In the following, the cutting problem is described in more detail. The manufacturer receives N orders {1, . . . , N}from different customers. Each order iis characterized by lengthli, widthwi, thicknessti, and qualityqi of the requested veneer pieces which have to be cut from the veneer strip that is peeled off the tree trunk. The length dimension of a veneer piece corresponds to the length of the tree trunk and the width dimension of a veneer piece corresponds to the footage that is peeled off the tree trunk. The (previously fitted) length of a tree trunk ranges from 800mm to 2000mm and, depending on the diameter of the tree trunk, up to 150 meters of veneer strip can be peeled off the tree trunk. Length, width, and thickness are given in millimeters, the quality ranges from very bad (quality 10) to very good (quality 1). For the quality, certain wood characteristics such as wood color, vain, or knotholes are taken into account. The quality of the veneer strip can be determined by the veneer peeling machine during the cutting process.
Furthermore, each order specifies the number of required piecesni and a deadlinedi. The main goal of the manufacturer is to find cutting patterns on each day that fulfill all orders with deadline at this day and minimize the wood offcut. Further objectives are described in Section 5.3.2.
Consequently, we are dealing with a multi-objective optimization problem. The man-ufacturer provided preferences for the different objectives, from which we were able to deduce reasonable weights. Therefore, we applied the concept of weighted-sum scalar-ization and obtained a single-objective optimscalar-ization problem as described in Section 5.3.
In contrast, in Section 5.4.2, a multi-objective simplified version of the cutting problem is considered.
For a better understanding of the problem and the notion of a cutting pattern, the production process is described in the following: Depending on the orders, the tree trunks are cut to lengths and the bark is removed. Subsequently, the tree trunk is peeled into a thin veneer strip which is then cut down to the required veneer pieces. This process is visualized in Figure 5.1. In this phase, only cuts in the width dimension can be made.
Thus, the length of the pieces cannot be changed. If necessary, the length is manually cut to size in an additional working step. In order to plan the production process, it has to be decided which tree trunk lengths are used and how the veneer strips are cut down to pieces, both in the width and the length dimension. Each length corresponds to a cutting scheme and due to certain characteristics of the production process, the number of different cutting schemes per day is limited to Cmax. Furthermore, there is a limit ofPmaxon the volume of wood in cubic meters that can be processed on each day.
The difficulty of the problem is further increased by the following two aspects:
Figure 5.1: The tree trunk is peeled into a thin veneer strip and cut into veneer pieces.
c Becker KG
First of all, the quality of the wood is uncertain. Wood of European beech (bot. Fa-gus sylvatica), which is commonly used by the manufacturer, has several characteristics which have to be taken into account when producing rotary cut veneer. These character-istics have an influence on optical and mechanical properties such as color and bending strength. In our case, the most important characteristics are heartwood and knottiness, which will be briefly explained in the following.
Red heartwood labels the appearance of red colored wood in the center of the beech tree trunk. Unlike other species, such as oak, where all trees have heartwood, not all beech trees have a heartwood area. If they do, this area is colored red to reddish-brown.
Another difference to, for example, oak wood, is that heartwood of beech wood does not have other mechanical properties like strength. There have been some investigations (see, for example. (Ràcz, 1961)) but not much can be said about the likelihood of the appearance of heartwood in beech. It can be said though that heartwood appears more often with increasing age and diameter at breast height (cf. (Lohmann, 2003)).
Knottiness describes the frequency of occurrence of knots in round wood and timber.
Apart from the frequency, form and type of knots is important. A forest tree can be divided into three parts: the lower part of the tree which is more or less free of external branches (ground tree trunk), the middle part with thick dead branches and branch stubs, and the top end of the tree trunk with living branches. For the production of veneer mostly the ground tree trunk is used. Even though it is free of external branches, it contains inner knots. During the growing of the tree it builds branches which are dying and falling off very early. Knotless wood is then growing over these branch stubs, so that the ground tree trunk wood is more or less free of knots except for its inner part. The number and size of these knots cannot be estimated easily before cutting the tree. Beech trees have so-called “Chinese beards”, oval marks on the bark, which allow making rough estimations about inner knots (cf. (Lohmann, 2003)).
Due to these problems in determining the quality of the wood, the distribution of the different qualities can only be estimated and is therefore an uncertain parameter in
orders cutting pattern order li wi ti qi ni li wi ti qi ni
1 400 390 1.5 2 50 800 390 1.5 1,2 25
2 400 470 1.5 4 200 800 470 1.5 3,4 100
3 800 490 1.1 5 200 800 490 1.1 5 200
800 450 1.1 >5 400 Table 5.1: Orders and a cutting pattern for length 800mm.
the problem formulation. In Section 5.3, a distribution of the wood quality provided by the manufacturer based on historical values is used to model the uncertainties. In Section 5.4, we take into account the uncertainty of this quality distribution and, due to the requirements of the manufacturer, the concept of robust optimization is chosen to be applied to a simplified version of the problem. Furthermore, orders can always be satisfied with veneer pieces of a higher quality than requested. Obviously, satisfying orders with higher qualities than requested is a loss of profit for the manufacturer and is, therefore, to be minimized.
Secondly, the manufacturer has the possibility to manually cut down the veneer pieces to the appropriate size. For example, there is an order for 100 pieces of length 800mm and width 350mm and another order for 200 pieces with length 300mm and width 390mm.
The manufacturer could schedule to cut 200 pieces of length 800mm and width 390mm, and cut down the width from 390mm to 300mm for the first order and the length from 800mm to two pieces with length 300mm for the second order. Of course, the manual cutting incurs additional working time, therefore the number of manually cut pieces is limited for each day.
In Table 5.1, three exemplary orders and an appropriate cutting pattern are given.
Note that, due to the distribution of the wood quality in the tree trunk, certain amounts of each quality have to be used. Therefore, there is an additional line in the cutting pattern, covering all qualities worse than five: the first three lines are sufficient to meet the three orders on the left, but, in order to deal with qualities worse than five, which are inevitably peeled off the tree trunk as well, the last line is added. Furthermore, qualities one and two are used for order one, and qualities three and four are used for order two.
Finally, orders one and two are cut with length 800mm and then manually cut down to 400mm. Therefore, only half the number of pieces of orders one and two is needed.
This example is supposed to give a brief glimpse of the planning problem of the manufacturer. The real-world problem is comprised of several hundreds of available orders with up to 50 different lengths and widths, 4 thicknesses and 10 qualities.
5.2.1 Classification of the Cutting Problem
The problem presented in this work is basically a cutting stock problem. The ordered ve-neer pieces are characterized by length and width, leading to a two-dimensional problem.
Furthermore, all ordered veneer pieces have to be assigned to a selection of tree trunks of different lengths and thicknesses. Finally, there are several different tree trunks and many ordered veneer pieces of many different shapes. Therefore, according to Dyckhoff’s typology for cutting and packing problems (Dyckhoff, 1990), the problem is classified as 2/V/D/M. But, in addition, we have to deal with heterogeneous tree trunks, i.e., each tree trunk consists of different qualities. Furthermore, deadlines add a temporal dimen-sion to the problem and certain production restrictions, such as a limited production for each day, increase the difficulty of the problem.
According to the classification of cutting and packing problems by Wäscher et al.
(2007), we are faced with a two-dimensional input minimization problem where all or-dered veneer pieces (strongly heterogeneous) have to be accommodated to several tree trunks for which one dimension is considered as a variable, resulting in an open dimen-sion problem. Still, we have to deal with the aforementioned additional aspects of the problem, making it significantly more difficult. To the best of our knowledge, the cutting problem as described above is not discussed in the literature. The basic cutting stock problem on a strip, which is the core of our problem, is for example considered by Benati (1997); Zhiping et al. (1997).