that incorporates several improved

66

heuristic procedures. Through a decomposition mechanism, Mao et. al [5]

5

67

proposed a Lagrangian relaxation (LR) approach based on machine capacity

68

relaxation. To solve the SCCSP contains several refining sub-stages, Li et.

69

al [6] presented an effective fruit fly optimization algorithm (FOA) and a de-

70

coding scheme with a forward list scheduling (FLS) method. Assuming the

71

cast sequence in the SCCSP is unknown, Pan [7] proposed a cooperative co-

72

evolutionary artificial bee colony (CCABC) algorithm in which a sub-swarm

73

was used to solve sub-problems.

74

In the dynamic scheduling problem, all parameters also are determinis-

75

tic, but some of them may be interrupted by random disturbances or un-

76

foreseen events. The algorithm for solving this type of problem is always

77

event-trigged. Considering the feasibility of an initial SCC schedule is af-

78

fected by operation time delays, Yu and Pan [8] proposed three heuristic

79

rescheduling policies including batch splitting, forward scheduling and back-

80

ward scheduling. Considering machine breakdowns or new charges arrival,

81

Tang et. al [9] proposed an improved differential evolution (DE) algorithm

82

with an incremental mechanism. The DE algorithm re-optimized the charge

83

sequence, machine assignment, and timetable of the SCCSP under dynamic

84

environments. In case of a machine breakdown or processing time variation,

85

Mao et. al [10] suggested an effective Lagrangian relaxation (LR) approach

6

86

to solve the rescheduling problem in the SCC manufacturing system. For

87

tackling the rescheduling problem in SCC systems, Yu et. al [11] developed

88

a heuristic algorithm with a quick-response.

89

In the uncertain scheduling problem some parameters are assumed to be

90

unknown. The algorithm for solving this type of problem always provides a

91

preventive solution. Considering possible machine breakdowns in the SCCSP,

92

Worapradya and Thanakijkasem [12] proposed a robust predictive scheduling

93

algorithm via a minimax genetic algorithm. Taking into account both effi-

94

ciency and effectiveness, we introduced a novel concept of a ”soft schedule”,

95

and proposed some optimization algorithms for solving uncertain SCCSPs

96

[13] [14] [15] in different scenarios. This idea will be detailedly described in

97

following subsection.

98

1.2. Motivation

99

For a typical SCCSP, all charges in the optimal solution must be se-

100

quenced and timetabled. It is very different from a typical HFSSP in which

101

only the job sequence in the first stage is determined [16]. However, since

102

numerous uncertainties are existed in the SCC system, the constraints of a

103

SCCSP are easily violated. Consider an example in which an optimal so-

104

lution with deterministic processing times (shown in Fig.2(a)), known as a 7

105

static schedule is released into the shop floor. If the realized processing time

106

of a operation is longer than its standard value, the continuity constraint

107

of its cast is broken (shown in Fig.2(b)), which is called cast-break. If the

108

realized processing time of a operation is shorter than its standard value, the

109

waiting constraint of charge is violated (shown in Fig.2(c)), which is called

110

over-waiting.

111

As stated above, a SCCSP in practical environments has to consider

112

not only common objectives like waiting time, flow time, and makespan,

113

but also penalties caused by constraint violations, like cast-break and over-

114

waiting. Therefore, a SCCSP under practical environments can be considered

115

as a uncertain multi-objective optimization problem (MOP). To the best of

116

our knowledge, insofar challenging problem has not been reported in the

117

literature.

118

To address uncertainties in the practical SCCSP, we have introduced the

119

novel concept of soft schedule in our previous works [13] [14] [15]. The pro-

120

posed concept includes two parts: 1) critical decisions, which are used to

121

determine the performance of the entire SCC solution, for example the par-

122

tial solution of the bottleneck stages or machines; 2) characteristic indicators,

123

which are used to build a mapping between the production system states and

8

124

125

126

127

128

the realized schedule. There are two benefits for this approach: • It is can be used to resist some uncertainty factors in the feasibility and optimality of a initial solution. • It provides more flexibility for decision-making online, such as temperature regulation and cast-break protection.

129

To tackle another challenge comes from the multiple objectives in the

130

practical SCCSP, we adopt a novel way utilizing the preference informa-

131

tion specified by a decision maker. Most methods for solving the MOP are

132

finding its non-dominate solution set on the Pareto frontier. Nevertheless,

133

it is too highly complex for the SCC manufacturing system to satisfy the

134

response time requirement. An preference-inspired approach can ease the

135

multi-objective complexity in uncertain environments, which allows the de-

136

cision maker to define his most preferred solution according to the problem

137

characteristics [17].

138

Focusing on a practical SCCSP considering uncertain factors and multi-

139

ple objectives, we introduce a novel decision-making method based on the

140

soft schedule, which is significantly different from other works in the lit-

141

erature, and develop a preference-inspired multi-objective optimization ap-

142

proach,which is not involved in our previous works. The remainder of this 9

143

paper is organized as follows: Section 2 provides the mathematic model of

144

the practical SCCSP, introduce a soft-form schedule and a multi-objective

145

decision method. In Section 3, based on the problem-specific characteristics,

146

a preference-inspired chemical reaction optimization (PICRO) algorithm is

147

proposed to solve the scheduling problems associated with practical SCC

148

manufacturing system (hereafter referred to as practical SCCSPs). Sections

149

4-5 presents and analyzes the experimental results for synthetic and indus-

150

trial instances, respectively . Finally, Section 6 provides some conclusions

151

and suggestions on further studies.

152

2. Problem Statement

153

2.1. Deterministic formulation

154

2.1.1. Parameter List

155

j = 1, 2, ..., n: Charge index, where n is the total number of charges.

156

l = 1, 2, ..., h: Cast index, where h is the total number of casts.

157

i = 1, 2, ..., g: Stage index, where g is the total number of stages.

158

(l, r): The rth Charge in cast l, where r = 1, 2, ..., nl and nl is the total

159

160

number of charges in cast l. ni : Total number of machines in stage i.

10

161

rj : Release time of charge j.

162

pi,j : Processing time of charge j in stage i .

163

tri1 ,i2 :Transport time from stage i1 to stage i2 .

164

jh(l): The first charge index of cast l.

165

jt(l): The last charge index of cast l .

166

Su(l): Setup time of cast l.

167

Qt: Maximum waiting time for each charge.

168

L: A sufficiently lager number.

169

170

171

172

173

174

175

176

2.1.2. Decision Variables xi,j,k : 0/1, i < g. If charge j is allocated on machine k in stage i , it equals 1; otherwise, it equals 0. yi,j1 ,j2 : 0/1, i < g. If charge j1 comes prior to charge j2 to be processed in stage i , it equals 1; otherwise, it equals 0. zl1 ,l2 ,k : 0/1, i = g. If cast l1 is prior to cast l2 that is allocated on machine k in stage i , it equals 1; otherwise, it equals 0. si,j : Starting time of charge j in stage i .

11

177

2.1.3. Constraint Equations (1) Each charge must reach the steelmaking and refining stage, and it must be processed by exactly one machine at each stage. ni X

xi,j,k = 1, i = 1, 2, ..., g − 1, j = 1, 2, ..., n

(1)

k=1

(2) For any two different operations in the same stage, there exists one and only precedence relationship.

yi,j1 ,j2 + yi,j2 ,j1 = 1, i = 1, 2, ...g − 1, j1 , j2 = 1, 2, ..., n, j1 6= j2

(2)

(3) For two consecutive operations in a charge, the next operation can be started only after the previous one is completed and the charge is transported to the next stage.

si,j + pi,j + tri,i+1 ≤ si+1,j , i = 1, 2, ...g − 1, j = 1, 2, ..., n

(3)

(4) For any two operations allocated to the same machine, only when the

12

preceding operation has been finished, the next one can be started.

si,j1 + pi,j1 − si,j2 − L × (3 − xi,j1 ,k − xi,j2 ,k − yi,j1 ,j2 ) ≤ 0 (4) i = 1, 2, ...g − 1, j1 , j2 = 1, 2, ..., n, j1 6= j2 , k = 1, 2, ...ni

(5) Any operation in the first stage is only started after its charges is released. rj − s1,j ≤ 0, j = 1, 2, ..., n

(5)

(6) In the casting stage, a cast is allocated on only one machine. ng X

zl1 ,l2 ,k = 1, l1 , l2 = 1, 2, ..., h, l1 6= l2

(6)

k=1

(7) For any two different casts, there exists one and only precedence relationship in the same CC.

zl1 ,l2 ,k + zl1 ,l2 ,k ≤ 1, l1 , l2 = 1, 2, ..., h, l1 6= l2 , k = 1, 2, ..., ng

(7)

(8) For a pair of the casts (l1 , l2 ) allocated on the same CC, a setup time

13

of the second cast must be provided.

sg,jt(l1 ) + pg,jt(l1 ) + sul2 −sg,jh(l2 ) − L × (2 − zl1 ,l2 ,k − zl2 ,l1 ,k ) ≤ 0 (8) l1 , l2 = 1, 2, ..., h, l1 6= l2 , k = 1, 2, ..., ng

(9) Any two adjacent charges in the same cast must be processed continuously in the casting stage.

sg,(l,r) + pg,(l,r) = sg,(l,r+1) , l = 1, 2, ..., h, r = 1, 2..., nl

(9)

(10) The waiting time in between-stages of each charge is limited due to its temperature reasons.

si+1,j − si,j − pi,j − tri,i+1 − Qt ≤ 0, i = 1, 2, ..., g − 1, j = 1, 2, ..., n (10)

178

2.1.4. Objective Function

179

Under static environments, the aim of a SCC schedule is to minimize

180

waiting times of all charges, because it contributes to reduce the temper-

181

ature drop and boost production efficiency. The objective function of the

14

182

deterministic SCCSP studied in this paper is formulated as follows.

(P )

min f =

n X

sg,j −

j=1

183

g−1 X

! (pi,j + tri,i+1 ) − rj

(11)

i=1

2.2. Complexity Analysis

184

It is reported by Gupta [18], a simple HFSSP with two stages, even

185

only one stage has two machines, is NP-hard. Because this problem is a

186

simplified case for the SCCSP, we can conclude the SCCSP is also NP-hard

187

[5]. To quantitatively analysis its complexity, we calculate the number of

188

binary variables (NBV), the number of continuous variables (NCV), and the

189

number of constraint equations (NCE) in a formulated SCCSP with the same

190

problem size.

191

• NCV: n × g .

192

• NBV: n

193

• NCE: n(n − 1)

194

NCV only increase with n. NBV increase with n and h. NCE increase

195

with n, h and nl . It can be seen that the SCCSP more complex than the

196

typical HFSSP where NCV, NBV and NCE only are only related to n.

Pg−1 i=1

ni + n(n − 1)(g − 1) + h2 × ng . Pg−1 i=1

ni + h(h − 1)(2ng + 1) + (n + 1)(ng − n + 1)g − h.

15

197

2.3. Uncertainty Assumption

198

Under practical environments, some parameters in above-mentioned for-

199

mulation cannot be precisely estimated. In this study, we assume that the

200

processing time pi,j is a stochastic variable with known expectation E(pi,j )

201

and variance D(pi,j ). Because the continuity (9) and waiting (10) constraints

202

are easily violated, a practical SCCSP minimizes not only the expected value

203

of primal objective (11) but also the expected penalties caused by constraint

204

violations. Therefore, a practical SCCSP can be formulated as a MOP as

205

follows:

     E(f1 ) = E         (SP ) minE (f ) E(f2 ) = E            E(f3 ) = E 

n X

sg,j −

j=1

g−1 X

!! (pi,j + tri,i+1 ) − rj

i=1

h n l −1 X X

! sg,(l,r+1) − sg,(l,r) − pg,(l,r)




l=1 r=1 n g−1

XX

! max{sg,j − s1,j − pi,j − tri,i+1 − Qt, 0}

j=1 i=1

(12) 206

Eq.(12), E(f1 ) represent the expected value of waiting time (see eq.(11)),

207

E(f2 ) and E(f3 ) respectively represent the expected penalty caused by cast-

208

break and over-waiting (see eq.(9) and (10)). Then, given an aspiration value

209

vector u∗ = {u∗1 , u∗2 , u∗3 }, the problem SP is changed into the following single16

210

objective optimization problem with reference point-based method [17].

0

(SP )

    max  min asf = d=1,2,3    

{λd (E(fd ) −

u∗d )}

+ρ

3 X

[λd (E(fd ) − u∗d )] ,

d=1

s.t. (1) ∼ (11) (13)

211

where λd is the weight coefficient, and ρ is the augmentation coefficient.

212

With this expected value model, it should be noted that even though a

213

particular solution is preferred over other solutions with the small expected

214

value for all scenarios, it may be a very bad solution with high objective

215

and penalty values in certain scenarios. Therefore, the expected objective

216

model is not always considered as good measures for the optimization prob-

217

lem under uncertain environments [19]. Therefore, we employ the β-efficient

218

measure proposed by Kataoka [20] and the concept of ordinal optimization

219

to reformulate SP as a chance constrained optimization problem.

0

0

SP (β)

    max  min ASF = d=1,2,3

{λd (ud −

u∗d )}

+ρ

3 X d=1

    s.t. P r (E(f ) ≤ u) ≥ β

[λd (ud − u∗d )] ,

and (1) ∼ (11) (14)

17

220

where β = {β1 , β2 , β3 } is the probability level vector.

221

3. Proposed Algorithm

222

Chemical reaction optimization (CRO) is a novel meta-heuristic algorithm

223

invented by Lam and Li [21], which simulates the chemical reaction process in

224

which molecules start from high-energy states and terminate at a low-energy

225

state via a sequence of molecular reactions. Generally, a molecule in CRO

226

represents a solution in the optimization problem, which has two types of

227

energies: potential energy (PE) and kinetic energy (KE). PE corresponds to

228

the objective value of a solution while KE represents its ability of escaping

229

from a local minimum.

230

The optimization problem formulated in Section 2 is nonlinear, multi-

231

dimensional, multi-objective and stochastic. To the best of our knowledge,

232

no classical algorithm can directly solve this type of problem. In this section,

233

based on specific characteristics of the studied practical SCCSP, we develop

234

a multi-objective soft scheduling (MOSS) approach in which all solutions are

235

represented by soft-form schedules and optimized by the PICRO algorithm.

18

236

3.1. Overview of the MOSS

237

The main procedure of MOSS is described in Algorithm 1. First, the

238

algorithm initializes the population with a random manner, and determine a

239

user-specified reference point. Second, the initialized solutions are iteratively

240

optimized by CRO variations. Finally, the optimized population is further

241

ranked and selected using the clean-up procedure proposed by Butler et al

242

[22].

243

3.2. Encoding and Decoding Scheme Based on the problem characteristics stated in the previous sections, most special constraints are imposed in the casting stage, and only the partial decisions in the casting stage (sg , j) influence the objective of the entire SCC solution (see Eq.11). Therefore, the casting stage can be identified as the bottleneck in the SCC system. According the theory of drum-buffer-rope (DBR) [23], we should utilize buffer times to protect the constraints and performance in the casting stage. Following these assumption, a feasible soft schedule can be represented with the following two-part vector:

$ = [pr1 , pr2 , ..., prh |wr1 , wr2 , ..., wrh ] ,

19

where the first part represents cast priorities, which are critical decisions and used to determine the processing sequence of all charges in the casting stage, and the second part represents waiting ratios of casts, which are characteristic indicators and used to determine the buffer size before the last stage. The waiting ratio of the charges in the cast l is defined as follows:

wrl = Pg−1 i=1

Bufl , ∀j ∈ Ωl (pi,j + tri,i+1 )

(15)

244

where Bufl is the buffer size. Because both prl and wrl are real-value vari-

245

ables, the SCCSP represented by the above vector is a continuous optimiza-

246

tion problem.

247

Based on the cast priorities and waiting ratios listed in $ , we can con-

248

struct a executable partial schedule in the casing stage using the iterative

249

backward list scheduling (IBLS) described in Algorithm 2.

250

3.3. Population Initialization

251

To initialize the first population, the values of waiting ratios are generated

252

with a uniform distribution rand(0.0,0.5). In the proposed algorithm, because

253

the casting stage is identified as bottleneck, the workload of each charge in

254

the last stage may larger than its value in upstream stages after insert buffer 20

255

time. According to the johnson’s rule for flow shop scheduling problem, the

256

cast priorities are produced by an initial sequence determined by the long

257

processing time (LPT) rule. The initial procedure is illustrated in Algorithm

258

3.

259

3.4. Multi-objective Evaluation

260

In the proposed algorithm, we use a ASF (Eq.14) to evaluate the waiting

261

time objective, cast-break and over-waiting penalty of each solution $. At

262

each iteration of the PICRO, uidl and unad are chosen from the current pop-

263

ulation Π, and updated by suggestion by Luque et.al [24] in the following

264

way: • uidl d is the optimal objective for the following problem

min ud $∈Π

(16)

s.t. P r{E (fd ($)) ≤ ud } ≥ 0.50

• unad is the optimal objective for the following problem d

min ud $∈Π

(17)

s.t. P r{E (fd ($)) ≤ ud } ≥ 0.99

21

Based on risk minimization principle, the we specified the reference point u∗ using the local information about the ideal and nadir points.

nad ∗ idl ∗ idl u∗1 = (uidl 1 + u1 )/2, u2 = u , u3 = u .

265

However, the reference point in PICRO algorithm is changed at each

266

iteration, while the weights are kept unchanged during the all iterations.

267

Then the weights can be set as follows:

λd =

268

unad d

1 − uidl d

(18)

3.5. CRO-based Variation

269

In a standard CRO, there are four elementary reactions. Since the op-

270

timization problems in this study are real-coded, CRO-based variations are

271

t modified using the method recommended by Lam et al [25]. We use $p,r

272

represents the rth element of the pth molecule in the population Π(t) , and

273

we implements the variations as follows:

274

• On-wall reaction

275

t+1 t $p,r ← $p,r + N (0, 1) , where N (0, 1) is a random number generate

276

by standard Normal distribution. 22

277

• Decomposition reaction

278

t First, $pt+1 ← $pt+1 ← $p,r . 1 ,r 2 ,r

279

Second, if rand(0, 1) > 0.5, $pt+1 ← $pt+1 ; else $pt+1 ← $pt+1 . 1 ,r 2 ,r 2 ,r 1 ,r

280

281

282

283

• Inter-molecular reaction t+1 t+1 If rand(0, 1) > 0.5, $p,r ← $pt 1 ,r ; else $p,e ← $p2 ,r .

• Synthesis reaction t+1 t+1 If rand(0, 1) > 0.5, $p,r ← $pt 1 ,r ; else $p,r ← $p2 ,r .

284

The real-valued variables in vector $ are usually bounded. In other

285

words, the soft solution is only assigned a value in the interval . When

286

constructing new solutions, a value that lies outside the boundaries is not

287

allowed. In PICRO, a boundary constraint handling method reported in [25]

288

is adopted.

289

t+1 t t+1 t+1 $p,r ← $p,r + rand() × ($+ − $− ), if $p,r > $+ or $p,r < $− ,

290

where $+ and $− represent the upper and lower bound of each element

291

,respectively.

292

3.6. Rough Estimation

293

In uncertain environments, to estimate expected objective E (f ) of each

294

solution with its mean vector f¯ and variance vector S 2 (f ) , a number of 23

295

simulation based on Monte Carlo sampling must be performed. To simulate

296

the manufacturing process, we develop a discrete event simulation (DES)

297

algorithm with stochastic processing times, which is presented in detail in

298

Algorithm 4.

299

After performed τ simulation replications, the PE corresponding to the

300

achievement scalarizing function (ASF) value of each solution is estimated.

301

Then, probability constraints of problem SP (β) can be reformulated as fol-

302

lows:

¯  fd − E(fd ) f¯d − ud √ ≥ √ P r (E(fd ) ≤ ud ) = βd ⇒ P r = βd S(fd )/ τ S(fd )/ τ   f¯d − ud √ ⇒ P r t(τ − 1) ≤ = 1 − βd S(fd )/ τ  ¯  fd − ud √ ⇒ϕ = 1 − βd S(fd )/ τ

(19)

⇒ ud = f¯d − ϕ−1 (1 − βd , τ − 1)S(fd )

where φ is the probability distribution function of a standard normal distribution, φ−1 is its inverse function. Using the above T-test method, Then a

24

new ASF can be defined as follows: 



  max λd f¯d + φ−1 (βd )S(fd ) − ud  d=1,2,3  0   ASF (β) =  3  X 
  +ρ λd f¯d + φ−1 (βd )S(fd ) − u∗d 
∗



(20)

d=1

303

3.7. Knowledge-based Local Search

304

It is clear that waiting ratios in the soft solution controls the values of

305

cast-break and over-waiting. Based on this knowledge, we generate three

306

type neighborhood for $.

307

• N (1): Iclr(l), randomly select a cast l, and set wrl ← rand(wrl , wr+ ).

308

• N (2): Dclr(l), randomly select a cast l, and set wrl ← rand(wr− , wrl ).

309

• N (3): Swap(l1 , l2 ), randomly select two casts l1 and l2 , and swap the

310

position of prl1 andprl2 .

311

Utilizing these operators, we implement a KLS procedure for the top e%

312

solutions to enhance the convergence ability of PICRO. The detailed steps

313

of the KLS are given in Algorithm 5.

314

3.8. Clean-up Strategy

315

In the procedure of PICRO, we use a T-test method to roughly identify

316

the best solutions, however the simulation replication of each solution is ex25

317

tremely limited to precisely estimate their expectation. In this section, we

318

employ a indifference-zone(IZ)-based ranking & selection procedure called

319

clean-up to select the best solution in the population optimized by the CRO.

320

This procedure proposed by Butler et al [22] includes two stages. The first

321

stage is the subset selection, which eliminate clearly inferior solutions. The

322

second stage is ranking, which determine the additional simulation repli-

323

cations and select the best solution. The clean-up procedure embedded in

324

MOSS is detailed in Algorithm 6.

325

4. Computational Experiments

326

In this section, we report the experimental results to verify the perfor-

327

mance of the proposed MOSS approach. All test algorithms are coded by

328

Visual C++ 2012 in Windows 7 environment and run on a personal computer

329

with Intel Core i7 3.60GHz CPU and 8GB RAM.

330

4.1. Test Instance Generation

331

To test our algorithm, we synthesize test instances characterized by the

332

shop configuration of the SCC manufacturing system and the size of schedul-

333

ing problem. We use the number of machines per stage to represent shop

334

configuration, and use the number of casts and the number of charges in each 26

335

cast to represent problem size. With the following three different shop con-

336

figurations and six problem sizes, 18 problem instances are generated (shared

337

at https://github.com/janason/Soft-Scheduling/tree/master/MOSS).

338

339

340

341

• Shop configuration: {A : 3 × 4 × 3, B : 3 × 4 × 4, C : 4 × 4 × 3} . • Problem size: {1 : 6 × 10, 2 : 6 × 15, 3 : 8 × 10, 4 : 8 × 15, 5 : 10 × 10, 6 : 10 × 15}.

342

The parameters in each instance are generated with following ways:

343

• Standard processing time in different stages: p1,j = 30, p2,j ∼ U (20, 40),

344

and p3,j ∼ U (25, 35)

345

• Transport time in two adjacent stages: tri,i+1 = 5

346

• Interval of arrival time for each charge: ∆(r) ∼ Expo(30)

347

• Setup time for each cast: Sul = 60

348

• Maximum waiting time for charges: Qt = 30

349

To simulate the uncertain factors in these synthetic instances, we use

350

the noise level ηi,j = D(pi,j )/E(pi,j ) to represent the degree of parameter

351

disturbance. Because the steel grades of the charges in a cast always are

352

same, we assume that the noise levels of the charges in same stage and cast are

353

equal. Therefore, the noise level ηˆi,l is generated with uniform distributions 27

354

in the following manners:

355

• In the steelmaking sage: ηˆ1,l ∼ U (0.0, 0.1)

356

• In the refining stage: ηˆ2,l ∼ U (0.1, 0.3)

357

• In the casting stage: ηˆ3,l ∼ U (0.0, 0.15)

358

4.2. Parameter Setting

359

It is generally known that a suitable parameter setting is able to enhance

360

the performance of a meta-heuristic algorithm. The parameters of the MOSS

361

can be divided into three categories: preference parameters, CRO parameters

362

and KLS parameters.

363

364

The preference parameters containing the weight vector and the augmentation coefficient are set as follows: 1 unad −uidl d d

365

• λd =

366

• ρ = 10−3

367

For CRO parameters, we employ the adaptive mechanism suggested by

368

Yu et al. [26] to set initial molecular kinetic energy (iniKE), initial buffer

369

(iniBuf ), loss rate (losRate), decomposition threshold (decT hres) and syn-

370

thesis threshold (synT hres). The population size N P , collision rate (colRate)

371

and average simulation replication τ are undetermined.

28

372

373

The KLS parameters including local rate (e%) and maximum iteration number (Imax ) are also undetermined.

374

Before tuning above five undetermined parameters, we have applied a

375

design of experiments (DoE) approach to chose values from the recommended

376

list (as shown in Table 1) which is user-defined based on a number of trials.

377

The full factorial design (including 54 = 625 groups) is almost impractical in

378

this case. Therefore, we use the Taguchi design method with the orthogonal

379

array L16 (54 ), which implies that only 16 orthogonal instances have to be

380

tested. An 8 × 10 instance in A-type SCC system is adopted for testing and

381

its result data (S/N ratios, i.e. signal-to-noise ratios) is shown in Figure 3

382

(output by the Minitabr software). As the figure suggests, the values of

383

other undetermined parameters in MOSS are set as follows: N P = 5 × |$| ,

384

colRate = 0.1, τ = 10, e% = 10%, Imax = 10.

385

4.3. Performance Analysis on the MOSS

386

To verify the effectiveness of the MOSS, some basic component described

387

in Section 3 is tested by solving an (8 × 10)-size instance for above three

388

SCC systems. The basic CRO algorithm combines the KLS and clean-up

389

procedure. Then, four different algorithms are formed, and the computa-

390

tional results are obtained by each algorithm with 1000 repeated simulations 29

391

(as shown in Figure 4). According to the experimental data, both the KLS

392

and clean-up procedures contribute to improving the basic CRO (with rates

393

about 15.69% and 14.94% respectively ). The combination of the KLS and

394

clean-up in the CRO results in the most improvement (with rates about

395

27.44%).

396

4.4. Comparison with the State-of-the-art

397

In this section, we will use the proposed MOSS to solve different-sized un-

398

certain SCCSP instances in three shop configuration. The algorithm is com-

399

pared with the soft-decision algorithm (SDA) which utilizes particle swam

400

optimization (PSO) as the main optimizer. Because there are some difference

401

between their problem characteristics, we implement SDA with the following

402

considerations. First, the SDA uses the soft schedule including cast priority

403

and workload, which has a similar form of MOSS. Second, PSO is a well-

404

known meta-heuristic algorithm, which an provide a baseline for comparison

405

with MOSS.

406

The parameters of PSO in the SDA are set as follows: the swam size

407

N P = 10 ∗ |$|, the cognitive and social coefficients c1 = c2 = 0.1, the inertia

408

weight w = 0.98.

409

For the comparisons to be meaningful, we set a common stop criterion 30

410

with a computational time limitation for both the algorithms. Generally,

411

the time limitation should be allocated according to the size of a problem

412

instance. In the following experiments, the time limitation allocated to a

413

h × nl -sized instance is calculated as T = 1.0 × (h × nl ) .

414

Tables 2-4 report the mean (f¯) and variance (S 2 (f )) value for each ob-

415

jective (waiting cost, cast-break penalty, over-waiting penalty) of the best

416

solution with 1000 simulation runs. Three SCC system with different config-

417

urations (A,B,C) are tested.

418

Based on the results presented in Table 2-4 , the mean vectors obtained

419

from the MOSS and SDA are combined and plotted in Figure 5 for a quali-

420

tative comparison. Although both the results fall approximately in the same

421

range, the results obtained from MOSS are close to the border area of the

422

feasible space and the solutions generated are more concentrated.

423

To quantitatively compare optimality on objectives, the relative percent-

424

age deviations (RPD) of fd− (d = 1, 2, 3) between the MOSS and the SDA are

425

p calculated and listed in Table 5 where fd− = f¯d − 1.96 ∗ S/ (1000) and

RP D(fd− )

426

SDA(fd− ) − M OSS(fd− ) = × 100%. SDA(fd− )

The average values of RPD results are calculated in the bottom of Table 31

427

5. The box plots of RPD results are draw in Fig. 6. These results show

428

that the optimality of MOSS is better than SDA, especially on f2 and f3 .

429

When the steelmaking stage is the bottleneck (B-type SCC system), the

430

improvement on penalty objectives (f2 , f3 ) is most non-obvious, but it is

431

opposite on waiting objective (f1 ). The main reason for this is that the last

432

stage easily suffers from starvation where steelmaking stage can not supply

433

enough charges, then cast-break frequently happens. To avoid these risks a

434

larger buffer time should be inserted.

435

To observe convergence of the algorithms, we plot the convergence curves

436

for MOSS and SDA in Figure 7, which output by running on a 8×10 instance

437

in A-type shop configuration. These curves show that the MOSS converges

438

rapidly to lower objective values than SDA.

439

It can be concluded that MOSS results in more superior solutions in most

440

instances. Its superiority results from the following aspects: 1) estimation

441

with T-test; 2) enhancement by KLS procedure and 3) further ranking and

442

selection mechanism with clean-up procedure.

32

443

5. Further Discussion

444

5.1. Sensitivity Analysis

445

For the scheduling problems studied in this paper, their uncertainties are

446

defined by the noise levels on processing times in the steelmaking, refining and

447

casting stage. In the experiments mentioned in above section, they are fixed

448

for all synthetic instances. But they may be changed along with dynamic

449

production environments. In the following, we design two uncertain scenarios

450

of instance 1# to examine the effect of these variables on the minimized

451

objectives.

452

In the first scenario, we fix ηˆ2,l = 0.2 and ηˆ3,l = 0.1, and the value of ηˆ1,l

453

varies from 0.0 to 0.1 (10 steps). In the second scenario, we fix ηˆ1,l = 0.1 and

454

ηˆ2,l = 0.2, and the value of ηˆ3,l varies from 0.0 to 0.1 (10 steps). Under each

455

value of ηˆi,l (i = 1, 3) we run the proposed MOSS with the same parameter

456

settings as stated before. The results are displayed in Figure 8. From the

457

figure, we see an rising trend in the value of each objective as the noise level.

458

The values of cast-break and over-waiting under the first scenario are smaller

459

than their values under the second scenario.

33

460

5.2. Industrial Verification

461

To test MOSS algorithm under the real-world industrial environment,

462

we directly take 10 instances with the practical production data within 10

463

different periods, which comes from a SCC manufacturing system of a large

464

iron & steel company in China. The SCC system has 3 BOFs, 4 LFs, and

465

3 CCs. The problem size within one-day period, includes about 10 casts

466

and around 150 charges. The noise levels are set with statistic data from its

467

database.

468

Table 6 presents the results output by the MOSS, SDA and RSH which

469

denotes the realized solution by heuristic in the real-world SCC manufactur-

470

ing system. With respect to expected objective (f¯1 ), the MOSS performs

471

similarly with the SDA, and it is much better than RSH. With respect to

472

penalty objectives, the MOSS obtains considerable improvements compared

473

other two algorithms under real-world uncertain environments. This illus-

474

trates that proposed algorithm is inclined to decrease the value of cast-break

475

and over-waiting just as the way of RSH applied in practice, because the

476

cast-break and over-waiting are more crucial to the production quality and

477

stability.

34

478

6. Conclusion and Future Works

479

In this paper, an uncertain SCCSP with continuity and waiting con-

480

straints is studied. Based on the problem-specific characteristics, we con-

481

structed a soft-form schedule in which the cast priorities are treated as crit-

482

ical decisions and the waiting ratios are treated as characteristic indicators.

483

To solve an uncertain SCCSP considering multiple objectives, we propose a

484

PICRO algorithm. In the initialization phase, the preference is determined

485

by the local information. In the iterative phase, we propose a T-test method

486

to roughly estimate each solution and a KLS algorithm to accelerate con-

487

vergence, use a clean-up procedure accurately select a near-optimal solution.

488

Finally, the computational results verify the effectiveness and efficiency of

489

the MOSS approach for synthetic and industrial instances . To improve the

490

performance of MOSS, future studies should include the following aspects:

491

(1) The computational results showed that there are big gaps in the penal-

492

ties caused by cast-break and over-waiting in the different types of SCC sys-

493

tem. Therefore, the preference information should be extracted from the

494

shop configuration.

495

496

(2) It will be useful to deploy the proposed algorithm in a real-world SCC system and extend it to solve other types of scheduling problems. 35

497

References

498

[1] S. W. Missbauer H, Hauber W, A scheduling system for the steelmaking-

499

continuous casting process. a case study from the steel-making industry,

500

International Journal of Production Research 47 (2009) 4147–4172.

501

[2] L. Tang, P. B. Luh, J. Liu, L. Fang, Steel-making process scheduling us-

502

ing lagrangian relaxation, International Journal of Production Research

503

40 (2002) 55–70.

504

[3] A. Atighehchian, M. Bijari, H. Tarkesh, A novel hybrid algorithm for

505

scheduling steel-making continuous casting production, Computers &

506

Operations Research 36 (2009) 2450–2461.

507

[4] Q. K. Pan, L. Wang, K. Mao, J. H. Zhao, An effective artificial bee

508

colony algorithm for a real-world hybrid flowshop problem in steelmak-

509

ing process, IEEE Transactions on Automation Science & Engineering

510

10 (2013) 307–322.

511

[5] K. Mao, Q. K. Pan, X. Pang, T. Chai, A novel lagrangian relaxation

512

approach for a hybrid flowshop scheduling problem in the steelmaking-

513

continuous casting process, European Journal of Operational Research

514

236 (2014) 51–60. 36

515

[6] J. Q. Li, Q. K. Pan, K. Mao, P. N. Suganthan, Solving the steelmak-

516

ing casting problem using an effective fruit fly optimisation algorithm,

517

Knowledge-Based Systems 72 (2014) 28–36.

518

[7] Q. K. Pan, An effective co-evolutionary artificial bee colony algorithm

519

for steelmaking-continuous casting scheduling, European Journal of Op-

520

erational Research 250 (2015) 702–714.

521

[8] Y. U. Sheng-Ping, Q. K. Pan, A rescheduling method for operation

522

time delay disturbance in steelmaking and continuous casting production

523

process, International Journal of Iron and Steel Research 19 (2012) 33–

524

41.

525

[9] L. Tang, Y. Zhao, J. Liu, An improved differential evolution algorithm

526

for practical dynamic scheduling in steelmaking-continuous casting pro-

527

duction, IEEE Transactions on Evolutionary Computation 18 (2014)

528

209–225.

529

[10] K. Mao, Q. K. Pan, X. Pang, T. Chai, An effective lagrangian relax-

530

ation approach for rescheduling a steelmaking-continuous casting pro-

531

cess, Control Engineering Practice 30 (2014) 67–77.

532

[11] S. Yu, T. Chai, Y. Tang, An effective heuristic rescheduling method for 37

533

steelmaking and continuous casting production process with multirefin-

534

ing modes PP (2016) 1–14.

535

[12] K. Worapradya, P. Thanakijkasem, Worst case performance scheduling

536

facing uncertain disruption in a continuous casting process, in: IEEE In-

537

ternational Conference on Industrial Engineering and Engineering Man-

538

agement, pp. 291–295.

539

[13] J. Hao, M. Liu, S. Jiang, C. Wu, A soft-decision based two-layered

540

scheduling approach for uncertain steelmaking-continuous casting pro-

541

cess, European Journal of Operational Research 244 (2015) 966–979.

542

[14] S. L. Jiang, M. Liu, J. H. Lin, H. X. Zhong, A prediction-based on-

543

line soft scheduling algorithm for the real-world steelmaking-continuous

544

casting production, Knowledge-Based Systems 111 (2016) 159–172.

545

546

[15] S. Jiang, M. Liu, J. Hao, A two-phase soft optimization method for the uncertain scheduling problem in the steelmaking industry (2016) 1–16.

547

[16] A. A. Juan, B. B. Barrios, E. Vallada, D. Riera, J. Jorba, A simheuristic

548

algorithm for solving the permutation flow shop problem with stochastic

549

processing times, Simulation Modelling Practice & Theory 46 (2014)

550

101–117. 38

551

[17] M. Luque, K. Miettinen, P. Eskelinen, F. Ruiz, Incorporating preference

552

information in interactive reference point methods for multiobjective

553

optimization , Omega 37 (2009) 450–462.

554

555

556

557

558

559

[18] J. N. Gupta, Two-stage, hybrid flowshop scheduling problem, Journal of the Operational Research Society (1988) 359–364. [19] J. R. Birge, F. Louveaux, Introduction to stochastic programming, Springer Science & Business Media, 2011. [20] S. Kataoka, A stochastic programming model, Econometrica 31 (1963) 181–196.

560

[21] A. Y. S. Lam, V. O. K. Li, Chemical-reaction-inspired metaheuristic

561

for optimization, IEEE Transactions on Evolutionary Computation 14

562

(2010) 381–399.

563

564

[22] B. L. Nelson, Using ranking and selection to ”clean up” after simulation optimization, Operations Research 51 (2003) 814–825.

565

[23] V. Sirikrai, P. Yenradee, Modified drum-buffer-rope scheduling mecha-

566

nism for a non-identical parallel machine flow shop with processing-time

39

567

variation, International Journal of Production Research 44 (2006) 3509–

568

3531.

569

[24] M. Luque, F. Ruiz, J. M. Cabello, A synchronous reference point-

570

based interactive method for stochastic multiobjective programming,

571

OR Spectrum 34 (2012) 763–784.

572

[25] A. Y. S. Lam, V. O. K. Li, J. J. Q. Yu, Real-coded chemical reaction op-

573

timization, IEEE Transactions on Evolutionary Computation 16 (2012)

574

339–353.

575

[26] J. J. Q. Yu, A. Y. S. Lam, V. O. K. Li, Adaptive chemical reaction

576

optimization for global numerical optimization, in: Evolutionary Com-

577

putation.

40

Algorithm 1 MOSS using PICRO Input: Population size N P , Output: The best soft solution $best 1:

t←0

2:

P (0) ← Initialize P oplulation()

3:

R ← Specif y P ref erence(P (0))

4:

while Not Termination Criterion do

5:

Q (t) ← CRO V ariation (P (t))

6: 7:

P (t + 1) ← Rough Evaluation (P (t + 1)) S P (t + 1) ← P (t) Q (t) /*merge parent and child population */

8:

P (t + 1) ← KLS (P (t + 1))

9:

P (t + 1) ← Selection P opulation (P (t + 1)) /* |P (t + 1)| = N P */

10:

/*knowledge-based local search*/

t←t+1

11:

end while

12:

$best ← clean up (P (t))

13:

return $best

/*generate new solutions*/

41

Algorithm 2 IBLS procedure for decoding Input: A representation of soft solution, $ . Initialize: Set the release time of each machine: M Ri,k ← ∞, and set total time conflict: ϕ ← 0. Output: The starting time of each cast: Sl . 1. Casting stage scheduling (1) According to the priority values in π, allocate casts to the machines in the casting stage with the earliest available machine (EAM) rule. (2) Calculate starting time of each charge in the casting stage: sg,j 2. Steelmaking & Refining stage scheduling For i = g − 1 to 1 (1) Calculate the latest completion time of each charge in stage i: LCi,j , LCi,j ← si+1,j − wrl × (tri,i+1 + pi,j ). (2) Sort charge set Gj in this stage with descending order by LCi,j . (3) For each j in Gj (a) Allocate charge j on machine k ∗ with the latest available machine rule, where k ∗ = arg max (M Ri,k ), M Ri,k∗ ← min (M Ri,k∗ , LCi,j ) − E(pi,j ), k=1,...,ni

and si,j ← M Ri,k∗ (b) Calculate the conflict value of the arrival time ϕ

←

max (rj − s1,j , 0) (c) If (ϕ > 0), let lj represents the cast of charge j, right shift all charges in cast lj and the succeeding casts with ϕ time units, go to step 2. End for 3. Right shift (a) Let s0 = min (s1,j , 0), and N S ← max (|s0 | , ϕ) . (b) Right shift the starting times of all operations with ϕ time units. 42

Algorithm 3 Initialize population Input: The population size, N P .  0 Output: The initial population, Π0 = $10 , $20 , ..., $q0 , ..., $N P . 1:

Generate an initial cast sequence π with LPT rule.

2:

Generate N P/10 solutions with the following method:

prl

=

1.0/ (π (l) + 1)), and wrl = rand(0.0, 0.5). 3:

N P ← N P − 10.

4:

while N P > 0 do

5: 6: 7: 8: 9: 10: 11:

if rand() > 0.5 then Generate a random wait ratio vector and 10 cast priority vectors. else Generate a random cast priority vector and 10 wait ratio vectors. end if N P ← N P − 10. end while

43

Algorithm 4 DES procedure for evaluation Input: Critical decisions, Sl ; Characteristic Indices, wrl . Output: The simulated objective vector fˆ. 1:

Release all arrived charges to the SCC manufacturing system.

2:

Generate messages ψ of the first operation on the arrived charges, put them into Ψ .

3:

while |Ψ| > 0 do

4:

ψ ← P OP (Ψ)

5:

if ψ.type = machine release then if ψ.machine is idle then

6:

(a) Select an eligible operation for current machine, and set

7:

the machine status to busy. 8:

end if

9:

if ψ.machine is busy then (a) Complete the operation in process, and set the operation

10:

in the

next stage to arrive. (b) Set the machine status to idle.

11: 12:

end if

13:

end if

14:

if ψ.type = operation arrive then

15: 16: 17:

put the psi.operation into the waiting list before the stage. end if end while

44

Algorithm 5 KLS procedure Input: The incumbent solution, $. Output: The optimized solution $opt .. 1:

Define reference variables, vd = λd (ud − u∗d )

2:

Set neighbor N ← 0 .

3:

if v2 > max(v1 , v3 ) then

4: 5: 6: 7: 8: 9:

LS ← N (1) else if v3 > max(v1 , v2 ) then LS ← N (2) else LS ← N (3) end if

10:

set it ← 0.

11:

while it > Tmax do

12:

Set new solution $new ← LS ($)

13:

if Asf ($new ) < asf ($) then

14:

$ ← $new

15:

end if

16:

it ← it + 1.

17:

end while

45

Algorithm 6 Clean-up Procedure Input: The final population, P (t). Output: The best solution $best . 1:

Determine the desired overall confidence level 1 − α and the indifferencezone parameter δ > 0. Set the allowable error corresponding to the screening procedure (as ) and the selection procedure (al ), and 1 − αs = √ 1 − αl = 1 − α

2:

Based on the ASFs obtained from the repeated simulations, Fp,r , p = 1, 2, ..., N P, r = 1, 2, ..., R. Calculate the sample mean and the sample P PR 2 ¯ 2 variance F¯p = R p=1 Fp,r /R and S = r=1 (Fp,r − Fp,r ) . The screening threshold Wp1 ,p2 can be calculated as follows: Wp1 ,p2 = (t2p1 ∗ Sp21 )/Rp1 + (t2p2 ∗ Sp22 )/Rp2

3:

Select the sub-population P sub ,  P sub = p1 : 1 < p1 < N P

4:


1/2

&& F¯p1 ≥ F¯p2 − Wp1 ,p2



Calculate the Rinott’s constant % = %(2, (1 − αl )1/N P −1 , R), and then determine the additional replications for each solution. (

ρ ∗ Sp2 Np = max np , d δ 5:



2 ) e

For each solution in P sub perform (Np − R) additive repeated simulation. return The best solution with minimum mean value of ASF

46

Figure 1: Schematic of the SCC manufacturing system

47

1

Sojourn time

3

Steelmaking

2

Waiting time Transfer time

4 1

3

Refining

2

4 1

2

3

4

Casting

(a) optimal solution in the static enviroment 1 Steelmaking

3 2

4 1

3

Refining

2

4 1

2

3

4

Casting

cast-break

(b) cast-break in the dynamic environment 1 Steelmaking

3 2

4 1

Refining

3

over-waiting

2

4 1

2

3

4

Casting

(c) over-waiting in the dynamic environment Figure 2: Gantt Chart for SCC solutions in different scenario

48

Figure 3: Average S/N ratio at each level of the parameters

Figure 4: Validation of the components in MOSS

49

3000

12000 10000

2500

f3

f3

8000 2000

6000 1500

4000

1000 80

2000 1500 60

400

MOSS SDA

800

200

20 f2

1000

1000

300

40

MOSS SDA

100 0

0

f1

(a) A-type System

600

500

400 200 0

f2

0

f1

(b) B-type System

1400 1200

f3

1000 800 600 400 60 80

40

60 40

20

MOSS SDA

20 f2

0

0

f1

(c) C-type System Figure 5: Combined results of MOSS and SDA

30 25

1000

250

800

200

600

150

10

RPD(%)

15

RPD(%)

RPD(%)

20

400

100

5 0

200

50

0

0

−5 −10 SCC System

(a) f1

SCC System

(b) f2 Figure 6: Box-plots for RPD results

50

SCC System

(c) f3

40

1650 MOSS SDA

1600

40 MOSS SDA

35

MOSS SDA

35

1550 30

30

25

25

f1

f2

1450 1400

f2

1500

20

20

15

15

1350 1300 10

1250

10

5

1200 0

10

20

30

40 time (S)

50

60

70

5 0

80

10

20

(a) f¯1

30

40 time (S)

50

60

70

80

0

10

20

(b) f¯2

30

40 time (S)

50

60

70

80

(c) f¯3

Figure 7: The convergence curves of MOSS and SDA

1300

12 scenario 1 scenario 2

1250

40 scenario 1 scenario 2

11

30

mean of f1

1150

mean of f1

9

1200 mean of f1

scenario 1 scenario 2

35

10

8 7

25 20 15

6

1100

10

5 1050

5

4 3

1000 0

0.02

0.04 0.06 noise level

0.08

0 0

0.1

(a) impact on f¯1

0.02

0.04 0.06 noise level

0.08

0.1

(b) impact on f¯2

0

0.02

0.04 0.06 noise level

(c) impact on f¯3

Figure 8: The impact of ηˆ1,l and ηˆ1,l

Table 1: Parameter levels for MOSS

Parameter NP colRate τ e% Imax

levels 5 × |$| 0.1 10 2.5 10

10 × |$| 0.2 15 5.0 20

51

15 × |$| 0.3 20 7.5 30

0.08

20 × |$| 0.4 25 10.0 40

0.1

52

2026.20

2006.77

1726.80

1730.50

1510.40

2380.20

2420.90

2158.60

A4-2#

A4-3#

A5-1#

A5-2#

A5-3#

A6-1#

A6-2#

A6-3#

1283.17

A3-1#

2502.00

1905.15

A2-3#

A4-1#

1651.20

A2-2#

1130.29

1669.82

A2-1#

A3-3#

1178.72

A1-3#

1224.70

1123.87

A1-2#

A3-2#

1069.97

f¯1

A1-1#

No.

193.58

264.67

219.77

205.68

89.79

134.69

135.44

256.65

147.04

276.32

235.95

168.95

152.70

188.14

247.42

249.09

76.98

172.17

S(f1 )

18.30

26.60

32.90

14.10

35.10

19.60

32.00

12.00

42.40

4.06

10.10

16.25

43.38

5.30

4.73

18.72

30.07

29.77

f¯2

10.76

40.71

44.22

25.37

39.41

26.67

49.51

22.95

53.87

12.27

21.83

34.28

44.33

10.41

11.09

63.18

45.28

46.38

S(f2 )

MOSS

10.10

33.40

53.00

16.90

33.80

20.50

20.31

47.80

61.30

15.12

25.00

18.67

26.00

26.40

36.36

21.39

12.20

15.87

f¯3

11.62

79.67

50.46

28.82

22.00

17.72

21.14

92.57

68.59

61.30

55.92

41.20

23.56

31.06

75.03

56.11

17.84

65.80

S(f3 )

242

286

289

189

179

186

214

248

228

160

165

138

155

183

163

121

114

116

CPU(s)

2199.88

2591.19

2456.77

1526.25

1750.71

1745.61

2184.44

2369.53

2694.80

1173.11

1184.74

1431.94

2129.91

1578.19

1659.61

1189.13

1269.13

1168.24

f¯1

732.36

630.92

509.42

218.59

256.55

112.44

399.45

425.56

475.18

259.85

241.73

171.88

705.04

192.91

145.44

227.14

182.36

122.30

S(f1 )

Table 2: The computational results for the A-type SCC system

147.28

165.27

75.57

69.50

88.09

98.62

143.19

72.69

23.82

31.02

25.63

48.95

106.21

26.42

23.95

47.04

72.45

72.22

f¯2

130.53

154.20

130.44

129.76

145.94

72.29

132.00

95.92

252.86

42.60

48.32

64.12

155.34

49.96

38.01

110.71

86.25

68.16

S(f2 )

SDA

24.63

40.46

42.40

19.03

29.46

30.05

43.43

72.11

57.84

21.59

36.59

38.74

30.51

23.41

48.32

36.78

42.50

20.84

f¯3

58.32

60.17

68.03

39.74

33.51

33.14

40.21

120.08

46.90

57.05

64.65

56.36

22.51

20.04

48.06

42.01

69.05

24.79

S(f3 )

242

286

291

189

176

187

216

249

230

162

164

140

156

183

161

118

114

110

CPU(s)

53

8075.60

6112.70

4549.10

3743.17

3764.80

9224.50

8813.60

10373.70

B4-2#

B4-3#

B5-1#

B5-2#

B5-3#

B6-1#

B6-2#

B6-3#

3683.90

B3-1#

7842.80

4676.09

B2-3#

B4-1#

3886.42

B2-2#

3161.82

4279.50

B2-1#

B3-3#

2239.00

B1-3#

3356.69

2109.80

B1-2#

B3-2#

2648.60

f¯1

B1-1#

No.

1243.95

1623.32

1220.29

398.05

247.02

541.75

1082.11

1360.78

998.03

767.29

354.13

232.86

439.36

627.22

340.80

130.43

135.46

82.97

S(f1 )

385.30

461.60

588.90

268.40

256.42

268.00

454.10

707.60

667.60

339.00

286.62

360.00

127.45

116.17

150.80

69.77

58.50

54.20

f¯2

208.97

318.77

279.49

120.59

148.99

145.80

146.94

227.42

156.04

231.58

118.87

94.58

125.28

145.12

83.79

97.56

102.47

84.04

S(f2 )

MOSS

1195.40

758.30

467.40

234.30

160.67

164.50

190.00

92.70

97.33

104.36

114.31

77.40

166.91

106.25

127.60

78.92

86.00

77.40

f¯3

466.03

487.64

308.04

86.25

182.07

168.56

69.55

62.85

61.81

79.54

53.43

68.07

59.04

67.28

82.80

99.26

86.74

96.62

S(f3 )

327

275

290

189

209

223

275

252

229

181

177

162

180

181

174

124

111

135

CPU(s)

10966.04

10636.40

10734.24

4464.21

3627.39

4677.99

8052.88

9295.68

9321.05

3971.45

3858.70

3828.20

4812.33

4063.05

4588.36

2197.45

2348.89

2785.88

f¯1

1261.77

2157.02

1787.37

705.80

533.68

539.70

1854.76

1670.18

1704.67

746.05

550.03

487.02

794.66

864.32

563.93

249.55

70.43

173.52

S(f1 )

Table 3: The computational results for the B-type SCC system

733.87

741.34

918.17

610.08

440.45

653.70

665.17

897.48

812.98

451.80

370.55

531.24

158.89

164.87

188.89

87.72

88.09

79.96

f¯2

236.71

223.61

201.20

271.22

161.95

284.04

228.37

260.02

241.26

192.54

212.48

161.87

153.98

151.87

126.37

102.01

60.11

75.17

S(f2 )

SDA

1127.82

806.62

435.72

252.04

146.20

169.98

202.00

95.73

101.76

94.84

128.33

88.52

195.27

114.22

116.19

83.43

107.12

85.63

f¯3

296.16

139.80

167.75

68.09

77.97

44.46

45.90

40.66

90.14

68.48

84.86

69.33

93.45

93.88

54.69

40.60

124.81

78.96

S(f3 )

322

271

288

183

210

217

280

255

233

181

174

165

183

183

176

122

104

132

CPU(s)

54

1029.86

987.95

897.69

881.19

833.26

1107.42

1212.80

1016.12

C4-2#

C4-3#

C5-1#

C5-2#

C5-3#

C6-1#

C6-2#

C6-3#

695.00

C3-1#

946.11

822.33

C2-3#

C4-1#

716.48

C2-2#

650.90

707.19

C2-1#

C3-3#

619.70

C1-3#

669.91

408.40

C1-2#

C3-2#

494.50

f¯1

C1-1#

No.

446.37

312.31

583.56

378.35

421.12

327.36

199.46

352.86

365.51

345.76

289.08

279.70

328.72

343.32

181.51

179.29

124.44

130.93

S(f1 )

12.02

22.44

6.65

9.91

4.94

1.69

13.03

15.86

8.58

1.10

0.91

2.50

16.05

11.75

14.33

3.80

1.93

0.67

f¯2

34.59

50.64

18.11

25.77

15.80

5.27

47.55

67.41

24.06

3.14

2.70

4.40

34.68

33.79

30.68

9.35

7.22

2.31

S(f2 )

MOSS

15.50

11.46

18.88

11.22

5.13

6.54

7.25

17.93

10.68

5.60

19.00

9.30

8.92

5.30

16.00

9.20

9.73

12.67

f¯3

41.90

36.34

45.76

43.40

25.27

21.66

33.98

40.42

56.25

27.12

46.81

24.54

33.97

40.38

43.84

28.64

14.30

19.87

S(f3 )

265

292

256

177

173

195

172

195

216

128

112

124

143

163

157

99

96

92

CPU(s)

1062.94

1254.19

1213.68

887.61

902.32

879.88

983.46

1060.05

1017.12

669.40

697.85

726.17

830.57

758.59

712.28

647.77

419.31

442.78

f¯1

278.22

399.62

407.67

315.32

246.46

221.10

274.48

282.11

260.34

203.96

200.66

207.54

357.81

261.04

245.72

197.59

112.92

181.05

S(f1 )

Table 4: The computational results for the C-type SCC system

38.42

44.62

65.25

21.82

25.13

12.77

40.89

28.89

46.75

9.09

7.50

14.41

26.84

25.51

30.05

11.13

6.92

2.79

f¯2

52.12

37.42

76.95

39.05

38.29

31.36

59.17

34.58

50.64

22.79

13.95

36.11

60.07

34.04

35.34

23.29

26.29

15.34

S(f2 )

SDA

23.08

13.13

23.35

18.14

9.50

10.35

8.32

21.78

14.89

6.73

24.39

13.74

13.63

7.12

22.66

11.45

10.53

14.26

f¯3

97.84

30.49

127.86

46.32

36.02

13.64

36.31

15.47

42.11

21.28

32.26

30.19

57.54

33.83

74.90

30.89

9.80

16.46

S(f3 )

255

283

249

170

165

188

173

192

214

125

110

124

141

165

154

97

92

88

CPU(s)

55 8.07 1.17 0.57 1.00 2.48 6.14 0.37 4.66

13#

14#

15#

16#

17#

18#

AVG.

-3.33

8#

12#

11.68

7#

16.56

10.05

6#

11#

-4.47

5#

6.92

-0.24

4#

10#

1.01

3#

3.94

12.40

2#

9#

9.57

f1

1#

No.

317.56

689.37

546.73

123.76

390.58

142.04

424.54

366.65

531.00

-79.14

760.12

158.78

218.41

137.70

401.06

434.16

171.40

146.13

152.81

f2

A

56.34

124.05

29.05

-23.44

9.61

-15.58

44.30

115.46

53.74

-3.71

59.48

51.31

118.70

18.64

-9.43

42.99

90.80

244.50

63.70

f3

10.93

5.74

20.54

16.12

18.19

-3.58

2.86

31.30

15.03

18.44

26.04

14.69

3.50

2.46

4.21

6.93

-2.19

11.57

4.98

f1

55.89

93.15

64.65

58.46

127.37

74.13

145.63

46.30

27.09

21.29

35.49

27.98

47.18

24.78

45.05

24.35

27.74

61.78

53.70

f2

B

5.84

-4.89

9.60

-5.13

8.24

-5.37

8.55

7.25

4.96

2.86

-8.89

10.88

15.09

16.07

6.19

-7.89

11.19

23.27

13.06

f3

Table 5: The RPD results for all SCC systems

3.27

5.79

3.01

10.94

7.19

3.74

-1.28

-0.94

3.43

8.40

4.34

5.13

5.26

0.80

6.79

0.16

4.43

2.90

-11.27

f1

367.36

256.31

119.16

994.17

133.37

474.56

694.09

269.17

128.96

515.22

747.98

793.47

446.49

66.30

142.35

124.16

200.78

256.87

249.11

f2

C

43.81

31.87

22.08

-3.86

79.00

103.93

82.87

17.99

34.99

70.71

38.07

39.08

52.58

47.68

79.58

35.65

28.43

12.20

15.75

f3

Table 6: The comparison results on industrial instances

MOSS

No.

SDA

RSH

f1

f2

f3

f1

f2

f3

f1

f2

f3

1#

2571.7

2728.9

2891.2

55.3

61.8

74.2

71.9

78.4

89.5

2#

2694.8

2664.3

2707.5

44.7

46.9

48.5

43.5

52.1

69.7

3#

2488.6

2607.5

2649.8

41.6

44.8

68.3

64.6

62.4

75.7

4#

2323.4

2555.7

2634.5

49.3

59.0

70.4

70.5

61.3

69.6

5#

2651.1

2716.2

2851.3

34.3

38.9

44.9

53.9

64.6

86.3

6#

2626.4

2789.0

2847.8

40.2

45.4

52.7

56.1

67.4

62.5

7#

2751.6

2626.8

2954.4

50.1

46.0

57.5

44.1

53.5

54.7

8#

2522.8

2729.1

2867.4

32.5

34.3

48.9

61.7

64.9

70.7

9#

2391.0

2510.1

2616.7

39.0

48.1

50.3

54.5

61.6

85.3

10#

2772.2

2819.4

2946.6

34.5

36.1

49.7

47.6

55.7

62.8

AVE.

2579.4

2674.7

2796.7

42.2

46.1

56.5

56.8

62.2

72.7

56

Acknowledgments We would like to thank the anonymous reviewers and the editors for their constructive and pertinent comments. This work is supported by the National Natural Science Foundation of China (No. 51474044), the Key Projects of Chongqing Science and Technology Research Projects of China (No. CSTC2011AB3053), the Fundamental Research Funds for the Central Universities (No. 106112017CDJXY).

Highlights
An uncertain scheduling problem arising from practical SCC production is studied.
A soft-form schedule is introduced to tackle uncertainties.
A preference-inspired method is proposed to address multiple objectives.
A CRO algorithm with clean-up is proposed to solve the scheduling problem.