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Weekly UpdatesIn the literature on rational design of furniture, numerical optimization algorithms for searching for the best solution in the field of dimensioning of elements and constructional nodes are applied fairly commonly. Despite the discussions on productivity in wood machining (Ratnasingam et al. 1999), however, there are no reports concerning the utilization of numerical methods for optimizing the technology of furniture manufacture. Studies on the subject which have been published so far primarily focus on the utilization of experimental research conducted for different processing conditions (Anguilera et al. 2003, Ratnasingam and Scholz 2004). On the other hand, a look at the literature devoted to other branches of industry shows that there are a number of papers which focus on the development of mathematical models or computer programs which make it possible to conduct rationalization of technological operations (Grieve and Griffiths 1984, Armarego et al. 1993, Wang 1998, Li et al. 2004).
From the point of view of industrial practice, increased diversity and quantity of different products manufactured by furniture factories creates the need to develop computer tools which would allow cooperation with integrated systems of the MRP-II/ERP (Manufacturing Resource Planning/Economical Resource Planning) class, in order to make it possible to improve the determination of technological routes (i.e., the detailed sequence of operations with associated workstations and technological parameters) for individual items. To make the preparation of such tools possible, it is essential to develop algorithms based on expert knowledge or optimization methods allowing the determination of rational processing parameters.
Research objective
The purpose of this study was to develop a mathematical model and computer program for the optimization of a selected fragment of the technological process in a furniture factory. The developed optimization process should focus on obtaining the best processing parameters which in turn would ensure minimization of the processing time of a given operation and the technical cost of manufacturing. The result of the computer program implementation should allow the automation of the generation of technological routes in the MRP system, especially through the assessment of operation time standards. Another objective of the research was to conduct optimization for an example input data set obtained on the basis of observations made in a furniture factory.
Research methodology
As the subject of the optimization exercise, rough mill processing was chosen. The examined technology included the following three operations: cross cutting of sawn timber, longitudinal separation of blocks on a multirip saw, and four-sided planing. Cross cutting of sawn timber is usually done on circular saws with manual feed or on high-yielding automatic devices. In the case of manual feed, the actual output is of random character depending on the skill, as well as physical and mental condition, of the worker. In the case of automatic devices, the analysis of technical specifications of some widely used machines leads to the conclusion that both velocity of the conveyor and time required to make the cut are constant. From the point of view of a technologist, the impact on the parameters and productivity of this operation is negligible: therefore, it was decided that it would be excluded from the decision-making process.
There is a relationship between the width of the strip prior to its four-sided planing, b', and the final strip width, b, which can be expressed as:
b' = b + 2[a.sub.b] [1]
where:
[a.sub.b] = width allowance at four-sided planing.
On the other hand, the nominal strip thickness prior to planing, h', is identical with the thickness of sawn timber, [h.sub.t], selected from the standard thickness series of types, [H.sub.t], currently in force, basing on the thickness after planing, h, and the value of thickness allowance, [a.sub.h], during planing:
h' = min{[h.sub.t] [member of] [H.sub.t]: [h.sub.t] [greater than or equal to] h + 2[a.sub.h]} [2]
Maximal chip volume per second machined during longitudinal cutting can be determined on the basis of the following dependence:
[V.sub.I] = [u.sub.1][b.sub.s] (h' + [e.sub.t])[q.sub.s] [3]
where:
[u.sub.1] = feed velocity at sawing,
[b.sub.s] = thickness of the saw tooth,
[e.sub.t] = maximal positive deviation of the timber thickness, and
[q.sub.s] = number of saws in a set.
According to the EN 1313-2 standard, the [e.sub.t] value may reach 4 mm in the case of sawn timber 32 mm thick. In practice, however, due to maximization of raw material utilization, sawmills attempt to minimize the excess thickness of sawn timber. That is why the value specified in the cited standard should be rarely expected in practice. On the other hand, the increase of the discussed parameter beyond the practically expected limit would decrease the accuracy of optimization results. Additionally, considering that standards change worldwide, the unified, universal value of [e.sub.t] can hardly be specified. That is why most practitioners should determine this parameter on the basis of observations of real conditions or local requirements. For many years, sawmills in Poland were obliged to follow the requirements of currently out-of-date PN-72/D-96002 standard, according to which the [e.sub.t] value should be 1 mm. The use of such a relatively small maximal positive deviation of timber thickness is consistent with the natural tendency to use the raw material in the most rational way. According to some practitioners, however, the adoption of EU standards did not significantly change the considered property of timber in Poland. On that basis, the author assumed that the [e.sub.t] value of 1 mm is much closer to practice than that proceeding from EN 1313-2 standard and decided to use the value of 1 mm in further investigations. This value should be reconsidered when applying the model to actual conditions.
As a result of sawing, a surface whose theoretical irregularity can be determined from the dependence adopted from the literature (Staniszewska and Zakrzewski 1997) is obtained:
[R.sub.z] = [A.sub.zl] tg [chi] sin [[theta].sub.2] [4]
where:
[[DELTA].sub.zl] = feed per tooth at sawing,
[chi] = one-sided angle of swelling of saw teeth in the frontal plane, and
[[theta].sub.2] = cutting edge outlet angle from material (Fig. 1).
It was assumed that planing of the element takes place with the assistance of four planing spindles--one per each side. Maximum thicknesses of the layers planed by horizontal spindles, [d.sub.H], and vertical spindles, [d.sub.V], can be determined from the formulas:
[d.sub.H] = h' - h + [e.sub.t]/2 [5]
[d.sub.V] = b' - h/2 + [R.sub.z] [6]
The maximal chip volume per second machined by horizontal [V.sub.2H] and vertical [V.sub.2V] spindles of the four-sided planer are, therefore, described by the following dependences:
[V.sub.2H] = [u.sub.2] b [d.sub.H] [7]
[V.sub.2V] = [u.sub.2] h [d.sub.V] [8]
The theoretical surface waviness, f after the planing machine is described by the formula (Staniszewska and Zakrzewski 1997):
f = [d.sub.2]/2 (1 - cos 1/[d.sub.2]/[[DELTA].sub.z2] + [z.sub.2]/[pi]) [9]
where:
[d.sub.2] = diameter of the planning head,
[[DELTA].sub.z2] = feed per tooth at planing, and
[z.sub.2] = number of cutting edges in the planing head.
[FIGURE 1 OMITTED]
It was assumed that the selection of production means was conducted in the most rational way, whereas the developed model should support decisions at the stage of the preparation of technological routes. At that moment the general technology adopted in the factory is being particularized with regard to a specific semifinished article or element.
Keeping in mind the above considerations, it was decided that the decision variables of the optimization process would include feed velocity during longitudinal cutting and four-sided planing: [u.sub.1] and [u.sub.2]. Both decision variables were subject to direct limitations resulting from the parameters of the machine tool feed mechanism:
[U.sub.i,min] [less than or equal to] [u.sub.i] [less than or equal to] [u.sub.i,max,] i = 1, 2 [10]
The remaining limiting conditions resulted from the following factors:
* rated power of motors driving the spindles of machine tools,
* chip volume contained in the saw gullet of the circular saw, and
* allowable level of the theoretical surface irregularity after planing.
The condition requirement of not exceeding the rated power on the motor shaft has the following form:
[P.sub.ij] [less than or equal to] [P.sub.mij] [11]
where:
[P.sub.ij] = planing power on the spindle motor shaft j of the machine tool i and
[P.sub.mij] = rated power on the spindle motor shaft j of the machine tool i.
Measured power on the motor shaft during planing is determined from the formula:
[P.sub.ij] = [V.sub.ij] [k.sub.ij]/[[eta].sub.mij] [12]
where:
[V.sub.ij] = chip volume per second machined by the spindle j of the machine tool i,
[k.sub.ij] = specific work of cutting for the spindle j of the machine tool i, and
[[eta].sub.mij] = efficiency of the machine mechanical system of the spindle j of the machine tool i.
In order to determine the specific work of wood cutting, the Manshoz method was employed using its synthetic description prepared by Orlicz (1970). The method relies on the determination of the specific cutting force for basic cutting conditions and, then, correcting the result using the product of coefficients dependent on real cutting conditions. The basic formula for calculating the specific work of wood cutting using the Manshoz method has the following form:
FEED