外文翻译--多级下料问题的建模 英文版

Modeling multistage cutting stock problemsEugene J. Zak*Majiq Inc., 8343 154th Avenue NE, Redmond, WA 98052, USAReceived 12 June 2000； accepted 28 August 2001AbstractIn multistage cutting stock problems (CSP) the cutting process is distributed over several successive stages. Everystage except the last one produces intermediate products. The list of intermediate products may be given or arbitrary.The goal is to minimize the total amount of material taken out of stock to cut nished products sucient to meetcustomer demands. If the intermediate sizes are given, the column generation technique can be applied to multistagecutting problems. If the intermediate sizes are not given then another dimension is added to the problem complexity.We propose a special procedure for this case that dynamically generates both rows (intermediate sizes) and columns(patterns). We refer to this method as row-and-columngeneration. The method uses an auxiliary problem embedded intothe frame of the revised simplex algorithm. It is a non-linear knapsack problem that can be solved eciently. In contrastto the column generation method the developed technique cannot guarantee the optimal solution. However, the resultsof computational experiments are very promising and prove that the method is a valuable addition to the set of tools formodeling and solving multistage CSPs. C211 2002 Elsevier Science B.V. All rights reserved.Keywords: Linear programming； Multistage cutting stock problem； Large-scale optimization； Row-and-column generation1. IntroductionA one-dimensional cutting stock problem (CSP) has an important practical generalization when acutting process is distributed over several successive stages. This multistage CSP includes not just cuttingpatterns and their activities, but also the intermediate products and their quantities produced at every stageof the cutting process except the last one, and consumed at every stage of the cutting process except the rstone. These intermediate products are cut to produce smaller intermediate sizes or nished sizes. The in-termediate products are both output and input during the cutting process. This kind of problem occurs inalmost every industry where a classic CSP takes place: paper, lm, leather, steel, etc. Although the articlesresults are equally applicable to any industry where multistage cutting takes place, for the purpose ofsubject area illustration, we will use terminology accepted in the paper industry. In particular, we will referEuropean Journal of Operational Research 141 (2002) 313327/locate/dsw*Tel.: +1-452-881-7100； fax: +1-452-881-5084.E-mail address: (E.J. Zak).0377-2217/02/$ - see front matter C211 2002 Elsevier Science B.V. All rights reserved.PII: S0377-2217(02)00127-3to a roll as a product geometrically dened by its width. Roll diameter, total length of unrolled paper, andpaper caliper are not relevant for the current investigation.Fig. 1 illustrates a three-stage cutting process. In this example three types of stock rolls S1, S2, and S3 areused to produce nine types of nished rolls (F1F9). It is interesting to note that stock rolls can supply anystage of the process. Here, S1 goes to the rst stage, S2 goes to the second, and S3 goes to the third.Conversely, the nished rolls can be produced at any stage. Three types of intermediate rolls I1, I2, and I3are cut at the rst two stages. Clearly, we may see proliferation of stock types, intermediate products, andeven cutting stages in real-world problems.Multistage is an overloaded word, particularly in the Operations Research and CSP areas. Gilmoreand Gomory 7 refer to a two-dimensional CSP solved by cutting strips along rst, then by cutting thestrips themselves across, as a multistage problem. In our case all cuts are along (lengthwise) and theproblem is a one-dimensional CSP. Dyckho 3 introduced multistage for a so-called one-cut modelwith multistage cutting. The one-cut model is an extreme case opposed to a usual and well-known case of aone-stage problem with unlimited cuts. It is interesting to note that those one-cut and multi-cut modelsare just two dierent formulations of the same problem, while in a real-world situation the one-stage CSP isoften a relaxation of the multistage CSP. The one-stage relaxation provides a reasonable lower bound forthe original multistage CSP.Several researchers have attacked a multistage CSP. Haessler 9 addresses a two-stage problem with awinder at the rst stage and a production rewinder at the second. He explores an LP approach with patterngeneration either beforehand or during simplex iterations using a column generation technique. He rep-resents a winder pattern in terms of nished rolls and checks whether the pattern can be broken up into acombination of legitimate intermediate rolls. If such a combination is permissible, the pattern is admittedinto the problem matrix. Although the approach is workable, it has some drawbacks. It, potentially, in-cludes a large number of dierent intermediate rolls, and dening whether a pattern can be partitioned is acomplicated bin-packing problem. Also, the approach does not scale easily to cutting processes with morethan two stages.Ferreira and others 4 also investigate the two-stage problem, which they refer to as a two-phasedproblem. The authors adapted Haesslers sequential heuristic procedure 8, initially developed for a classicCSP, to a two-stage cutting process. At every step of the sequential procedure they are trying to nd a set ofgood intermediate rolls ensuring a good pattern for the rst stage and good patterns for the second. If thepattern for the rst stage is accepted, a residual problem in terms of nished rolls is updated, reducing theFig. 1. Three stage cutting process: Stock rolls: S1, S2, and S3； Intermediate rolls: I1, I2, and I3； Finished rolls: F1, F2, F3, F4, F5, F6,F7, F8, and F9.314 E.J. Zak / European Journal of Operational Research 141 (2002) 313327ordered quantity by the amount dened by the pattern and its activity. Apparently, this heuristic resemblesa manual procedure for solving a two-stage CSP. The main diculty with this heuristic is generating a set ofgood intermediate rolls.Carvalho and Rodrigues 1 follow an LP approach. Their problem, however, is subject to a techno-logical restriction nished rolls of one width should comprise every intermediate roll. The restrictionallows for predening a list of possible intermediate rolls. The authors reformulate an initial LP problemposed in terms of nished rolls into a LP problem in terms of intermediate rolls. A column generationtechnique with a regular knapsack as an auxiliary problem is applied.The idea of an intelligent generation of intermediate rolls emerged in 10 when a two-stage system skiving and cutting was investigated. In the present paper we crystallize the idea and propose arow-and-column generation technique for solving a multistage one-dimensional CSP. The technique is ageneralization of the seminal column generation technique suggested by Gilmore and Gomory 5,6 forsolving a classic CSP, or, in our notation, a single-stage CSP. For a multistage problem, a more compli-cated auxiliary problem may result in column-candidates entering the basis, together with a combination ofrows corresponding to new intermediate rolls. We expand both rows and columns in the LP matrix. A nitenumber of iterations of simplex algorithm lead either to an optimal or a near-optimal solution.In the next sections we will formulate two basic models for the two-stage CSP, present the row-and-column generation method, and then analyze the computational experiments. Some of the results areborrowed from the authors paper 12.2. Model with a given list of intermediate rollsThere are three lists of roll sizes: List of stock sizes. List of intermediate sizes. List of nished sizes.Please see Fig. 2(a), which shows the relationship between these three types of rolls. For stock sizes theavailable amounts are known. Stock sizes can potentially be consumed at every stage of the cutting process,and can be cut into intermediate or nished rolls. Intermediate rolls are both input and output. Thetechnological constraints for intermediate rolls are strict: for every size the total number of consumed rollscannot exceed the produced amount. Ideally there should be a total balance； otherwise, some excessiveamount of unclaimed intermediate rolls should go to stock in a warehouse, if there is storage spaceavailable. But this is another question of tradeo between cost associated with material waste and ware-housing cost, which is beyond the scope of the current investigation. So we assume that we are consideringan open problem with inequality constraints, and we regard unclaimed intermediate rolls as waste. For anished roll we have an ordered amount that should be satised.Here we consider a two-stage CSP with one width of stock roll that is to be cut at the rst stage intoseveral intermediate rolls (Fig. 2(b). Finished rolls are produced at the second stage by cutting the in-termediate rolls. We assume that the width of an intermediate roll coming out of the rst stage and going tothe second one satises minimummaximum restrictions. The width of every intermediate roll should alsoincorporate a minimum edge to be trimmed at the second stage.Let w and y be vectors of the nished and intermediate roll widths, respectively. Cutting patterns of therst stage and the second stage are represented by matrices A11and A22, respectively. To make up a full LPmatrix we dene another matrix A12showing the relations between the two. Every column j of the con-necting matrix A12is a vector 0； .；0；C01；0； .；0T, where exactly one non-zero element )1 appears inE.J. Zak / European Journal of Operational Research 141 (2002) 313327 315the position i corresponding to the intermediate roll i that should be cut according to the cutting patterndened by column j in matrix A22.Now we can formulate an LP model for the multistage CSP:Minimize 1T0TC0C1x1x2C18C191subject toA11A120 A22C18C19x1x2C18C19P0bC18C19； 2x1；x2P0: 3Here vectors x1and x2are pattern activities for the rst and the second stages, respectively； b is a vector ofcustomer demands on nished rolls.Fig. 2. Graphs depicting roll relations: (a) general case； (b) case under consideration.316 E.J. Zak / European Journal of Operational Research 141 (2002) 313327The objective function (1) is to minimize the number of used stock rolls that is dened by the term 1Tx1.Constraints (2) ensure that consumption of intermediate rolls A12x2at the second stage should not exceed their production A11x1atthe rst stage, and customer demand of nished rolls b should be met.Notice that the overall LP matrix has a special structure. There are two diagonal blocks A11and A22representing cutting patterns for two stages, connecting block A12, and a block of 0s in the lower-leftcorner. The right-hand side is made up by 0-demand on intermediate rolls and the vector b. Model (1)(3) isa LP presentation of a two-stage CSP that explicitly involves intermediate rolls. The LP matrix might be fullif the problem is relatively small. In this case all patterns for all stages can be presented in the matrix.Otherwise, on-line column generation is an appropriate technique for column selection. But in both cases,whether we generate all columns in advance, or use column generation, the number of rows in the matrixremains unchanged, since the list of possible intermediate sizes is given.2.1. Dual problemLet us consider the dual problem associated with (1)(3):Maximize 0TbTC0C1u1u2C18C194subject toAT110AT12AT22C18C19u1u2C18C19610C18C19； 5u1；u2P0: 6Here vectors u1and u2are vectors of dual variables corresponding to intermediate rolls and nished rolls,respectively.The dual problem (4)(6) leads to two types of auxiliary problem that should be solved at the columnselection step of the revised simplex algorithm.2.2. Column generationThe rst type of auxiliary problem is generation of a cutting pattern related to the rst stage of thecutting process. The list of intermediate rolls remains unchanged. Clearly, this type is associated with therst group of constraints (5) of the dual problem, and the auxiliary problem is essentially the sameknapsack problem as we have in a classic or a single-stage CSP. The auxiliary problem can be stated as thefollowing knapsack problem:Knapsack I Maximize uT1asubject to yTa6w0；aP0；and integer:Hereu1isavectorofobjectivefunctioncoecientsthatarevaluesofthedualvariablesofthemasterproblem；y is a vector of intermediate roll widths； w0is the stock roll width and vector a is a vector of variables.If the objective function value exceeds 1.0, a new column a for the rst stage is generated. This conditionimmediately follows from the rst group of the constraints (5) which can be presented as uT1A1161T. Thesolution vector enters as a column into matrix A11.E.J. Zak / European Journal of Operational Research 141 (2002) 313327 317The second type of auxiliary problem is associated with cutting patterns generated for nished rollsutilizing existing intermediate rolls. This type is associated with the second group of constraints (5). Foreach intermediate roll yjwe should solve the following knapsack problem:Knapsack II Maximize uT2a；subject to wTa6yjC0 emin；aP0；and integer:Here u2is a vector of objective function coecients that are values of the dual variables of the masterproblem； w is a vector of nished sizes； eminis a mandatory minimal edge, eminP0； yjis width of inter-mediate roll j and vector a is a vector of variables.Let u1jbe a value of the dual variable corresponding to the intermediate size j. If the objective functionvalue exceeds u1j, a new column a for the second stage is generated. This condition immediately followsfrom the second group of the constraints (5) which can be presented as uT2A226 C0uT1A12. Given the matrixA12structure, the right-hand side boils down to a vector of dual variables corresponding to intermediaterolls.The solution vector enters as a column into matrix A22.If we do not have uncertainty in the intermediate rolls, two types of knapsacks shown above KnapsackIand Knapsack II are sucient to solve the problem optimally.3. Model with unknown intermediate rollsIf intermediate rolls are unknown, we face a more complicated situation. We are free to pick any suitableintermediate size y from a given range ymin；ymax. Since every intermediate roll and a nished roll is as-sociated with a row of the LP matrix, uncertainty in the LP matrix stretches in both directions: columns androws. For this reason, a column generation technique alone cannot solve the problem. On the other hand,the potentially huge number of intermediate rolls may preclude generating all possibilities beforehand. Wecan estimate the potential number of dierent intermediate rolls in real life situations. Let Dy be the rangeof intermediate roll widths. Parameter Dy is machine-dependent, but usually Dy C25 800 mm. Let d be theleast roll width accuracy. Normally, d 0:5 mm. Thus the number of dierent rolls n C25 Dy=d. This for-mula gives us an estimate n C25 1600. Certainly for particular instances of the multistage CSP the estimatemay be much less. Nevertheless the full size of the LP matrix tends to be very large.Another argument that is against advanced intermediate roll generation is an operational issue. Thegreat diversity in intermediate rolls at a time slows down the material ow, complicates roll-tracking tasks,and provides less exibility in cutting operations. Invariably a paper mill tends to operate with the leastnumber of dierent intermediate roll sizes.Needless to say, it would be highly desirable to have an intelligent approach generating intermediate rollsas can be done for cutting patterns only when needed. Next, we will present the row-and-column gen-eration technique for the two-stage problem (1)(3).3.1. Row-and-column generationAlong with Knapsack I, and Knapsack II, we may face a third type of auxiliary problem associated withsimultaneous generation of new intermediate rolls and cutting patterns. We should remember here thatrows of the LP matrix present intermediate and nished rolls. (If we had constraints on stock rolls wewould include stock rolls as additional rows as well.) The columns of the LP matrix are patterns for cuttingstock rolls into intermediate rolls and intermediate rolls to nished ones.318 E.J. Zak / European Journal of Operational Research 141 (2002) 313327Here we are trying to adapt the revised simplex to a task that is not typical for it. It is known that onevery step of the revised simplex a column is entering the basis and replacing another column that is leavingthe basis. If columns are not known in advance we use a column generation technique, expanding the LPmatrix in the column direction. Nothing in the revised simplex gives us the ability to generate unknownrows. The question is how can unknown intermediate rolls be generated using a step of the revised simplex?Let us restrict the intermediate roll search by generating not more than one new intermediate roll onevery step of the revised simplex. Thus we are supposed to generate a new row of the LP matrix, and, in thenon-degenerate case, the rank of the LP matrix will be eectively increased by 1. The basis matrix shouldalso be expanded by one row and one column, and since only one column leaves the basis during a pivotingstep, we conclude that two new columns should actually be generated during a simplex step. One of themshould be added up to the basis matrix unconditionally, and the other one should replace the existing one ina pivoting step.Let vector y be intermediate roll sizes that have been in the model so far, variable z be a new intermediateroll size, and v be a corresponding dual variable. We face the following nonlinear integer programmingproblem:Knapsack III Maximize uT1a11 va 7subject to yTa11 za6w0； 8wTa226z C0 emin； 9uT2a22 v； 10vP0； 11ymin6z6ymax； 12a11；a22P0；and integers: 13The variables here are two new columns: column a11for the rst stage and column a22for the second stage,intermediate roll width z, the dual variable for the new row v, and the number of rolls a for the new in-termediate width in the cutting pattern of the rst stage. Note that constraints (10) have an equality form.As a result of solving Knapsack III we will nd a new intermediate roll size, and two cutting patterns for therst and the second stages, respectively.Theoretically, the limitation imposed above restricts our choices for generating intermediate rolls. In-deed, a situation may occur when there exist no new cutting patterns involving just one new intermediateroll. At least two new intermediate rolls should be included in a cutting pattern on the rst stage. We canimagine that this situation is more probable at the beginning of the revised simplex, when the initial set ofintermediate rolls is scarce. As soon as the procedure generates more intermediate rolls, this situationbecomes less probable. As we show later, the one intermediate roll at a time restriction is justied for awide range of real world CSPs.3.2. A modied column selection in the revised simplex algorithmLet us prove the following proposition rst.Proposition 1.LetBbeanon-singularmatrixofm C2 m,and Babeitsextensionformedbyaddingonerowandone column as shown below:BaB a0TC01C18C19； 14E.J. Zak / European Journal of Operational Research 141 (2002) 313327 319where a is a vector of dimension m, and 0 is a vector of zeros of dimension m. Then, an inverse matrix BC01aexists and is defined asBC01aBC01BC01a0TC01C18C19: 15Proof. It is not dicult to verify that BaBC01a I, where I is an identity matrix. C3The modied column selection step of the revised simplex algorithm is as follows:Step1. Solve Knapsack I. If the optimal value of the functional exceeds 1.0, solution a11is a column toenter the basis at the pivoting step of the revised simplex algorithm. Otherwise go to the next step.Step2. Solve Knapsack II for each existing intermediate roll yj, j 1； .；k. If the optimal value for theproblem j exceeds u1jthen solution a22is a column to enter the basis at the pivoting step of the revisedsimplex algorithm. Otherwise go to the next step.Step 3. Solve Knapsack III. If the optimal value of the functional exceeds 1.0, we should expand the basis matrix by one row and one column, and pick the other column as a column to enter the basis at the pivoting step of the revised simplex algorithm.It does not matter which column should be added explicitly rst. In our implementation we add the columncorresponding to the cutting pattern of the second stage rst.The basis matrix is expanded according to (14), where B is a basis matrix found so far, and a is a newadded column. The expanded inverse matrix retains the current inverse matrix BC01as its sub-matrix and it isedged appropriately by zeros in the row and by BC01a elements in the column as shown in (15).If the optimal solution of Knapsack III does not exceed 1.0, and the reduced costs of slack variables arenon-negative, then the current solution is optimal.The column selection looks more complex than in the case of a classic column generation. Actually twoadditional steps 2 and 3 are involved. Now we investigate the auxiliary problem itself.3.3. A nonlinear Knapsack problemWhile the rst two types of an auxiliary problem Knapsack Iand Knapsack II are traditional andwell covered by the literature (see 11, or 2), the third type Knapsack III requires a special investigation.Knapsack III is a non-linear knapsack with several additional constraints (7)(13). The non-linearity isstipulated by two terms: va in the objective function (7) and za in the constraint (8).Proposition 2. It is sufficient to consider only those intermediate rolls whose width may be presented as alinear combination of finished roll widths plus minimum edge emin:z eminwTx； 16where xP0 is a vector of integers.Proof. Let us assume that we have found a solution of the two-stage problem (1)(3), and at least oneintermediate roll width z cannot be presented as (16). Then we can safely drop its width z to the nearestvalue, identied either by a linear combination (16), or by ymin. Such reduction does not violate anyproblem constraints. All patterns at both stages involving the intermediate roll z keep their validity. Moretrim loss at the rst stage will be compensated by less trim loss at the second stage. The objective functionkeeps it value. C3320 E.J. Zak / European Journal of Operational Research 141 (2002) 313327We suggest an economical method for solving Knapsack III. The idea is to partition Knapsack III intoseveral simpler sub-problems, solve them separately, and then pick the best solution among them. Theobvious candidate for partitioning is variable a. Its possible values are integers in the range from 1 tobw0C0 emin=yminc. Note that the case a 0 is out of consideration since it reduces Knapsack III toKnapsack I.Now, Proposition 2 and constraints (9) and (10) allow for the elimination of variables z and v from theproblem model using the linear presentations of variables a22. The modied problem is presented by thefollowing knapsack problem:Knapsack III0 Maximize uT1a11uT2a22a 17subject to yTa11wTa22a6w0C0 emina； 18ymin6eminwTa226ymax； 19a11；a22P0； and integers: 20Notice that the element a is a parameter here, a 1；2； .；bw0C0 emin=yminc. Introducing new parameters adjusted widths w0i wia, and adjusted values u02i u2ia, Knapsack III0is getting a form of a regularknapsack problem but with an additional two-sided constraint (19).Next, we outline a branch-and-bound procedure for solving Knapsack III0.3.4. Lexicographic algorithm for the Knapsack problemThe classic Knapsack problem isMaximize cTxsubject to aTx6b；xP0； and integers:All problem parameters fc；a；bg, objective function coecients and single-constraint parameters, are as-sumed to be positive. The lexicographic algorithm proposed by Gilmore and Gomory 6 in the Chvatal 2notation is the following:Step 1. Arrange the items in the descending order of ratios: ci=ai, i 1； .；m, where m is the vectorsdimension. Without loss of generality, we may assume that c1=a1Pc2=a2P C1C1C1Pcm=am. Initialize an indexof the branching variable, k 0, a record value of the objective function fC3 0, and a working parameterof the algorithm a remaining (unlled) part of the knapsack brest b. (You cannot trace this workingparameter in the original paper, but it emerges inevitably as soon as you start coding the algorithm.)Step 2. Find the most promising extension of the current branch. Iteratively from i k 1tom set:xi brest=aibc；andbrest brestC0 aixi:Step 3. Is an improved solution obtained? Calculate the value of the objective function f cTx andcompare it with the record. If f fC3, then update the record solution xC3with x, and the record value fC3with f.Step4. Backtrack to the next branch. If k 1, then stop, otherwise decrement k iteratively by 1 until therst k with xk 0 encounters. If k 1, then set:xk xkC0 1；E.J. Zak / European Journal of Operational Research 141 (2002) 313327 321andbrest brest ak:Step 5. Is the branch worth exploring? The branch potential is estimated by an upper bound of theknapsack tail functional. The knapsack tail starts with index k 1, and its optimal non-integer solution isxk1 brest=ak1, xk2C1C1C1xm 0. So ifPki1cixi ck1brest=ak1 fC3, then the branch is worth of ex-ploring. Go to step 2. Otherwise, go to step 4.This is a basic lexicographic algorithm. For Knapsack III0, step 2 is complemented by checking thevalidity of the two-sided constraint (19). If the checking fails go to step 4.3.5. Finding a good initial solutionIt is desirable to start with a feasible solution close to the optimal. Looking back at Proposition 2 above,we conclude that an initial list of intermediate rolls should at least include yminsince it may not be presentedby a linear combination of nished roll widths. Besides that to provide a warm starting we generate aninitial list of intermediate rolls using the following procedure.Start. The initial list is Y fy1g, where y1 ymin.Iteration. For each nished roll width wkwe identify an intermediate roll width ykasakymaxC4C0 emin=wkC5； 21yk emin wkak: 22Obviously yk6ymax.IfykPymin, and it diers from the roll widths that are already in the list Y, then we willadd roll width ykto the list: Y : Y yk.For every intermediate roll width ykdened by (21), (22) we identify a cutting pattern0C1C1C1C01C1C1C10C1C1C1aiC1C1C10 for nished roll i, where aibykC0 emin=wic and )1 is associated with inter-mediate roll width yk. In a similar way we identify initial patterns for the rst stage. The overall proceduremay be regarded as a generalization of the greedy procedure suggested by Chvatal 2.4. A sample problemSuppose there are four nished rolls to be cut. Two machines perform two-stage cutting consecutively:the rst machine cuts stock rolls into intermediate rolls, and the second one cuts the intermediate rolls intonished rolls. The input data are given in Tables 1 and 2.To start with, we will generate an initial solution. Let us restrict the initial list of intermediate rolls byone obligatory roll: y1 ymin 1200 mm. So we have one pattern for the rst stage producing 1200 mmrolls and four patterns of the second stage for every nished roll.The initial basis matrix is highlighted in Table 3. The objective function value is 60.777.Table 1Stock and nished rollsiwi(mm) biRoll type0 5000 Stock1 320 72 Finished2 340 88 Finished3 450 150 Finished4 500 108 Finished322 E.J. Zak / European Journal of Operational Research 141 (2002) 313327Table 4 shows the problem-solving dynamic. It is interesting to track how the algorithm generates newcolumns (patterns) and new rows (intermediate sizes). The problem was started with initial matrix seen inthe bold frame. Then patterns 9 and 10 were generated during column generation steps. Next an inter-mediate size of 1900 mm was generated along with two new patterns: pattern 11 and pattern 12. Then thealgorithm takes advantage of the new size and generates new columns 1316. Then a new intermediate sizeof 1690 mm was generated along with two new patterns 17 and 18, and so on.Table 2MachinesStage ymin(mm) ymax(mm) emin(mm) Maximum rolls out1 032 1200 1900 50 5Table 3Initial basis matrixTable 4The problem-solving dynamicE.J. Zak / European Journal of Operational Research 141 (2002) 313327 323The algorithm nds an optimal solution with 36 sets of the rst stage, in other words, 36 stock rolls areneeded to cut the ordered amount. How can we prove the optimality? We can calculate a lower bound byallowing all nished rolls to be cut directly at the rst stage. The optimal solution of the relaxed problem,that is a single-stage CSP, is also 36 sets.It is interesting to note that the solution is a degenerate solution since only six non-zero basic variablesare in the basis of nine elements. The basic patterns are highlighted in Table 4. There are four intermediaterolls of 1200, 1390, 1710, and 1900 mm in the optimal solution.Next we will show how to improve the operational quality of the solution.5. Intermediate rolls reductionCeteris paribus, a schedule with the least number of dierent intermediate sizes is the most desirable. Thesituation is analogous to pattern reduction in the single-stage problem. The exact algorithms for solving thisproblem are still waiting for attention from the Operations Research community. What we possess now area few heuristics that conduct post-optimal analysis trying iteratively to replace one intermediate roll withanother one that is already in the solution, or to replace two existing intermediate rolls by a new one, orthree existing intermediate rolls by two new ones, etc.Here we demonstrate how two-to-one conversion works. Let us look at the sample we just investi-gated. In the solution there are four intermediate rolls: 1200, 1390, 1710, and 1900 mm. In the patterns ofthe rst stage we can notice that the only appearance of rolls of 1390 and 1710 mm is pattern 22, and bothrolls have factor 1. Let us attempt the following substitution:w w1 w2=2: 23According to (23) we will dene a new intermediate roll size 1550 mm 1390 mm 1710 mm=2. Sopattern 22 will be almost intact with one exception: instead of one appearance of 1390 and 1710 mm eachwe will get two appearances of 1550 mm. Now we also should transform patterns of the second stageassociated with rolls of 1390 mm and 1710 mm. These patterns are 23 and 25. Please notice that these twopatterns have an identical number of sets 22. Now we can replace these two patterns by one pattern of1550 mm 50 mm 320 mm 2 C1 340 mm 500 mm of 44 sets.So we have not degraded a solutions eciency, but we have reduced the total number of intermediaterolls: instead of four dierent sizes we have three dierent sizes. It is a big improvement from an operationalpoint of view.Intermediate roll reduction techniques as demonstrated above proved to be simple but eective tools inimproving quality of solutions.6. ExperimentsThe main goals we pursued during the experiment were: to identify whether the one intermediate roll at a time rule provides a decent quality of solutions, to estimate how many intermediate sizes are in solutions, and to estimate the eect of warm starting.We implemented the revised simplex algorithm with two extensions: column generation method, androw-and-column generation method using Microsoft Visual C+. We have developed a generator ofrandom two-stage problems. In addition, we ran a relaxation of each trim problem eliminating the secondcutting stage by allowing all nished rolls to be cut directly at the rst stage. The problem relaxations weresolved with the regular column generation technique.324 E.J. Zak / European Journal of Operational Research 141 (2002) 313327The computational results are presented in Table 5, and Figs. 3 and 4. The computational experimentsbrought several surprises.1. The solutions found by row-and-column generation almost perfectly match its lower bounds denedby one-stage problems. We ran the program on randomly generated problems with sizes between one orderand 50 orders. The exact parameters of the random generator are in Table 6. We calculated the gap as100%f C0 fest=fest； 24where f is a functional value, and festis the optimal value of the functional of the estimate problem.On a sample of 1000 randomly generated problems the maximum gap is 11.1%. It has been reached ontwo instances only, and both are tiny problems with three orders. Only eight instances had a gap over 0.5%.So in a few cases only optimality of the solutions is questionable.Table 5Computational results, minimum edge50 mmNumber of orders Number of patterns generated Number of iterations CPU time (seconds)10 119 191 2.28120 256 668 5.32830 326 1339 7.71940 504 2967 20.34350 720 3271 46.68760 813 6633 74.78170 1103 10726 205.93780 1107 10667 234.79790 944 8842 182.843100 1011 9717 231.641Fig. 3. Performance as a function of the number of orders.Fig. 4. Number of intermediate rolls in a solution as a function of the number of orders.E.J. Zak / European Journal of Operational Research 141 (2002) 313327 3252. The number of intermediate rolls in a solution is not a monotonic function of number of orders. Whenthe number of orders is small it increases reaching some modest values and after a certain point it startsdeclining. For large-scale problems the number of dierent intermediate sizes that are in solutions tends tobe a very small. The best explanation of this phenomenon is the following: as the number of orders grows,the second-stage patterns get so diverse that only a few intermediate sizes are required to provide higheciency of the second-stage cutting. The high eciency of the rst stage cutting is guaranteed by choosingintermediate sizes to be well t to the stock sizes. For example, in many instances only two intermediatesizes for 5000 mm stock sizes have been generated: 1585 and 1830 mm providing a perfect rst stagepattern: 2 C1 1585 mm 1830 mm. Of course, the problem may have solutions with a big number of in-termediate rolls but the advantage of the devel