- Email: [email protected]
A MTSP Model for Hot Rolling Scheduling in Baosteel Complex
A MTSP MODEL FOR HOT ROLLING SCHEDULING IN BAOSTEE...
14th World Congress ofIFAC
N-7b-05-1
Copyright ~ 1999 IF;\C 14th Triennial World Congr~ss, Bcijing, P.R. China
A MTSP MODEL FOR HOT ROLLING SCHEDULING IN BAOSTEEL COMPLEX *
Lixin Tang
1
2
t+
Aiying Rong
2
and
Zihou Yang 1
Institute o/S}'stems Engineering~ Northeastern University, Shen;;ang) P. R. China
Departnlent of.Auto/nation, Beijing Institute ofMachinery Industry. Bejing, P.R. China
Abstract: T'his article has investigated the hot rolling scheduling in the iron & steel plants. The task of a hot rolling scheduling is to detennine a sequence of orders (bars or slabs) on hot rolling finishing mill. Unlike previously considered series strategy_~ we proposed the parallel strategy to simultaneously produce M turns on same shift from global optitnal viewpoint. U sing this strategy; the model to hot rolling scheduling is developed as MTSP Inodel and then this model is converted into single TSP model. A modified genetic algorithm (MGA) is constructed to obtain near-optimal solutions to TSP. This procedure has been embedded in the practical hot rolling scheduling system by combining MTSP \vith man-machine interactive methods. Copyright © 19991FAC, Keyw'ords: hot rolling scheduling, Multiple Traveling
1.
INTRODUCTION
algorithm.
in actual production efficiency, finished product rate incorporation profit and product quality as well as great reducti on in production cost. The hot strip mi 11 is considered the "bottleneck" to overall iron and steel production and the production sequencing of steel orders through the hot rolling mill becomes the key to the hot strip mill production scheduling. The scheduling observes the constraints of the mill and attempts to obtain optjmal results by arranging the sequence for rolling the individual orders. Not only does it recognize constrains associated with specific orders and the number of jobs in the turn(round), but j
lligh production quality, low cost and higher production rates are three important objectives that today's iron and steel industry is trying to achieve. One of the possible ways to reach these targets is through optimal production scheduling. As is well knO\VTl, umeasonable scheduling will result in signific,ant loss in production. It is very important to seek an effective way to quantify scientifically production scheduling because benefits of utilizing such an approach involve considerable improvement
or
Sa]esman~ genetic
The Project 1S Supported by National Natural SC2ence Foundation vfChina Through Approved No 79700006, and National 863JCIMS of China Through Approved No. 863-511-708-009.
l To 'whom correspondence should be addressed.
E-Inail: [email protected]
7056
Copyright 1999 IFAC
ISBN: 0 08 043248 4
A MTSP MODEL FOR HOT ROLLING SCHEDULING IN BAOSTEE...
also satisfi.es the requirement for snlooth thickness or gauge transitions from order to order and the requirement for a generally decreasing width pattern in the rolling sequence. The hot strip rolling scheduling aillls at minimizing operating costs (such as transition costs between width groups in a sequence). In this paper, 'hre proposed the parallel strategy to simultaneously produce M turns on same shift from global optimal view. Using proposed parallel strategy, the model to hot rolling scheduling is developed as MTSP model based on actual production constrains. When solutions to MTSP model are found, the MTSP nlodel is converted into single TSP model. A mod ified genetic algoritrnn (MGA) is constructed to obtain near-optimal solutions to TSP. The ne"' crossover operations (seed based crossover) have been introduced to the MGA framework. This procedure has been embedded in the practical hot rolling scheduling system which is implemented and ran in Shanghai Baosteel Complex by combining proposed MTSP with man-machine interactive methods.
2.
14th World Congress of IFAC
part of the turn and quality of the turn dependents on combination and sequence of orders in "staple matericl In this paper, the term of turn discussed below in this paper presents the staple material section of that turn. The staple material rolling section should meet the follo\~ling requirements: Ca) the total length (or weight) of the staple materials is limited to a given quantity; (b) each staple order is no V\-"ider than the one that preceded it and vi/idth jump is small; (c) width, gauge and hardness jumps are not permitted to occur simultaneously; (d) order gauge which is not allowed to jump repeatedly should change smoothly; (e) hardness should change gently, gradually increasing or decreasing; and (t) ~'hen the change of hardness, gauge and width competes against each other'} the order of priority is: hardness, gauge and width. H
•
3.
CHARACTERISTICS AND MODELING STRATEGY OF HOT ROLLING SCHEDULING
3.1 Characteristics ofhot rolling scheduling
HOT ROLLING SCHE·DULING CONSTRAINTS
In hot rolling plant at Baosteel Complex, every \vorkday can be divided into three shifts and schedulers make throughput hot rolling scheduling of next shift on previous shift. This hot rolling scheduling general1y includes about 5-7 turns \vhich is basjc unit of hot rolling. 'fhere are three main considcratlons in the hot rolling scheduling: (I) product quality~ (2) roIler replacement cost; and (3) roller wear. The cost of roller replacelnent is so great that it is necessary to organize as many orders as possible within the range of maximal rolling length constraint of a turn to reduce the replacement cost. The impact of rolling order on the roller leads to roller wear: (1) the harder the rolling orders, the greater the hardness jump between from order to order, the greater the impact on the roller. (2) the larger the hardness difference [rolu order to order~ the greater the impact on the roller. (3) Gouging wilJ result at the edge of productroller contact area if too many orders with the same \vidth are rolled continuously. In order to guarantee product quality, the rolling scheduJes of staple materials are arranged so that each order is no w-ider than the one that preceded it while the first few orders during each turn increase in 'Vvidth to heat the mill, such an approach is called "warm up roller. Generally, the rolling job ,",vidth of a complete rolling turn appears as lI coffin profile". The H warm up" material section takes on the fonn of outgoing coffin profile while the staple material ro lling section assumes the fonn of inward coffin profile. The warm up" is minor part of turn and it is easjly detennined by schedulers. However, nstaple materiar' is major 11
T1
Characteristics of hot rolling scheduling are gi yen as fo llo\vs: Unlike general sequencing problems, (1 ) production sequencing in the steel rolling mill is affected greatly by the process constraints, which requires to specify the sequence of rolling orders as well as the combination of rolling orders. This kind of sequencing and like-product aggregation is referred to as sequential c ]ustering problem. (2) Sequence-dependent setups. Setup costs occurred by changcover bet,;yeen adjacent orders differ from that of another two adjacent orders.
3.2 Modeling strategies ofhot rolling scheduling
1) As was analyzed above, hot TOl1ing scheduling target should satisfy the following requirement:( I) to insure the minimal jump in hardness~ (2) to insure the minimal jump in gauge; and (3) to insure the minimal jump in width. Orders in the hot strip mill are known in advance and schedules are created only after sufficient orders have been taken since penalties are totally dependent on the physical properties of two adjacent orders. Penalties can be obtained from order to order, by direct computation. Now the problem may be described as finding the shortest path bet\veen the starting order and the ending order. Width, gauge and hardness jumps may be quantified by a penalty structure whi ch reflects the conditions in the hot strip mill. The sequencing with the lowest penalty will result in lower damage to the rollers, and in turn, higher product quaJity, hence, better profit. A hot
7057
Copyright 1999 IFAC
ISBN: 0 08 043248 4
A MTSP MODEL FOR HOT ROLLING SCHEDULING IN BAOSTEE...
strip mill turn with lower penalty is regarded as the better sequencing. Therefore, the scheduling objective becomes minimization of the total penalties. Penalties are only assigned from order to order and there is no penalty to sequential schedule orders of identical width, gauge and hardness. However, note that the gauge penalty from order to order may be asymmetricaJ. 2) In traditional system of Baosteel Complex, each turn on same shift is produced sequentially in the series strategy by schedulers. In previous research, the single TSP model proposed by Balas and Martin (1991), Kosiba, et ol. (1992) simulated schedulerts idea and also used series strategy to arrange only one turn for hot rolling scheduling. The series strategy is essentially similar to the Greedy idea that earlier arrangjng turns are probably well and later arranging turns are relatively poor because the number of candidate orders in order pool decreases as the number of produced tUITI increases. Therefore, the series strategy will suffer from the disadvantage of local optimization. When actual rolling scheduling is created, it is conunon to work out M rolling turn schedules from N orders in the preseletion pool. From global optimal viewpoint, \ve proposed the parallel strategy to simultaneously produce M turns on same shift. The hot rolling scheduling problem using parallel strategy may be reduced to MTSP \vhich is the expansion and continuity ofa single TSP. MTSP which is known to be the NP-complete problem has found extensive applications in vehicle schedule, robot operations research, economics, management and communications network.
4.
14th World Congress of IFAC
(1) A feasible tour of every man for MTSP is a closed circuit problem. This is that for anyone of M salesmen, if he starts from the point i, then finally he must returns to the point i. Thus, a feasible tours of MTSP include M closed circuits. However, the actual hot rolling scheduling probJeITI is an open path~ that is, each production order is rolled exactly once. If a hot rolling scheduling includes M turns, then there exit only M open paths. (2) For MTSP ~ all the salesmen set out from the same fixed city R and finally come back to the starting city R to insure the minimal total traveling distances. Hovvever, for the actual hot rolling scheduling problem, starting order and ending order of each turn differs from that of other turns. This is that there aren't same order between any two turns.
4.3 To convert hot rolling scheduling into a normal lyfT5}P
We introduce M dummy nodes though two steps to convert hot rolling scheduling problem into MTSP. First step is: an additional dummy node (order) is introduced into hot rolling scheduling to ensure that all turns star from that dummy node and end at dummy node. This dummy node acts as both source node and destination node to make closed circuits. Second step is: M-I additional dummy node (order) are introduced into hot rolling scheduling problem to ensure that M closed circuits are made and every node should be visited by exactly one salesman, e. q. each production order is rolled exactly once.
MTSP MODEL TO ROLLING SCHEDULING 4.4 kfTSP "lodel ofthe hot rolling scheduling
4.1 The general description of MTSP MTSP in discussion may be stated as follo\\'s: Given N cities and M salesmen. All the salesmen set out from the same fixed city and finally come back to the starting city to insure the minimal total traveJing distances. It is required that each city should be visited by exactly one salesman and each salesman should visit at least one city. The MTSP which arise are then transfonned to single TSPs following Lenstra and Rinnooy Kan (1975).
Assume that N orders are rolled in M rolling turns on one shift. N orders may be vieVt~ed as N nodes and M turns may be regarded as M traveling salesmen. As long as M extra dummy nodes: N+], N+2, .... , N+M, are added, the hot rolling scheduling problem may be reduced to single TSP, that is, one traveling salesman visits N + M cities. Thus the mathematical model may be represented as follows: (P) N+Jv! ],{+M
min.
L::: Le J==]
4.2- The difference behl'een MTSP and hot rolling scheduling As was analyzed in Setction 3.2, a hot rolling scheduling problem may be reduced to MTSP. However, there are two aspects of differences between hot rolling scheduling problem and general MTSP:
ij
X ij
(1)
j=1
subject to N+lt.I
L
Xij=l,
jE{1,2,.o,N+M},
(2)
i=1
7058
Copyright 1999 IFAC
ISBN: 0 08 043248 4
A MTSP MODEL FOR HOT ROLLING SCHEDULING IN BAOSTEE...
14th World Congress of IFAC
N+M
L,
iE{I, 2, ... ~ N+M},
Xij=l,
(3)
J==l
X
Jj
==
{l
o
L, L, ieS
Xij'::;
IS:-1,
S c {l,~~~, N+M},
produced at the turn i - N othenvise;
for iE{N+l, ...,N+M },jE{I, ... ,N}
jES\{i}
(4)
X~i {l =
X
if order j is first to be
{O~
ij E
I}
1,jE{I, ... , N+M},
(5)
\~'here
N - the number of production orders to be rolled
on a shift, M - the number of rolling turns to be rolled on a shift ,
o
if order j is last to be produced at the turn i - N otherwise;
Objective function (1) mInImIzes the total changeover costs. Constraints (2) represent that exactly one task is Toned before order j. Constraints (3) represent that exactly one order is rolled after order i. Constraints (4) avoid subtours: they prohibit subtours and many uncormected subtours to occur in the set of feasible solutions. Constraint (5) represents that variables are integer of 0 or 1.
C i j = additional penalty cost when production of
order i is directly changed into that of order j. The C i j
5*
is further defined in details as follows (formulations
MODIFIED GENETIC ALGORlTHM (MGA) FOR SOLVING TSP
(6) -(10)): 5.1 The tveaknesses of crossover scherna of general genetic algorithms
where
pWij , pgij,
and phij represent penalty for width,
gauge and hardness jumps from order to order respectively, that is
pWij
can insure the minimal jump
in hardness; pBij can insure the minimal jump in gauge; and p\j can insure the minimal jump in width. Cij=o,
iE{I~
... ,N},jE{N+l, ... ,N+M},
(7)
C ij
iE{ N+l, ... ,N+M },jE{ 1, ... , N},
(8)
i,jE{N+l, ... ,N+M},
(9)
=
0,
Cij=CO,
C i i:--
00
~
i
E
{I, "0' N+M},
(10)
The variables are given as follows: for i ,jE{l, ...,
Xij
for
={lo
N}~
i *j,
if order j immediately"
follo~"s
order i at the same turn other"\vise;
iE {] "... ~N},j E {N+l~"4,N+M},
The nwnber of studies have reported that perfonnance of general GA is poor. For general genetic algorithms" crossover scheme for scheduling problems have at least the following three weaknesses. l) Blindness* In these crossaver methods, crossover position of two parents is generated" at random and genes of the generated children are also random~ Thus it is difficult to guarantee that the generated child chromosomes are better than those of their t\\lO parents. 2) Insufficient inheritances. Generating new chromosomes through random crossover destroys the parent chromosomes, the good segment of genes in the parent chromosomes may not be inherited. As a result, """hen the iteration approaches the end~ the chromosome evolution cannot become stable as desired. 3) According the experiment observation for the iterative process of general genetic algorithms, we have found that: (1) Lovv iterative effective: The fitness function of best chromosome of each generation is slo"vly improved as generation number increases, and when it reaches to certain level, improvement becomes very difficulty . (2) Mean population objective function: For mean value of objective function of all population in each generation improvement level is small and undulate as generation number increases.
7059
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress of IFAC
A MTSP MODEL FOR HOT ROLLING SCHEDULING IN BAOSTEE...
5.2 Theframework ofne"'1/ crossover scheme
5.4.2 Generation ofproblem instances When using GA to solve an optimization problem, \ve want it converges quickly. On the other hand, we do not want to see pre-mature convergence (trapped in local optimum). It is often difficult for the general GA procedure to keep a good balance between the 1:\vo (computation time and solution quality). We propose a modified GA in this section for a better trade offbetween the two conflict criteria. For general genetic algorithms, t\vo parents chromosomes which are used to make crossover are generated at random. The main difference of the new crossover operator with general genetic algorithms is that a couple of two parents chromosomes consist of that one is chosen at random, another is detennine which js best chromosome so far. The couple of nvo parents chromosomes produces offspring still using ex. This operator is called seed based crossover (Se).
5.3 The analysis o/new crossover schema
In the selection step of GA, solutions are selected w~ith
probability. Although the probabilities are so assigned that good solutions have better chance to be selected, it is no guarantee that the best solution ~vi]J be selected. To increase the chance for the optimal solution being approached quickly~ an additional step is introduced here to the GA procedure. The best solution recorded so far, among the reasons of ne\\o~ crossover scheme are: (1) according to GA convergence theorem in the general sense, if best so lution wi 11 be he Id in each generation, then the GA converges to optimal solution when N ~ ~; (2) according to knowledge of genetic evolution, if an excellent individual is put into a population, evolution of the population will be improved; (3) the child offspring's produced by two parents chromosomes among which one is best chromosome better with larger probability than that of two child offspring chromosomes produced by both bad parents chrolnosornes.
5.4
Implelnenting of MGA experiencefor Ta.,;p
and
computational
5.4. I Parameter setting We have implemented an MGA for TSP based on the structure described above. In this MGA, the representation of solutions, fitness function, and mutation operation are all chosen jn the same way as in the GA described by Goldberg (1989). The parameters are set as follows. Maximum number of generations = 6000 Population size = 100 Probability of crossover = 0.99 Probability of mutation = 0.010
For TSP, number of cities was chosen at five levels: 50, 75,100, 125, 150. Each level corresponds to one problem instance set which includes ten problem instances. Totally, 50 problem instances were generated and used in the experiment. The distance both any cities were randomly generated from unifonn distributions of which range "vas set to be from 1 to 500 for all problem instances.. 5.4.3 Computational results To evaluate the performances of the MGA, computational experiment has been done on a large nwnber of carefully selected representative problem instances. Two criteria are used for the comparison and evaluation - optimality and computation time~ Volgenant & lanker's (1982) exact algorithm were used for comparison. We used a ratio of CIC* as optimalit~y measurement where C* is objective function of optinlal solutions produced by Volgenant & lanker's exact algorithm and C is the result for the instance given by the MGA algorithm. The average optimality performances and computational time of MGA are ShO\V11 in table I.
Table 1 The optima1jty perfonnance and computational time ofMGA for TSP
Optimality performance No.of cities
50 75 100 125 150
V&J
MGA
1.0000 1.0000 1.0000 1.0000 1.0000
1.0272 1.0343 1.0613 1.0639 1.0727
Computational time (second) V&J 72 76 183 406 886
MGA 159 263 396 559 745
From the figures in the tables~ we can see that (1) MGA is very closed to exact algorithm in optimality performance. The average deviation of MGA from exact algorithm is \vith 50/0. (2) The computation time of both MGA and exact algorithm increases with increasing problem size. However~ increasing speed of computational effort with increasing problem of MGA is less than that of exact algorithm. All the time shown are in seconds on a Pentium PC.
5_ 5
An application and irnplernenting
MGA is employed to recreate hot strip roHing scheduling in the Shanghai Baoshan Iron & Steel Complex, China. The scheduling objective becomes
7060
Copyright 1999 IFAC
ISBN: 0 08 043248 4
A MTSP MODEL FOR HOT ROLLING SCHEDULING IN BAOSTEE...
to minimize the total penalty. A typical structure for \vidth, hardness and gauge change is proposed by Kosiba, et al. (1992) We have taken this typical structure as penalty cost in hot rolling system implemented by us. In this experiment the total penalties using MTSPlMGA decrease by 25.53350/0 (=(368-274)/64338)] over the manual scheduling.
CONCLUSIONS
This paper studied the development of hot rolling scheduling system through establishing MTSP model and using MGA. MTSP model using parallel strategy is employed to layout Inultiple lot rolling turns within a shift. "ve quantify the qualitative indexes of the model accordjng to actual process conditions~ analyze the structure of model parameters and represent them in a specific way. The new crossover operations (seed based crossover) have been introduced to the MGA framework to solve MTSP~ Unlike pl-eviously considered algorithms~ thjs MGA procedure has been implemented in the practical hot rolling scheduling system of Baosteel complex. This system has been successfully ran for one year and it show's two benefits: (1) improve hot rolling scheduling quality. Hot rolling schedules created by ne\\' system are superior to that produced from made-manual system. Average improvement of hot rolling scheduling quality of ne",,~ system in comparison with old system (made-manual system) is of at least 20%. (2) Reduce the time taken by created hot rolling scheduling. New scheduling system takes about 40 minutes to produce final scheduling, while old system takes about 4 hours to produce final scheduling.
14th World Congress ofIFAC
REFERENCES Balas, E., and C. Martin (1991). Combinatorial optimization in steel rolling, In: The DIMACSIRUTCOR Workshop on Combinatorial Optimization in Science and Technology:o Rutgeres University:New Brunswick, NJ, April. Kosiba, E. D., J. R. Wright. and A. E. Cobbs (1992). Discrete event sequencing as a traveJing salesman problem. COlnputers in Industry~ 19, 317-327. Lenstra, J. K. and A.H.G. Rinnooy Kan (1915). Some simple applications of the traveling problem. Operational Research Quarter, 26, 7] 7-733. Goldberg, D.E. (1989). Genetic algorithms in search} optimization} and machine learning. Addison WesJey, Reading, Mass. Volgenant A., and R. Jonker (1982). A branch and bound algorithm for the symmetric traveling salesman problem based on the l-tree relaxation, European Journal of Operational Research, 9, 83-89.
7061
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress ofIFAC
N-7b-05-1
Copyright ~ 1999 IF;\C 14th Triennial World Congr~ss, Bcijing, P.R. China
A MTSP MODEL FOR HOT ROLLING SCHEDULING IN BAOSTEEL COMPLEX *
Lixin Tang
1
2
t+
Aiying Rong
2
and
Zihou Yang 1
Institute o/S}'stems Engineering~ Northeastern University, Shen;;ang) P. R. China
Departnlent of.Auto/nation, Beijing Institute ofMachinery Industry. Bejing, P.R. China
Abstract: T'his article has investigated the hot rolling scheduling in the iron & steel plants. The task of a hot rolling scheduling is to detennine a sequence of orders (bars or slabs) on hot rolling finishing mill. Unlike previously considered series strategy_~ we proposed the parallel strategy to simultaneously produce M turns on same shift from global optitnal viewpoint. U sing this strategy; the model to hot rolling scheduling is developed as MTSP Inodel and then this model is converted into single TSP model. A modified genetic algorithm (MGA) is constructed to obtain near-optimal solutions to TSP. This procedure has been embedded in the practical hot rolling scheduling system by combining MTSP \vith man-machine interactive methods. Copyright © 19991FAC, Keyw'ords: hot rolling scheduling, Multiple Traveling
1.
INTRODUCTION
algorithm.
in actual production efficiency, finished product rate incorporation profit and product quality as well as great reducti on in production cost. The hot strip mi 11 is considered the "bottleneck" to overall iron and steel production and the production sequencing of steel orders through the hot rolling mill becomes the key to the hot strip mill production scheduling. The scheduling observes the constraints of the mill and attempts to obtain optjmal results by arranging the sequence for rolling the individual orders. Not only does it recognize constrains associated with specific orders and the number of jobs in the turn(round), but j
lligh production quality, low cost and higher production rates are three important objectives that today's iron and steel industry is trying to achieve. One of the possible ways to reach these targets is through optimal production scheduling. As is well knO\VTl, umeasonable scheduling will result in signific,ant loss in production. It is very important to seek an effective way to quantify scientifically production scheduling because benefits of utilizing such an approach involve considerable improvement
or
Sa]esman~ genetic
The Project 1S Supported by National Natural SC2ence Foundation vfChina Through Approved No 79700006, and National 863JCIMS of China Through Approved No. 863-511-708-009.
l To 'whom correspondence should be addressed.
E-Inail: [email protected]
7056
Copyright 1999 IFAC
ISBN: 0 08 043248 4
A MTSP MODEL FOR HOT ROLLING SCHEDULING IN BAOSTEE...
also satisfi.es the requirement for snlooth thickness or gauge transitions from order to order and the requirement for a generally decreasing width pattern in the rolling sequence. The hot strip rolling scheduling aillls at minimizing operating costs (such as transition costs between width groups in a sequence). In this paper, 'hre proposed the parallel strategy to simultaneously produce M turns on same shift from global optimal view. Using proposed parallel strategy, the model to hot rolling scheduling is developed as MTSP model based on actual production constrains. When solutions to MTSP model are found, the MTSP nlodel is converted into single TSP model. A mod ified genetic algoritrnn (MGA) is constructed to obtain near-optimal solutions to TSP. The ne"' crossover operations (seed based crossover) have been introduced to the MGA framework. This procedure has been embedded in the practical hot rolling scheduling system which is implemented and ran in Shanghai Baosteel Complex by combining proposed MTSP with man-machine interactive methods.
2.
14th World Congress of IFAC
part of the turn and quality of the turn dependents on combination and sequence of orders in "staple matericl In this paper, the term of turn discussed below in this paper presents the staple material section of that turn. The staple material rolling section should meet the follo\~ling requirements: Ca) the total length (or weight) of the staple materials is limited to a given quantity; (b) each staple order is no V\-"ider than the one that preceded it and vi/idth jump is small; (c) width, gauge and hardness jumps are not permitted to occur simultaneously; (d) order gauge which is not allowed to jump repeatedly should change smoothly; (e) hardness should change gently, gradually increasing or decreasing; and (t) ~'hen the change of hardness, gauge and width competes against each other'} the order of priority is: hardness, gauge and width. H
•
3.
CHARACTERISTICS AND MODELING STRATEGY OF HOT ROLLING SCHEDULING
3.1 Characteristics ofhot rolling scheduling
HOT ROLLING SCHE·DULING CONSTRAINTS
In hot rolling plant at Baosteel Complex, every \vorkday can be divided into three shifts and schedulers make throughput hot rolling scheduling of next shift on previous shift. This hot rolling scheduling general1y includes about 5-7 turns \vhich is basjc unit of hot rolling. 'fhere are three main considcratlons in the hot rolling scheduling: (I) product quality~ (2) roIler replacement cost; and (3) roller wear. The cost of roller replacelnent is so great that it is necessary to organize as many orders as possible within the range of maximal rolling length constraint of a turn to reduce the replacement cost. The impact of rolling order on the roller leads to roller wear: (1) the harder the rolling orders, the greater the hardness jump between from order to order, the greater the impact on the roller. (2) the larger the hardness difference [rolu order to order~ the greater the impact on the roller. (3) Gouging wilJ result at the edge of productroller contact area if too many orders with the same \vidth are rolled continuously. In order to guarantee product quality, the rolling scheduJes of staple materials are arranged so that each order is no w-ider than the one that preceded it while the first few orders during each turn increase in 'Vvidth to heat the mill, such an approach is called "warm up roller. Generally, the rolling job ,",vidth of a complete rolling turn appears as lI coffin profile". The H warm up" material section takes on the fonn of outgoing coffin profile while the staple material ro lling section assumes the fonn of inward coffin profile. The warm up" is minor part of turn and it is easjly detennined by schedulers. However, nstaple materiar' is major 11
T1
Characteristics of hot rolling scheduling are gi yen as fo llo\vs: Unlike general sequencing problems, (1 ) production sequencing in the steel rolling mill is affected greatly by the process constraints, which requires to specify the sequence of rolling orders as well as the combination of rolling orders. This kind of sequencing and like-product aggregation is referred to as sequential c ]ustering problem. (2) Sequence-dependent setups. Setup costs occurred by changcover bet,;yeen adjacent orders differ from that of another two adjacent orders.
3.2 Modeling strategies ofhot rolling scheduling
1) As was analyzed above, hot TOl1ing scheduling target should satisfy the following requirement:( I) to insure the minimal jump in hardness~ (2) to insure the minimal jump in gauge; and (3) to insure the minimal jump in width. Orders in the hot strip mill are known in advance and schedules are created only after sufficient orders have been taken since penalties are totally dependent on the physical properties of two adjacent orders. Penalties can be obtained from order to order, by direct computation. Now the problem may be described as finding the shortest path bet\veen the starting order and the ending order. Width, gauge and hardness jumps may be quantified by a penalty structure whi ch reflects the conditions in the hot strip mill. The sequencing with the lowest penalty will result in lower damage to the rollers, and in turn, higher product quaJity, hence, better profit. A hot
7057
Copyright 1999 IFAC
ISBN: 0 08 043248 4
A MTSP MODEL FOR HOT ROLLING SCHEDULING IN BAOSTEE...
strip mill turn with lower penalty is regarded as the better sequencing. Therefore, the scheduling objective becomes minimization of the total penalties. Penalties are only assigned from order to order and there is no penalty to sequential schedule orders of identical width, gauge and hardness. However, note that the gauge penalty from order to order may be asymmetricaJ. 2) In traditional system of Baosteel Complex, each turn on same shift is produced sequentially in the series strategy by schedulers. In previous research, the single TSP model proposed by Balas and Martin (1991), Kosiba, et ol. (1992) simulated schedulerts idea and also used series strategy to arrange only one turn for hot rolling scheduling. The series strategy is essentially similar to the Greedy idea that earlier arrangjng turns are probably well and later arranging turns are relatively poor because the number of candidate orders in order pool decreases as the number of produced tUITI increases. Therefore, the series strategy will suffer from the disadvantage of local optimization. When actual rolling scheduling is created, it is conunon to work out M rolling turn schedules from N orders in the preseletion pool. From global optimal viewpoint, \ve proposed the parallel strategy to simultaneously produce M turns on same shift. The hot rolling scheduling problem using parallel strategy may be reduced to MTSP \vhich is the expansion and continuity ofa single TSP. MTSP which is known to be the NP-complete problem has found extensive applications in vehicle schedule, robot operations research, economics, management and communications network.
4.
14th World Congress of IFAC
(1) A feasible tour of every man for MTSP is a closed circuit problem. This is that for anyone of M salesmen, if he starts from the point i, then finally he must returns to the point i. Thus, a feasible tours of MTSP include M closed circuits. However, the actual hot rolling scheduling probJeITI is an open path~ that is, each production order is rolled exactly once. If a hot rolling scheduling includes M turns, then there exit only M open paths. (2) For MTSP ~ all the salesmen set out from the same fixed city R and finally come back to the starting city R to insure the minimal total traveling distances. Hovvever, for the actual hot rolling scheduling problem, starting order and ending order of each turn differs from that of other turns. This is that there aren't same order between any two turns.
4.3 To convert hot rolling scheduling into a normal lyfT5}P
We introduce M dummy nodes though two steps to convert hot rolling scheduling problem into MTSP. First step is: an additional dummy node (order) is introduced into hot rolling scheduling to ensure that all turns star from that dummy node and end at dummy node. This dummy node acts as both source node and destination node to make closed circuits. Second step is: M-I additional dummy node (order) are introduced into hot rolling scheduling problem to ensure that M closed circuits are made and every node should be visited by exactly one salesman, e. q. each production order is rolled exactly once.
MTSP MODEL TO ROLLING SCHEDULING 4.4 kfTSP "lodel ofthe hot rolling scheduling
4.1 The general description of MTSP MTSP in discussion may be stated as follo\\'s: Given N cities and M salesmen. All the salesmen set out from the same fixed city and finally come back to the starting city to insure the minimal total traveJing distances. It is required that each city should be visited by exactly one salesman and each salesman should visit at least one city. The MTSP which arise are then transfonned to single TSPs following Lenstra and Rinnooy Kan (1975).
Assume that N orders are rolled in M rolling turns on one shift. N orders may be vieVt~ed as N nodes and M turns may be regarded as M traveling salesmen. As long as M extra dummy nodes: N+], N+2, .... , N+M, are added, the hot rolling scheduling problem may be reduced to single TSP, that is, one traveling salesman visits N + M cities. Thus the mathematical model may be represented as follows: (P) N+Jv! ],{+M
min.
L::: Le J==]
4.2- The difference behl'een MTSP and hot rolling scheduling As was analyzed in Setction 3.2, a hot rolling scheduling problem may be reduced to MTSP. However, there are two aspects of differences between hot rolling scheduling problem and general MTSP:
ij
X ij
(1)
j=1
subject to N+lt.I
L
Xij=l,
jE{1,2,.o,N+M},
(2)
i=1
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A MTSP MODEL FOR HOT ROLLING SCHEDULING IN BAOSTEE...
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N+M
L,
iE{I, 2, ... ~ N+M},
Xij=l,
(3)
J==l
X
Jj
==
{l
o
L, L, ieS
Xij'::;
IS:-1,
S c {l,~~~, N+M},
produced at the turn i - N othenvise;
for iE{N+l, ...,N+M },jE{I, ... ,N}
jES\{i}
(4)
X~i {l =
X
if order j is first to be
{O~
ij E
I}
1,jE{I, ... , N+M},
(5)
\~'here
N - the number of production orders to be rolled
on a shift, M - the number of rolling turns to be rolled on a shift ,
o
if order j is last to be produced at the turn i - N otherwise;
Objective function (1) mInImIzes the total changeover costs. Constraints (2) represent that exactly one task is Toned before order j. Constraints (3) represent that exactly one order is rolled after order i. Constraints (4) avoid subtours: they prohibit subtours and many uncormected subtours to occur in the set of feasible solutions. Constraint (5) represents that variables are integer of 0 or 1.
C i j = additional penalty cost when production of
order i is directly changed into that of order j. The C i j
5*
is further defined in details as follows (formulations
MODIFIED GENETIC ALGORlTHM (MGA) FOR SOLVING TSP
(6) -(10)): 5.1 The tveaknesses of crossover scherna of general genetic algorithms
where
pWij , pgij,
and phij represent penalty for width,
gauge and hardness jumps from order to order respectively, that is
pWij
can insure the minimal jump
in hardness; pBij can insure the minimal jump in gauge; and p\j can insure the minimal jump in width. Cij=o,
iE{I~
... ,N},jE{N+l, ... ,N+M},
(7)
C ij
iE{ N+l, ... ,N+M },jE{ 1, ... , N},
(8)
i,jE{N+l, ... ,N+M},
(9)
=
0,
Cij=CO,
C i i:--
00
~
i
E
{I, "0' N+M},
(10)
The variables are given as follows: for i ,jE{l, ...,
Xij
for
={lo
N}~
i *j,
if order j immediately"
follo~"s
order i at the same turn other"\vise;
iE {] "... ~N},j E {N+l~"4,N+M},
The nwnber of studies have reported that perfonnance of general GA is poor. For general genetic algorithms" crossover scheme for scheduling problems have at least the following three weaknesses. l) Blindness* In these crossaver methods, crossover position of two parents is generated" at random and genes of the generated children are also random~ Thus it is difficult to guarantee that the generated child chromosomes are better than those of their t\\lO parents. 2) Insufficient inheritances. Generating new chromosomes through random crossover destroys the parent chromosomes, the good segment of genes in the parent chromosomes may not be inherited. As a result, """hen the iteration approaches the end~ the chromosome evolution cannot become stable as desired. 3) According the experiment observation for the iterative process of general genetic algorithms, we have found that: (1) Lovv iterative effective: The fitness function of best chromosome of each generation is slo"vly improved as generation number increases, and when it reaches to certain level, improvement becomes very difficulty . (2) Mean population objective function: For mean value of objective function of all population in each generation improvement level is small and undulate as generation number increases.
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A MTSP MODEL FOR HOT ROLLING SCHEDULING IN BAOSTEE...
5.2 Theframework ofne"'1/ crossover scheme
5.4.2 Generation ofproblem instances When using GA to solve an optimization problem, \ve want it converges quickly. On the other hand, we do not want to see pre-mature convergence (trapped in local optimum). It is often difficult for the general GA procedure to keep a good balance between the 1:\vo (computation time and solution quality). We propose a modified GA in this section for a better trade offbetween the two conflict criteria. For general genetic algorithms, t\vo parents chromosomes which are used to make crossover are generated at random. The main difference of the new crossover operator with general genetic algorithms is that a couple of two parents chromosomes consist of that one is chosen at random, another is detennine which js best chromosome so far. The couple of nvo parents chromosomes produces offspring still using ex. This operator is called seed based crossover (Se).
5.3 The analysis o/new crossover schema
In the selection step of GA, solutions are selected w~ith
probability. Although the probabilities are so assigned that good solutions have better chance to be selected, it is no guarantee that the best solution ~vi]J be selected. To increase the chance for the optimal solution being approached quickly~ an additional step is introduced here to the GA procedure. The best solution recorded so far, among the reasons of ne\\o~ crossover scheme are: (1) according to GA convergence theorem in the general sense, if best so lution wi 11 be he Id in each generation, then the GA converges to optimal solution when N ~ ~; (2) according to knowledge of genetic evolution, if an excellent individual is put into a population, evolution of the population will be improved; (3) the child offspring's produced by two parents chromosomes among which one is best chromosome better with larger probability than that of two child offspring chromosomes produced by both bad parents chrolnosornes.
5.4
Implelnenting of MGA experiencefor Ta.,;p
and
computational
5.4. I Parameter setting We have implemented an MGA for TSP based on the structure described above. In this MGA, the representation of solutions, fitness function, and mutation operation are all chosen jn the same way as in the GA described by Goldberg (1989). The parameters are set as follows. Maximum number of generations = 6000 Population size = 100 Probability of crossover = 0.99 Probability of mutation = 0.010
For TSP, number of cities was chosen at five levels: 50, 75,100, 125, 150. Each level corresponds to one problem instance set which includes ten problem instances. Totally, 50 problem instances were generated and used in the experiment. The distance both any cities were randomly generated from unifonn distributions of which range "vas set to be from 1 to 500 for all problem instances.. 5.4.3 Computational results To evaluate the performances of the MGA, computational experiment has been done on a large nwnber of carefully selected representative problem instances. Two criteria are used for the comparison and evaluation - optimality and computation time~ Volgenant & lanker's (1982) exact algorithm were used for comparison. We used a ratio of CIC* as optimalit~y measurement where C* is objective function of optinlal solutions produced by Volgenant & lanker's exact algorithm and C is the result for the instance given by the MGA algorithm. The average optimality performances and computational time of MGA are ShO\V11 in table I.
Table 1 The optima1jty perfonnance and computational time ofMGA for TSP
Optimality performance No.of cities
50 75 100 125 150
V&J
MGA
1.0000 1.0000 1.0000 1.0000 1.0000
1.0272 1.0343 1.0613 1.0639 1.0727
Computational time (second) V&J 72 76 183 406 886
MGA 159 263 396 559 745
From the figures in the tables~ we can see that (1) MGA is very closed to exact algorithm in optimality performance. The average deviation of MGA from exact algorithm is \vith 50/0. (2) The computation time of both MGA and exact algorithm increases with increasing problem size. However~ increasing speed of computational effort with increasing problem of MGA is less than that of exact algorithm. All the time shown are in seconds on a Pentium PC.
5_ 5
An application and irnplernenting
MGA is employed to recreate hot strip roHing scheduling in the Shanghai Baoshan Iron & Steel Complex, China. The scheduling objective becomes
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A MTSP MODEL FOR HOT ROLLING SCHEDULING IN BAOSTEE...
to minimize the total penalty. A typical structure for \vidth, hardness and gauge change is proposed by Kosiba, et al. (1992) We have taken this typical structure as penalty cost in hot rolling system implemented by us. In this experiment the total penalties using MTSPlMGA decrease by 25.53350/0 (=(368-274)/64338)] over the manual scheduling.
CONCLUSIONS
This paper studied the development of hot rolling scheduling system through establishing MTSP model and using MGA. MTSP model using parallel strategy is employed to layout Inultiple lot rolling turns within a shift. "ve quantify the qualitative indexes of the model accordjng to actual process conditions~ analyze the structure of model parameters and represent them in a specific way. The new crossover operations (seed based crossover) have been introduced to the MGA framework to solve MTSP~ Unlike pl-eviously considered algorithms~ thjs MGA procedure has been implemented in the practical hot rolling scheduling system of Baosteel complex. This system has been successfully ran for one year and it show's two benefits: (1) improve hot rolling scheduling quality. Hot rolling schedules created by ne\\' system are superior to that produced from made-manual system. Average improvement of hot rolling scheduling quality of ne",,~ system in comparison with old system (made-manual system) is of at least 20%. (2) Reduce the time taken by created hot rolling scheduling. New scheduling system takes about 40 minutes to produce final scheduling, while old system takes about 4 hours to produce final scheduling.
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REFERENCES Balas, E., and C. Martin (1991). Combinatorial optimization in steel rolling, In: The DIMACSIRUTCOR Workshop on Combinatorial Optimization in Science and Technology:o Rutgeres University:New Brunswick, NJ, April. Kosiba, E. D., J. R. Wright. and A. E. Cobbs (1992). Discrete event sequencing as a traveJing salesman problem. COlnputers in Industry~ 19, 317-327. Lenstra, J. K. and A.H.G. Rinnooy Kan (1915). Some simple applications of the traveling problem. Operational Research Quarter, 26, 7] 7-733. Goldberg, D.E. (1989). Genetic algorithms in search} optimization} and machine learning. Addison WesJey, Reading, Mass. Volgenant A., and R. Jonker (1982). A branch and bound algorithm for the symmetric traveling salesman problem based on the l-tree relaxation, European Journal of Operational Research, 9, 83-89.
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