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A mathematical programming model for scheduling steelmaking-continuous casting production1
European Journal of Operational Research 120 (2000) 423±435
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Theory and Methodology
A mathematical programming model for scheduling steelmaking-continuous casting production 1 Lixin Tang b
a,*
, Jiyin Liu b, Aiying Rong c, Zihou Yang
d
a Department of Systems Engineering, Northeastern University, Shenyang, People's Republic of China Department of Industrial Engineering and Engineering Management, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, People's Republic of China c Department of Automation, Beijing Institute of Machinery Industry, Beijing, People's Republic of China d Department of Systems Engineering, Northeastern University, Shenyang, People's Republic of China
Received 1 November 1997; accepted 1 November 1998
Abstract This paper presents a mathematical model, based on the just-in-time (JIT) idea, for solving machine con¯icts in steelmaking-continuous casting (SCC) production scheduling in the computer integrated manufacturing system (CIMS) environment. The model is developed as a non-linear model based on actual production situations, considering both punctual delivery and production operation continuity. It is then converted into a linear programming model which can be solved using standard software packages. An example demonstrating the application of the proposed method is given. The paper also describes the implementation of an SCC production scheduling system in which the proposed model is used as an eective method to optimize production continuity and product delivery while eliminating machine con¯icts. Ó 2000 Elsevier Science B.V. All rights reserved. Keywords: Iron and steel industries; Steelmaking-continuous casting; Production scheduling; Machine con¯ict; Linear programming
1. Introduction Modern iron and steel corporations are moving towards continuous, high-speed and automated
*
Corresponding author. E-mail: [email protected] The project is supported by National Natural Science Foundation of China through approved No. 79700006 and by National 863/CIMS of China through approved No. 863-511708-009. 1
production process with large devices. The focus is placed on high quality, low cost, just-in-time (JIT) delivery and small lot with dierent varieties. In order to enhance their competitive power, many international iron and steel corporations are devoted to developing computer integrated manufacturing systems (CIMS) which can improve productivity of large devices, shorten waiting-time between operations, reduce material and energy consumption, and cut down production costs. Production scheduling is a key component of
0377-2217/00/$ ± see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 9 9 ) 0 0 0 4 1 - 7
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CIMS. Its task is to determine the starting times and the ending times of jobs on the machines so that a chosen measure of performance is optimized. Steelmaking-continuous casting (SCC) production scheduling problems are to determine in what sequence, at what time and on which device molten steel should be arranged at various production stages from steelmaking to continuous casting. Unlike general production scheduling in machinery industry, SCC production scheduling problems have to meet special requirements of steel production process. In the SCC process, the products being processed are handled at high temperature and converted from liquid (molten steel) into solid (drawn billets). There are extremely strict requirements on material continuity and ¯ow time (including processing time on various devices and transportation and waiting time between operations). The study of this paper is investigating the SCC production scheduling problem and aims at developing a computerized scheduling system for generating optimal schedules. The project uses the steelmaking plant in Shanghai Baoshan Iron and Steel Complex as the study background. The whole scheduling process consists of four steps: (1) cast sequencing which determines cast sequences and charge sequence in each cast; (2) sub-scheduling which ful®lls the scheduling of individual charge sets; (3) rough scheduling which merges sub-schedules; and (4) optimal scheduling which eliminates machine con¯icts. While the ®rst three steps consider mainly the operation relationships and can be done relatively easily, the last step needs to consider resource constraints to ensure practical feasibility of the resulting schedule. This paper emphasizes this optimal scheduling step and proposes a mathematical programming model to eliminate machine con¯icts. In the following sections, we ®rst brie¯y review related previous work (Section 2). Then the SCC production process and the overall scheduling problem are described in Section 3. Section 4 presents the mathematical programming model for solving the machine con¯ict problem. The nonlinear model is then transformed into a linear programming (LP) model in Section 5. Section 6
gives an application example and discusses the issues of using the model in the implementation of a practical SCC scheduling system. Section 7 concludes the study. 2. Literature review Solving production operational management problems in SCC production through mathematical models and computers is an important new research topic and has recently been widely explored. Various production planning and scheduling techniques for SCC production have been reported in literature. According to the information from International Steel Production Management Conference in 1993 and reports in the latest literature, some advanced steel incorporations have started to carry out extensive research on production operational management which has become a hot subject in the academic circle of international iron and steel incorporations. Most approaches to scheduling continuous casters treat them as stand-alone facilities that are coupled only loosely with further processing such as rolling. These approaches attempt to improve productivity of the continuous casters while reducing operating costs and guaranteeing on-time delivery. All the methods used for SCC production scheduling can be classi®ed into two categories ± mathematical programming based methods and expert system based methods. 2.1. Mathematical programming based methods An example of o-line scheduling for steel production using mathematical programming was provided by Redwine and Wismer [1]. Dynamic programming was used to solve the model. Petersen et al. [2] have developed a mathematical programming model for a steel production scheduling problem to optimally schedule the slabs through the reheating furnace and the rolling mill. The model was solved heuristically. Lally et al. [3] established a simple mixed-integer linear programming solution to the problem of caster scheduling. They considered a simple model of a steel plant in
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which steel was started at an electric arc furnace, held in a ladle, and cast on a continuous caster. However, the model did not consider all the complexities of a real continuous caster. Tong et al. [4] constructed a complex mixed-integer linear programming model and solved it using heuristic techniques for twin strand continuous slab caster scheduling problem at LTV and Geneva Steel Works. The Geneva system is a state-of-theart application program that schedules a continuous caster while considering business constraints and capacity constraints of the various facilities upstream and downstream. LTV caster scheduling model [5,6] was implemented in 1983, in support of the ®rst continuous caster installed at LTV Cleveland Works. The model was intended to schedule caster production from customer order while optimizing several key objectives such as maximizing caster productivity. Meil and Lee [7] described the primary production scheduling procedure for the two-strand casters at LTV Cleveland Steel Works. The scheduling system, developed by IBM Research and ATM, exploited a fast sequencing algorithm for schedules satisfying global constraints. At Stahl Linz GmbH, Neuwirth [8] reported a linear programming model with machine con¯icts and provided key modeling factors of SCC scheduling and charge allocation scheme in the furnace, but the mathematical representation of the model was not given.
GmbH, Stohl and Spopek [15] established a hybrid co-operative expert system modeling to solve SCC scheduling problems, but they were unable to construct an optimized mathematical model. Epp et al. [16] described an interactive scheduling system developed using AI method for an SCC facility at Inland Steel Corporation. Hamada et al. [17] presented a framework for solving complex steelmaking scheduling problems and then combined rule-based expert system and genetic algorithm to produce ecient schedules. However, no systematic research has been realized so far on the general structure, model and algorithm which can be applied to steelmaking plants.
2.2. Expert system based methods
EAF BF BOF LMF Tundish LF Cluster
An expert system was used by Jimichi et al. [9] to determine parameters and operational conditions to match slab production with customer orders. Another example of using expert system techniques for iron and steel production scheduling is provided by Sato et al. [10]. Numao and Morishita [11±14] described an expert system application to perform co-operative scheduling in which the schedule was modi®ed by the scheduler using a graphical user interface. The main justi®cation for the use of expert systems came from reducing waiting time from charge to charge and minimizing energy consumption. Signi®cant bene®ts have been achieved as reported in the literature. At Stahl Linz
3. SCC production process and scheduling problem 3.1. SCC production process To describe the SCC production process, we ®rst introduce the following special terms: Billet Bloom Machine
Grade Heat Slab Strand Charge
A steel piece with square cross section, smaller than a bloom A steel piece with square cross section, larger than a billet A production device which performs one operation at a time. The machines for performing identical operations are called alternative machines Electric Arc Furnace Blast Furnace Basic Oxygen Furnace Ladle Metallurgical Facility A receptacle at top of caster Ladle Furnace Production between caster turnarounds Steel with a speci®ed metallurgical composition Furnace-load of steel A steel piece with elongated rectangular cross section Stream of steel from a caster A unit of production that consists of a sequence of operations on a heat
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Charge set CAST
Unit
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A set of charges that produces the same products, i.e., that have almost identical operations A set of charges that continually casts on the same continuous caster and has a similar chemical composition An operation that speci®es the machine, the starting time, and the ending time
Fig. 1 is a diagrammatic representation of the above terms. Here, the horizontal axis stands for time, and each line represents a machine. One charge path consists of units and handling times, which are expressed by lines connecting the units. Waiting time is illustrated using dashed lines before the units. In the iron and steel production from iron ore input to steel product output, there are three major manufacturing processes: ironmaking, steelmaking, and rolling. Steelmaking re®nes pig iron into steel and casts it into slabs, blooms, or billets. The SCC production process is illustrated in Fig. 2. The steelmaking process starts with the charge of crude steel and scrap iron in one of the EAFs. Liquid iron, tapped from BF, will be transported to the steelmaking shop where BOFs and/or EAFs are located. BOF and EAF burn out the excessive carbon, sulphur, silicon, and other impurities from liquid iron and re®ne it to steel with desired con-
tents. The ®lling of one furnace is called a charge (heat) and already contains the main alloying elements. The furnaces have dierent capacities from 17±55 t. The duration of the melting process depends on the ingredients as well as external factors. Because of voltage peaks, as many as 5 hours can be required for melting, but usually 2±2.5 hours are enough. A ®xed set up and a maintenance interval of 20 minutes are included in this duration. The melted steel is poured into ladles that are transported by a crane to an LF. If the proceeding heat has a long processing time in the LF, the current heat must wait. This slack time cannot exceed 2 hours. The next step is a heat treatment in the LF, where the ®ne alloying takes place. The duration is usually about the same as the melting. Then, special treatment may be performed in the Ladle-re®ning equipment (VOD or VD) to eliminate impurities from molten steel or add alloy ingredients to the molten steel in ladles to make high-grade steel. Finally, a continuous caster casts molten steel continuously into slabs, blooms, or billets. If the casting format needs to be altered, a set-up time must be considered. 3.2. Framework of SCC production operational management system Fig. 3 describes the overall structure of SCC production operational management. In this
Fig. 1. Process ¯owsheet of a CAST including three charges.
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Fig. 2. The SCC production process.
Fig. 3. Overall structure of production management system for SCC production.
structure, the original ``customer order data'' is converted into ``production order data'' through ``production target quality'' (PT/Q) design and ``production target plan'' (PT/P) design. Later, the order plan is produced based on production capacity of the main processes. SCC production lot plan includes charge design and cast design. The production lot plan is formulated through optimal combination of orders according to the contents of ``production order information data'' and ``order plan'' to meet process constraints and dierent order requirements. On the basis of lot plan, SCC production scheduling is to determine in what sequence, at what time and on which machine molten steel should be arranged to enter various
production stages from steelmaking to continuous casting.
3.3. Basic structure of SCC production scheduling design A ¯owchart for SCC production scheduling is presented in Fig. 4. The whole SCC production scheduling problem is handled in four steps. First, cast sequencing is arranged. Sub-schedules and rough schedule are then established. Finally the SCC production schedule is formed considering availability of machines at all stages. A
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(4) Elimination of machine con¯icts Because only the machine operation conditions in one cast are taken into account, machine con¯icts often exist in the rough schedule. Only when machine con¯icts are completely eliminated can a feasible and practical schedule be established. This step is to produce an optimal schedule in which all machine con¯icts are eliminated. The three steps mentioned above are realized using man±computer interaction. The following discussion emphasizes on the elimination of machine con¯icts using an optimization model. 4. Mathematical model for eliminating machine con¯icts in optimized scheduling
Fig. 4. Flowchart for the SCC production scheduling module.
brief description for each of these steps is given below. (1) Cast sequencing Cast sequences on the casters and charge sequence in each cast are determined in this step based on their delivery times. These can be considered as single machine sequencing problems without resource constraints. (2) Establishment of sub-schedules After a cast has been established, a cast job timetable called sub-schedule is formed for each cast in this step according to time progress of the operations including steelmaking, re®ning and continuous casting in each charge. A sub-schedule has been shown in Fig. 1. (3) Establishment of a rough schedule In this step the sub-schedules with relative times are combined to produce a rough job schedule (superposition of sub-schedules) with physical time.
This model is proposed here to eliminate machine con¯icts in optimized scheduling within the overall scheduling framework described in the last section. In this section we ®rst give a brief narrative description of the model for easy understanding of the practical meanings. Then the notations to be used in the model are introduced. After these preparations the model formulation is ®nally presented. 4.1. Overview of the model (1) Basic assumptions The model is developed under the following assumptions which are made based on practical requirements in the SCC production process and the scheduling system structure. (a) The sequence of sub-schedules remains unchanged. (b) At most one job can be processed on a machine at a time. (c) Only the three kinds of machines ± steelmaking furnace, re®ning furnace and continuous caster ± are taken into account in the model. Other resources such as intermediate transportation cranes are neglected based on the fact that their capacities are large enough. However, transportation times are considered. (d) All steel products follow the same production process route: steelmaking, re®ning and continuous casting. If some steel products do
L. Tang et al. / European Journal of Operational Research 120 (2000) 423±435
not need re®ning, then their processing times for re®ning are considered to be zero. (2) The decisions to be made The machines used for each operation of each charge are ®xed in the ®rst three steps of the scheduling framework. The decisions to be made in this model are the time points at which the processing of each charge i on machine j begins. (3) The objective function The objective chosen for this model is to ensure continuity of the production process and just-intime delivery of ®nal products. This is achieved through minimizing a cost function consisting of the following terms. (a) Cast break loss penalties exerted to ensure that charges in the same cast are cast as continuously as possible. (b) Molten steel temperature drop cost in terms of waiting time from operation to operation. (c) Earliness/tardiness penalty used to ensure that blooms or billets in each charge are delivered as punctually as possible. (4) Constraints considered The following constraints are considered in the model to guarantee that there will be no machine con¯icts in the schedule generated. (a) For the two contiguous operations for the same charge, only when the preceding operation has been completed, the immediately next one can be started. (b) For two contiguous charges processed on the same machine, only when the preceding charge has been completed, the immediately next one can be started. (c) Setup time and interval time are required from cast to cast on the same continuous caster. 4.2. Notations The following notations are used for de®ning the problem parameters and variables in the model. Parameters X the set of all charges, X f1; 2; . . . ; N g, where N is the total number of production charges
Xk
Nk P Pi U
SI(i, j) SP(i, j) di C1k C2ij C3i
C4i Tij tjm Skj l
429
the set of all charges in the kth cast, k 2 f1; 2; . . . ; Mg, where M is the total number of casts. Xj \ Xk ;, for any j, k 2 f1; . . . ; Mg and j 6 k. X1 [ X2 [ [ XM X the total P number of charges in the kth cast. M k1 Nk N the set of all machines, P f1; 2; . . . ; J g, where J is the total number of machines the set of all machines used for the ith charge, Pi P the set of all continuous casters, jUj C, where C is the total number of continuous casters. Note that the casters are the machines used at the ®nal stage of the SCC production and therefore U P the immediate successor charge of charge i processed on machine j the immediate successor machine of machine j for processing charge i order delivery time of charge i coecient of cast break loss penalty for cast k coecient of penalty cost for the waiting time of charge i after ®nishing process on machine j coecient of penalty cost for the production of charge i being ®nished before the delivery time speci®ed in the production order coecient of penalty cost for the production of charge i being late with respect to the delivery time processing time of charge i on machine j transportation time from machine j to machine m setup time of cast k on machine j interval time from cast to cast
Decision variables starting time of charge i on machine j Xij 4.3. The model formulation Using the above notations, the mathematical model for the optimal scheduling step of the SCC scheduling problem is formulated as follows:
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(P)
transported to the next machine, the immediately next operation on the charge can be started. Constraint (4) ensures enough setup time and interval time required between casts. As can be seen, this is a non-linear model. For easier solution it is transformed into a linear programming model in the next section.
minimize : M X X C1k XSI i;j;j ÿ Xij ÿ Tij Z k1 i2Xk
j2Pi \U SI i;j2Xk
N X X i1
ÿ
C2ij Xi;SP i;j ÿ Xij ÿ Tij ÿ tj;SP i;j
j2Pi
SP i;j2Pi
N X X
5. Solution strategy
C3i min 0; Xij Tij ÿ di
i1 j2Pi \U
N X X
C4i max 0; Xij Tij ÿ di
1
i1 j2Pi \U
subject to: XSI i;j;j ÿ Xij P Tij ;
i 2 X; j 2 Pi ; SI i; j 2 X; 2
Xi;SP i;j ÿ Xij P Tij tj;SP i;j ;
i 2 X; j 2 Pi ;
SP i; j 2 Pi ;
3
XSI i;j;j ÿ Xij P Tij SSI i;j;j l; i 2 Xk ; j 2 Pi \ U; SI i; j 2 Xp ; k; p 1; . . . ; M; and k 6 p;
4
Clearly, the non-linearity of the model comes from the last two terms in the objective function, i.e., the penalties for earliness and lateness in ®nal product delivery. Note that these two terms and associated variables are only related to the casters (the machines used at the last stage of production ± members of U). In order to deal with the nonlinearity, we ®rst separate the set of machines used for charge i into two subsets, the casters fjjj 2 Pi \ Ug and the other machines fjjj 2 Pi ; j 62 Ug, and accordingly break each relevant term in the objective function and each relevant constraint into two. After this treatment, the model changes to the following form: (TP1) minimize :
Xij P 0;
i 2 X; j 2 Pi :
5
It is not dicult to match this mathematical formulation with the overview of the model in Section 4.1. The four terms in the objective function represent the cast break loss penalty, charge waiting time, penalty cost, earliness penalty cost and lateness penalty cost, respectively. The objective function is in line with the JIT idea, both in terms of ®nished product delivery and in terms of operations on semi-products. Constraint (2) ensures that for two contiguous charges processed on the same machine, only when the preceding charge is ®nished, the immediately next one can be started. Constraint (3) ensures that for contiguous operations of the same charge, only when the preceding operation is completed and the charge is
Z
M X X
C1k XSI i;j;j ÿ Xij ÿ Tij
k1 i2Xk
j2Pi \U SI i;j2Xk
N X X i1
SP i;j2Pi SP i;j62U
N X X i1
ÿ
C2ij Xi;SP i;j ÿ Xij ÿ Tij ÿ tj;SP i;j
j2Pi
C2ij Xi;SP i;j ÿ Xij ÿ Tij ÿ tj;SP i;j
j2Pi
SP i;j2Pi \U
N X X
C3i min 0; Xij Tij ÿ di
i1 j2Pi \U
N X X
C4i max 0; Xij Tij ÿ di
i1 j2Pi \U
6
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subject to:
(TP2)
XSI i;j;j ÿ Xij P Tij ; i 2 X; SI i; j 2 X;
minimize :
j 2 Pi ; and j 62 X;
7
Z
M X X
j2Pi \U SI i;j2Xk
8
Xi;SP i;j ÿ Xij P Tij tj;SP i;j ;
ÿ TSI i;j;j dSI i;j ÿ Yij Zij ÿ di
i 2 X; SP i; j 2 Pi ; and SP i; j 62 U; j 2 Pi ;
9 i 2 X; 10
Now the parts in the model related to the casters are the ®rst and last three terms in the objective function and constraints (8), (10) and (11). We de®ne two sets of new variables related to the casters:
Yij max 0; Xij Tij ÿ di ;
i 2 X; j 2 Pi \ U; i 2 X; j 2 Pi \ U:
It is clear that Yij and Zij are nonnegative and Yij ÿ Zij Xij Tij ÿ di ;
i 2 X; j 2 Pi \ U;
or Xij Yij ÿ Zij ÿ Tij di ;
SP i;j2Pi \U
N X X
C3i Zij
N X X
XSI i;j;j ÿ Xij P Tij ;
i 2 X; SI i; j 2 X; j 2 Pi ; and j 62 U;
Substituting all the Xij , i 2 X, j 2 Pi \ U, in model (TP1) with Yij and Zij using the above relations, the model can be transformed into the following form:
13
14
YSI i;j;j ÿ ZSI i;j;j ÿ Yij Zij P TSI i;j;j ÿ dSI i;j di ; i 2 X; SI i; j 2 X; j 2 Pi \ U;
15
Xi;SP i;j ÿ Xij P Tij tj;SP i;j ; i 2 X; SP i; j 2 Pi; and SP i; j 62 U; j 2 Pi ;
16
Yi;SP i;j ÿ Zi;SP i;j ÿ Xij P Tij tj;SP i;j Ti;SP i;j ÿ di ; i 2 X; SP i; j 2 Pi \ U; j 2 Pi ;
17
YSI i;j;j ÿ ZSI i;j;j ÿ Yij Zij P SSI i;j;j l TSI i;j;j ÿ dSI i;j di ; i 2 Xk ; j 2 Pi \ U; SI i; j 2 Xp ; k; p 1; . . . ; M; and k 6 p;
i 2 X; j 2 Pi \ U:
C4i Yij
i1 j2Pi \U
subject to:
12
Zij ÿ min 0; Xij Tij ÿ di ;
C2ij Yi;SP i;j ÿ Zi;SP i;j
j2Pi
i1 j2Pi \U
11
i 2 X; j 2 Pi :
SP i;j2Pi SP i;j62U
ÿ Ti;SP i;j di ÿ Xij ÿ Tij ÿ tj;SP i;j
i 2 Xk ; j 2 Pi \ U; SI i; j 2 Xp ; k; p 1; . . . ; M; and k 6 p;
C2ij Xi;SP i;j ÿ Xij ÿ Tij ÿ tj;SP i;j
j2Pi
N X X i1
XSI i;j;j ÿ Xij P Tij SSI i;j;j l;
Xij P 0;
N X X i1
SP i; j 2 Pi ; \U; j 2 Pi ;
C1k YSI i;j;j ÿ ZSI i;j;j
k1 i2Xk
XSI i;j;j ÿ Xij P Tij ; i 2 X; SI i; j 2 X; j 2 Pi \ U;
Xi;SP i;j ÿ Xij P Tij tj;SP i;j ;
431
18
Xij P 0;
i 2 X; j 2 Pi ;
19
Yij P 0;
i 2 X; j 2 Pi \ U;
20
Zij P 0;
i 2 X; j 2 Pi \ U:
21
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The model now becomes a linear programming model which can be solved by standard linear programming software packages.
X f1; 2; 3; 4; 5; 6g; X1 f1; 2g; X2 f3; 4g; X3 f5; 6g;
6. Applications
P2 f2; 3; 6g; P3 f1; 4; 5g;
6.1. An example
P4 f2; 3; 5g; P5 f1; 3; 5g; P6 f2; 4; 5g; U f5; 6g; jUj 2:
P f1; 2; 3; 4; 5; 6g; P1 f1; 4; 6g;
Fig. 5 presents an example of SCC production system which is based on a system used in Shanghai Baoshan Iron and Steel Complex, China. In this system there are two converter furnaces, two re®ning furnaces and two continuous casters. Consider three casts and six charges to be processed in this system. Basic model parameters for these jobs are given in Table 1. Sub-schedules and a rough schedule are presented in Fig. 6. In this ®gure, horizontal axis represents time in seconds, each horizontal line stands for a machine, the value of ``N'' in each series ``NNNNNN'' indicates a charge number and the length of the series represents the processing time of that charge, and ``#'' depicts machine con¯ict point. Based on the rough schedule given in Fig. 6, the related sets in the model can be de®ned as follows:
Self-de®ning connection relationships between the charges and the machines are shown in Table 2. The model for eliminating machine con¯icts in this example problem was built using the formulation presented in the last two sections and was solved in 20 seconds using the simplex method on an IBM/586 computer. The computation results are presented in Table 3 and the corresponding schedule is illustrated in Fig. 7. 6.2. Application of the model in implementing the SCC scheduling system Combining the proposed model with man-machine interactive methods, an SCC scheduling
Fig. 5. The steelmaking-casting process routes for the example.
Table 1 Basic model parameters Cast No Charge No
Processing time(s) 1
1 2 3
1 2 3 4 5 6
8 8 6
2
3
9
9
8
7 7
8
4
Due date d(s) 5
7 6 8
8 8 8 8
6 8 8
42 50 34 42 54 62
Penalty coecient
Transport times (s)
C1
C2
C3
C4
Machine±Machine
20
30 15 20 28 21 24
25 20 30 15 19 14
30 10 35 18 20 18
t1;3 5 t1;4 3 t2;3 3 t2;4 5 t3;5 4 t3;6 4 t4;5 2 t4;6 3 Setup time: S4;5 3 Interval time: l 2
30 40
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Fig. 6. Rough schedule with machine con¯icts for the example. Table 2 Self-de®ning connection relationship between charges and machines Immediate successor item of item i on machine j
Immediate successor machine of item i on machine j
SI(1,1) 5, SI(2,2) 6, SI(3,1) 1, SI(4,2) 2, SI(5,1) 0, SI(6,2) 0,
SP(1,1) 4, SP(2,2) 3, SP(3,1) 4, SP(4,2) 3, SP(5,1) 3, SP(6,2) 4,
SI(1,4) 6, SI(2,3) 5, SI(3,4) 1, SI(4,3) 2, SI(5,3) 0, SI(6,4) 0,
SI(1,6) 2 SI(2,6) 0 SI(3,5) 4 SI(4,5) 5 SI(5,5) 6 SI(6,5) 0
SP(1,4) 6, SP(2,3) 6, SP(3,4) 5, SP(4,3) 5, SP(5,3) 5, SP(6,4) 5,
SP(1,6) 0 SP(2,6) 0 SP(3,5) 0 SP(4,5) 0 SP(5,5) 0 SP(6,5) 0
Table 3 Computation results Charge 1 2 3 4 5 6 Objective value of
Starting time of all of charges on corresponding machine (s) X11 13 X14 24 X23 29 X22 17 X34 16 X31 5 X43 20 X42 9 X53 38 X51 27 X64 47 X62 34 optimal solution OPT 232
system was developed with Client/Server structure on Novell network. The development was done using the MS C 6.0 language and SYBASE database system. The system was designed such that the users can ®rst obtain initial solutions for their SCC scheduling problem using man±machine in-
Y16 0 Y26 0 Y35 0 Y45 0 Y55 3 Y65 3
Z16 0 Z26 0 Z35 2 Z45 2 Z55 0 Z65 0
X16 34 X26 42 X35 24 X45 32 X55 49 X65 57
teractive method. The system then generates optimal reference solutions by solving the mathematical model built in. Final solutions are produced by adjusting the optimal reference solutions, using man±machine interactive method again. The structure of the system follows the
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Fig. 7. Final schedule without machine con¯icts for the example.
design in Section 3.3. Major steps for the system to produce solutions for SCC scheduling are as follows. Step 1. Determine cast and charge sequences. Step 2. Establish sub-schedules. Step 3. Form a rough schedule. Step 4. Generate optimal reference solutions using the model presented in Sections 4 and 5. Step 5. Obtain the ®nal schedule by adjusting optimal reference solutions using man-machine interactive method. Repeat Steps 4 and 5 if necessary. To facilitate man±machine interaction, the system displays production schedules in Gantt chart form. A full-screen, visually oriented user interface allows the user to directly manipulate all features of the schedule using his experience and intuition. To explore improvement potentials, the scheduler typically makes global changes. To modify the schedule, the scheduler can select a unit using the mouse. Then the system displays a menu of operations that the scheduler may require. By selecting menu items, he can shift the unit on the same machine or move the unit to an alternative machine. He can also select and shift a charge set both as a whole and as individual units. In this way the scheduler, in cooperation with the built-in
scheduling engine, interactively modi®es an existing schedule through the user interface until the schedule is satisfactory. The system was implemented and tested in Shanghai Baoshan Iron and Steel Complex at the end of 1995. The average waiting time was reduced from 20 to 14 minutes per charge. This translates into a saving of about $0.5 million a year in production costs and an increase of about 5% in throughput. 7. Conclusions This paper studied the SCC scheduling problem in the CIMS environment. A just-in-time based non-linear mathematical model for solving machine con¯icts was developed, considering both punctual delivery and production operations continuity. It was then converted into a linear programming model which can be solved using standard software packages. An example has been given, demonstrating the application of the proposed model. Combining the model and man-machine interactive methods, an SCC production scheduling system was developed in which the proposed model was used as an eective method to
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optimize production continuity and product delivery while eliminating machine con¯icts. The system has been tested in a steelmaking plant. The average production waiting time was reduced dramatically, implying signi®cant economic bene®ts.
[10]
References
[11]
[1] C.N. Redwine, D.A. Wismer, A mixed integer programming model for scheduling order in a steel mill, Journal of Optimization Theory and Applications 14 (3) (1974) 305± 318. [2] C.M. Petersen, K.L. Sorensen, R.V.V. Vidal, Inter-process synchronization in steel production, International Journal of Production Research 30 (6) (1992) 1415±1425. [3] B. Lally, L. Biegler, H. Henein, A model for sequencing a continuous casting operation to minimize costs, Iron & Steelmaker 10 (1987) 63±70. [4] X. Tong, J. Silverman, S. Clausen, A caster sequencing optimization program for Geneva Steel's Facility Modernization project, presented at Annual AISE Convention, September 1994. [5] R.E. Box, D.G. Herbe, A scheduling model for LTV steel's Cleveland works' twin-strand continuous slab caster, Interface 18 (1) (1988) 42±56. [6] H.S. Lee, S.S. Murthy, S.W. Haider, D.V. Morse, Primary production scheduling at steelmaking industries, IBM Jorunal of Research and Development 40 (2) (1996) 231± 252. [7] D. Meil, H.S. Lee, Scheduling steel production on two twin-strand casters, in: Proceedings of the International Manufacturing Productivity Symposium, East Fishkill, NY, 1993. [8] J. Neuwirth, A production scheduling system at the Stahl Linz Gmbh, in: D.K. Baik (Ed.), International Conference
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on Computerized Production Control in Steel Plant, Korea, 1993, pp. 342±350. Y. Jimichi, A. Hori, H. Ishihara, T. Odira, An expert system of automatic slab assignment for hot strip mill, ISIJ International 30 (1990) 155±160. S. Sato, T. Yamaoka, Y. Aoki, T. Ueda, Development of integrated production scheduling system for iron and steel works, International Journal of Production Research 15 (1977) 539±552. M. Numao, S. Morishita, SCHEPLAN ± A scheduling expert for steel-making process, in: Proceedings of the International Workshop on Arti®cial Intelligence for Industrial Applications, Hitachi, 1988, pp. 467±472. M. Numao, S.-i. Morishita, A scheduling environment for steel-making processes, in: The Fifth Conference on Arti®cial Intelligence Applications, Washington, DC, USA, 1989, pp. 279±286. S. Morishita, M. Numao, Y. Tozawa, A scheduling expert for steel-making process by cooperative scheduling, Journal of Japanese Society for Arti®cial Intelligence 5 (2) (1990) 184±193. M. Numao, S.I. Morishita, Cooperative scheduling and its application to steelmaking processes, IEEE Transactions on Industrial Electronics 38 (2) (1991) 150±155. K. Stohl, W. Spopek, VAI-Schedex: A hybrid expert system for co-operative production scheduling in steel plant, in: D.K. Baik (Ed.), International Conference on Computerized Production Control in Steel Plant, Korea, 1993, pp. 207±217. H.P. Epp, M.G. Kalin, D.P. Miller, K.R. Karwan, J.R. Sweigart, An interactive, adaptive, real-time scheduler for steel-making, in: Proceedings of the Third International Conference. Expert Systems and the Leading Edge in Production and Operations Management, Hilton Head Island, SC, USA, 1989, pp. 495±504. K. Hamada, T. Baba, K. Sato, M. Yufu, Hybridizing a genetic algorithm with algorithm with rule-based reasoning for production planning, IEEE Expert 10 (1995) 60±67.
www.elsevier.com/locate/orms
Theory and Methodology
A mathematical programming model for scheduling steelmaking-continuous casting production 1 Lixin Tang b
a,*
, Jiyin Liu b, Aiying Rong c, Zihou Yang
d
a Department of Systems Engineering, Northeastern University, Shenyang, People's Republic of China Department of Industrial Engineering and Engineering Management, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, People's Republic of China c Department of Automation, Beijing Institute of Machinery Industry, Beijing, People's Republic of China d Department of Systems Engineering, Northeastern University, Shenyang, People's Republic of China
Received 1 November 1997; accepted 1 November 1998
Abstract This paper presents a mathematical model, based on the just-in-time (JIT) idea, for solving machine con¯icts in steelmaking-continuous casting (SCC) production scheduling in the computer integrated manufacturing system (CIMS) environment. The model is developed as a non-linear model based on actual production situations, considering both punctual delivery and production operation continuity. It is then converted into a linear programming model which can be solved using standard software packages. An example demonstrating the application of the proposed method is given. The paper also describes the implementation of an SCC production scheduling system in which the proposed model is used as an eective method to optimize production continuity and product delivery while eliminating machine con¯icts. Ó 2000 Elsevier Science B.V. All rights reserved. Keywords: Iron and steel industries; Steelmaking-continuous casting; Production scheduling; Machine con¯ict; Linear programming
1. Introduction Modern iron and steel corporations are moving towards continuous, high-speed and automated
*
Corresponding author. E-mail: [email protected] The project is supported by National Natural Science Foundation of China through approved No. 79700006 and by National 863/CIMS of China through approved No. 863-511708-009. 1
production process with large devices. The focus is placed on high quality, low cost, just-in-time (JIT) delivery and small lot with dierent varieties. In order to enhance their competitive power, many international iron and steel corporations are devoted to developing computer integrated manufacturing systems (CIMS) which can improve productivity of large devices, shorten waiting-time between operations, reduce material and energy consumption, and cut down production costs. Production scheduling is a key component of
0377-2217/00/$ ± see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 9 9 ) 0 0 0 4 1 - 7
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CIMS. Its task is to determine the starting times and the ending times of jobs on the machines so that a chosen measure of performance is optimized. Steelmaking-continuous casting (SCC) production scheduling problems are to determine in what sequence, at what time and on which device molten steel should be arranged at various production stages from steelmaking to continuous casting. Unlike general production scheduling in machinery industry, SCC production scheduling problems have to meet special requirements of steel production process. In the SCC process, the products being processed are handled at high temperature and converted from liquid (molten steel) into solid (drawn billets). There are extremely strict requirements on material continuity and ¯ow time (including processing time on various devices and transportation and waiting time between operations). The study of this paper is investigating the SCC production scheduling problem and aims at developing a computerized scheduling system for generating optimal schedules. The project uses the steelmaking plant in Shanghai Baoshan Iron and Steel Complex as the study background. The whole scheduling process consists of four steps: (1) cast sequencing which determines cast sequences and charge sequence in each cast; (2) sub-scheduling which ful®lls the scheduling of individual charge sets; (3) rough scheduling which merges sub-schedules; and (4) optimal scheduling which eliminates machine con¯icts. While the ®rst three steps consider mainly the operation relationships and can be done relatively easily, the last step needs to consider resource constraints to ensure practical feasibility of the resulting schedule. This paper emphasizes this optimal scheduling step and proposes a mathematical programming model to eliminate machine con¯icts. In the following sections, we ®rst brie¯y review related previous work (Section 2). Then the SCC production process and the overall scheduling problem are described in Section 3. Section 4 presents the mathematical programming model for solving the machine con¯ict problem. The nonlinear model is then transformed into a linear programming (LP) model in Section 5. Section 6
gives an application example and discusses the issues of using the model in the implementation of a practical SCC scheduling system. Section 7 concludes the study. 2. Literature review Solving production operational management problems in SCC production through mathematical models and computers is an important new research topic and has recently been widely explored. Various production planning and scheduling techniques for SCC production have been reported in literature. According to the information from International Steel Production Management Conference in 1993 and reports in the latest literature, some advanced steel incorporations have started to carry out extensive research on production operational management which has become a hot subject in the academic circle of international iron and steel incorporations. Most approaches to scheduling continuous casters treat them as stand-alone facilities that are coupled only loosely with further processing such as rolling. These approaches attempt to improve productivity of the continuous casters while reducing operating costs and guaranteeing on-time delivery. All the methods used for SCC production scheduling can be classi®ed into two categories ± mathematical programming based methods and expert system based methods. 2.1. Mathematical programming based methods An example of o-line scheduling for steel production using mathematical programming was provided by Redwine and Wismer [1]. Dynamic programming was used to solve the model. Petersen et al. [2] have developed a mathematical programming model for a steel production scheduling problem to optimally schedule the slabs through the reheating furnace and the rolling mill. The model was solved heuristically. Lally et al. [3] established a simple mixed-integer linear programming solution to the problem of caster scheduling. They considered a simple model of a steel plant in
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which steel was started at an electric arc furnace, held in a ladle, and cast on a continuous caster. However, the model did not consider all the complexities of a real continuous caster. Tong et al. [4] constructed a complex mixed-integer linear programming model and solved it using heuristic techniques for twin strand continuous slab caster scheduling problem at LTV and Geneva Steel Works. The Geneva system is a state-of-theart application program that schedules a continuous caster while considering business constraints and capacity constraints of the various facilities upstream and downstream. LTV caster scheduling model [5,6] was implemented in 1983, in support of the ®rst continuous caster installed at LTV Cleveland Works. The model was intended to schedule caster production from customer order while optimizing several key objectives such as maximizing caster productivity. Meil and Lee [7] described the primary production scheduling procedure for the two-strand casters at LTV Cleveland Steel Works. The scheduling system, developed by IBM Research and ATM, exploited a fast sequencing algorithm for schedules satisfying global constraints. At Stahl Linz GmbH, Neuwirth [8] reported a linear programming model with machine con¯icts and provided key modeling factors of SCC scheduling and charge allocation scheme in the furnace, but the mathematical representation of the model was not given.
GmbH, Stohl and Spopek [15] established a hybrid co-operative expert system modeling to solve SCC scheduling problems, but they were unable to construct an optimized mathematical model. Epp et al. [16] described an interactive scheduling system developed using AI method for an SCC facility at Inland Steel Corporation. Hamada et al. [17] presented a framework for solving complex steelmaking scheduling problems and then combined rule-based expert system and genetic algorithm to produce ecient schedules. However, no systematic research has been realized so far on the general structure, model and algorithm which can be applied to steelmaking plants.
2.2. Expert system based methods
EAF BF BOF LMF Tundish LF Cluster
An expert system was used by Jimichi et al. [9] to determine parameters and operational conditions to match slab production with customer orders. Another example of using expert system techniques for iron and steel production scheduling is provided by Sato et al. [10]. Numao and Morishita [11±14] described an expert system application to perform co-operative scheduling in which the schedule was modi®ed by the scheduler using a graphical user interface. The main justi®cation for the use of expert systems came from reducing waiting time from charge to charge and minimizing energy consumption. Signi®cant bene®ts have been achieved as reported in the literature. At Stahl Linz
3. SCC production process and scheduling problem 3.1. SCC production process To describe the SCC production process, we ®rst introduce the following special terms: Billet Bloom Machine
Grade Heat Slab Strand Charge
A steel piece with square cross section, smaller than a bloom A steel piece with square cross section, larger than a billet A production device which performs one operation at a time. The machines for performing identical operations are called alternative machines Electric Arc Furnace Blast Furnace Basic Oxygen Furnace Ladle Metallurgical Facility A receptacle at top of caster Ladle Furnace Production between caster turnarounds Steel with a speci®ed metallurgical composition Furnace-load of steel A steel piece with elongated rectangular cross section Stream of steel from a caster A unit of production that consists of a sequence of operations on a heat
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Charge set CAST
Unit
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A set of charges that produces the same products, i.e., that have almost identical operations A set of charges that continually casts on the same continuous caster and has a similar chemical composition An operation that speci®es the machine, the starting time, and the ending time
Fig. 1 is a diagrammatic representation of the above terms. Here, the horizontal axis stands for time, and each line represents a machine. One charge path consists of units and handling times, which are expressed by lines connecting the units. Waiting time is illustrated using dashed lines before the units. In the iron and steel production from iron ore input to steel product output, there are three major manufacturing processes: ironmaking, steelmaking, and rolling. Steelmaking re®nes pig iron into steel and casts it into slabs, blooms, or billets. The SCC production process is illustrated in Fig. 2. The steelmaking process starts with the charge of crude steel and scrap iron in one of the EAFs. Liquid iron, tapped from BF, will be transported to the steelmaking shop where BOFs and/or EAFs are located. BOF and EAF burn out the excessive carbon, sulphur, silicon, and other impurities from liquid iron and re®ne it to steel with desired con-
tents. The ®lling of one furnace is called a charge (heat) and already contains the main alloying elements. The furnaces have dierent capacities from 17±55 t. The duration of the melting process depends on the ingredients as well as external factors. Because of voltage peaks, as many as 5 hours can be required for melting, but usually 2±2.5 hours are enough. A ®xed set up and a maintenance interval of 20 minutes are included in this duration. The melted steel is poured into ladles that are transported by a crane to an LF. If the proceeding heat has a long processing time in the LF, the current heat must wait. This slack time cannot exceed 2 hours. The next step is a heat treatment in the LF, where the ®ne alloying takes place. The duration is usually about the same as the melting. Then, special treatment may be performed in the Ladle-re®ning equipment (VOD or VD) to eliminate impurities from molten steel or add alloy ingredients to the molten steel in ladles to make high-grade steel. Finally, a continuous caster casts molten steel continuously into slabs, blooms, or billets. If the casting format needs to be altered, a set-up time must be considered. 3.2. Framework of SCC production operational management system Fig. 3 describes the overall structure of SCC production operational management. In this
Fig. 1. Process ¯owsheet of a CAST including three charges.
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Fig. 2. The SCC production process.
Fig. 3. Overall structure of production management system for SCC production.
structure, the original ``customer order data'' is converted into ``production order data'' through ``production target quality'' (PT/Q) design and ``production target plan'' (PT/P) design. Later, the order plan is produced based on production capacity of the main processes. SCC production lot plan includes charge design and cast design. The production lot plan is formulated through optimal combination of orders according to the contents of ``production order information data'' and ``order plan'' to meet process constraints and dierent order requirements. On the basis of lot plan, SCC production scheduling is to determine in what sequence, at what time and on which machine molten steel should be arranged to enter various
production stages from steelmaking to continuous casting.
3.3. Basic structure of SCC production scheduling design A ¯owchart for SCC production scheduling is presented in Fig. 4. The whole SCC production scheduling problem is handled in four steps. First, cast sequencing is arranged. Sub-schedules and rough schedule are then established. Finally the SCC production schedule is formed considering availability of machines at all stages. A
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(4) Elimination of machine con¯icts Because only the machine operation conditions in one cast are taken into account, machine con¯icts often exist in the rough schedule. Only when machine con¯icts are completely eliminated can a feasible and practical schedule be established. This step is to produce an optimal schedule in which all machine con¯icts are eliminated. The three steps mentioned above are realized using man±computer interaction. The following discussion emphasizes on the elimination of machine con¯icts using an optimization model. 4. Mathematical model for eliminating machine con¯icts in optimized scheduling
Fig. 4. Flowchart for the SCC production scheduling module.
brief description for each of these steps is given below. (1) Cast sequencing Cast sequences on the casters and charge sequence in each cast are determined in this step based on their delivery times. These can be considered as single machine sequencing problems without resource constraints. (2) Establishment of sub-schedules After a cast has been established, a cast job timetable called sub-schedule is formed for each cast in this step according to time progress of the operations including steelmaking, re®ning and continuous casting in each charge. A sub-schedule has been shown in Fig. 1. (3) Establishment of a rough schedule In this step the sub-schedules with relative times are combined to produce a rough job schedule (superposition of sub-schedules) with physical time.
This model is proposed here to eliminate machine con¯icts in optimized scheduling within the overall scheduling framework described in the last section. In this section we ®rst give a brief narrative description of the model for easy understanding of the practical meanings. Then the notations to be used in the model are introduced. After these preparations the model formulation is ®nally presented. 4.1. Overview of the model (1) Basic assumptions The model is developed under the following assumptions which are made based on practical requirements in the SCC production process and the scheduling system structure. (a) The sequence of sub-schedules remains unchanged. (b) At most one job can be processed on a machine at a time. (c) Only the three kinds of machines ± steelmaking furnace, re®ning furnace and continuous caster ± are taken into account in the model. Other resources such as intermediate transportation cranes are neglected based on the fact that their capacities are large enough. However, transportation times are considered. (d) All steel products follow the same production process route: steelmaking, re®ning and continuous casting. If some steel products do
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not need re®ning, then their processing times for re®ning are considered to be zero. (2) The decisions to be made The machines used for each operation of each charge are ®xed in the ®rst three steps of the scheduling framework. The decisions to be made in this model are the time points at which the processing of each charge i on machine j begins. (3) The objective function The objective chosen for this model is to ensure continuity of the production process and just-intime delivery of ®nal products. This is achieved through minimizing a cost function consisting of the following terms. (a) Cast break loss penalties exerted to ensure that charges in the same cast are cast as continuously as possible. (b) Molten steel temperature drop cost in terms of waiting time from operation to operation. (c) Earliness/tardiness penalty used to ensure that blooms or billets in each charge are delivered as punctually as possible. (4) Constraints considered The following constraints are considered in the model to guarantee that there will be no machine con¯icts in the schedule generated. (a) For the two contiguous operations for the same charge, only when the preceding operation has been completed, the immediately next one can be started. (b) For two contiguous charges processed on the same machine, only when the preceding charge has been completed, the immediately next one can be started. (c) Setup time and interval time are required from cast to cast on the same continuous caster. 4.2. Notations The following notations are used for de®ning the problem parameters and variables in the model. Parameters X the set of all charges, X f1; 2; . . . ; N g, where N is the total number of production charges
Xk
Nk P Pi U
SI(i, j) SP(i, j) di C1k C2ij C3i
C4i Tij tjm Skj l
429
the set of all charges in the kth cast, k 2 f1; 2; . . . ; Mg, where M is the total number of casts. Xj \ Xk ;, for any j, k 2 f1; . . . ; Mg and j 6 k. X1 [ X2 [ [ XM X the total P number of charges in the kth cast. M k1 Nk N the set of all machines, P f1; 2; . . . ; J g, where J is the total number of machines the set of all machines used for the ith charge, Pi P the set of all continuous casters, jUj C, where C is the total number of continuous casters. Note that the casters are the machines used at the ®nal stage of the SCC production and therefore U P the immediate successor charge of charge i processed on machine j the immediate successor machine of machine j for processing charge i order delivery time of charge i coecient of cast break loss penalty for cast k coecient of penalty cost for the waiting time of charge i after ®nishing process on machine j coecient of penalty cost for the production of charge i being ®nished before the delivery time speci®ed in the production order coecient of penalty cost for the production of charge i being late with respect to the delivery time processing time of charge i on machine j transportation time from machine j to machine m setup time of cast k on machine j interval time from cast to cast
Decision variables starting time of charge i on machine j Xij 4.3. The model formulation Using the above notations, the mathematical model for the optimal scheduling step of the SCC scheduling problem is formulated as follows:
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(P)
transported to the next machine, the immediately next operation on the charge can be started. Constraint (4) ensures enough setup time and interval time required between casts. As can be seen, this is a non-linear model. For easier solution it is transformed into a linear programming model in the next section.
minimize : M X X C1k XSI i;j;j ÿ Xij ÿ Tij Z k1 i2Xk
j2Pi \U SI i;j2Xk
N X X i1
ÿ
C2ij Xi;SP i;j ÿ Xij ÿ Tij ÿ tj;SP i;j
j2Pi
SP i;j2Pi
N X X
5. Solution strategy
C3i min 0; Xij Tij ÿ di
i1 j2Pi \U
N X X
C4i max 0; Xij Tij ÿ di
1
i1 j2Pi \U
subject to: XSI i;j;j ÿ Xij P Tij ;
i 2 X; j 2 Pi ; SI i; j 2 X; 2
Xi;SP i;j ÿ Xij P Tij tj;SP i;j ;
i 2 X; j 2 Pi ;
SP i; j 2 Pi ;
3
XSI i;j;j ÿ Xij P Tij SSI i;j;j l; i 2 Xk ; j 2 Pi \ U; SI i; j 2 Xp ; k; p 1; . . . ; M; and k 6 p;
4
Clearly, the non-linearity of the model comes from the last two terms in the objective function, i.e., the penalties for earliness and lateness in ®nal product delivery. Note that these two terms and associated variables are only related to the casters (the machines used at the last stage of production ± members of U). In order to deal with the nonlinearity, we ®rst separate the set of machines used for charge i into two subsets, the casters fjjj 2 Pi \ Ug and the other machines fjjj 2 Pi ; j 62 Ug, and accordingly break each relevant term in the objective function and each relevant constraint into two. After this treatment, the model changes to the following form: (TP1) minimize :
Xij P 0;
i 2 X; j 2 Pi :
5
It is not dicult to match this mathematical formulation with the overview of the model in Section 4.1. The four terms in the objective function represent the cast break loss penalty, charge waiting time, penalty cost, earliness penalty cost and lateness penalty cost, respectively. The objective function is in line with the JIT idea, both in terms of ®nished product delivery and in terms of operations on semi-products. Constraint (2) ensures that for two contiguous charges processed on the same machine, only when the preceding charge is ®nished, the immediately next one can be started. Constraint (3) ensures that for contiguous operations of the same charge, only when the preceding operation is completed and the charge is
Z
M X X
C1k XSI i;j;j ÿ Xij ÿ Tij
k1 i2Xk
j2Pi \U SI i;j2Xk
N X X i1
SP i;j2Pi SP i;j62U
N X X i1
ÿ
C2ij Xi;SP i;j ÿ Xij ÿ Tij ÿ tj;SP i;j
j2Pi
C2ij Xi;SP i;j ÿ Xij ÿ Tij ÿ tj;SP i;j
j2Pi
SP i;j2Pi \U
N X X
C3i min 0; Xij Tij ÿ di
i1 j2Pi \U
N X X
C4i max 0; Xij Tij ÿ di
i1 j2Pi \U
6
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subject to:
(TP2)
XSI i;j;j ÿ Xij P Tij ; i 2 X; SI i; j 2 X;
minimize :
j 2 Pi ; and j 62 X;
7
Z
M X X
j2Pi \U SI i;j2Xk
8
Xi;SP i;j ÿ Xij P Tij tj;SP i;j ;
ÿ TSI i;j;j dSI i;j ÿ Yij Zij ÿ di
i 2 X; SP i; j 2 Pi ; and SP i; j 62 U; j 2 Pi ;
9 i 2 X; 10
Now the parts in the model related to the casters are the ®rst and last three terms in the objective function and constraints (8), (10) and (11). We de®ne two sets of new variables related to the casters:
Yij max 0; Xij Tij ÿ di ;
i 2 X; j 2 Pi \ U; i 2 X; j 2 Pi \ U:
It is clear that Yij and Zij are nonnegative and Yij ÿ Zij Xij Tij ÿ di ;
i 2 X; j 2 Pi \ U;
or Xij Yij ÿ Zij ÿ Tij di ;
SP i;j2Pi \U
N X X
C3i Zij
N X X
XSI i;j;j ÿ Xij P Tij ;
i 2 X; SI i; j 2 X; j 2 Pi ; and j 62 U;
Substituting all the Xij , i 2 X, j 2 Pi \ U, in model (TP1) with Yij and Zij using the above relations, the model can be transformed into the following form:
13
14
YSI i;j;j ÿ ZSI i;j;j ÿ Yij Zij P TSI i;j;j ÿ dSI i;j di ; i 2 X; SI i; j 2 X; j 2 Pi \ U;
15
Xi;SP i;j ÿ Xij P Tij tj;SP i;j ; i 2 X; SP i; j 2 Pi; and SP i; j 62 U; j 2 Pi ;
16
Yi;SP i;j ÿ Zi;SP i;j ÿ Xij P Tij tj;SP i;j Ti;SP i;j ÿ di ; i 2 X; SP i; j 2 Pi \ U; j 2 Pi ;
17
YSI i;j;j ÿ ZSI i;j;j ÿ Yij Zij P SSI i;j;j l TSI i;j;j ÿ dSI i;j di ; i 2 Xk ; j 2 Pi \ U; SI i; j 2 Xp ; k; p 1; . . . ; M; and k 6 p;
i 2 X; j 2 Pi \ U:
C4i Yij
i1 j2Pi \U
subject to:
12
Zij ÿ min 0; Xij Tij ÿ di ;
C2ij Yi;SP i;j ÿ Zi;SP i;j
j2Pi
i1 j2Pi \U
11
i 2 X; j 2 Pi :
SP i;j2Pi SP i;j62U
ÿ Ti;SP i;j di ÿ Xij ÿ Tij ÿ tj;SP i;j
i 2 Xk ; j 2 Pi \ U; SI i; j 2 Xp ; k; p 1; . . . ; M; and k 6 p;
C2ij Xi;SP i;j ÿ Xij ÿ Tij ÿ tj;SP i;j
j2Pi
N X X i1
XSI i;j;j ÿ Xij P Tij SSI i;j;j l;
Xij P 0;
N X X i1
SP i; j 2 Pi ; \U; j 2 Pi ;
C1k YSI i;j;j ÿ ZSI i;j;j
k1 i2Xk
XSI i;j;j ÿ Xij P Tij ; i 2 X; SI i; j 2 X; j 2 Pi \ U;
Xi;SP i;j ÿ Xij P Tij tj;SP i;j ;
431
18
Xij P 0;
i 2 X; j 2 Pi ;
19
Yij P 0;
i 2 X; j 2 Pi \ U;
20
Zij P 0;
i 2 X; j 2 Pi \ U:
21
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The model now becomes a linear programming model which can be solved by standard linear programming software packages.
X f1; 2; 3; 4; 5; 6g; X1 f1; 2g; X2 f3; 4g; X3 f5; 6g;
6. Applications
P2 f2; 3; 6g; P3 f1; 4; 5g;
6.1. An example
P4 f2; 3; 5g; P5 f1; 3; 5g; P6 f2; 4; 5g; U f5; 6g; jUj 2:
P f1; 2; 3; 4; 5; 6g; P1 f1; 4; 6g;
Fig. 5 presents an example of SCC production system which is based on a system used in Shanghai Baoshan Iron and Steel Complex, China. In this system there are two converter furnaces, two re®ning furnaces and two continuous casters. Consider three casts and six charges to be processed in this system. Basic model parameters for these jobs are given in Table 1. Sub-schedules and a rough schedule are presented in Fig. 6. In this ®gure, horizontal axis represents time in seconds, each horizontal line stands for a machine, the value of ``N'' in each series ``NNNNNN'' indicates a charge number and the length of the series represents the processing time of that charge, and ``#'' depicts machine con¯ict point. Based on the rough schedule given in Fig. 6, the related sets in the model can be de®ned as follows:
Self-de®ning connection relationships between the charges and the machines are shown in Table 2. The model for eliminating machine con¯icts in this example problem was built using the formulation presented in the last two sections and was solved in 20 seconds using the simplex method on an IBM/586 computer. The computation results are presented in Table 3 and the corresponding schedule is illustrated in Fig. 7. 6.2. Application of the model in implementing the SCC scheduling system Combining the proposed model with man-machine interactive methods, an SCC scheduling
Fig. 5. The steelmaking-casting process routes for the example.
Table 1 Basic model parameters Cast No Charge No
Processing time(s) 1
1 2 3
1 2 3 4 5 6
8 8 6
2
3
9
9
8
7 7
8
4
Due date d(s) 5
7 6 8
8 8 8 8
6 8 8
42 50 34 42 54 62
Penalty coecient
Transport times (s)
C1
C2
C3
C4
Machine±Machine
20
30 15 20 28 21 24
25 20 30 15 19 14
30 10 35 18 20 18
t1;3 5 t1;4 3 t2;3 3 t2;4 5 t3;5 4 t3;6 4 t4;5 2 t4;6 3 Setup time: S4;5 3 Interval time: l 2
30 40
L. Tang et al. / European Journal of Operational Research 120 (2000) 423±435
433
Fig. 6. Rough schedule with machine con¯icts for the example. Table 2 Self-de®ning connection relationship between charges and machines Immediate successor item of item i on machine j
Immediate successor machine of item i on machine j
SI(1,1) 5, SI(2,2) 6, SI(3,1) 1, SI(4,2) 2, SI(5,1) 0, SI(6,2) 0,
SP(1,1) 4, SP(2,2) 3, SP(3,1) 4, SP(4,2) 3, SP(5,1) 3, SP(6,2) 4,
SI(1,4) 6, SI(2,3) 5, SI(3,4) 1, SI(4,3) 2, SI(5,3) 0, SI(6,4) 0,
SI(1,6) 2 SI(2,6) 0 SI(3,5) 4 SI(4,5) 5 SI(5,5) 6 SI(6,5) 0
SP(1,4) 6, SP(2,3) 6, SP(3,4) 5, SP(4,3) 5, SP(5,3) 5, SP(6,4) 5,
SP(1,6) 0 SP(2,6) 0 SP(3,5) 0 SP(4,5) 0 SP(5,5) 0 SP(6,5) 0
Table 3 Computation results Charge 1 2 3 4 5 6 Objective value of
Starting time of all of charges on corresponding machine (s) X11 13 X14 24 X23 29 X22 17 X34 16 X31 5 X43 20 X42 9 X53 38 X51 27 X64 47 X62 34 optimal solution OPT 232
system was developed with Client/Server structure on Novell network. The development was done using the MS C 6.0 language and SYBASE database system. The system was designed such that the users can ®rst obtain initial solutions for their SCC scheduling problem using man±machine in-
Y16 0 Y26 0 Y35 0 Y45 0 Y55 3 Y65 3
Z16 0 Z26 0 Z35 2 Z45 2 Z55 0 Z65 0
X16 34 X26 42 X35 24 X45 32 X55 49 X65 57
teractive method. The system then generates optimal reference solutions by solving the mathematical model built in. Final solutions are produced by adjusting the optimal reference solutions, using man±machine interactive method again. The structure of the system follows the
434
L. Tang et al. / European Journal of Operational Research 120 (2000) 423±435
Fig. 7. Final schedule without machine con¯icts for the example.
design in Section 3.3. Major steps for the system to produce solutions for SCC scheduling are as follows. Step 1. Determine cast and charge sequences. Step 2. Establish sub-schedules. Step 3. Form a rough schedule. Step 4. Generate optimal reference solutions using the model presented in Sections 4 and 5. Step 5. Obtain the ®nal schedule by adjusting optimal reference solutions using man-machine interactive method. Repeat Steps 4 and 5 if necessary. To facilitate man±machine interaction, the system displays production schedules in Gantt chart form. A full-screen, visually oriented user interface allows the user to directly manipulate all features of the schedule using his experience and intuition. To explore improvement potentials, the scheduler typically makes global changes. To modify the schedule, the scheduler can select a unit using the mouse. Then the system displays a menu of operations that the scheduler may require. By selecting menu items, he can shift the unit on the same machine or move the unit to an alternative machine. He can also select and shift a charge set both as a whole and as individual units. In this way the scheduler, in cooperation with the built-in
scheduling engine, interactively modi®es an existing schedule through the user interface until the schedule is satisfactory. The system was implemented and tested in Shanghai Baoshan Iron and Steel Complex at the end of 1995. The average waiting time was reduced from 20 to 14 minutes per charge. This translates into a saving of about $0.5 million a year in production costs and an increase of about 5% in throughput. 7. Conclusions This paper studied the SCC scheduling problem in the CIMS environment. A just-in-time based non-linear mathematical model for solving machine con¯icts was developed, considering both punctual delivery and production operations continuity. It was then converted into a linear programming model which can be solved using standard software packages. An example has been given, demonstrating the application of the proposed model. Combining the model and man-machine interactive methods, an SCC production scheduling system was developed in which the proposed model was used as an eective method to
L. Tang et al. / European Journal of Operational Research 120 (2000) 423±435
optimize production continuity and product delivery while eliminating machine con¯icts. The system has been tested in a steelmaking plant. The average production waiting time was reduced dramatically, implying signi®cant economic bene®ts.
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