Steelmaking-Continuous Casting Scheduling Problem with Interval Type 2 Fuzzy Random Due Dates M. H. Fazel Zarandi Department of Industrial Engineering Amir-Kabir University of Technology Tehran, Iran
[email protected]
F. Dorry
F. Shabany Moghadam
Department of Industrial Engineering Amir-Kabir University of Technology Tehran, Iran F.dorry @aut.ac.ir
Department of Industrial Engineering Amir-Kabir University of Technology Tehran, Iran f.sh.m @aut.ac.ir
Abstract—In this paper we propose a steelmaking-continuous casting scheduling (SCCS) model, where due dates are interval type 2 fuzzy random variables. Our proposed model is based on just-in-time (JIT) idea. We assign satisfaction level to charge’s completion time according to interval type 2 membership functions. Support position of membership functions depend upon the expected due dates, which are exponentially distributed random variables. In this scheduling model, limit for total waiting time between processing and waiting time before casting stage are considered as interval type 2 fuzzy sets. The interval type 2 fuzzy random optimization problem is converted to a crisp and nonlinear optimization problem by using the symmetric approach. Finally an example is presented to show the application of proposed model.
The whole scheduling process can be divided into two steps [2]:
Keywords—Steelmaking and continuous casting scheduling; interval type 2 fuzzy random variable; symmetric approach;
Recently due dates in some SCCS problems have been formulated as uncertain values in order to make the model closer to real word. Wang et al. [4] proposed a SCCS problem with fuzzy processing and delivery time. They proposed an improved genetic algorithm to solve the problem. Hornig [5] developed two MILP models for scheduling of steelmaking-continuous casting process. The objective is makespan minimization. In addition, he developed a MILP modeling approach for fuzzy overall constraint satisfaction maximization. In this model he used type 1 fuzzy sets for due dates, transit times and job continuity constraints.
I.
INTRODUCTION
Steel industry is an important part of any industrial economy which provides primary materials for other industries. Since it is capital and energy intensive, steel companies should mainly rely on the new integrated production processes to improve productivity, reduce energy consumption, and maintain competitiveness in the market. The influence of an efficient process control on the cost and energy reduction and environmental effects in iron and steel industry, makes the process control one of the main issues of this industry. In modern iron and steel corporations the focus is placed on high quality, low cost, just-in-time (JIT) delivery and small lot with different varieties [1]. Steelmaking-continuous casting scheduling (SCCS) problems should determine in what sequence, at what time, and on which device, the molten steel should be arranged at various production stages from steelmaking to continuous casting [2]. Atighehchian et al. [2] investigated the SCCS problem. They proposed a novel hybrid algorithm to solve the problem. In their proposed algorithm, ant colony optimization and nonlinear optimization methods are hybridized efficiently.
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1. Cast sequencing which determines job sequence in each casting machine. 2. Sequencing and scheduling of steelmaking process and timing of the jobs on continuous casting machines. In many SCCS problems, due dates are crisp values. Tang et al. [1] produced a crisp mathematical programming model for scheduling steelmaking-continuous casting production. The model was developed as a nonlinear model, considering both punctual delivery and production operation continuity. Sun et al. [3] introduced a surrogate subgradient algorithm for lagrangian relaxation to solve the SCCS problem.
When the degree of vagueness of information is high interval type 2 fuzzy is a more suitable approach than type 1 to model complicated problems. Fazel zarandi et al. [6] developed an interval type 2 fuzzy multi agent based expert system for scheduling of steel production. In their system variables that gained from manufacturing experts, are presented by interval type 2 fuzzy membership functions. In some real situations of decision making, there exists uncertainty that can't be described only by fuzziness. These environments include both fuzziness and randomness. It is proper to use fuzzy random variables in order to formulate such a situation [7]. Itoh et al. [7] proposed an n-job, one machine scheduling model, where due dates for jobs are fuzzy random variables. Wang et al. [8] developed a
methodology for parallel machines scheduling problem with fuzzy random due dates. They designed a genetic algorithm to solve the model. In steel industries, goals, objectives and operative constraints are obtained by polling a group of domain experts. These experts will not necessary be in agreement, therefore we use interval type 2 fuzzy for presenting different viewpoints of domain experts. Because of the stochastic nature of production processes in steel industries, this environments includes both fuzziness and randomness and it is proper that we assume due dates are stochastic. These due dates which we call them expected due dates aren't deadlines and we are allowed to produce products over them in some degree because they are uncertain. As the best of our knowledge, there is no SCCS problem in literature that considers both interval type 2 fuzziness and randomness in one model. In this paper we extend the concept of fuzzy random due date proposed by Itoh et al. [7] to interval type 2 fuzzy random due date and formulate a SCCS problem with interval type 2 fuzzy random due dates. The interval type 2 fuzzy random due date is an interval type 2 fuzzy due date whose upper and lower membership functions fluctuate stochastically. Like the work of Itoh et al. [7] we assume that expected due dates are exponentially distributed random variables. In this work, we extend the type 1 fuzzy scheduling model proposed by Horing to type 2 fuzzy random one. Our model is based on just in time (JIT) idea and both the early and tardy completion of charges would decrease the satisfaction levels. In our proposed scheduling model limit for waiting time before casting stage is considered as interval type 2 fuzzy set. Finally we solve the interval type 2 fuzzy random scheduling problem by converting it to a crisp and nonlinear optimization problem. The rest of this paper is organized as follows: In section II, problem definition is presented. In section III, first we give a short description of crisp problem and then we formulate the interval type 2 fuzzy random scheduling problem. The solution procedure is presented in section IV. Then a numerical example is given in section V. Finally, conclusions and future work are presented in section VI. II.
PROBLEM DEFINITION
Tundish, the input unit of a continuous caster (CC), for casting. Depending on the grade, the charge should not spend more than a certain time before casting in order to keep the charge temperatures within a required range. There is also a maximum limit on the total waiting time between processing; otherwise the production of a grade may fail and the product may not meet its quality requirements without additional costs [2]. A cast is a sequence of charges that are consecutively cast on the same continuous caster. No setup time is required on the caster between adjacent charges in the same cast. However, a relatively long set-up time is required between two casts in the same caster [2]. In the continuous casting stage, the existence of waiting time between two consecutive processed charges in one cast will bring lots of economic loss for the continuous cast machine. Less breaking cast between the consecutive process charges in one cast, more economic benefits will be got [3]. B. Assumptions We assume that there is only one machine at each stage, i.e. we have one EAF, one LF and one CC machine. The sequence of charges on CC machine is defined at the planning level. Because of the existence of only one machine at steelmaking stage and one machine at refining stage, the order of processing the charges in these two stages is the same as continuous casting stage. SCCS is then to assign the starting time of each charge at each stage, such that: • Casting break is minimized • Waiting time constraints are met • Due dates are met The setup time of steelmaking stage and refining stage aren't considered. III.
PROBLEM FORMULATION
The crisp scheduling problem will first be presented in subsection B and the formulation will be extended to interval type 2 fuzzy random formulation in subsection C. A. Notations In this subsection we introduce some of the notations proposed by Hornig [5] and Tang et al. [9]: N: number of casts. ni : number of charges in cast i, i=1,…,N.
A. Steelmaking- continuous casting process and scheduling problem In this section a brief description of SCCS problem is given.
n: total number of charges to be processed (n= ∑Ni=1 ni .
The steelmaking process consists of three stages: steelmaking, refining and continuous casting. In the steelmaking stage the impurities of molten iron are reduced to desirable levels by burning with oxygen in an electric arc furnace (EAF). The basic unit of steelmaking production is a charge, which is defined as a “job” in SCCS [2]. Each charge is defined through its related gauge and Grade. Grade is a product quality description including both chemical and physical properties of the charge. The molten steel from the steelmaking stage is poured into ladle furnaces (LF) for refining [2]. After refining, molten steel is poured into a
Ωi : set of all charges in cast i, i=1,…,N.
Ω: set of all ordered charges in ordered casts, Ω={1,…,n}. The sequence is defined at the planning level. The charges of set Ω are ordered such that the first charges are those of cast 1 and are in set Ω1 , the next charges are those of cast 2 and are in set Ω2 and so on up to the last charges of cast N that are in set ΩN [5]. Pjk : processing time of job j at stage k; j=1,…,n; k=1,2,3.
Si : set up time before first charge of cast i at stage 3; i=1,…,N.
dj : due date for job j; j=1,…,n. si(p) : pth charge in cast i; p=1,…,ni. t(k,k+1): transportation time of a job from stage k to stage k+1; k=1,2. TWBP: maximum limit for total waiting time between processing. WBC: maximum limit for waiting time before casting stage. Decision variables: xjk : starting time of job j on stage k; j=1,…,n; k=1,2,3. Relations: cj =xj,3 +Pj,3 Wj =xj,3
j=1,…,n;
xj,2
TWj =xj,3
Pj,2
xj,1
Bp 1 =xsi(p
t 2,3
Pj,1 xsi(p),3
1) ,3
t 1,2
C. Interval type 2 fuzzy random problem formulation based on just-in-time (JIT) idea In introduction we mentioned that the nature of due dates, limit for total waiting time between processing and limit for waiting time before casting stage in SCCS problems are (1) uncertain. In this subsection we suppose due dates Dj are interval type 2 fuzzy random due dates. According to charge's (3) completion time cj , satisfaction levels are assigned for it based on the lower and upper membership functions μ cj (2)
j=1,…,n;
Pj,2
transported to that stage. Constraint (7) ensures that for two contiguous charges processed at the same stage, only when the preceding charge has finished, the immediately next one can be started. Constraint (8) ensures enough setup time between two consecutive casts on continuous caster. Constraint (9) ensures that due dates are met. Constraint (10) sets an upper limit for total waiting time between processing. Constraint (11) that is barrowed from Atighehchian et al [2] sets an upper limit for the waiting time before casting operations. Constraint (12) ensures the non-negativity of start times.
t 2,3
j=1,…,n;
Dj
Psi(p),3
and μD cj . Membership functions are shown in Fig. 1. j Since the experts will not necessary be in agreement in i=1,…,N; p=1,…,ni 1; (4) satisfaction levels, we use interval type 2 fuzzy set. We define the lower and upper membership functions as is In above cj is completion time of job j; Wj is waiting time shown in (13) and (14). before casting stage of job j; TWj is total waiting time We propose to develop schedules that complete each between processing of job j; and Bp 1 is break casting charge j at or near its expected due date dj . In this scenario, time of charge p+1; p=1,…,ni-1; both the early and tardy completion of charges would B. Crisp problem formulation decrease the satisfaction levels. The crisp scheduling model is presented below. This scheduling model is mainly based on model proposed by Hornig [5] with a little change that considers only one machine at each stage and considers waiting time constraint before casting too. n
1
min J= ∑Ni=1 ∑pi=1 xsi(p
xj,k+1 ≥xj,k +Pj,k +t(k,k+1) xj+1,k ≥xj,k +Pj,k i
xj,3 +Pj,3 ≤dj xj,3
xj,1
(5)
j=1,…,n; k=1,2;
(6)
1; k=1,2,3;
(7)
j=1,…,n
xsi(n ),3 +Psi(n ),3 +Si+1 ≤xsi i
Psi(p),3
xsi(p),3
1) ,3
1(1) ,3
i=1,…,N
Pj,2
(8) (9)
j=1,…,n; Pj,1
1;
Fig. 1. Type 2 fuzzy random due date
0
t 1,2
1
t(2,3)≤TWBP xj,3 xj,k ≥0
xj,2
Pj,2
(10)
j=1,…,n; t(2,3)≤WBC
j=1,…,n; k=1,2,3;
j=1,…,n;
(11) (12)
The objective is to schedule the charges while minimizing total break casting time. Constraint (6) ensures the precedence relationships of the contiguous operations for a charge, so that the process at one stage can start only if the charge has finished on the previous stage and was
μ
Dj
cj =
q1j
cj
