The cutting stock problem for large sections in the iron and steel industries

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European $ o = ~ a l of Operational Research 22 ¢1985) 280-292 North-Holland

Case Study

The cutting stock problem for large sections in the iron and steel industries H. T O K U Y A M A and N. U E N O Control and OR Technology Department, Sumitomo Metal lndust-ies, Amagasaki, Japan

Abstract: The characteristics of a cutting stock problem for large sections in the iron and steel industries are as follows: (1) There is ~ variety of criterions swzh as a~aximizing yield and increasing effeciency of production lines. (2) A cutting stock problem is accompanied by an optimal stock selection problem. A two-phase algorithm is developed, using an heuristic methcd. This algorithm gives nearly optimal solutions in real time. It is applied to both batch-solving and on-line solving of one-dimensional cutting of large section. The new algorithm has played an important role in a large-section production system to increase the yield by approximately 2.5~. Keywnrds: Cutting stock proble~ z. heuristics, production

I. introducfi,m In Sumitomo Metal Industries. large scale corni, :+or systems have been built up and have greatly :~ ibuted to keep high productivity and material i ! etc. (see Figure 1). The Order Entry Comi: ~ System in the Head Office receives orders i :a customers and decides which work is better ~, prod,ace them, regarding order's specification and work's load. The production planning and management system in each work combines orders c,f similar specifications and delivery dates into a lot that is adequate to the production units of processes and gives the production instructions. A large section is one of the steel products; the average length is about 10 m. and a cross section is H-shaped. Orders are cut out from an as-roll 1he autt" ~rs ~,,tsh to thank Dr. Iri for his useful suggestions and ~hank t ~.e persons concerned in Sumitemo Metal Industries for their useful disct,,=sions. Received .~anua~ 1985

len,,4h+ which means the length of a material after rolling. n this paper, an heuristic but practical algorithm using a combinatorial method will be de~.eloped. This algorithm gives nearly optimal soh,tions in real time. It is applied to both batchsob,ing and on-line solving of one-dimensional cut-ing of large sections. The new algorithm has played an important r o k in the large-section production system to increase the yield by approximately 2.5%, etc.

2. Cutting stock problem of large section 2.1. Preliminaries Orders are grouped into order groups to sectional dimensions, standards and tions. Stocks are grouped into stock groups to sectional shape, material grade and tions.

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Let N and M be the number of order groups and the number of stock groups, respectively. Other symbo|s concerning with o ~ e r and stock information are as follows: n~ = number of orders in order group g, S~ = material specification for order group g, l~. = length of the i-th order in order group g, P~, = required number of the /-th order for order group g, %, = weight per unit length of the i-th order in order group g, m h = number of stocks in stock groups h. Z~, = material specification for s'~ock group h, Yh = stack number for stock group h, Wj,, = weight of the j-th stock in stock group h, L~ = length of the j-th stock in stock group h. 'In ti~e following ~ction, let g, be the i-th order of order group g and h~ be the j-th stock of stock group h.

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2. 4. Objectives and constraints

Let i'~~'. be the number of orders g, which are cut out of stock h,. Then a cutting pattern of stock h is denoted symbolically by

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roll-chance, for purpose of improving the production efficiency. Further it is assumed that the sequence of roll-chance is predetermined by the order delivery, dates a n d / o r the production planning. Therefore, the problem is to find the optimal sequence of the order group for each roll-chance. (3) Specification matching between order and material. An order group consists of several orders which specifications are the same, and a stock group consists of several stocks which specifications are the same, too. Thus, specification matching between order aad material is equally replaced by specification matching between an order group and an stock group. (4) Workability condition on the piling bed. The number of the different order lengths included in one cutting pattern should be fewer than those of the available piling beds, to improve workability on the piling bed:

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~b) Loading eff/ciency of ~win=. machines and co~ling bed. In Figure 3 the relation between cutting patterns and loading efficiency of sa~,'ing machines and a cooling bed is shown. For cutting pattern G-1 in the figure, a product of the as-roll length is cut into 3 pieces by No. 1 hot saw (No. 1 HS) at each po-;nt, where is marked 'v'. Then. each piece is cut into two by No. 2 HS at the peints ,marked "v" and go forward on the cooling b ~ in two rows. So, the number of cutting by No. 1 and No. 2 HS is respectively 4 and 3. On the other hay.d, for cutting pattern B-l, a product ~| the same as-roll length as G-1 is cut into 4 pie:.~ by No. 1 HS at the points marked "v' and then :he last piece of cutting pattern is cut by No. 2 HS SO, the number of cutting by No. 1 and No. 2 for this catting pattern is respectively 5 and 1. There a:-e some vacancies in the cooling bed. Therefore, ioad balancing of two HS machine~ and workaoitit~ on the ccoling bed depend upon the cutting secuence of the order within a cutting pattern. Le S~ and S~, be the number of cuttings in the j-th ,zutting pattern of the h-th stock group by the two HS machines, respectively. Let YI~, and Y~ be the lengths of a pair of products which proceed on the cooling bed in two rc~ws In tie case that the sum of Y~, and Y~ is longer lhal~ the width of the c~)ling bed, each product should proceed in one row. and Yg~ must be zero. In order to prevent the HS machine~ No. 1 anti No. 2 from load unbalancing, the following inequalities must be satisfied: I~,', - S~, I ~ Sc, for I ~
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order group g within roll-chance and (3) a cutting pattern { P~j } satisfying the constraints ¢1)-(5) for which the objectives (1)-(3) are minimal. The cutting stock problem has 3 features. Firstly: the problem is a complicated combination of a combinatorial problem and a scheduling problem. Secondly: the problem is a large-scale problem. The number of all possible cutting patterns ,~ill turn out to be enormous. The number of combinations, which is generated by selecting npn orders within n , orders included in one order group, is ,,C,, ¢ Assuming that the number of cuttings in one cutting pattern is A, i.e., E,P~¢' = A, the number of cutting patterns is (npB) :t. So, the number of candidates amounts to

For example, in the case that n~ = 50, np~ = 3, and A = 10, this number is

mC3 × 31'~ 10 '~Thirdly, the problem has 3 objectives. Since the weight among them depends on demand, production level, sale's policy and so on, it is impractical to unify 3 objectives int.o one stationally. Therefore~ a flexible algorithm ~hould be developed.

3. Algorithm It has been tempting to tackle the cutting stock problem [1-8] from a combinatorial point of view. Gilmore znd Gomo~' [1,2] solved the required number of stocks by linear programming. Tilanus and Gerhardt [3] presented an alogrithm to solve a cutting stock problem based on dynamic programming. However, the problem in this paper is a large-scale one, and involves a stock-selection problem and a roll-scheduling problem. A two-phase algorithm applied to the newly derived cutting stock problem of large sections is developed.

2.5. t~m~blem [ormulatton

3.1. Basic idea The cutting stock problem of large sections can be described as the problem of finding (1) An assignment pair I g. h), (2) a roll sequence of the

The 5 constraints prevealed in section 2.4.B can be classified to be satisfied on the basis of order

H. Tokuyama, N. Ueno / The cutting stock problem

group or order itself from the standpoint of minimum unit for each condition. The constraints (2) rob scheduling conditions and (3) specification matching condition must be satisfied on the basis of order group a n d / o r stock group. Next, the constraints (4) the workability condition on the piling bed, and (5b) the loading efficiency of sawing machines and cooling bed, are satisfied on the basis of the order's length and the order's number. Lastly, the constraints (1) fulfillment of the required quantity for each order, and (5a) limitation of stock length, are satisfied not only on the basis of order and stock but also on the basis of order group and stock group. Therefore, by using the structure of the problem, we developed a two-phase algorithm shown in Figure 4. At Phase I, the problem that is on the basis of order group and stock group is solved. The material allotment problem is to find the best assignment pairs between order group and stock group. Further, the roll scheduling problem is to find the optimal .sequence of the order group. Next, at Phase lI, as a result of the best solution, a cutting stock problem arises for each order group, It is a one-dimensional combinatorial prob. lem, and is on the basis of order and stock, q-he problem is to find the optimal cutting pattern {I~' } ~,atisfying the constraints (i), (4~, t5a) and t5b). Though this two-phase algorithm is to find near optimal solution sequentially by heuristic approach, it is sufficiently available for real time

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processing in the production line. In the problem the as-roll length of a stock. from which a bloom is rolled, measures 90-100 m long, and the average order-length is about 10 m. In the situation that about 5-10 products are cut out of the as-roll length of a bloom, it is experimentally knov, n that at phase II sufficiently nearly optimal solutions can be found by a pattern generation discussed in the next section. 3.2. Material allotment and roll scheduling problem 3.2.1. Problem formulation at Phase I Let us consider the constraint (I) fulfillment of the required quantity, which is satisfied not oral5 on the basis of order and stock but also on the basis of order group and stock group. How the amount of crop wilt be derived from each bloom is not estimated accurately until the problem at Phase I! will be solved. So, for the problem at Phase I. the previous condition can be transformed to the fulfillment of the required weight aL defined as t

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the handling operation in the material yard. Hence, the procedure to solve the latter one is expiained under the assumption that all order groups in one roll-chance could be rolled by the same caliber and any sequences among them could be allowed. One of LP solutions is shown in Step C(t) of Figure 5. Order groups S~ and S,~ are divided into the st{~k groups B~ and B:~. In this case it is better to modify this LP solution into the one shown in Step C(2) to minimize the number of partitionings of stock groups under the guarantee of avoiding the over-grade matching. The roll scheduling problem is solved along the flow shown in Figure 6. This procedure is described as follows: (i) For each order group allotted to two or more stock groups, a partial sub-roll .schedule which we will call sub.chain is determined. The partitioned stock group (B3) is permutated to be ended in this sub-chain as shown in Figures 6 a), b). (ii) Each stock group is connected adequately as shown in Figure 6 c) and finally the roll schedule is obtained in Figure 6 d). Since the goodness of the roll schedu)e obtained depends targely on the LP solution, column permutations of the LP tableau are performed repeatedly in order to get the other rdl schedules. Step D. Select the best solution where the number of partitioned stock groups is minimum. For example, there are two partitioned stock groups

(B 2 and B4) in Step C(3) and none in Step C'(3), as shown in Figure 5. The consumed time (IBM 370/158) attained about one second to solve the problem with 25 order groups and 40 st(xak groups, 3. 3. Cutting sto{ k problem 3. 3. I. Prr~blem ]orrnukttton at Phase 11 As a result of the best solution of the previous, material allo~inent and roll scheduling problem, a one-dimensional cutting stock problem aris,e~ for each order group g. It is a special version of a general combinatorial problem, The problem is as follows:

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Secondly, an approach to improve a local optimal solution toward a total optimal solution is discussed. The workability condition restricts the number of orders to be combined in a cutting pattern. Once orders are selected, order combination continues until the required number of orders is completely combined. This restriction tightens the computational load, but, on the other hand, the restriction and a sequentially solving procedure with an heuristic approach affects the total yield of the roll-chance. Because the yield of each cutting pattern decreases remarkably when there exist a few orders to be combined, at the near end of the computation. In order to avoid yield decreasing at the near end of computation, it is known experimentally that the orders which are included in the worstcutting pattern of the proceeding solution may preferably be combined and then the next solving procedure may be conducted. These solving procedures are repeated during a specified number of times, and the best solution whose yield of the chance is maximal is selected.

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where Sc and E are adequate parameters. Compared with other author [1-8], the cutting stock problem of large sections in this paper hz.~ features as follows: (a) the rolling sequence of bloom is predeterr,ained, (b} there are blooms of various as-roll lengths and not-standard lengths, because of the various weights of the blooms, (c) the number of different order lengths in a cutting pattern is restricted, and is less than that of the piling beds because of their workability condition, (d) the cutting sequence in a cutting pattern affects workability on the piling beds, and loading efficiency of sawing machines and cx}oling bed. 3. 3. 2. Basic idea Two ideas are deviced for the problem at Phase II. One is pattern generation by an enumeration method to find a lot of cutting patterns sequen. tially, and the other one is an approach to improve a l o c # optimal solution. Firstly, the constraints (1I-2)-0I-5) are all on the basis of each bloom and the roll sequence of a blocm is predetermined. So, without (II-IL minimizing the total amount of crop for an order group is equally replaced by minimizing the total amount for each bloom. Further in the situation that about 5-10 products are cut out of ~he mat,.'rial, a cutting pattern satisfying (II-4) and {II-.¢) at nearly miximal yield can be obtained. The:efore, all cutting patterns which yield is larger thar~ the target yield A, for example 98.5%, ~atisfying 1II-4) and (II-5) are obtained by an er.umeration method at high speed. Then the cutting pattern satisfying (I!-4) and (II-5) at the maximal yield is selected. Even if no cutting pattern satisfying (II-4) and (II-5) is obtained, A is reset by A - AA, where AA denotes a very little increment, then all the cutting pattern of which yield ~s larger than A - AA is generated. This procedure is continued until the cutting pattern s~.tisfying (1I-4) and (II-5) is obtained.

3. 3. 3. Solution procedure

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The ~lution of cutting .~tock problem depends on the sequence of the order combination during the computation. The algorithm shown in Figure 7 i~ as follows: Step A (Setting the &gree of preference). The highest degree of preference is assigned to the orders of which length are long and/or nu nber of pieces are large, because they should be pr;ferably combined at the beginning of comput-.tion and should not be combined at the near end of computation. Step B (Main procedure of cutting stock). (i) Select one bloom tn accordance w=th the roll schedule and calculate its a~-roiJ length. (ii) Select as many orders as the number of piling beds according to their degree of preference. If the orders which are not completely combined to the previous bloom remain, select them first. (iii} Combine orders and generate all the cutting patterns which yield is larger than the predetermined value A by an enumeration method (Pattern generation). If no cutting pattern with yield larger than A and satisfying the constraints is generated, reset A by A - AA, and repeat the

H. Tokuyama, N. Ueno / The cutting stock problem

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above pattern generation until a cutting pattern is obtained. {iv) Select one of the cutting patterns whose yield is maximal subject to the above constraint.'-,. If all orders are completely combined, then proceed to Step C. Else return to Step B(i). Step C (Modification of the degree of preference). (i) The highest degrees are assigned to the orders which are in the worst-cutting paltern at minimal yield of the proceeding solution. Then the next solving procedure is conducted using the modified degrees of preference. (it) If these solving procedures (Step C and Step B) are performed until a specified number times, for example, ten times, then proceed to Step D. Else, return to Step C, Step D. Select the best solution whose yield of the roll-chance is maximal.

3.3.4. Computational results Table 2 shows an example of a cutting stczk problem, which corresponds to one order groLp. Blooms in the list are arranged in order of the

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63 68 75 85

95 8O

( b ) Stock i n f o r m a t i o n

of 81

No. I roiling KI

KZl .3l I K4 K5 K6

__y7

~ i i

"

' 8z.SB= 84.7 3 i 22 8708

1

89.45 q_|.77 ~4.t2

I .5 13, I 3 i

.... 2.:

i-

4

-J

2~J

!t. Tak~,m,r,a, N. Utno / The cumng stack ~obk~

]~le 3

An actual ~ f u ~ o~ a oataag ~tockproblemunder the (x=ditioaof two p i ~ bed, R~uired -. m" tEs, Umted [ C.mlbked order r~. Yield Bmo _r~a[os.mtI length ~ Wt W2 W3 W4 W5 W6 (%1 13k:X)ITin~ 998 I 2 K~ i 82.38(m) 1 2 8 84.73 : 4 6 99.4 t 2 K2 99.0 4 K3 87.08 6 4 I 10 99.8 5 K4 89.43 1 91.77 K5 3 I 5 99.4 91,77 13 99.4 2 K6 94.12 I 12 99.1 3 K6 K'7 j 94.t 2 8 8 99.1 8 i_. 99.1 K7 94.1 2 5 4 I i 8 7 94.1 2 99.1 ! K7 K7 99.1 I i 94.12 i 6 8_ K7 99.1 I I 8 6 i 94.1 2 6 I I K7 99.1 I i 94.12 K7 : 94.12 5 3 99.1 9 94.12 K7 7 89.5 I . K7 94.12 . 575.6 I

|i Order of Cu.ing po.etn I~. rollir~ '

I

I

2

3 4 5 6 7 8 9 I0

t

i {

II

!

I2

t3

14

{

!

15 16

i L

r;,aa

r

number

~:~ 68 75 85 95 BE)

Estimated Total

Yie.!d

= 98.5%

I

as-roll lengths, which are calculated from the estimated weight. ~Pae s<~Iution of the problem presented in Table 2 is shown in Table 3 under the condition of two piling bexis and an example of it (No. 1) is illustrated in Figure 8. We have developed a computer program

(Sumitomo Cutting Stock Programming _Package: SCOOP) based on this algorithm. The algorithm was tested on 5 problems. The results are presented in Table 4. qhe required time (IBM 370/158) varied from 2 to 5 .~onds.

4. Applications

c,o. -'lt'k:~W4] "ff41 '#5 t WSiW5 t W5iW51W5]W51W .,~ Estimated

as-roll

length

Figure 8. ,An example of cutting pattern (cutting pattern No. 1)

Table 4 Computational resuhs Nu,,o~, of orders

Number of bloom~

Yield

no. 1 2 3 4 5

11 7 15 9 7

24 31 6?

97.3 97.8 99.1 99.2 99.2

4_~

69

The algorithm developed in this paper has been applied to both batch-solving and on-line solving in a large section cutting stock system since 1975 as ;~'own in Figtre 9. And about 1800 products are cutted out of 1800 blooms every day. The actual application is as follows: (i) On the da 3 before rolling, batch solving is executed in acco-dance with order informations and stock infatuations. In this step the as-roll length of a bloom is estimated from its dimensions. (ii) According to the result, instructions of char~ng blooms into a heat furnace are submitted. (iii) Just before charging, the actual weight of "each bloom is measured. Then on-line solving is e~ecuted every one or two hours using the actual weight. If the estimated weight differs from the

H. Tokuyama, I~: Ueno / The cutting stock problem

Bo td~ solving) Stock Order informction information number k"

x45

~-'-~i

x 25

,~

~rd.

Stock yard

I .+i

", x 3 3

~.~

'--L

X

291

one stock /

Conditioning information

~

I'

,

grouping )

gro,pigg I

Stock

I~ n g

Instruction

Moteriol allotment t, roll scheduling rder Group ] ~ Stock Group

i

Heating f u r n a c e ~+,rnea surernent,~

x'°t+l:=r

Actual weight

B

lls2

, ,,

...........

,,, ,

Si Combinotion for every"'chonce

L

}.,- Estimored as-roll lenglll-~ .... ,' • ,',' j +, ""=~ im

....... j

+

J

I

Roll sequence

'L+

Surplus :

/

I

~-7-,i

Rollin_g

~iJ u

...........

Iniormotion I of batch-solving

On -'-'] ine cot~binetion Re I

:~L-'-- i ....... [

LI_~ + ? ] , .et,+h,

,equencet

_

I~I

I

.... I

~

Ks.a

J

J _-/¢-

l' -'-'1" I l I /'j) Estlmoted ........... _ ,_ weight

..... J

e_~

~ - - ~

i

,........ r--'-T--"J ,, Actuol

;~r ?~ J

llmachin.__ _..

i_

t

information!

~"~:~

I

i

J

Piling

L¢-_'_/

Figure 9. Large ~ectioncatting stock ~ystem

1

"

i

292

H, Tokuyama~ N, Uena / The cutting ztock problem

"fable 5 Benefit~ hems

Effects

Improvem.'mt in yield Decrea.~ in surplus ratio Increase in operating ratio 1ncregs¢ in e ffk--qency

2.5 (~c) 4,1 ~ ) 0.5 (~) 5 ~T/H)

actual one, the cutting pattern of on-fine solving may differ from that of batch solving.

5. Conclusion

The algorithm has played an important role in the Large Section Production System of Kashima Works in Sumitomo Metal Industries. In spite of some changes of production technique, the twop h a ~ algorithm contributes to the increase of yield as shown in Table 5. References {1] Gilmore, PC., and Gomory. R,E,. "A linear programming approach I> ~he ~,utfin~ ~tc~k problem", Operationu Rewurch 9 ¢196!) 849=859.

[2] G i l ~ e , P.C.~ and Gomory. P-,.E, "A linear programminz ~gp~o~h to the cutting st~-~ probler~., Pan II", O~ralions Rese~ch I I {19~3~ ~ - ~ 8 . [31 Tilanus, C.B,. and GefharL C.. "An apFl/ca6on of cu¢fing stock in the Steel lndtt~tr/es'. Opermi~'u2! R.e~ezzrch ~'15 (Proc~dLng5 IFORS 7:h Ce~_-.~), No~h-Ho!!m~d. Amsterdam. 1976. [4] Eisemann. K., "The trim problem". Management Science 3 (1957) 279-284. 15] Haessler. 1LW.. "'An heuristic programming solution to a turn.linear cutting stock problem". Management ~czence 17 (1971) IEI-793-I~-802. [6] Haessler. R.W. "Ccmtrotling cutting pattern changes in one-dimensional trim problems". Operations Research 23 (1975) 483-493. 17] Coverdale, !., and Wharton. F., "An impr~-ed heuristic procedure for a non-linear cutting stock problem", Afanagement Science 23 (1976) 78-86. [8] Staintom R.S., "The cutting stock problem for the stockholder of steel r c i n f o r ~ t bars". Operational Re~earch Quarterly 28 (1977) 139-149, [9] Tol~uyama. H,. and Ueno, N., "The cutting stock problems in the Iron and Steel industries". Operational Research '~,' (Proceedings !FORS 9th Congress}, North-Holland. Amsterdam. 1982. [10] Tokuyama. H., Ueno. N. and Toyoda, T., "'The cutting stock problems in the Iron and Steel Industries". umitomo ~ r c h 26 (1981; 27-39. [11] Toyoda. T., Tokuyama. H, et al., "Production control ~y~tem for a large shape mill", Computer Applications in Japanese Industry. Academic Pres~. New York. 1985,