Mathematical Model of Optimisation

Each construction is described explicitly by the parameters of shape, type of material, exchange, stress, deformation, load, functionality, colour, etc. If any of these features can be attributed to a certain value, then the whole construction can be
described by a set of n numbers, where n is a number of characteristics that describe the proposed construction. Therefore, construction can be treated as a certain point x in an N-dimensional space R. This point is a mathematical model of the con­struction, which we can write as follows:

x = (x1,X2,.. .xN),

X Є RN,

where

xn decision variables.

Construction parameters written in the form of a vector can be divided into three groups:

– determined parameters, e. g. by the designer or recipient,

– imposed parameters, e. g. by the technological requirements,

– parameters chosen by the manufacturer.

Construction parameters chosen by the manufacturer are called decision variables:

x (x1; x2; ; xn; xn+1; ; xN) ;

P = xn – xn,

where

x decision variables,

P parameters.

There can be one or many decision variables. They can be expressed by numbers or constitute functions of other variables. Depending on the type and number of decision variables, different optimisation procedures are used. Mathematical rela­tions in the model of optimisation can take the form of inequality that can be written in the structured form:

Ui(xn, P)<0 і = 1,…, k

and equations

Wj (xn, P) = 0 j – 1,…, s.

They constitute certain conditions imposed on decision variables, having the character of restrictions. Example of restrictions of inequality type are strength conditions. So there are restrictions of maximum stresses to the values not exceeding destructive stresses, or conditions limiting maximum deflection/displacements to the
values specified in the applicable normative acts. Examples of equalities are obvious. When building a mathematical model, the number of restrictions should be taken into consideration. The number of inequality restrictions can be arbitrarily large, with one reservation that this set is not empty. On the other hand, the number of equality restrictions must be smaller than the number of decision variables, since these restrictions reduce the dimensionality of the permissible area Ф created in an N-dimensional space by a set of points fulfilling the construction conditions. When searching for the optimal construction solution, such an element from the set of permissible decisions should be selected, which corresponds to this optimal solution. This choice depends on the optimisation criterion. In the literature devoted to optimisation (Brdys and Ruszczynski 1985; Dziuba 1990; Findeisen et al. 1980; Golinski 1974; Lesniak 1970; Ostwald 1987; Pogorzelski 1978; Smardzewski 1989, 1992), there are three groups of criteria. These are as follows:

– criteria concerning minimum construction costs,

– criteria of the maximum stiffness or smallest pliability,

– criteria for evening the stresses.

The criterion for minimum construction costs results from the natural human aspiration to ensure the maximum effects for minimum cost. There are two most commonly used optimisation criteria resulting in minimum volume and minimum weight. If we assume that construction costs are proportional to the volume of material, the minimum cost criterion is equivalent to the criterion of minimum volume or minimum weight for all furniture constructions. Criteria for the maxi­mum stiffness or smallest pliability concern elastic deformations of pliable systems. These criteria express the belief of the dominant importance of pliability of con­struction in the assessment of its value. It mainly concerns case furniture. According to the criterion of evening the stresses in an optimal construction, the stress is low under a specified load in all points where this is possible. Meeting the condition of evening the stresses in all directions is possible only in certain constructions, e. g. in three articulated side frames of chairs. In other constructions, meeting the condition of evening the stresses is possible only in certain areas of construction or only on certain surfaces.

Optimisation criterion is expressed as a function of purpose, which is a function of decision variables:

Q = f (xj, x2,..xn) = optimum. (6.259)

Decision variables should be selected in such a way that the function of purpose reaches the maximum value. The optimal solution is therefore a point at which the value of the function of purpose is the optimal value. This point, representing the optimal construction, should be written as follows:

(6.261)

When building a mathematical model, the manufacturer should formulate the optimisation problem in such a way that its mathematical model is adequate to the actual solution to the problem. This is particularly important in the case of the use of computers to solve optimisation tasks. They allow us to solve virtually insoluble problems through traditional calculation methods. But computer is only a machine, and when solving the intended problem, it must be presented to it in an under­standable way.

Mathematical models of construction due to the nature of parameters can be classified as follows (Golinski 1974; Ostwald 1987):

– deterministic models, where all the parameters are known and constant, i. e. every possible decision corresponds to one and only one value of the function of

purpose,

– probabilistic models, where one or more parameters are random variables with known distribution of probability,

– statistical models, where one or more parameters are random variables with unknown distribution of probability or the distribution of parameter in the function of time is known (stochastic process),

– discrete models, where one or more parameters can assume one of many pos­sible values, while the set of these possible values is in most cases known.

The construction of a mathematical model of optimisation of construction should include the following:

– determining decision variables,

– determining the permissible area (N-dimensional cube of decision variables in Euclidean space),

– determining the criterion and function of purpose.

Below, we will discuss some of the deterministic and random optimisation methods.