蓋冒墊片落料拉深復(fù)合模具設(shè)計(jì)【說(shuō)明書(shū)+CAD】

蓋冒墊片落料拉深復(fù)合模具設(shè)計(jì)【說(shuō)明書(shū)+CAD】,說(shuō)明書(shū)+CAD,蓋冒墊片落料拉深復(fù)合模具設(shè)計(jì)【說(shuō)明書(shū)+CAD】,墊片,落料拉深,復(fù)合,模具設(shè)計(jì),說(shuō)明書(shū),仿單,cad Modeling multistage cutting stock problems Eugene J. Zak * Majiq Inc., 8343 154th Avenue NE, Redmond, WA 98052, USA Received 12 June 2000; accepted 28 August 2001 Abstract In multistage cutting stock problems (CSP) the cutting process is distributed over several successive stages. Every stage 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 meet customer demands. If the intermediate sizes are given, the column generation technique can be applied to multistage cutting 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 into the frame of the revised simplex algorithm. It is a non-linear knapsack problem that can be solved eciently. In contrast to the column generation method the developed technique cannot guarantee the optimal solution. However, the results of computational experiments are very promising and prove that the method is a valuable addition to the set of tools for modeling 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 generation 1. Introduction A one-dimensional cutting stock problem (CSP) has an important practical generalization when a cutting process is distributed over several successive stages. This multistage CSP includes not just cutting patterns and their activities, but also the intermediate products and their quantities produced at every stage of the cutting process except the last one, and consumed at every stage of the cutting process except the rst one. 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 in almost every industry where a classic CSP takes place: paper, lm, leather, steel, etc. Although the articles results are equally applicable to any industry where multistage cutting takes place, for the purpose of subject area illustration, we will use terminology accepted in the paper industry. In particular, we will refer European Journal of Operational Research 141 (2002) 313327 * 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)00127-3 to a roll as a product geometrically dened by its width. Roll diameter, total length of unrolled paper, and paper 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 are used to produce nine types of nished rolls (F1F9). It is interesting to note that stock rolls can supply any stage 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 I3 are cut at the rst two stages. Clearly, we may see proliferation of stock types, intermediate products, and even cutting stages in real-world problems. Multistage is an overloaded word, particularly in the Operations Research and CSP areas. Gilmore and Gomory 7 refer to a two-dimensional CSP solved by cutting strips along rst, then by cutting the strips themselves across, as a multistage problem. In our case all cuts are along (lengthwise) and the problem is a one-dimensional CSP. Dyckho 3 introduced multistage for a so-called one-cut model with multistage cutting. The one-cut model is an extreme case opposed to a usual and well-known case of a one-stage problem with unlimited cuts. It is interesting to note that those one-cut and multi-cut models are just two dierent formulations of the same problem, while in a real-world situation the one-stage CSP is often a relaxation of the multistage CSP. The one-stage relaxation provides a reasonable lower bound for the original multistage CSP. Several researchers have attacked a multistage CSP. Haessler 9 addresses a two-stage problem with a winder at the rst stage and a production rewinder at the second. He explores an LP approach with pattern generation 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 a combination of legitimate intermediate rolls. If such a combination is permissible, the pattern is admitted into 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 a complicated bin-packing problem. Also, the approach does not scale easily to cutting processes with more than two stages. Ferreira and others 4 also investigate the two-stage problem, which they refer to as a two-phased problem. The authors adapted Haesslers sequential heuristic procedure 8, initially developed for a classic CSP, to a two-stage cutting process. At every step of the sequential procedure they are trying to nd a set of good intermediate rolls ensuring a good pattern for the rst stage and good patterns for the second. If the pattern for the rst stage is accepted, a residual problem in terms of nished rolls is updated, reducing the Fig. 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) 313327 ordered quantity by the amount dened by the pattern and its activity. Apparently, this heuristic resembles a manual procedure for solving a two-stage CSP. The main diculty with this heuristic is generating a set of good 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 restriction allows for predening a list of possible intermediate rolls. The authors reformulate an initial LP problem posed in terms of nished rolls into a LP problem in terms of intermediate rolls. A column generation technique 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 a row-and-column generation technique for solving a multistage one-dimensional CSP. The technique is a generalization of the seminal column generation technique suggested by Gilmore and Gomory 5,6 for solving 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 of rows corresponding to new intermediate rolls. We expand both rows and columns in the LP matrix. A nite number 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 are borrowed from the authors paper 12. 2. Model with a given list of intermediate rolls There 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 the available 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. The technological constraints for intermediate rolls are strict: for every size the total number of consumed rolls cannot exceed the produced amount. Ideally there should be a total balance; otherwise, some excessive amount of unclaimed intermediate rolls should go to stock in a warehouse, if there is storage space available. 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 considering an open problem with inequality constraints, and we regard unclaimed intermediate rolls as waste. For a nished 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 into several 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 to the second one satises minimummaximum restrictions. The width of every intermediate roll should also incorporate 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 the rst stage and the second stage are represented by matrices A 11 and A 22 , respectively. To make up a full LP matrix we dene another matrix A 12 showing the relations between the two. Every column j of the con- necting matrix A 12 is a vector 0; .;0;C01;0; .;0 T , where exactly one non-zero element )1 appears in E.J. Zak / European Journal of Operational Research 141 (2002) 313327 315 the position i corresponding to the intermediate roll i that should be cut according to the cutting pattern dened by column j in matrix A 22 . Now we can formulate an LP model for the multistage CSP: Minimize 1 T 0 T C0C1x 1 x 2 C18C19 1 subject to A 11 A 12 0 A 22 C18C19 x 1 x 2 C18C19 P 0 b C18C19 ; 2 x 1 ;x 2 P0: 3 Here vectors x 1 and x 2 are pattern activities for the rst and the second stages, respectively; b is a vector of customer 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) 313327 The objective function (1) is to minimize the number of used stock rolls that is dened by the term 1 T x 1 . Constraints (2) ensure that consumption of intermediate rolls A 12 x 2 at the second stage should not exceed their production A 11 x 1 at the 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 A 11 and A 22 representing cutting patterns for two stages, connecting block A 12 , and a block of 0s in the lower-left corner. The right-hand side is made up by 0-demand on intermediate rolls and the vector b. Model (1)(3) is a LP presentation of a two-stage CSP that explicitly involves intermediate rolls. The LP matrix might be full if 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 matrix remains unchanged, since the list of possible intermediate sizes is given. 2.1. Dual problem Let us consider the dual problem associated with (1)(3): Maximize 0 T b T C0C1u 1 u 2 C18C19 4 subject to A T 11 0 A T 12 A T 22 C18C19 u 1 u 2 C18C19 6 1 0 C18C19 ; 5 u 1 ;u 2 P0: 6 Here vectors u 1 and u 2 are 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 column selection step of the revised simplex algorithm. 2.2. Column generation The rst type of auxiliary problem is generation of a cutting pattern related to the rst stage of the cutting process. The list of intermediate rolls remains unchanged. Clearly, this type is associated with the rst group of constraints (5) of the dual problem, and the auxiliary problem is essentially the same knapsack problem as we have in a classic or a single-stage CSP. The auxiliary problem can be stated as the following knapsack problem: Knapsack I Maximize u T 1 a subject to y T a6w 0 ; aP0;and integer: Hereu 1 isavectorofobjectivefunctioncoecientsthatarevaluesofthedualvariablesofthemasterproblem; y is a vector of intermediate roll widths; w 0 is 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 condition immediately follows from the rst group of the constraints (5) which can be presented as u T 1 A 11 61 T . The solution vector enters as a column into matrix A 11 . E.J. Zak / European Journal of Operational Research 141 (2002) 313327 317 The second type of auxiliary problem is associated with cutting patterns generated for nished rolls utilizing existing intermediate rolls. This type is associated with the second group of constraints (5). For each intermediate roll y j we should solve the following knapsack problem: Knapsack II Maximize u T 2 a; subject to w T a6y j C0 e min ; aP0;and integer: Here u 2 is a vector of objective function coecients that are values of the dual variables of the master problem; w is a vector of nished sizes; e min is a mandatory minimal edge, e min P0; y j is width of inter- mediate roll j and vector a is a vector of variables. Let u 1j be a value of the dual variable corresponding to the intermediate size j. If the objective function value exceeds u 1j , a new column a for the second stage is generated. This condition immediately follows from the second group of the constraints (5) which can be presented as u T 2 A 22 6 C0u T 1 A 12 . Given the matrix A 12 structure, the right-hand side boils down to a vector of dual variables corresponding to intermediate rolls. The solution vector enters as a column into matrix A 22 . If we do not have uncertainty in the intermediate rolls, two types of knapsacks shown above Knapsack Iand Knapsack II are sucient to solve the problem optimally. 3. Model with unknown intermediate rolls If intermediate rolls are unknown, we face a more complicated situation. We are free to pick any suitable intermediate size y from a given range y min ;y max . 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 and rows. 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. We can estimate the potential number of dierent intermediate rolls in real life situations. Let Dy be the range of intermediate roll widths. Parameter Dy is machine-dependent, but usually Dy C25 800 mm. Let d be the least 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 estimate may 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. The great 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 least number of dierent intermediate roll sizes. Needless to say, it would be highly desirable to have an intelligent approach generating intermediate rolls as 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 generation Along with Knapsack I, and Knapsack II, we may face a third type of auxiliary problem associated with simultaneous generation of new intermediate rolls and cutting patterns. We should remember here that rows of the LP matrix present intermediate and nished rolls. (If we had constraints on stock rolls we would include stock rolls as additional rows as well.) The columns of the LP matrix are patterns for cutting stock rolls into intermediate rolls and intermediate rolls to nished ones. 318 E.J. Zak / European Journal of Operational Research 141 (2002) 313327 Here we are trying to adapt the revised simplex to a task that is not typical for it. It is known that on every step of the revised simplex a column is entering the basis and replacing another column that is leaving the basis. If columns are not known in advance we use a column generation technique, expanding the LP matrix in the column direction. Nothing in the revised simplex gives us the ability to generate unknown rows. 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 on every step of the revised simplex. Thus we are supposed to generate a new row of the LP matrix, and, in the non-degenerate case, the rank of the LP matrix will be eectively increased by 1. The basis matrix should also be expanded by one row and one column, and since only one column leaves the basis during a pivoting step, we conclude that two new columns should actually be generated during a simplex step. One of them should be added up to the basis matrix unconditionally, and the other one should replace the existing one in a pivoting step. Let vector y be intermediate roll sizes that have been in the model so far, variable z be a new intermediate roll size, and v be a corresponding dual variable. We face the following nonlinear integer programming problem: Knapsack III Maximize u T 1 a 11 va 7 subject to y T a 11 za6w 0 ; 8 w T a 22 6z C0 e min ; 9 u T 2 a 22 v; 10 vP0; 11 y min 6z6y max ; 12 a 11 ;a 22 P0;and integers: 13 The variables here are two new columns: column a 11 for the rst stage and column a 22 for 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 the rst 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 intermediate roll. At least two new intermediate rolls should be included in a cutting pattern on the rst stage. We can imagine that this situation is more probable at the beginning of the revised simplex, when the initial set of intermediate rolls is scarce. As soon as the procedure generates more intermediate rolls, this situation becomes less probable. As we show later, the one intermediate roll at a time restriction is justied for a wide range of real world CSPs. 3.2. A modied column selection in the revised simplex algorithm Let us prove the following proposition rst. Proposition 1.LetBbeanon-singularmatrixofm C2 m,and B a beitsextensionformedbyaddingonerowand one column as shown below: B a B a 0 T C01 C18C19 ; 14 E.J. Zak / European Journal of Operational Research 141 (2002) 313327 319 where a is a vector of dimension m, and 0 is a vector of zeros of dimension m. Then, an inverse matrix B C01 a exists and is defined as B C01 a B C01 B C01 a 0 T