VRP-Based Model and Algorithm for Hot Rolling Lot Planning

ELSEVIER

Copyright © IFAC New Technologies for Automation of Metallurgical Industry, Shanghai , P.R. China, 2003

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VRP-BASED MODEL AND ALGORITHM FOR HOT ROLLING LOT PLANNING

Sbixin Liu., Jianbai Soni, Mengguang Wang l 1. Northeastern University, Shenyang 110004, China 2. Shanghai Baosight Software Company, Ltd. Shanghai 201900, China

Abstract: According to the process programs of hot rolling strips production in an iron & steel enterprise, a VRP (Vehicle Routing Problem) based model for hot rolling lot planning is presented, an ACO (Ant Colony Optimization) algorithm is designed to solve it. Computational study for practical instances has been made, the hot rolling lot plans obtained with the new model are compared with that made by senior planner through human-machine coordination, results show that the former is more effective and efficient. Copyright © 2003 IFAC Keywords: Hot rolling lot planning; Hot rolling turn; VRP; ACO

production process, the suitability of rolling lot plan affects product quality, roller maintenance cost and

1. INTRODUCTION Hot rolling strips are essential products in many iron

rolling lot plan

& steel enterprise. The main raw materials used in

rolling turn

hot strips rolling process are blooming rolling slabs and continuous casting slabs. After reheating, roughing rolling, finishing rolling, cooling, rolling up and polishing process the slabs are processed into strip coils. The quality of hot rolling strips is mainly guaranteed in finishing rolling process. The finishing rolling mill set generally includes six or seven continuous rolling mill stands, each stand consists of two work rollers and two backup rollers. Because of high temperature, high speed and heavy wear and tear, work rollers and backup rollers on each stand have to be replaced from time to time to ensure the strips quality. Slabs rolled between consecutive changes of work rollers are called a rolling turn, the multiple rolling turns rolled between consecutive changes of backup rollers are called a rolling lot. Fig. t illustrates the relation between rolling lot plan and rolling turns.

NO.k

replacement of backup roller

Fig.t Relation between rolling lot plan and rolling turns

production efficiency directly. The optimizing of rolling lot planning has been attracting many attentions of researchers and practitioners recently. Kosiba and Wright (1992) developed a traveling salesman problem model for rolling turn planning optimizing; Tang et al. (2000) presented a multiple traveling salesman problem model for rolling lot planning optimizing, and designed a modified genetic algorithm to solve it; Qian (1997) formulated a VRP model for rolling lot planning optimizing, and

One key job of hot rolling strips production operation management is making suitable rolling turns plans, and combining multiple rolling turns into a rolling lot plan. Because of high complexity in hot strips

133

designed a tabu search algorithm to solve it. But all the above models have not considered following process programs constraints which are jump limit of finishing temperature and rolling up temperature, and the length limit of strips rolled consecutively in same width within a rolling turn. In this paper, the above process programs constraints are further considered, a VRP based model for hot rolling lot planning is established, and an ACO algorithm is designed to solve it.

are to minimize the total cost caused by these jumps between neighbour slabs, and to minimize the number of roller replacements, so as to guarantee the product quality, reduce production cost and to improve rolling production efficiency. . 1sectIOn warm up matena Jl

. 1 sectIOn staPle matena

------~

en .... 0

2. MODEL FOR ROLLING LOT PLANNING

.s "0 .~

2.1 Process Programs Constraints Production

Of Hot

Rolling

A whole rolling turn consists of two section slabs, which are warm up materials and staple materials respectively. Hot rolling strip width sequence of a complete rolling turn appears as a 'coffin profile', which illustrates as Fig. 2. The warm up materials section takes on the form of outgoing coffin profile while the staple materials section assumes the form of inward coffm profile. The warm up materials section is used for heating the rollers, and is a minor part of a turn. Because of the unsymmetrical expansion of rollers, the strips quality is no guaranteed during early rolling process, so that usually only low-quality requirement products are arranged in this section. Therefore this part of turn can be easily arranged by schedulers, and is not taken into account for optimizing in this paper.

rolling turn Fig. 2 The structure of rolling turn 2.2 VRP Based Model

VRP is a kind of classic operations research problem (Clarke and Wright, 1964; Laporte, 1992). Let G = (N, A, d) be a complete weighted directed graph, where N = {nO,nl, ... ,nn} is the set of nodes,

A={(i,j)li,j=O,l, ... ,n,i:;tj} is the set of arcs, and each arc (i, j) has an associated weight di} ~ 0 which represents the distance between n; and n J

•

Node no represents a depot, where M

vehicles are located, each one of capacity D, while the other nodes represent customer locations. A demand d ; ~ 0 is associated to each customer n;

The strips quality of a turn depends on the combination and sequence of slabs in the staple materials section, the major part of the turn. According to the hot rolling process programs, the staple materials section should meet the following requirements: (1) strips gauge should change smoothly and not jump repeatedly; (2) strips hardness should change gently, gradually increasing or decreasing; (3) each strip is no wider than the one that precedes it and the width jump should be small, the length of strips rolled consecutively in same width is limited; (4) the total length of strips within a turn is limited; (5) width, gauge and hardness jumps are not permitted to occur simultaneously; (6) when changes in hardness, gauge and width compete against each other, the order of priority is: hardness, gauge and then width; and (7) the jumps of fmishing temperature and rolling up temperature is limited.

( do

= 0). The

objective is to find minimum cost vehicle routes and minimum number of vehicles dispatched such that (1) every customer is visited exactly once by exactly one vehicle; (2) for every vehicle the total demand does not exceed the vehicle capacity D; (3) every vehicle starts and ends its tour in the depot. Let no represent dummy slab, nodes {n l ,···, nn } represent slabs {1,2, ... , n} , d; represent the strip rolling length of slab i , D represent the total strip rolling length limit of a turn. The weight d i} of arc (i,

j) corresponds to the jump penalty cost from consecutive slab i to slab j within a turn.

d Above hot rolling process programs are developed to guarantee the strips quality and reduce roller wear and tear brought by the jumps of width, gauge, hardness. Rolling lot planning is a clustering and sequencing process of raw slabs that conforms to hot rolling programs, namely clustering the raw slabs into several rolling turns, and sequencing the raw slabs within each rolling turn ordering by width, gauge, hardness, finishing temperature and rolling up temperature. The objectives of hot rolling planning

I

2

3

4

5

i} = Pi} + Pi} + Pi} + Pi} + Pi} I

2

(1)

3

where Pi}, Pi} and Pi} represent penalty for width, gauge and hardness jumps from slab i to j respectively (Kosiba and Wright 1992).

p~

P:

and

represent penalty for jumps of finishing

temperature and rolling up temperature from slab i to j respectively, the values are set in a similar approach. The penalties for jumps from dummy slab 0 to other 134

slabs are do; = d;o = 0, i e {1,2, .. . ,n}. Let s be a solution of the VRP, U be the number of vehicles dispatched in s, the customers (slabs) serviced by each vehicle make a turn, then rolling lot planning problem are transformed into a VRP, which has the following objective function

f(s) ~ Min{tf,;v~' +C't,I. +C' ou} where

probability to make a random choice, O~qo ~l;

Evaporation coefficient of trail; The ant that made the globally best route

(2)

.

Then in the algorithm, an ant k located at n; chooses to go to node

rolling turn in s, Cl is penalty cost for violation of the length limit of strips rolled consecutively in same 2

since the start of the algorithm; The set of nodes visited by ant kgb

Tg/J

Vu is the slab sequence within the u-th

width, I" is violation times, C of work rollers.

qo the higher the

A parameter, the smaller

n.

is replacement cost

J

= { ar

g

nj

with a probability given by

If

max {[Tjzt . [17jzt}

",£M.(,)

0

q ~ qo (3)

Otherwise

where 0 is a node selected randomly from

Mic (i)

according to probability Plc (i, 0) :

3. ACO ALGORITHM

3.1 Algorithm Frame Pk(i,O)

P [T;ot . [17iol P L.J[Tjzt . [17iZ]

~

=

{ ",£M. (i)

The rolling lot planning model in section 2.2 is a modified VRP. Based on the model characters an ACO algorithm is designed to solve it, see (Bonabeau, et al., 2000; Bullnheimer, et al., 1999). In the algorithm, the construction of vehicle routes is done as follows: the artificial ants successively choose customers to visit, until each customer has been visited. Whenever the choice of another customer would lead to an infeasible solution for reasons of vehicle capacity, the ant returns to the depot and starts a new tour.

o

If

no

(5)

where ~ 'fij is the trail reinforcement computed with the following formula: 6.Tij

=

{~19bo

If

(i, j) E route done by ant Icgb (6)

Otherwise

where W is a parameter,

19 b

is the objective

function value of the solution made by ant kgb • The algorithm flow is reported as following: l. Initialize: 'fij := 'fo,

i,j = O,l, ... ,n, i '* j;

Iter = O·, Tgb''= ., 2. Set starting node as no for each ant:

;

Mic (i) Set of eligible nodes that ant

Tic := {no} ;

For k:=1 To m Do For 1:=1 To n Do If Mic (Id) = AND I < n

Introduces the following notations: 'fij The intensity of trail on edge (i,j);

An ant whose task is to make a route; The number of ants in the algorithm;

(4)

Otherwise

'fij+-(l-p)·rij+~'fij

by 'lij'

The visibility from n; to n j

k 1

iteration:

whereas the latter is the local heuristic function, this measure of desirability, called visibility, is denoted

'lij

M (.)

is updated with the following rule after every

'fij

For the selection of customers that have not yet been served, two aspects are taken into account: how good was the choice of that customer in previous iterations, and how promising is the choice of that customer in general. The former information is stored in the pheromone trail 'fij associated with each arc (i, j),

k m

E

Tic := Tic v{no} Choose next node n/cl with rule (3) and (4);

Then

Tic :=TIc v {n/cl} ; Recalculate Mic (Id) ;

k located at

n; can visit in next step;

a

Influence of pheromone trial intensity 'fij ;

p

Influence of visibility 'lij;

q

A value chosen randomly with uniform probability in [0, I];

End-for End-for 3. Apply local search algorithm to improve the route made by each ant 4. For k:=1 To m Do Compute objective function value for each route; End-for If The globally best route is improved 135

Table 1 Comparison of plans obtained by different approaches

Then Update Tgb ;

For every (i,j), f'ij ~(l-p)'f'ij +~f'ij; i. If

Iter ~ MAXIter Then OutputTgb;

Else GOTO Step 2

Turn

Tk = RI

U

R2

U ... u

Ru ' where Ru is the u-th

rolling turn. The A. - exchange operation is as following: exchanging the sequence of A. slabs in

Ru' c Tk with another sequence of A. slabs in Ru. ~ Tk (R u' and Ru. may be identical or not) , if the hot rolling process programs are violated after exchanging, restore and exchange again, else it make a new route. If the objective function value has not been improved after LocIter iterations, stop the local search process.

58.7

83

59.9

2

73

63 .1

61

63 .9

3

76

66.6

72

64.6

This research is financially supported by National 863/CIMS Scheme of China through approval No.2002AA412010 and Natural Sciences Foundation of Liaoning province in China through Approval NO.2002I0ll. REFERENCES Bonabeau, E., M. Dorigo and G. Theraulaz (2000). Inspiration for optimization from social insect behaviour. Nature, 406, 39-42. Bullnheimer, B., R.F. Hartl and C. Strauss (1999). An improved Ant System algorithm for the Vehicle Routing Problem. Annals of OperatiOns Research, 89, 319-328. Clarke, G. and J.W. Wright (1964). Scheduling of vehicles from a depot to a number of delivery points. OperatiOns Research, 12, 568-581. Kosiba, E.D. and J.R. Wright (1992). Discrete event sequence as a traveling salesman problem. Computers in Industry, 19, 317-327. Laporte, G. (1992). The vehicle routing problem: an overview of exact and approximate algorithms. European Journal of Operational Research, 59, 345-358. Qian, X. (1997). Study of modelling and algorithms for rolling production lot planning. Dissertation, Northeastern University, China. Tang, L., J. Liu, A. Rong and Z. Yang (2000). A multiple traveling salesman problem model for hot rolling scheduling in Shanghai Baoshan Iron & Steel Complex. European Journal of Operational Research, 124, 267-282

In this study, the parameters are set as follows: The number of ants is equal to n, the number of slabs,

= P =5 ,

=0.25,

a

p

nodes

np .. . , nn ' set

W = 10,

MaxIter = 2n, and set the visibility as 7]IJ..

qo

=0.9

Rolling strip length (km)

ACKNOWLEDGEMENTS

To test the effectiveness and efficiency of the model and algorithm, the model and algorithm are applied to a practical instance of an iron & steel enterprise in China, the rolling lot plan obtained using the new approach are compared with that made by senior planner through human-machine coordination. The instance includes 216 slabs, the maximum rolling strip length of a turn is 80km, the maximum rolling strip length of staple materials is 66km. The algorithm has been coded in Java language under Windows 2000 system and run on a Pentium IIII IGI128M personal computer.

at the

Number of slabs

Computational results show that the effectiveness and efficiency has been improved obviously. Future work should focus on development of integrated model which combines mUltiple objective, such as product quality, production cost and efficiency. Another promising direction is improving the algorithm efficiency by embedding senior planner's know-how into it.

4. COMPUTATIONAL EXPERIMENT

initially placed

New approach

67

J.2 Local Search Algorithm

Local search algorithm is developed based on A.!xchange approach, see (Qian, 1997). Let a route nade by ant k have the following format:

Human-machine coordination Number Rolling strip length (km) of slabs

,

= 11 d IJ'

The computational results are shown in table 1. The plans obtained by two different approaches both consist of three rolling turns, but the objective function value of plan obtained by the new approach is superior to the other obviously, which are 24875 and 35753 respectively. For the efficiency, it took 8 minutes by the new approach, and the other took about one hour.

5. CONCLUSIONS In this paper, the hot rolling lot planning problem in iron & steel enterprise is studied, and a VRP-based model and ACO algorithm are developed. 136