Computer-implemented methods for efficiently calculating an optimization problem, and control device, computer system, computer-readable memory medium and computer program product each for carrying out such methods

Abstract

The invention relates to a computer-implemented method for efficiently calculating an optimization problem using a computer system (100) which comprises a control device (110) and a plurality of calculation devices (120), each coupled to the control device (110) for data transmission, the method comprising the steps of: providing input data in a control-device memory (111) of the control device (110), the input data relating to multiple objects associated with the optimization problem; accessing the control-device memory (111) using a control-device processor (112) of the control device (110) in order to extract the following extraction data from the input data: an object number, a plurality of attribute values corresponding to a property attribute of the objects, an optimization goal corresponding to the optimization problem; and depositing the extraction data in the control-device memory (111).

Description

[0001] Computer-implemented methods for the efficient computation of an optimization problem, as well as control device, computer system, computer-readable storage medium, and computer program product, each for carrying out such methods.

[0002] The invention relates to computer-implemented methods for the efficient calculation of an optimization problem, as well as a control device, a computer system, a computer-readable storage medium and a computer program product, each for carrying out such methods.

[0003] Computer-implemented methods for solving optimization problems are, according to the state of the art, executed iteratively, i.e., in multiple stages. To solve or compute such an optimization problem, or task, for example, to calculate or perform a matrix-matrix multiplication, this task is typically divided into several subtasks or optimization subproblems. The individual optimization subproblems are then computed, in particular using multiple computing units of a computer system. For this purpose, at least one initial optimization subproblem is computed, yielding a partial solution or optimization result. Usually, at least one further optimization subproblem, for example, a second one, is then computed.The optimization subproblem is calculated using the partial result of the first optimization subproblem as the basis or input. The calculation of the next optimization subproblem thus builds upon the solution of at least one previously calculated optimization problem. Therefore, to calculate the next optimization subproblem, the first optimization subproblem must be solved and this solution provided to a unit, in particular a computing device, that performs the calculation of the next optimization subproblem. Especially when using multiple computing devices, it is necessary for these devices to communicate with each other, exchange data, and be synchronized. In other words, the computing devices must be synchronized.

[0004] P10001 - where

[0005] March 1, 2026. The synchronized exchange of data, particularly data related to at least one optimization subproblem, between computing devices, for example via GPU-to-GPU communication, is a critical efficiency factor in the calculation of optimization problems, especially matrix-matrix multiplication. This is because this synchronized exchange, for example when using multi-GPU systems, represents a bottleneck for computing power. To exchange large amounts of data when calculating an optimization subproblem, partial data must be exchanged between the computing devices via interconnected data transmission links. For example, to multiply extremely large matrices that, due to memory limitations, do not fit into the VRAM of a single GPU, submatrices (so-called submatrix matrices) must be used.Tiles) can be exchanged between individual GPUs via connections such as NVLink or PCIe.

[0006] This data exchange requires time, appropriate hardware, and / or energy. Therefore, according to the current state of the art, it is necessary to accept inefficiencies in terms of time (i.e., long computing times), hardware (i.e., expensive and / or space-consuming computer systems), and / or energy (i.e., high energy consumption) in order to solve optimization problems. It is possible to solve optimization problems more efficiently. However, this involves using approximation methods, resulting in a less accurate solution. The invention therefore aims to enable reliable and efficient, i.e., accurate, fast, cost-effective, and resource-saving, solving of optimization problems.

[0007] The problem is solved by the subject matter of the independent patent claims. Advantageous further developments of the invention result from the features of the dependent patent claims, the following description, and the figures.

[0008] A first aspect of the invention relates to a computer-implemented method for the efficient calculation of an optimization problem. Efficient in this context means that the method can be performed accurately, quickly, cost-effectively, and / or resource-efficiently. The optimization problem to be calculated can also be described as a probability problem. The term probability P10001 -wo

[0009] March 1, 2026, means that the solution corresponds to an optimum with the highest probability. In other words, the solution to the optimization problem lies with the most probable of several possible solutions. Therefore, probabilities are specifically determined during the calculation of the optimization problem. The procedure is carried out using a computer system. The computer system comprises a control unit and several computing units, each coupled to the control unit for data transmission. The procedure includes the steps discussed below.

[0010] The method comprises the step of providing input data to a control device memory, wherein the input data relates to a variety of objects associated with the optimization problem. The control device memory can comprise volatile memory, such as RAM, and / or non-volatile memory, such as flash memory / SSD. Providing the input data includes, in particular, receiving this input data and storing it in the control device memory. Preferably, the input data is in the form of structured data and / or is transformed into a data structure suitable for computation of the optimization problem. The objects are related to the optimization problem.This means that the solution to the optimization problem is a specific property, or a particular value of such a property, that connects the objects. The objects are related to each other, or rather, are related to each other during the calculation of the optimization problem. The objects related in this way form an object arrangement. These objects are also referred to as Oi, where is an object-specific, in particular integer, index.

[0011] The procedure further includes the step of accessing the control device memory using a control device processor of the control device in order to extract the following extraction data from the input data: a number of objects, several attribute values ​​corresponding to a property attribute of the objects, and a function relevant to the optimization problem.

[0012] P10001 - where

[0013] March 1, 2026, corresponding optimization goal, and storage of the extracted data in the control device memory. The control device processor is communicatively coupled to the control device memory. Access to the control device memory serves to access the input data provided in the control device memory. The number of objects corresponds to the number of objects. The number of objects is also abbreviated as Z. In particular, a corresponding attribute value is extracted for each object, so that Z attribute values ​​result, corresponding to the number of objects. Preferably, there is a one-to-one correspondence between the attribute values ​​on the one hand and the objects on the other. Alternatively, an attribute value can be assigned to several objects, for example, two objects. The optimization goal is preferably designed as a minimum, maximum, or reference value, or corresponds to one of these.In particular, the optimization goal relates to a relationship between several attribute values ​​of several objects, for example, a relationship between two attribute values ​​of two objects. Such relationships can be referred to simply as attribute value relationships (see below). The corresponding optimization goal can also be described as, or include, a single-attribute-value-relation rule. Alternatively or additionally, the optimization goal can relate to the sums of several attribute value relationships. Such an optimization goal can also be described as, or include, a total attribute value relationship rule. In summary, it is preferably intended that the optimization goal includes a single-attribute-value-relation rule and / or a total attribute value relationship rule. Storing the extraction data in the control device memory can also be referred to as saving.

[0014] Furthermore, the method comprises the step of providing a relation matrix stored in the control device memory by means of the control device processor, wherein the relation matrix comprises a plurality of relation cells, wherein at least some of the relation cells each contain an attribute value relation between the attribute values ​​of two objects corresponding to the respective relation cell. Preferably, providing the relation matrix includes its determination by means of the control device processor using the extraction data object count and their P10001 -wo

[0015] March 1, 2026. Storing or saving the data using the control device processor in the control device memory. Alternatively or additionally, providing the relation matrix includes determining the attribute value relations using the control device processor with the extracted attribute values. The relation matrix can also be referred to as an arrangement representation. Preferably, the relation cells are each assigned to a relation row by a number of relation rows corresponding to the number of objects, and to a relation column by a number of relation columns corresponding to the number of objects. In other words, the objects are preferably related to each other column-wise and row-wise by means of the relation matrix. One object is derived from the column index and the other from the row index of the respective relation cell.In particular, each relation cell stores an attribute value relation. If a relation cell relates to a column with the same index as the corresponding row, a zero is preferably stored in that relation cell. This zero corresponds to the relation of the attribute value of the object in question with itself. For example, such a relation cell with column index three and row index three relates the attribute value of the object with index three to itself.Furthermore, the procedure includes the step of providing optimization data stored in the control device memory using the control device processor. This optimization data is assigned to several representation factors, at least one of which depends on the number of objects. The optimization data comprises several sub-data sets, each corresponding to one of several optimization sub-problems of the optimization problem. By dividing the optimization problem into several optimization sub-problems, the computation of the optimization problem, i.e., the individual optimization sub-problems, can be performed more efficiently overall. The optimization data represents an assignment rule with respect to the individual optimization sub-problems. Each sub-data set corresponds to a specific, concrete value of the representation factors and thus to an optimization sub-problem.Preferably, the partial data are each represented as a vector, with each coordinate being one.

[0016] P10001 - where

[0017] 01.03.2026 such vector corresponds to a specific form of the relevant representation factor.

[0018] Furthermore, the method comprises the step of providing, by means of the control device processor, a total result matrix comprising a plurality of total result cells in the control device memory, wherein the total result cells each initially contain empty data. Preferably, the total result matrix comprises several total result rows, in particular Z total result rows, and several total result columns, in particular Z total result columns. Preferably, each total result cell is assigned to both one of the total result rows and one of the total result columns. The total result cells are initially empty. This means that an empty total result matrix is ​​provided initially, i.e., at the beginning. In the further course of the method, the total result matrix, or rather its cells, are filled (see below).

[0019] Furthermore, steps i. to iii., discussed below, are executed once for each of the partial data points. "Once" in this context means that steps i. to iii. are executed until no more partial data points remain that can be assigned to one of the computing devices (step i., see below). "Once" also specifically means that the same partial data points are assigned to one of the computing devices only once (step i., see below).

[0020] The first step, also referred to as step i, involves assigning the partial data to one of the computing devices using the control device processor, provided that at least one of the following conditions is met: no partial data is assigned to the computing device, or an optimization sub-result of an optimization sub-problem corresponding to the partial data assigned to the computing device is or is stored in the control device memory.

[0021] Preferably, different partial data are assigned in parallel to several of the computing devices. If no partial data is assigned to the computing device in question (first condition), this means, in particular, that no partial data has yet been assigned to this computing device or that such a previous assignment is no longer traceable.

[0022] P10001 - where

[0023] On March 1, 2026, partial data will be assigned to the relevant computing device for the first time. If partial data has already been assigned to a computing device and new or different partial data is to be assigned, it is preferably provided that the previous assignment is overwritten or replaced. In other words, the old partial data is then overwritten by the new partial data. Particularly with regard to the assignment sequence, it is preferably provided that the assignment of the partial data is carried out using a loop construct. Each loop corresponds to one of the representation factors, with the individual representation factors corresponding to loop variables of the individual loops, which, starting from one, are incremented by one until a maximum of the respective representation factor is reached.By appropriately arranging the loops in relation to each other or nesting them, computing time can advantageously be saved. The second condition, namely that an optimization sub-result of an optimization sub-problem corresponding to the partial data assigned to the computing device is stored in the control device's memory, means that the optimization sub-problem in question has been completely computed or solved. This frees up computing resources of the respective computing device, so that a new, i.e., not yet computed, optimization sub-problem can be solved using this computing device. The second step, also known as step ii.The term "optimization subproblem," referring to the fact that partial data has been assigned to the computing device, encompasses a calculation according to a formula that is identical for all optimization subproblems. This formula applies to the optimization subproblem corresponding to the assigned partial data, taking into account these partial data and the relation matrix, and using a processor of the respective computing device to obtain a respective optimization sub-result. The formula is the same for all optimization subproblems and therefore the same for all computing devices. In other words, each computing device calculates its assigned optimization subproblem in the same way. This identical formula can also be called a unified formula. "Unified" or "identical" refers to how a specific value of the representation factors is handled in each case.Uniform or identical P10001 -wo.

[0024] However, March 1, 2026 does not mean that only the same representation factors, i.e., partial data, are always used. Preferably, the optimization subproblem is also calculated taking into account the attribute value relations stored in the respective relation cells.

[0025] The third step, also referred to as step iii, comprises transmitting the optimization partial result to the control device via the relevant computing device processor and storing the optimization partial result in the control device memory by additively adding the optimization partial result to the respective content of the relevant overall result cell(s). The optimization partial result transmitted during the third step is the optimization partial result obtained by means of the second step. The transmission is carried out, in particular, using the computing device processor of the relevant computing device. Preferably, the relevant partial data are not also transmitted when the optimization partial result is transmitted.In other words, it is preferably not necessary for the control device to store data or information that would allow a particular optimization sub-result to be assigned to the relevant partial data. Storing the optimization sub-result in the control device's memory is preferably done using the control device's processor. An optimization sub-result can be stored in multiple overall result cells, meaning it can be distributed across or relative to multiple overall results. Adding the optimization sub-result to the respective content of the relevant overall result cell(s) means that each optimization sub-result is added to the entry of the relevant overall result cell incrementally, like a counter. For example, a relevant overall result cell has the value one before the addition.Then the values ​​are added together, and the total result cell has the value two, meaning one has been added. This allows the writing of optimization sub-results into the total result matrix to occur independently of other writing operations. Specifically, the optimization sub-result is designed to be structured within the total result matrix, like a template, without a more precise assignment to individual P10001 cells.

[0026] 01.03.2026Total result cells or without having to select the relevant total result cells, can be added up.

[0027] Finally, it is provided that, in response to the fact that no more partial data can be assigned to one of the computing devices (120), the contents of at least some of the overall result cells form an overall result. In this process, all already initiated first to third steps, i.e., steps i. to iii., are still executed to completion. Preferably, the contents of all overall result cells constitute the overall result. In other words, the fully populated overall result matrix constitutes the overall result. This fully populated overall result matrix can also be referred to as an arrangement-representation rule (see below).

[0028] The above method makes it possible to reliably and efficiently calculate an optimization problem.

[0029] The invention also includes further developments that result in additional advantages.

[0030] Preferably, at least some, and in particular each, of the computing devices have a computing device memory, wherein the partial data assigned to a computing device are stored in its computing device memory. This enables fast and collision-free parallel access to the partial data during the computation of an optimization subproblem. The storage of the relevant partial data in the respective computing device memory preferably occurs during the first step.

[0031] It is also preferred that an empty individual result matrix, structurally corresponding to the overall result matrix, is stored in several, in particular each, of the computing device memories. Such an individual result matrix is ​​then filled with the optimization partial result and can subsequently be easily added to the overall result matrix during step iii. In other words, data is advantageously stored redundantly in the respective computing device memories. This avoids the need for redundant data storage during the calculation of the respective optimization results.

[0032] P10001 - where

[0033] 01.03.2026 Sub-problems the relevant computing device processors must access the control device memory in parallel.

[0034] Furthermore, it is preferably provided that the relations matrix and / or the optimization data are stored in at least some, and in particular all, of the computing device's memories. This allows more efficient access to this data during the computation of the optimization subproblem.

[0035] Preferably, it is provided that at least some, and in particular each, of the partial data are assigned exclusively to a single computing device.

[0036] This allows for a particularly efficient process.

[0037] Preferably, at least some, and in particular all, of the computing devices are not coupled to another computing device for data transmission. In other words, there are no data connections between the computing devices. This allows for a particularly efficient, especially resource- and energy-saving, method. This is ensured in particular by the fact that each optimization subproblem, which is calculated using one of the computing devices, is designed differently from a sub-computation task known from the prior art, such as one calculated using a GPU. Particularly with regard to machine learning, for example when training AI models, a considerable amount of exchange, especially data exchange, takes place between individual computing devices according to the prior art.In accordance with the above preferred embodiment, the computing devices are advantageously not coupled to each other, so that, unlike in prior art solutions, no exchange takes place between individual computing devices.

[0038] Preferably, at least one, and in particular each, of steps i. to iii. is performed in parallel with respect to at least some, and in particular each, of the computing devices. This allows for a particularly fast method.

[0039] P10001 - where

[0040] 01.03.2026 Preferably, it is provided that an object-n-tuple number is determined from different object-n-tuples, wherein the object-n-tuples each have a tuple content number of at least two objects, wherein the objects of an object-n-tuple are each distinct from one another, wherein the object-n-tuple number corresponds to a product of a number of factors corresponding to the tuple content number, wherein a first factor corresponds to the object number, wherein further factors each correspond to a difference between the object number as minuend and an integer, starting with the value one and increasing by one from factor to factor as subtrahend, wherein in the last factor the subtrahend corresponds to the tuple content number reduced by one, wherein each object-n-tuple has a number of object positions corresponding to the tuple content number, wherein for at least some, in particular each,In the object-n tuple, each object position is assigned an object position index, where the object position index takes integer values ​​in the range from one to the tuple content count. The tuple content count can also be denoted as n. The object position index can also be denoted as j. The object-n tuples can also be referred to as (Oi, ... Ok).

[0041] Furthermore, it is preferably provided that a relation sum matrix stored in the control device memory is provided by means of the control device processor, wherein the relation sum matrix comprises a plurality of relation sum cells, wherein in at least some, in particular each, of the relation sum cells a relation sum number of attribute value relation sums corresponding to the number of object n tuples is stored, wherein the attribute value relation sums are determined for at least some, in particular each, of the object n tuples, wherein the attribute value relation sums each correspond to a sum of attribute value relations assigned to a single object n tuple.wherein the attribute value relations assigned to a single object-n-tuple are determined pairwise between a first object of the object-n-tuple and a second object of the object-n-tuple, taking into account the relations matrix and the optimization goal, wherein the attribute value relations assigned to the single object-n-tuple are determined for at least some, in particular each, of the possible pairs of first objects and second objects, each consisting of two P10001 -wo,

[0042] March 1, 2026. Objects of the object n-tuple are generated, whose object position indices are adjacent and of which the first object has a lower object position index than the second object. The relation sum matrix is ​​transmitted along with the relation matrix. Preferably, the relation sum matrix is ​​also transmitted to and / or stored in the computing device's memory. Preferably, the relation sum matrix comprises several relation sum rows, in particular Z relation rows, and several relation sum columns, in particular Z relation sum columns. Furthermore, preferably, the relation sum rows are each assigned to one of the relation sum rows and one of the relation sum columns.Preferably, the attribute value relations assigned to a single object-n-tuple are determined pairwise, taking into account the attribute value of the first object and the attribute value of the second object. The attribute value relations assigned to a single object-n-tuple are determined, in particular, by considering an optimization goal that includes a single attribute value relation rule. The relevant attribute values ​​for determining the attribute value relations are extracted, in particular, from the relations matrix.

[0043] Furthermore, it is preferably provided that at least some, in particular each, of the partial data are designed as an arrangement representation, wherein the arrangement representations are determined such that at least some, in particular each, of the arrangement representations have a number of representation positions that corresponds to the number of objects increased by one, wherein for at least some, in particular each, of the arrangement representations, an object-n-tuple position index is assigned to each representation position, wherein the object-n-tuple position index takes on integer values ​​in a range from one to the number of representation positions, wherein each representation position is assigned an object-n-tuple, and wherein the representation position assigned to a first object-n-tuple is assigned a first object-n-tuple position index.the second object-n-tuple position index assigned to a representation position associated with a second object-n-tuple is reduced by one, where, in the case of pairs of a respective first object-n-tuple and a respective second object-n-tuple that are immediately,

[0044] P10001 - where

[0045] 01.03.2026 successive representation positions are assigned, the object position(s) of the second object-n-tuple with one or more second object position index(s) are assigned the same objects that are assigned to the object position(s) of the first object-n-tuple with one or more first object position index(s), wherein the second object position index(s) is / are reduced by one compared to the first object position index(s), wherein representation positions whose object-n-tuple position index is at least two are each assigned an object-n-tuple whose attribute value relation sum, taking into account the object(s) already predetermined by the immediately preceding representation position and assigned to the object position(s) in question, best satisfies the optimization goal. The object-n-tuple position index can also be referred to as Oti for short.The optimization goal used when assigning object-n-tuples to the relevant representation positions preferably includes the overall attribute value relation rule.

[0046] Furthermore, it is preferably provided that a corresponding number of arrangement representations is determined, wherein the number of arrangement representations corresponds to a product of a maximum of the first representation factor, which is four; a maximum of the second representation factor, which corresponds to the number of objects; a maximum of the third representation factor, which corresponds to the number of objects; and a maximum of the fourth representation factor, which is the number of objects less one. The first representation factor preferably corresponds to a formation rule. The formation rules lk-ikg, lk-ekg, rk-ikg, and rk-ekg discussed below are particularly suitable for this purpose. The formation rules can also be referred to as construction rules. The second representation factor preferably corresponds to a start object, i.e., the object with which the arrangement representation begins.The third representation factor preferably corresponds to an offset image. The fourth representation factor preferably corresponds to a first tuple. Advantageously, the loop construct used for assigning the sub-data has the first representation factor as its outer loop, the second representation factor as its second loop (i.e., the first inner loop), and the third loop (i.e., P10001 -wo).

[0047] 01.03.2026 the inner loop varies the fourth representation factor and as the fourth loop, i.e. the innermost loop, the third representation factor is varied or incremented.

[0048] A second aspect of the invention relates to a computer-implemented method for the efficient computation of an optimization problem, which can be carried out in addition to or as an alternative to the method according to the first aspect of the invention. The method according to the second aspect of the invention is also carried out using the computer system described above, i.e., using a computer system comprising a control device and several computing devices, each coupled to the control device for data transmission. Using the method according to the second aspect of the invention, a primary-target-object arrangement is determined. For this purpose, optimization subproblems of an optimization problem are assigned to individual computing devices in a manner similar to the method according to the first aspect of the invention.These optimization partial results and the optimization problem relating to the second aspect of the invention are also referred to as further optimization subproblems and a further optimization problem. In particular, it is provided that partial data are made available, each corresponding to an optimization subproblem. Preferably, the following steps are also performed once with respect to each of the partial data. First, the partial data are assigned by means of the control device processor to one of the computing devices for which at least one of the following conditions is met: no partial data is assigned to the computing device, or a partial optimization result of an optimization subproblem corresponding to the partial data assigned to the computing device is or is stored in the control device memory.Then, in response to the fact that partial data has been assigned to the computing device, the optimization subproblem corresponding to the assigned partial data is calculated according to a calculation rule that is the same for all optimization subproblems. This calculation takes into account these partial data and an overall result matrix, and uses the computing device's processor to obtain a respective optimization sub-result. Furthermore, if a global condition is met, the optimization sub-result is passed to the...

[0049] P10001 - where

[0050] On March 1, 2026, the control device is transmitted via the relevant computing device processor, and the optimization partial result is stored in the control device memory. The overall result matrix can be determined using the method according to the first aspect of the invention, whereby the method is subsequently continued with features relating to the second aspect of the invention. The optimization problem that leads to the overall result matrix is ​​then a different optimization problem than the one solved with reference to the second aspect of the invention. This optimization problem relating to the second aspect of the invention is then referred to as a further optimization problem. Alternatively, an overall result matrix can be provided, i.e., the method according to the second aspect of the invention is carried out independently of features relating to the first aspect of the invention.It is specifically intended that the optimization problem previously referred to as a further optimization problem will then only be referred to as an optimization problem. However, this reference is always context-dependent, whereby, depending on the perspective, in a method with features according to the first and second aspects of the invention, both individual optimization problems may also be referred to as a single, unified optimization problem. The above statements regarding the optimization problem(s) and their relation to the first and / or second aspect of the invention apply equally to the optimization subproblems, the partial data, and the optimization data.

[0051] It is intended that, in response to the fact that no more partial data can be assigned to one of the computing devices, the optimization sub-result stored in the control device memory constitutes a global result. The global result is thus the globally best primary-target-object arrangement out of potentially several primary-target-object arrangements determined during the execution of the procedure. This global result corresponds to a primary-target-object arrangement. In particular, this primary-target-object arrangement represents a level vector that is stored in the control device memory together with a current best value. Preferably, an optimization sub-result represents a level vector to which a current best value is assigned.

[0052] P10001 - where

[0053] March 1, 2026. Further features relating to the primary target object arrangement are described in particular below in connection with the fourth aspect of the invention. Preferably, the method further comprises the steps discussed below. The method preferably further comprises the step of providing, by means of the control device processor, a threshold matrix comprising a plurality of threshold cells in the control device memory, wherein the threshold cells each have an initially empty content. From the input data, a threshold is extracted as extraction data. The threshold is related to a frequency or a probability. In particular, the threshold lies in a range that extends at least between a smallest and a largest probability of occurrence.Preferably, the threshold matrix is ​​n-dimensional. For example, the threshold matrix for object-2 tuples is 2-dimensional. Furthermore, for example, the threshold matrix for object-3 tuples is 3-dimensional.

[0054] In response to the fact that the overall result has been formed, or using an overall result as described above, the following steps are also preferably carried out with respect to at least some, in particular each, of the overall result cells whose respective contents form the overall result.

[0055] A preferred subsequent step is to compare the content of the overall result cell with the threshold, which can also be referred to as the frequency limit (see below). A further subsequent step, in response to the fact that the content of the overall result cell exceeds the threshold, is to set the content of the threshold cell corresponding to the overall result cell to a positive value. A pair consisting of a relevant overall result cell and a relevant threshold cell corresponding to the relevant overall result cell, in particular, has the same column and row index with respect to the overall result matrix and threshold matrix, respectively. For this purpose, it is specifically provided that the threshold matrix has threshold rows, in particular as many as there are overall result rows, and that the threshold matrix has threshold columns P10001 -wo

[0056] The cell must show as many as the total result columns are provided for on March 1, 2026. It is assumed that the content of the respective total result cell corresponds to the frequency of an order representation rule. Therefore, if the aforementioned content indicates that a particular sub-result occurs frequently enough, a positive value results for the respective total result cell. The value one is used as an example of a positive value.

[0057] Preferably, in response to the fact that all contents of the relevant overall result cells have been compared with the threshold, the primary target object arrangement is determined using the threshold matrix and further optimization data, wherein the further optimization data comprises several additional sub-data, each corresponding to one of several further optimization sub-problems of a further optimization problem. The threshold matrix is, in particular, at least partially, and preferably completely, filled.

[0058] Preferably, it is provided that, in response to the fact that the content of the overall result cell does not exceed the threshold, the content of the threshold cell corresponding to the overall result cell is set to a negative value. The value zero is used as an example of a negative value.

[0059] Preferably, a search parameter vector is provided by the control device processor and stored in the control device memory. Advantageously, the search parameter vector comprises one coordinate for each of the aforementioned levels, in particular as many coordinates as there are objects, i.e., Z coordinates. Furthermore, each coordinate of the search parameter vector advantageously comprises several level parameters, in particular a maximum of Z - 1 level parameters. The level parameters can together form, in particular, a segment list.

[0060] As an example, the level parameters are passed to a relevant computing device. For instance, if a level with index 3, which can also be called the third level, is to be calculated, the following are passed to the relevant computing device: the object-n-tuple to be fixed for a first level, and the object-n-tuple to be fixed for a second level.

[0061] This preferably results in which, namely in particular all possible, object- P10001 -where

[0062] On March 1, 2026, n-tuples for the relevant level, for example the third level, will be tried.

[0063] In particular, it is provided that, based on a number of level parameters passed to the relevant computing device, it is determined at which level, i.e., at which level index, the process should continue. Preferably, the process continues at the level whose index corresponds to the number of passed level parameters + 1.

[0064] It is further preferably provided that the control device transmits to the computing devices the coordinate of the search parameter vector corresponding to the level currently being calculated. It is also preferably provided that a current segment list is transmitted, containing level parameters up to (but not for) the level from which the level parameter is to be further varied. In particular, the current segment list is an arrangement representation of object-n tuples.

[0065] Even more preferably, at least some, and in particular all, of the computing devices have a level vector of dimension 1 x Z. The level vectors are initially empty, i.e., initially filled with zeros. In particular, the level value of each level, starting with level 1, is stored in the level vector. For example, the level vector for level 2 is defined as follows: Level vector = (1, 2, 000000).

[0066] Even more preferably, each computing device has a level vector of dimension 1 x Z. The level vectors are also initially empty, meaning they are initially filled with zeros. In particular, for each level, starting with level 1, which is fixed, the level parameter value is stored in the corresponding level vector. The level parameter value corresponds to the order of each level. For example, for level 1: level parameter value = 1, and for level 2: level parameter value = 2. The level vector comprises the coordinates (Oi, Oj, Ok, ... ), where i + j + k. An example level vector has the following form: level vector = (03, 01, Oz, ...).

[0067] Preferably, the determination or selection of the coordinates or segments of the respective level vector includes, in at least some,

[0068] P10001 - where

[0069] 01.03.2026 especially anyone who meets at least some of the following test conditions:

[0070] The first test condition concerns the connection condition of a segment from the preceding level with an object-n-tuple of the current level that is to be checked. The first test condition specifically concerns an overlap (see below).

[0071] A second check condition concerns object assignment. Specifically, it is checked whether the last object in the object-n-tuple of the current level has not yet been assigned. This excludes objects in different object-n-tuples of the level vector that are identical due to the connection condition (see first check step) – such objects may therefore already be assigned in other segments.

[0072] A third test condition concerns a threshold condition. Here, the content of a threshold cell corresponding to the segment of the preceding level, as well as, if applicable, further contents of threshold cells corresponding to segments of further preceding levels, is compared with the content of a threshold cell corresponding to the object-n-tuple of the current level being tested, and it is checked whether at least one threshold condition is met.

[0073] Preferably, it is provided that, in response to a level vector being completely filled (for example, level vector = (1, 2, 3, Z - 1, Z)), a value of the associated level vector is evaluated. The completely filled level vector represents a possible primary target object arrangement, which is potentially the best current arrangement, and potentially even the best global arrangement (the global result, see above). This evaluation is performed specifically with respect to a length and / or an optimization value of the level vector in question.It is further preferably provided that, in response to the fact that the current value is better than a previously determined value (also referred to as the current best value), the previously determined value is overwritten with the current, i.e., better, value, particularly using the computing device (this then forms a new current best value), and, in particular, this better value is further transmitted to the control device. More preferably, the...

[0074] P10001 - where

[0075] 01.03.2026 Control device in response to receiving a better value, transmits this better value to at least some, in particular each, of the other computing devices.

[0076] It is further preferred that a previous arrangement representation of a computing device be overwritten with a current arrangement representation if the current arrangement representation has a previously best value.

[0077] Preferably, the content of a threshold cell corresponding to the object-n-tuple of the current level and the content of a threshold cell corresponding to the segment in the preceding level, and optionally in at least one further preceding level, jointly satisfy at least one of the following linking conditions. The linking conditions can also be referred to as threshold conditions. It may be provided that the linking conditions or threshold conditions are satisfied cumulatively or alternatively, at least with respect to some, and in particular each, of the linking conditions. By way of example, linking conditions for the contents of threshold cells corresponding to two consecutive coordinates of the level vector are described below.This refers to a tuple (relevant content of a threshold cell with reference to the previous level, relevant content of a threshold cell with reference to the current level):.

[0078] A first example linking condition is: (1, 1) (0, 1) (1, 0) -> yes, if (0, 0) discard the solution path from the current level, increase the number of discarded (incomplete) solution paths by (Z - a - 1)1. Continue in the same level if other segments are still present, otherwise jump to the next level.

[0079] A second example linking condition is: (1 , 1 ) -> yes, if (1 , 0) -> discard the solution path from the current level, increase the number of discarded (incomplete) solution paths by (Z - a - 1)1. Continue in the same level if other segments are still present, otherwise jump to the next level.

[0080] A third example linking condition is: (0, 0) yes if (0, 1) -> discard solution path from current level, number of discarded (incomplete) P10001 -wo

[0081] March 1, 2026: Increase the solution paths by (Z - a - 1)!. Continue in the same level if other segments are present; otherwise, skip to the next level.

[0082] According to another example, the following describes linking conditions for the contents of threshold cells that correspond to three consecutive coordinates of the level vector. This refers to a tuple (cell value of the level two levels prior, cell value of the previous level, cell value of the current level):

[0083] A fourth example linking condition is: (1, 1, 1) -> yes, if (1, 1, 0) -> discard the solution path from the current level, increase the number of discarded (incomplete) solution paths by (Z - a - 1)!. Continue in the same level if other segments are still present, otherwise jump to the next level.

[0084] A fifth example linking condition is: (0, 0, 0) yes, if (0, 0, 1) -> discard the solution path from the current level, increase the number of discarded (incomplete) solution paths by (Z - a - 1)!. Continue in the same level if other segments are still present, otherwise jump to the next level.

[0085] A sixth example linking condition is: (0, 1, 1) or (1, 0, 1) -> yes, if (0, 1, 0) or (1, 0, 0) -> discard the solution path from the current level, increase the number of discarded (incomplete) solution paths by (Z - a - 1)!. Continue in the same level if other segments are still present, otherwise jump to the next level.

[0086] The following discusses an exemplary method, comprising preferred steps, for determining an arrangement-representation rule according to a third aspect of the invention. It is preferably provided that this third aspect of the invention is combined with the first aspect of the invention discussed above and / or the second aspect of the invention discussed above, particularly also with regard to respective sub-aspects.

[0087] Regarding the third aspect of the invention, methods for determining an arrangement-representation rule in numerical space are known. So far, the solutions from a rule for finding a minimum lie within a factor of 3 / 2 of the minimum solution set. No method currently exists for determining an arrangement-representation rule that allows for the search

[0088] P10001 - where

[0089] 01.03.2026 either of an optimization goal, i.e. a minimum, a maximum or a reference value in numerical space, and which simultaneously enables an exact search for the solution.

[0090] The arrangement-representation rule preferably corresponds to a fully populated overall result matrix (see above regarding the first and / or second aspect of the invention). The method for determining an arrangement-representation rule is carried out taking into account a plurality of arrangement representations to be determined, which depend on an object arrangement corresponding to an arrangement of a plurality of object number (Z) of objects (Oi) and on attribute value relations between attribute values ​​of at least one property attribute to be selected or selected and assigned to the objects of the object arrangement, wherein each object (Oi) is assigned a specific object index (i), wherein the object index (i) takes on integer values ​​in a range from 1 to the number of objects (Z), and wherein several object n-tuples (Oi, ... Ok) are formed from several different objects (Oi).The procedure for determining an order representation rule comprises the following steps.

[0091] a) Specification of an individual attribute value relation rule with respect to the attribute value relations and an overall attribute value relation rule with respect to the attribute value relation sums,

[0092] b) Determining a respective attribute value assigned to the property attribute for, in particular, each, object,

[0093] c) Determining an object-n-tuple number from different object-n-tuples (Oi, ... Ok), wherein the object-n-tuples (Oi, ... Ok) each have a tuple content number (n) of at least two objects, wherein the objects of an object-n-tuple (Oi, ... Ok) are each distinct from one another, wherein the object-n-tuple number corresponds to a product of a number of factors corresponding to the tuple content number (n), wherein a first factor corresponds to the object number (Z), wherein further factors each correspond to a difference between the object number (Z) as the minuend and an integer, starting with the value one and increasing by one from factor to factor as the subtrahend, wherein in the last factor the subtrahend is the tuple content number (n) decreased by one.

[0094] 01.03.2026 corresponds, wherein each object-n-tuple (Oi, ... Ok) has a number of object positions corresponding to the number of tuple contents (n), wherein for, in particular, each, object-n-tuple (Oi, ... Ok) an object position index (j) is assigned to each object position, wherein the object position index (j) takes on integer values ​​in a range from one to the number of tuple contents (n),

[0095] d) Determining a number of relation sums of attribute value relation sums corresponding to the number of object n-tuples for, in particular, each object n-tuple (Oi, ... Ok), wherein the attribute value relation sums each correspond to a sum of the attribute value relations assigned to a single object n-tuple (Oi, ... Ok), wherein the attribute value relations assigned to a single object n-tuple (Oi, ... Ok) are determined pairwise between a first object (Oi) of the object n-tuple (Oi, ... Ok) and a second object (Oi) of the object n-tuple (Oi, ... Ok) taking into account the first attribute value assigned to the first object (Oi) and the second attribute value assigned to the second object (Oi) as well as the single attribute value relation rule, wherein the individual object n-tuple (Oi, ...The attribute value relations assigned to Ok are determined for all possible pairs of first objects and second objects that result from each of two objects of the object-n-tuple (Oi, ... Ok) whose object position indices (j) are adjacent and of which the first object (Oi) has a lower object position index (j) than the second object (Oi).

[0096] e) Determining a number of arrangement representations corresponding to a plurality of arrangement representations, wherein the number of arrangement representations corresponds to a product of a maximum of the first representation factor, which is four, a maximum of the second representation factor, which corresponds to the number of objects (Z), a maximum of the third representation factor, which corresponds to the number of objects (Z), and a maximum of the fourth representation factor, which is the number of objects (Z) less than one, wherein each arrangement representation has a number of representation positions, which corresponds to the number of objects (Z) increased by 1, wherein, in particular, each arrangement representation is assigned an object-n-tuple position index (Oti) to each representation position, wherein the object-n-tuple position index (Oti) is an integer

[0097] P10001 - where

[0098] 01.03.2026 Values ​​in a range from one to the number of representation positions, where each representation position is assigned an object-n-tuple (Oi, ... Ok), where the representation position assigned to the first object-n-tuple (Oi, ... Ok) is assigned a first object-n-tuple position index (Oti), which corresponds to the second object-n-tuple position index (Oti) assigned to the representation position assigned to the second object-n-tuple (Oi, ... Ok), reduced by one, where, in the case of pairs of a respective first object-n-tuple (Oi, ... Ok) and a respective second object-n-tuple (Oi, ... Ok) that are assigned to immediately consecutive representation positions, the object position or the object positions of the second object-n-tuple (Oi, ...Ok) with a second object position index (j) or with second object position indices (j) are assigned the same objects that are assigned to the object position or object positions of the first object-n-tuple (Oi, ... Ok) with a first object position index (j) or with first object position indices (j), wherein the second object position index (j) is / are reduced by one compared to the first object position index (j), and the second object position indices (j) are each reduced by one compared to the respective first object position indices (j), wherein representation positions whose object-n-tuple position index (Oti) is at least two each have one object-n-tuple (Oi, ...OK) is assigned, whose attribute value relation sum, taking into account the object(s) already predetermined by the immediately preceding representation position and assigned to the object position(s) in question, most closely satisfies the overall attribute value relation rule, and.

[0099] f) Determining an arrangement-representation rule, taking into account the arrangement representations, such that a frequency of occurrence of similar object n-tuples (Oi, ... Ok) in the arrangement representations is determined, and the attribute value relation sums assigned to the respective similar object n-tuples (Oi, ... Ok) are assigned to this frequency. The object arrangement contains objects from i = 1 to Z. Each object (Oi) is assigned a specific object index (i). The object index (i) is uniquely assigned to an object. In particular, the objects are each uniquely assigned to a

[0100] P10001 - where

[0101] The object index (i) pertaining to 01.03.2026 is assigned. Therefore, a one-to-one relationship preferably exists between the object indices (i) and the objects (Oi). The object index (i) takes integer values ​​in the range from 1 to the number of objects (Z). The object indices (i) with the values ​​i = 1 and i = Z also exist. The object position index (j) takes integer values ​​in the range from one to the number of tuple contents (n). The object position index (j) with the values ​​j = 1 and j = n also exist. Preferably, each object position index (j) is uniquely assigned to one object index (i). The object n-tuple position index (Oti) takes integer values ​​in the range from one to the number of representation positions. The object n-tuple position index (Oti) is also available, with the values ​​Oti = 1 and Oti = number of representation positions.

[0102] Several object-n-tuples (Oi, ... Ok) are formed from multiple different objects (Oi). Here, i of object (Oi) is not equal to k of object (Ok). Preferably, the single-attribute-value-relation rule states that the relevant attribute-value relations should have a minimum, a maximum, or a reference value and / or should approximate it as closely as possible. Furthermore, preferably, the overall attribute-value-relation rule states that the relevant attribute-value-relation sums should have a minimum, a maximum, or a reference value and / or should approximate it as closely as possible.

[0103] The number of object n-tuples corresponds to the maximum number of possible distinct object n-tuples (Oi, ... Ok). The number of object n-tuples is calculated according to the following formula, where Z is the number of objects (Z) and n is the number of tuple contents (n):

[0104]

[0105] Each individual object-n-tuple (Oi, ... Ok) is associated with a number of attribute value relations, where this number corresponds to the tuple content count (n) minus one. For an object-2-tuple, where the tuple content count (n) is two, the aforementioned number is therefore one, and for an object-3-tuple, where the tuple content count (n) is three, the aforementioned number is therefore two.

[0106] P10001 - where

[0107] 01.03.2026The first object (Oi) of the object-n-tuple (Oi, ... Ok) has an object position index j and the second object (Oi) of the object-n-tuple (Oi, ... Ok) has an object position index j+1.

[0108] The maximum of the first representation factor corresponds to the number of different design rules to be observed for the individual arrangement representations. Preferably, the maximum of the first representation factor is four. Corresponding design rules can be, for example: constructed from left to right and measured in the direction of construction (lk-ikg or Ik-ik or Ik-ig or left-ikg or Ik-Ig), constructed from left to right and measured against the direction of construction (lk-ekg or Ik-ek or Ik-eg or left-ekg or Ik-rg), constructed from right to left and measured in the direction of construction (rk-ikg or rk-ik or rk-ig or right-ikg or rk-lg), constructed from right to left and measured against the direction of construction (rk-ekg or rk-ek or rk-eg or right-ekg or rk-rg).

[0109] According to the construction rule from left to right, the object-n-tuples (Oi, ... Ok) of an arrangement representation are determined such that an object-n-tuple (Oi, ... Ok) following in the sense of the construction has an object-n-tuple position index (Oti) that is one higher than the object-n-tuple position index (Oti) of the object-n-tuple (Oi, ... Ok) immediately preceding the following object-n-tuple (Oi, ... Ok).

[0110] According to the construction rule from right to left, the object-n-tuples (Oi, ... Ok) of an arrangement representation are determined such that an object-n-tuple (Oi, ... Ok) following a subsequent object-n-tuple (Oi, ... Ok) in the sense of the construction has an object-n-tuple position index (Oti) that is one less than the object-n-tuple position index (Oti) of the object-n-tuple (Oi, ... Ok) immediately preceding the subsequent object-n-tuple (Oi, ... Ok).

[0111] According to the construction rule, constructed from left to right and measured in the direction of construction (lk-ikg) as well as constructed from right to left and measured against the direction of construction (rk-ekg), the attribute value relation sums are determined such that the respective first object (Oi) P10001 -wo

[0112] The object position index (j) assigned on 01.03.2026 is reduced by one compared to the object position index (j) assigned to the respective second object (Oi).

[0113] According to the construction rule, constructed from left to right and measured against the direction of construction (lk-ekg) as well as constructed from right to left and measured in the direction of construction (rk-ikg), the attribute value relation sums are determined in such a way that the object position index (j) assigned to the respective first object (Oi) is increased by one compared to the object position index (j) assigned to the respective second object (Oi).

[0114] The second representation factor addresses the number of objects (Z).

[0115] Preferably, each array representation includes a breakline. The breakline is the position between two immediately adjacent representation positions of an array representation, each of which is assigned the same object.

[0116] The third representation factor addresses the number of possible positions of a break edge in the various arrangement representations, which preferably corresponds to the number of objects (Z). Preferably, the object (Oi) to the left of the break edge corresponds to the object (Oi) to the right of the break edge.

[0117] The maximum of the fourth representation factor addresses the number of possible second objects (Oi) in the object arrangement that remain after selecting or defining the first object (Oi). Therefore, the maximum of the fourth representation factor corresponds to the number of objects (Z) minus one.

[0118] Representation positions whose object-n-tuple position index (Oti) is at least two are each assigned an object-n-tuple (Oi, ... Ok) whose attribute-value-relation sum, taking into account the object(s) already predetermined by the immediately preceding representation position and assigned to the object position(s) in question, most closely satisfies the overall attribute-value-relation rule. The condition that the attribute-value-relation sum in question most closely satisfies the overall attribute-value-relation rule means that the attribute-value-relation sum in question comes closest to the overall attribute-value-relation rule.

[0119] P10001 - where

[0120] 01.03.2026, for example, when it becomes minimal, maximum, or closest to a reference value.

[0121] The features described below are preferred, as also described below.

[0122] Object: The thing toward which the action is directed. Examples of an object include astronomical objects (planetary systems, star clusters, nebulae, galaxies, etc.), vehicles (cars, trucks, etc.), machines and equipment (milling machine, printing press, packaging machine, ladder, scaffolding, concrete mixer, etc.), sheets of paper (photographic paper, baking paper, colored paper, etc.), drops (water drops, pitch drops, etc.), bits in a binary string, atoms (carbon atom, hydrogen atom, oxygen atom, etc.).

[0123] Object Index: The object index of an object indicates what kind of object it is. For example, the natural numbers from 1 to Z can be used as object indices. Examples of the object index of an astronomical object are: object index of planetary system A is 1, object index of planetary system B is 2, object index of star cluster A is 3. Examples of the object index of a vehicle are: object index of passenger car A is 1, object index of passenger car B is 2, object index of truck A is 3. Examples of the object index of machines and equipment are: object index of milling machine is 1, object index of printing machine is 2, object index of packaging machine is 3. Examples of the object index of sheets of paper are: object index of photographic paper A is 1, object index of photographic paper B is 2, object index of baking paper A is 3, object index of colored paper is 4.Examples of the object index of drops are: object index of water droplet A is 1, object index of water droplet B is 2, object index of pitch droplet A is 3. Examples of the object index of bits in a string are: object index of the first bit in the string is 1, object index of the second bit in the string is 2, object index of the third bit in the string is 3. Examples of the object index of atoms in a molecule are: object index of carbon atom A is 1, object index of carbon atom B is 2, object index of hydrogen atom A is 3, object index of oxygen atom A is 4.

[0124] P10001 - where

[0125] March 1, 2026

[0126]

[0127] Objects that stand in a specific relationship to one another and form a system with a specific effect. In an object arrangement, it is known which objects stand in a specific relationship to one another. The label of an object indicates which object it is. An example of an object arrangement is outer space with its astronomical objects, which are arranged in a specific temporal and spatial relationship to one another and are characterized by their mass density and energy density. Another example is a vehicle fleet with vehicles that belong to an organizational unit, are of a specific design, and are used in specific areas. Another example is a machine park with machines and equipment whose temporal state, location, utilization, acquisition and current value, and maintenance and repair costs are known.Another example is a color swatch book with paper sheets used to demonstrate colors as a technical feature of a product. The sheets are arranged according to a color system, and their surface (matte, glossy, coated, or printed), density, and area determine the effect of the swatch book. Another example is a cloud of water droplets, whose spatial and temporal distribution, velocity, size, and condensation nucleus determine the cloud's effect. Another example is a string of bits, whose Boolean value (TRUE, FALSE) determines the string's effect. Another example is a molecule with atoms, whose species, covalent bonds via shared electron pairs, and temporal and spatial arrangement determine the molecule's effect.

[0128] Number of

[0129]

[0130] one

[0131]

[0132] : The number of objects (Oj) in an object arrangement is denoted by (Z) (j=1, ..., Z).

[0133] Property attributes: Property attributes are essential properties of the objects in an object arrangement. Examples of property attributes for astronomical objects include their coordinates in spacetime, their mass density, and their energy density. Another example is the property attributes of vehicles in a fleet: affiliation with which organizational unit, design, and area of ​​operation. Another example is the property attributes of machines and equipment in a machine park: temporal state, location, utilization, acquisition cost, and...

[0134] P10001 - where

[0135] March 1, 2026: Current value, maintenance and repair costs. Another example is the property attributes of the paper sheets in a color swatch: color, surface, density, and area. Another example is the property attributes of water droplets in a cloud: spatial and temporal distribution within the cloud, velocity, size, and type of condensation nucleus. Another example is the property attributes of a string of bits: the arrangement of the bits in the string and the Boolean values ​​of the bits (TRUE, FALSE). Another example is the property attributes of the atoms in a molecule: covalent bonding via shared electron pairs and their temporal and spatial arrangement within the molecule.

[0136] Attribute values: An attribute value describes the specific value within a predefined range for a given property attribute of an object and is a numerical value. If the property associated with an object's attribute is described with a numerical value, then the attribute value describing the given property attribute of the object is determined from the numerical value of the property associated with the object's attribute. If the property associated with an object's attribute is described with a Boolean value, then the numerical value associated with the Boolean value is first determined using a Boolean-numerical system. For example, the Boolean value "FALSE" is assigned the numerical value 0, and the Boolean value "TRUE" is assigned the numerical value 1.The attribute value, which describes the specified property attribute of an object, is then determined from the numerical value obtained using the Boolean numerical system from the Boolean value of the property assigned to the property attribute of an object.

[0137] If the property associated with an object's attribute is described using a string literal, the numerical value assigned to the string literal is first determined using a string literal numeric system. For example, the string literal "ABCD" is assigned the numeric value 1, and the string literal "EFGH" is assigned the numeric value 2. Then, the attribute value that describes the given property attribute of an object is extracted from the string literal using the string literal numeric system.

[0138] P10001 - where

[0139] 01.03.2026 The numerical value determined for the property attribute of an object is assigned to that property.

[0140] An attribute value depends on its range of values ​​and is numeric for all objects in an object array. For a given property attribute, the attribute value lies within the same predefined range of values ​​for all objects in an object array. Examples of attribute values ​​for astronomical objects include: the numerical coordinates in four-dimensional spacetime, the numerical values ​​of the mass density, and the numerical values ​​of the energy density.The attribute values ​​of the property attributes of the vehicles in a vehicle fleet are, for example, the numerical values ​​determined using the string literal-numeric system from the string literal of belonging to an organizational unit, the numerical values ​​determined using the string literal-numeric system from the string literal of the design type, and the numerical values ​​determined using the string literal-numeric system from the string literal of the areas of application.Examples of attribute values ​​for the properties of machines and equipment in a machine park include the numerical values ​​determined from the string literal representing the current state using a string literal-numerical system, the numerical values ​​of the location (GPS data), the numerical value of the utilization (in percent), the numerical value of the purchase price, the numerical value of the current value, the numerical value of the maintenance costs, and the numerical value of the repair costs. Examples of attribute values ​​for the properties of the paper sheets in a color fan deck include the numerical values ​​of the color system assignment determined from the string literal representing the color using a string literal-numerical system, the numerical values ​​determined from the string literal describing the surface area using a string literal-numerical system, the numerical value of the density, and the numerical value of the area.The attribute values ​​of the property attributes of the water droplets in a cloud are, for example, the numerical values ​​in four-dimensional spacetime, the numerical value of the velocity, the numerical value of the size, and the numerical values ​​determined from the string literals of the type of condensation nucleus using the string literal numeric system. The attribute values ​​of P10001 -wo.

[0141] March 1, 2026. Examples of property attributes of a string containing bits include the numerical values ​​of the bit's position in the string and the numerical values ​​derived from the bit's Boolean value (TRUE, FALSE) using the Boolean numerical system. Examples of attribute values ​​of atoms in a molecule include the numerical values ​​of the strength of the covalent bond via shared electron pairs and the numerical values ​​of four-dimensional spacetime.

[0142] Object position index in an object-n-tuple: n object position indices in an object-n-tuple in the range of values ​​from j = 1 to j = n, For each object-n-tuple the assignment between object position index and object index is unique.

[0143]

[0144] Two different objects 0 and Oj, the first object (Oi) with object index i and the second object (Oj) with object index j (i ≠ j), where the first object has object position index 1 and the second object has object position index 2.

[0145]

[0146] Three different objects Oi, Oj and Ok, the object Oi with object index i, the object Oj with object index j and the object Ok with object index k (i ≠ j ≠ k), where the first object has object position index 1, the second object has object position index 2 and the third object has object position index 3.

[0147]

[0148] n distinct objects Oj, ...Ok, the object Oi with the label i, the object Oj with the label j, and the object Ok with the label k (i ≠ j ≠ k). where the first object has the object position index 1, the second object the object position index 2, and the nth object the object position index n.

[0149]

[0150] -Number: Number of object n-tuples with at least one different object at at least one object position index ((Z * (Z - 1) *... * (Z - (n - 1)).

[0151]

[0152] Representational position occupied by an object-n-tuple. The object-n-tuple position index can take integer values ​​in the range of one to the number of representational positions.

[0153] The break edge object of the arrangement representation to the left of the break edge and to the right of the break edge is the same.

[0154] P10001 - where

[0155] 01.03.2026 Parameters for constructing an array representation: Position of the break edge, object to the left and right of the break edge, array representations for a given position of the break edge and a defined object to the left and right of the break edge, and construction rule.

[0156] Maximum representation factor for the breakline position: Number of ways to define the breakline position (Z)

[0157] Maximum representation factor for the object to the left and right of the break edge: number of possible objects to the left and right of the break edge (Z).

[0158] Maximum of the representation factor for the number of different arrangement representations with a fixed position of the break edge and a defined object to the left and right of the break edge: With a fixed position of the break edge and a defined object to the left and right of the break edge, the number of different arrangement representations is (Z - 1).

[0159] Maximum representation factor for construction rule: With a fixed position of the break edge, with a defined object to the left and right of the break edge, and with a defined arrangement representation, the number of possible construction rules is 4: Construction rule constructed from the left and measured in the construction direction, constructed from the left and measured against the construction direction, constructed from the right and measured in the construction direction, and constructed from the right and measured against the construction direction.

[0160] Arrangement Representation Number: Multiplicity of corresponding arrangement representations, where the arrangement representation number is a product of the maximum of the first representation factor for the position of the break edge (Z), the maximum of the representation factor for the object to the left and right of the break edge (Z), the maximum of the representation factor for the number of different arrangement representations when the position of the break edge is fixed and when the object to the left and right of the break edge is fixed (Z - 1), and the maximum of the representation factor for the construction rule.

[0161] Relationships between property attributes of all object-2-tuples of an object array: The property attributes for all objects of an object array are determined via the result of a numerical operation with the attribute values ​​) each

[0162] P10001 - where

[0163] On March 1, 2026, two objects Oi and Oj were related to each other. The result of a numerical operation with two numeric input values ​​is a numeric output value and describes the relationship between the property attributes of the objects Oi and Oj of the object-2 tuple.

[0164] Examples of relationships between the property attributes of two astronomical objects include: the difference in vector length in four-dimensional spacetime, the difference in mass density, and the difference in energy density.

[0165] Examples of relationships between the property attributes of two vehicles in a vehicle fleet include: the difference in numerical values ​​assigned to the organizational unit (organizational unit A ahead of organizational unit B), the difference in numerical values ​​assigned to the design type (design type C ahead of design type D), and the difference in numerical values ​​assigned to the areas of application (construction industry ahead of agriculture).

[0166] Examples of relationships between the property attributes of two machines and devices in a machine park include: the difference in numerical values ​​assigned to their temporal state (ready for use versus in need of maintenance), the GPS data of their location, the difference in utilization (10%), the difference in purchase price (100 euros), the difference in current value (90 euros), the difference in maintenance costs (200 euros), and the difference in repair costs (500 euros). Examples of relationships between the property attributes of two sheets of paper in a color fan deck include: the difference in numerical values ​​assigned to the colors in the color system, the difference in numerical values ​​assigned to the surface description ("smooth" versus "rough"), the difference in density, and the difference in area.

[0167] Examples of relationships between the property attributes of two water droplets in a cloud include: the difference in their vector in spacetime, the difference in their velocity, the difference in their size, and the difference in the numerical values ​​assigned to the type of condensation nucleus ("aerosol" versus "soot particle"). Examples of relationships between the property attributes of two bits in a string include the difference in the positions of the two bits in the string and the

[0168] P10001 - where

[0169] 01.03.2026 Absolute value of the difference of the numerical values ​​assigned to the boolean values.

[0170] The relationships between the property attributes of two atoms in a molecule are exemplified by the difference in the strength of the atomic bond via shared electron pairs and the difference of the two vectors in four-dimensional spacetime.

[0171] The method is applicable to objects of different dimensions in various N-dimensional spaces. An example is rectangular, colored paper sheets from a color fan deck in a color catalog. The color (corresponding to sensory perception) of the paper sheets is measured with a colorimeter and systematically ordered in the RAL Design color system according to hue (H), lightness (L), and chroma (C). The width and height of the paper sheets are measured with a ruler with an accuracy of 0.5 mm. The thickness of the paper sheets is measured with a micrometer screw gauge with an accuracy of 0.01 mm. The weight is measured with a paper scale with a resolution of 0.01 g.

[0172] Another example is astronomical objects in space. Another example is vehicles in a fleet. Another example is machines in a machine hall. Another example is droplets in a cloud. Another example is the atoms of a molecule.

[0173] An example of determining the property attribute "position" of an object relative to the Earth's surface, e.g., of near-Earth objects, is trigonometric parallax measurement using high-powered telescopes. An example of determining the property attribute "position" of an object relative to a reference point on the Earth's surface, e.g., of vehicles in a fleet, is the Global Positioning System (GPS). Here, the position of an object is determined with an accuracy of up to 1 meter. An example of determining the property attribute "position" of an object relative to the position of a laser scanner, e.g., of machines in a machine hall or of droplets in a cloud, is laser scanning. Here, the object's position is determined with an accuracy of up to 25 pm. An example of determining the property attribute "position" of an object, e.g., of atoms in a molecule in

[0174] P10001 - where

[0175] 01.03.2026With reference to a reference point molecule, electron beam microscopy with a resolution of 0.045 nm is used.

[0176] An example of the mechanical determination of an object's velocity attribute relative to its position, e.g., a vehicle on the Earth's surface, is the speedometer with a deviation of 1%. An example of the optical determination of an object's velocity attribute, e.g., droplets in a cloud, is optical position filter measurement technology, laser Doppler anemometry, or particle image velocimetry.

[0177] An example of determining the property attribute "force" between objects, such as the gravitational force between astronomical objects in space, is measuring the gravitational field above each of the astronomical objects using two satellites and a microwave rangefinder. An example of determining the property attribute "force" between objects, such as the intermolecular forces between droplets in a cloud, is measuring surface tension, capillary forces, and adhesive and cohesive forces. An example of determining the property attribute "force" between objects, such as the intermolecular forces between atoms of a molecule, is measuring electrostatic forces due to hydrogen bonds, ionic bonds, dipole-dipole bonds, van der Waals forces, and hydrophobic interactions.

[0178] When modeling the extension of an object arrangement by adding further objects, including the property attributes assigned to these additional objects and the relationships between identical property attributes between the additional objects and the objects already present before the extension, the problem often arises that existing object arrangements must first be captured in a computer model for the purpose of describing the extension. This is particularly relevant when adding an astronomical object to the astronomical objects in space, adding a droplet to the droplets in a cloud, or adding an atom to the atoms of a molecule. This effort is immense and, with an increasing number of objects in the arrangement, cannot be solved in polynomial time.

[0179] P10001 - where

[0180] March 1, 2026. When planning extensions to an object arrangement with additional objects, including the property attributes assigned to these additional objects and the relationships between identical property attributes between the additional objects and the objects already existing before the extension, the problem often arises that existing object arrangements, for the purpose of describing and planning the extension (e.g., when adding a vehicle to the vehicle fleet, adding machines to an engine room, or adding a rectangular, colored sheet of paper to a color swatch book, or similar), must first be captured in a computer model.

[0181] Currently, capturing data in a computer model is typically a time-consuming manual process, involving the manual creation of a model of the property attributes for each object in the object arrangement. Experience shows that the N-dimensional representation of an object arrangement changes significantly when the number of objects in the arrangement changes significantly, and / or when the property attributes of an object change, and / or when the relationships between identical property attributes of different objects change. This leads to exceptionally long planning times and considerably increases the costs associated with planning for changes in the number of objects in the object arrangement.

[0182] The following describes an exemplary use of a method relating to the third aspect of the invention.

[0183] Determining the relationships between property attributes of all object pairs in a

[0184] Determining an attribute value for each object in the object array relative to every other object in the object array based on the different property attributes of each object and relationships between identical property attributes of different objects.

[0185] In such a method for creating an N-dimensional representation of an object arrangement, in particular of Z objects with different property attributes and relationships between identical property attributes of different objects, especially of color charts in a color catalog, P10001 -wo

[0186] On March 1, 2026, each property attribute or each property attribute formed from several property attributes was first assigned a dimension j of the N-dimensional representation.

[0187] Property attributes of different objects that are assigned to the same dimension of the N-dimensional representation are called identical property attributes of different objects.

[0188] Property sub-attributes (EUA) of a given object, which are assigned to the same dimension of the N-dimensional representation, can be combined into one property attribute (EA) of the given object.

[0189] We combine the property attributes hue (H), lightness (L), and chroma (C) of the color charts into the property attribute hue. Similarly, we combine the property attributes width, height, and thickness of the color charts into the property attribute volume.

[0190] Each color card corresponds to an object (Oj). The property attributes of the color card are its color value, volume, and weight. The attribute values ​​assigned to the property attributes (color value, volume, and weight) are given in Table 1 as examples for 5 color cards (Z=5). The attribute values ​​assigned to each property attribute were determined for each property attribute (color value, volume, weight) for each color card and are listed in Table 1.

[0191] Table 1: Attribute values ​​of 5 color cards, which are assigned to the property attributes color value, volume and weight.

[0192]

[0193] P10001 - where

[0194] March 1, 2026. Then, for an object-2 tuple, the assigned attribute value relation for each property attribute is determined, taking into account the first attribute value assigned to the first object (Oi) and the second attribute value assigned to the second object (Oi), as well as the single attribute value relation rule. The single attribute value relation rule for the property attribute "color value" is recorded as the difference between the first attribute value assigned to the first object (Oi) and the second attribute value assigned to the second object (Oi) for the color value. Table 2 shows the assigned attribute value relation for the property attribute "color value" of object-2 tuples.

[0195] Table 2: Single-attribute value relation rule for the property attribute color value of object-2 tuples (Oi, Oj) formed from a set of 5 color cards.

[0196]

[0197] The single-attribute value relation rule for the property attribute "volume" is defined as the difference between the first attribute value assigned to the first object (Oi) and the second attribute value assigned to the second object (Oi) for volume. Table 3 shows the assigned attribute value relation for the property attribute "volume" of object-2 tuples.

[0198] Table 3: Single-attribute value relation rule for the property attribute volume of object-2 tuples (Oi, Oj) formed from a set of 5 color cards.

[0199]

[0200] P10001 - where

[0201] March 1, 2026: The single-attribute value relation rule for the property attribute "weight" is recorded as the difference between the first attribute value assigned to the first object (Oi) and the second attribute value assigned to the second object (Oi) for weight. Table 4 shows the assigned attribute value relation for the property attribute "weight" of object-2 tuples.

[0202] Table 4: Single-attribute value relation rule for the property attribute weight of object-2 tuples (Oi, Oj) formed from a set of 5 color cards.

[0203]

[0204] Then, using an object-n-tuple as an example, here using an object-3-tuple as an example, the corresponding number of relation sums of attribute value relation sums is determined for, in particular, each object-3-tuple (Oi, ... Ok), where the attribute value relation sums each correspond to a sum of the attribute value relations assigned to a single object-3-tuple (Oi, ... Ok), where the attribute value relations assigned to a single object-3-tuple (Oi, ... Ok) are determined pairwise between a first object (Oi) of the object-3-tuple (Oi, ... Ok) and a second object (Oi) of the object-3-tuple (Oi, ... Ok), taking into account the first attribute value assigned to the first object (Oi) and the second attribute value assigned to the second object (Oi), as well as the single attribute value relation rule, where the individual object-3-tuple (Oi, ...The attribute value relations assigned to object-3 tuple (Oi, Oj, Ok) are determined for all possible pairs of first and second objects that result from any two objects of the object-3 tuple (Oi, ... Ok) whose object position indices (j) are adjacent and of which the first object (Oi) has a lower object position index (j) than the second object (Oi). The attribute value relation sums of object-3 tuples (Oi, Oj, Ok) are listed in Table 5 for object-3 tuple (Oi, Oj, O1) and in Table 6 for...

[0205] P10001 - where

[0206] 01.03.2026 Object-3-tuple (0i,0j,02), shown in Table 7 for Object-3-tuple (Oi, Oj, O3), in Table 8 for Object-3-tuple (Oi, Oj,O4) and in Table 9 for Object-3-tuple (Oi, Oj,O5). Table 5: Attribute value relation sums, each assigned to a single Object-3-tuple (Oi, Oj, Ok) with 0k=01 of the color charts.

[0207]

[0208] Table 6: Attribute value relation sums are each assigned to a sum of the attribute value relations assigned to a single object-3 tuple (Oi, Oj, Ok) with Ok=O2 of the color cards.

[0209]

[0210] Table 7: Attribute value relation sums are each assigned to a sum of the attribute value relations assigned to a single object-3 tuple (Oi, Oj, Ok) with Ok=O3 of the color cards.

[0211]

[0212] P10001 - where

[0213] 01.03.2026 Table 8: Attribute value relation sums are each assigned to a sum of the attribute value relations assigned to a single object-3 tuple (Oi, Oj, Ok) with 0k=04 of the color cards.

[0214]

[0215] Table 9: Attribute value relation sums are each assigned to a sum of the attribute value relations assigned to a single object-3 tuple (Oi, Oj, Ok) with Ok=O5 of the color cards.

[0216]

[0217] Determining different arrangement representations for object-n-tuples

[0218] The determination of different arrangement representations for object-n-tuples is carried out as described in connection with Figures 1 to 14.

[0219] Determining an arrangement-representation rule for object-n-tuples

[0220] Determining an arrangement-representation rule for object-n-tuples is carried out by way of example as described in connection with Figs. 15 to 24.

[0221] Determining a target-object arrangement

[0222] From the arrangement-representation rules for object-2 tuples of 5 color cards, the target object arrangement of the 5 color cards is finally determined. The target object arrangement for the 5 color cards with a minimum sum of the attribute value relations for the color value is color card with object index 1 P10001 -wo

[0223] March 1, 2026 (letter A), color chart with object index 2 (letter B), color chart with object index 4 (letter D), color chart with object index 3 (letter C), and color chart with object index 5 (letter E). For the target object arrangement (A, B, D, C, E), the sum of the attribute value relations of the color values ​​is minimal and amounts to 152. The number of procedural steps required to determine the target object arrangement depends on the arrangement representation rule. The greater the frequency of the object n-tuples that are part of the desired target object arrangement compared to the frequency of the object n-tuples that are not part of the desired target object arrangement, the fewer steps are needed to determine the target arrangement. To a first approximation, the number of procedural steps scales with the time required to determine the target arrangement.For example, if the target arrangement is determined using a computer, the number of procedural steps for determining the target arrangement from the arrangement representation rule for object-2 tuples is 72 times smaller than the number of procedural steps for determining the target arrangement from the arrangement representation rule for object-3 tuples.The times in Table 10 are also compared as an example. These times were combined to determine the target arrangement from the arrangement-representation rule for all four design rules and were used for the four separate design rules: constructed from left to right and measured in the direction of construction (lk-ikg), constructed from left to right and measured against the direction of construction (lk-ekg), constructed from right to left and measured in the direction of construction (rk-ikg), and constructed from right to left and measured against the direction of construction (rk-ekg).

[0224] P10001 - where

[0225] 01.03.2026 Table 10: Time in s required to determine the target arrangement of 5 color cards from the arrangement-representation rule for various construction rules. In addition to the four construction rules described above, Table 10 presents a combination of these four construction rules. This combination corresponds, in particular, to an addition of the frequencies of the four construction rules (Ik-Ig, Ik-rg, rk-Ig, rk-rg) for each given object-n-tuple.

[0226]

[0227] Determining a target object arrangement of object-n-tuples, wherein the target object arrangement comprises a number of object-n-tuples corresponding to the number of objects (Z), where a minimum total sum corresponding to a sum of respective attribute value relations between the respective first object (Oi) of each object-n-tuple (Oi, ... Ok) and the respective second object (Oi) of the respective object-n-tuple (Oi, ... Ok) is minimal.

[0228] Preferably, several tuple content counts (n) are taken into account, whereby an arrangement-representation rule is determined for each of the several tuple content counts (n).

[0229] Preferably, the method for determining an arrangement representation rule includes the following additional steps:

[0230] i) Determining a respective Maximai object frequency as the highest frequency of the respective frequencies of occurrence of similar object-n-tuples (Oi, ... Ok) of the respective arrangement-representation rule, wherein a respective proportion of frequencies of occurrence of similar object-n-tuples (Oi, ... Ok) that do not fall below a respective frequency threshold dependent on the respective Maximai object frequency is determined, wherein the frequency threshold is preferably at least 0.6 times the Maximai object frequency, further preferably P10001 -wo

[0231] 01.03.2026 at least 0.7 times the Maximai object frequency, more preferably at least 0.8 times the Maximai object frequency, in particular at least 0.9 times the Maximai object frequency,

[0232] ii) Determining a respective maximum attribute value relation sum as the largest attribute value relation sum of the respective attribute value relation sums of the respective arrangement-representation rule, wherein a respective proportion of attribute value relation sums that do not exceed a respective maximum attribute value relation sum limit dependent on the respective maximum attribute value relation sum is determined, wherein the maximum attribute value relation sum limit is preferably at most 0.95 times the maximum attribute value relation sum, further preferably at most 0.9 times the maximum attribute value relation sum, more preferably at most 0.8 times the maximum attribute value relation sum, and even more preferably at most 0.7 times the maximum attribute value relation sum. at most 0.6 times the maximum attribute value relation sum, in particular at most 0.5 times the maximum attribute value relation sum, iii) selection of a selection arrangement representation rule from the arrangement representation rules such that the selection arrangement representation rule has a selection number of similar object n-tuples (Oi, ... Ok) that form a set of selection object n-tuples, and where at the same time their frequency of occurrence of similar object n-tuples (Oi, ... Ok) is greater than the frequency limit and their attribute value relation sum is less than the maximum attribute value relation sum limit, wherein the selection number of the selection arrangement representation rule is closest to the object number (Z) compared with the other arrangement representation rule(s),

[0233] iv) Determining a starting object-n-tuple from the selection object-n-tuples whose frequency of occurrence of similar object-n-tuples (Oi, ... Ok) is closest to the frequency limit,

[0234] v) Defining a selection / abort criterion taking into account a maximum attribute value relation of all possible pairs of first objects (Oi) of the object arrangement and second objects (Oi) of the object arrangement, and

[0235] P10001 - where

[0236] 01.03.2026vi) Determining a target object arrangement of object-n-tuples, wherein the target object arrangement comprises a number of object-n-tuples corresponding to the number of objects (Z), where a minimum total sum corresponding to a sum of respective attribute value relations between the respective first object (Oi) of each object-n-tuple (Oi, ... Ok) and the respective second object (Oi) of the respective object-n-tuple (Oi, ... Ok) is minimal.

[0237] The frequency limit can also be referred to as the threshold.

[0238] Preferably, the starting object-n-tuple is selected from the selection object-n-tuples whose frequency of occurrence of similar object-n-tuples (Oi, ... Ok) is closest to, and preferably corresponds to, the frequency limit.

[0239] Preferably, when determining a target object arrangement of object n-tuples, at least one, in particular several, and especially preferably all, of the following criteria are applied.

[0240] The first criterion is: If the attribute value relation sum is greater than or equal to the attribute value relation sum of Z-1 object-2 tuples, discard this object-n tuple (Oi, ... Ok) and restart the first criterion if not all first object-n tuples (Oi, ... Ok) from the multitude of object-n tuples (Oi, ... Ok) with the highest frequency and one of the attribute value relation sums that most closely satisfy the overall attribute value relation rule have yet been used.

[0241] The second criterion is: Determining another, second object-n-tuple (Oi, ... Ok) from the multitude of object-n-tuples (Oi, ... Ok) with the highest frequency and an attribute-value-relation sum that most closely satisfies the overall attribute-value-relation rule, where the n-1 objects with the object-position indices j=1 to j=n-1 of the first object-n-tuple (Oi, ... Ok) correspond to the n-1 objects of the second object-n-tuple (Oi, ... Ok) with the object-position indices j=2 to j=n.

[0242] The third criterion is: Determine the total attribute value relation subsumption of the corresponding different object-2 tuples contained in the first object-n tuple (Oi, ... Ok) and in the second object-n tuple (Oi, ... Ok) and compare it with the attribute value relation sums of Z-1 object-2 tuples.

[0243] P10001 - where

[0244] 01.03.2026The fourth criterion is: If the total attribute value relation subtotal is greater than or equal to the attribute value relation sum of Z-1 object-2 tuples, discard this second object-n-tuple (Oi, ... Ok) and return to c)ll, if not all possible second object-n-tuples (Oi, ... Ok) from the multitude of object-n-tuples (Oi, ... Ok) with the highest frequency and one of the attribute value relation sums that most closely satisfy the total attribute value relation rule have already been used.

[0245] The fifth criterion is: If the total attribute value relation subsumption is less than the attribute value relation sum of Z-1 object-2 tuples, determine another, third object-n tuple (Oi, ... Ok) from the multitude of object-n tuples (Oi, ... Ok) with the highest frequency and one of the attribute value relation sums that most closely satisfies the total attribute value relation rule, where the n-1 objects with the object position indices j=1 to j=n-1 of the second object-n tuple (Oi, ... Ok) correspond to the n-1 objects of the third object-n tuple (Oi, ... Ok) with the object position indices j=2 to j=n.

[0246] The sixth criterion is: Determine the total attribute value relation subsumption of the corresponding different object-2 tuples contained in the first object-n tuple (Oi, ... Ok), the second object-n tuple (Oi, ... Ok), and the third object-n tuple (Oi, ... Ok), and compare it with the attribute value relation sums of Z-1 object-2 tuples.

[0247] Preferably, after determining the starting object-n-tuple for at least one, in particular several, especially preferably all, of the following object-n-tuples, at least one, in particular several, especially preferably all, of the following criteria are applied.

[0248] The seventh criterion is: continuation of the procedure until the (Z-n+1) object-n-tuple (Oi, ... Ok) is reached.

[0249] The eighth criterion is: If the total attribute value relation subsumption of the corresponding different object-2 tuples contained in the first object-n tuple (Oi, ... Ok), second object-n tuple (Oi, ... Ok), third object-n tuple (Oi, ... Ok) up to (Z - n + 1 )th object-n tuple (Oi, ... Ok) is smaller than the attribute value relation sums of Z - 1 object-2 tuples, replace the attribute value relation sum of Z - 1 object-2 tuples with the total attribute value relation subsumption of P10001 -wo

[0250] 01.03.2026 corresponding to the different object-2-tuples contained in the first object-n-tuple (Oi, ... Ok), second object-n-tuple (Oi, ... Ok), third object-n-tuple (Oi, ... Ok) to (Z - (n - 1))-th object-n-tuple (Oi, ... Ok).

[0251] The ninth criterion is: Return to the corresponding point in the loop if not all object-n-tuples (Oi, ... Ok) have yet been used as (Z - (n - 1 ))th object-n-tuples (Oi, ... Ok).

[0252] The tenth criterion is: Return to the corresponding point in the loop if not all object-n-tuples (Oi, ... Ok) have yet been used as (Z - (n - 0))th object-n-tuples (Oi, ... Ok).

[0253] The eleventh criterion is: Return to the corresponding point in the loop if not all object-n-tuples (Oi, ... Ok) have yet been used as (Z - (n + 1 ))-th object-n-tuples (Oi, ... Ok).

[0254] The twelfth criterion is: Return to the corresponding point in the loop if not all object-n-tuples (Oi, ... Ok) have yet been used as (Z - (n + Z - n - 3))th object-n-tuples (Oi, ... Ok).

[0255] The thirteenth criterion is: Return to the seventh criterion if not all object-n-tuples (Oi, ... Ok) have yet been used as the second (Z - (n + Z - n - 2)) object-n-tuple (Oi, ... Ok).

[0256] The fourteenth criterion is: Return to the first criterion if not all object-n-tuples (Oi, ... Ok) have yet been used as the first (Z - (n + Z - n - 1 ))-th object-n-tuple (Oi, ... Ok).

[0257] The fifteenth criterion is: Determine the Z - 1 object-2 tuple, the last determined attribute value relation sum.

[0258] The sixteenth criterion is: Determine an arrangement of (Z) objects (Oi) corresponding to the object arrangement, where the attribute value relation sum of the Z - 1 object-2 tuples contained in the object arrangement most closely satisfies the overall attribute value relation rule, from the attribute value relation sum last determined from the Z - 1 object-2 tuples.

[0259] The following is an exemplary procedure, comprising preferred steps, for determining a primary target object arrangement according to a fourth

[0260] P10001 - where

[0261] March 1, 2026. Aspect of the invention discussed. It is preferably provided that this fourth aspect of the invention is combined with the first aspect of the invention discussed above, the second aspect of the invention discussed above and / or the third aspect of the invention discussed above, particularly also with regard to respective sub-aspects.

[0262] Regarding the fourth aspect of the invention, computer-implemented methods for precisely determining a target object arrangement in numerical space, which enable the search for either an optimization goal (i.e., a minimum, a maximum, or a reference value of the attribute values) with polynomial time expenditure (O(ZP) ... p is a natural number, Z is the number of objects (Z)), are not yet known. The time required for precisely determining the target object arrangement in numerical space currently scales with O(ZI). The implementation of a corresponding optimization problem in a computer model is generally carried out today in a time-consuming manual process, i.e., by manually creating a model of the property attributes for each object in the object arrangement.Experience shows that the N-dimensional representation of an object array changes completely when the number of objects (Z) changes and / or when the property attributes of an object and / or when the attribute value relationships between different objects change. This leads to particularly long planning times and significantly increases the costs of planning for changes in the number of objects in the object array.

[0263] The procedure for determining a primary target object arrangement is carried out taking into account a selection arrangement representation rule chosen from a multitude of arrangement representation rules, wherein the arrangement representation rules each depend on a multitude of attribute value relation sums, on an object arrangement corresponding to an arrangement of a multitude of object number (Z) of objects (Oi), and on attribute value relations between attribute values ​​of at least one property attribute to be selected or selected and assigned to the objects of the object arrangement, wherein each object (Oi) is assigned a specific object index (i), wherein the object index (i) takes integer values ​​in a range from 1 to the object number (Z), and wherein each of the following is derived from

[0264] P10001 - where

[0265] On March 1, 2026, several different objects (Oi) are used to form multiple object-n-tuples (Oi, ... Ok), where the arrangement-representation rules include a multitude of frequencies of occurrence of similar object-n-tuples (Oi, ... Ok) and the attribute-value relation sums assigned to these frequencies. The procedure for determining a primary-target object arrangement comprises the steps discussed below.

[0266] i) Determining a respective Maximai object frequency as the highest frequency of the respective frequencies of occurrence of similar object-n-tuples (Oi, ... Ok) of the respective arrangement-representation rule, wherein a respective proportion of frequencies of occurrence of similar object-n-tuples (Oi, ... Ok) that do not fall below a respective frequency threshold dependent on the respective Maximai object frequency is determined, wherein the frequency threshold is preferably at least 0.6 times the Maximai object frequency, further preferably at least 0.7 times the Maximai object frequency, more preferably at least 0.8 times the Maximai object frequency, and in particular at least 0.9 times the Maximai object frequency.

[0267] ii) Determining a respective maximum attribute value relation sum as the largest attribute value relation sum of the respective attribute value relation sums of the respective arrangement-representation rule, wherein a respective proportion of attribute value relation sums that do not exceed a respective maximum attribute value relation sum limit dependent on the respective maximum attribute value relation sum is determined, wherein the maximum attribute value relation sum limit is preferably at most 0.95 times the maximum attribute value relation sum, further preferably at most 0.9 times the maximum attribute value relation sum, more preferably at most 0.8 times the maximum attribute value relation sum, and even more preferably at most 0.7 times the maximum attribute value relation sum. at most 0.6 times the maximum attribute value relation sum, in particular at most 0.5 times the maximum attribute value relation sum, iii) selection of a selection-arrangement-representation rule from the arrangement-representation rules such that the selection-arrangement-representation rule represents a selection number of similar object n-tuples (Oi, ...,

[0268] P10001 - where

[0269] 01.03.2026Ok) exhibits a set of selection object n-tuples, and where simultaneously the frequency of occurrence of similar object n-tuples (Oi, ... Ok) is greater than the frequency limit and the attribute value relation sum is less than the maximum attribute value relation sum limit, where the selection number of the selection arrangement representation rule of the object number (Z) is closest to the other arrangement representation rule(s),

[0270] iv) Determining a starting object-n-tuple from the selection object-n-tuples whose frequency of occurrence of similar object-n-tuples (Oi, ... Ok) is closest to the frequency limit,

[0271] v) Defining a selection / abort criterion taking into account a maximum attribute value relation of all possible pairs of first objects (Oi) of the object arrangement and second objects (Oi) of the object arrangement,

[0272] vi) Determining a target object array of object-n-tuples, wherein the target object array comprises a number of object-n-tuples equal to the number of objects (Z), wherein a minimum total sum, corresponding to a sum of respective attribute value relations between the respective first object (Oi) of each object-n-tuple (Oi, ... Ok) and the respective second object (Oi) of the respective object-n-tuple (Oi, ... Ok), is minimal, wherein the target object array has a number of target object array positions equal to the number of objects (Z), wherein each target object array position is assigned a target position index (Ozi), wherein the target position index (Ozi) takes integer values ​​in a range from one to the number of objects (Z), and

[0273] vii) Determining a primary target object array having a number of primary target object array positions corresponding to the number of objects (Z), wherein each primary target object array position is assigned a primary target position index (Opzi), the primary target position index (Opzi) taking integer values ​​in a range from one to the number of objects (Z), wherein the primary target object array is determined such that for each object-n tuple of the target object array, the object having an object position index (j) of the value one is placed at the primary target object array position whose primary-

[0274] P10001 - where

[0275] 01.03.2026 Target Position Index (Opzi) corresponds to the Target Position Index (Ozi) of the relevant object n-tuple (Oi, ... Ok) of the target object arrangement.

[0276] The object arrangement contains objects from i = 1 to Z. Each object (Oi) is assigned a specific object index (i). The object index (i) is uniquely assigned to each object. In particular, each object is uniquely assigned to a specific object index (i). Therefore, there is preferably a one-to-one relationship between the object indices (i) and the objects (Oi). The object index (i) takes integer values ​​in the range from 1 to the number of objects (Z). The object indices (i) with the values ​​i = 1 and i = Z are also present.

[0277] The first object (Oi) of the object-n-tuple (Oi, ... Ok) has an object position index j and the second object (Oi) of the object-n-tuple (Oi, ... Ok) has an object position index j+1.

[0278] Preferably, the starting object-n-tuple is selected from the selection object-n-tuples whose frequency of occurrence of similar object-n-tuples (Oi, ... Ok) is closest to, and preferably corresponds to, the frequency limit.

[0279] Preferably, when determining a target object arrangement of object n-tuples, at least one, in particular several, and especially preferably all, of the following criteria are applied.

[0280] The first criterion is: If the attribute value relation sum is greater than or equal to the attribute value relation sum of Z-1 object-2 tuples, discard this object-n tuple (Oi, ... Ok) and restart the first criterion if not all first object-n tuples (Oi, ... Ok) from the multitude of object-n tuples (Oi, ... Ok) with the highest frequency and one of the attribute value relation sums that most closely satisfy the overall attribute value relation rule have yet been used.

[0281] The second criterion is: Determining another, second object-n-tuple (Oi, ... Ok) from the multitude of object-n-tuples (Oi, ... Ok) with the highest frequency and an attribute-value-relation sum that most closely satisfies the overall attribute-value-relation rule, where the n-1 objects with the object position indices j=1 to j=n-1 of the first object-n-tuple (Oi, ... Ok) are compared with the n-1 objects of P10001 -wo

[0282] 01.03.2026 second object n-tuple (Oi, ... Ok) with the object position indices j=2 to j=n.

[0283] The third criterion is: Determine the total attribute value relation subsumption of the corresponding different object-2 tuples contained in the first object-n tuple (Oi, ... Ok) and in the second object-n tuple (Oi, ... Ok) and compare it with the attribute value relation sums of Z-1 object-2 tuples.

[0284] The fourth criterion is: If the total attribute value relation subtotal is greater than or equal to the attribute value relation sum of Z-1 object-2 tuples, discard this second object-n-tuple (Oi, ... Ok) and return to c)ll, unless all possible second object-n-tuples (Oi, ... Ok) from the multitude of object-n-tuples (Oi, ... Ok) with the highest frequency and one of the attribute value relation sums that most closely satisfy the total attribute value relation rule have already been used.

[0285] The fifth criterion is: If the total attribute value relation subsumption is less than the attribute value relation sum of Z-1 object-2 tuples, determine another, third object-n tuple (Oi, ... Ok) from the multitude of object-n tuples (Oi, ... Ok) with the highest frequency and one of the attribute value relation sums that most closely satisfies the total attribute value relation rule, where the n-1 objects with the object position indices j=1 to j=n-1 of the second object-n tuple (Oi, ... Ok) correspond to the n-1 objects of the third object-n tuple (Oi, ... Ok) with the object position indices j=2 to j=n.

[0286] The sixth criterion is: Determine the total attribute value relation subsumption of the corresponding different object-2 tuples contained in the first object-n tuple (Oi, ... Ok), the second object-n tuple (Oi, ... Ok), and the third object-n tuple (Oi, ... Ok), and compare it with the attribute value relation sums of Z-1 object-2 tuples.

[0287] Preferably, after determining the starting object-n-tuple for at least one, in particular several, especially preferably all, of the following object-n-tuples, at least one, in particular several, especially preferably all, of the following criteria are applied.

[0288] P10001 - where

[0289] 01.03.2026The seventh criterion is: Continuation of the procedure until the (Z-n+1) object-n-tuple (Oi, ... Ok).

[0290] The eighth criterion is: If the total attribute value relation subsumption of the corresponding distinct object-2 tuples contained in the first object-n tuple (Oi, ... Ok), second object-n tuple (Oi, ... Ok), third object-n tuple (Oi, ... Ok) up to the (Z-n+1)th object-n tuple (Oi, ... Ok) is less than the attribute value relation sums of Z-1 object-2 tuples, replace the attribute value relation sums of Z-1 object-2 tuples with the total attribute value relation subsumption of the corresponding distinct object-2 tuples contained in the first object-n tuple (Oi, ... Ok), second object-n tuple (Oi, ... Ok), third object-n tuple (Oi, ... Ok) up to the (Z-(n-1 ))th object-n tuple (Oi, ... Ok).

[0291] The ninth criterion is: Return to the corresponding point in the loop if not all object-n-tuples (Oi, ... Ok) have yet been used as (Z-(n-1 ))th object-n-tuples (Oi, ... Ok).

[0292] The tenth criterion is: Return to the corresponding point in the loop if not all object-n-tuples (Oi, ... Ok) have yet been used as (Z-(n-0))th object-n-tuples (Oi, ... Ok).

[0293] The eleventh criterion is: Return to the corresponding point in the loop if not all object-n-tuples (Oi, ... Ok) have yet been used as (Z-(n+1))th object-n-tuples (Oi, ... Ok).

[0294] The twelfth criterion is: Return to the corresponding point in the loop if not all object-n-tuples (Oi, ... Ok) have yet been used as (Z-(n+Zn-3))th object-n-tuples (Oi, ... Ok).

[0295] The thirteenth criterion is: Return to the seventh criterion if not all object-n-tuples (Oi, ... Ok) have yet been used as the second (Z-(n+Zn-2)) object-n-tuple (Oi, ... Ok).

[0296] The fourteenth criterion is: Return to the first criterion if not all object-n-tuples (Oi, ... Ok) have yet been used as the first (Z-(n+Zn-1 ))th object-n-tuple (Oi, ... Ok).

[0297] The fifteenth criterion is: Determining the Z-1 object-2 tuple, the last determined attribute value relation sum.

[0298] P10001 - where

[0299] 01.03.2026The sixteenth criterion is: Determining an arrangement of (Z) objects (Oi) corresponding object arrangement, whereby the attribute value relation sum of the Z-1 object-2 tuples contained in the object arrangement most closely satisfies the overall attribute value relation rule, from the attribute value relation sum last determined from the Z-1 object-2 tuples.

[0300] Features relating to the fourth aspect of the invention are defined in the same manner as above with reference to the third aspect of the invention. An exemplary use of a method relating to the fourth aspect of the invention is described below.

[0301] Determining the relationships between property attributes of all object pairs in an object array

[0302] Determining an attribute value for each object in the object array relative to every other object in the object array based on the different property attributes of each object and relationships between identical property attributes of different objects.

[0303] In such a computer-implemented method for determining the target object arrangement of Z objects, especially objects with different property attributes and relationships between identical property attributes of different objects, particularly the property attribute "location," the attribute value "location coordinate" is assigned. As an example, we use a set of objects at locations (Z=13) that are arranged symmetrically. We denote this set of objects as SymmZ13. And we use another example: a set of objects at locations (Z=13) that are arranged asymmetrically. We denote this set of objects as AsymmZ13.Then, for an object-2 tuple of the object set SymmZ13 and the object set AsymmZ13, the assigned attribute value relation for the property attribute location is determined taking into account the first attribute value assigned to the first object (Oi) and the second attribute value assigned to the second object (Oi) as well as the single attribute value relation rule.

[0304] The single-attribute value relation rule for the property attribute "location" is recorded as the difference between the first attribute value P10001 -wo assigned to the first object (Oi).

[0305] 01.03.2026 and the second attribute value for location assigned to the second object (Oi). Table 11 shows the assigned attribute value relation for the property attribute "Location" of object-2 tuples from the object set SymmZ13. Table 11: Single attribute value relation rule for the property attribute "Location" of object-2 tuples (Oi, Oj) formed from the object set SymmZ13 of 13 locations.

[0306]

[0307] The single-attribute value relation rule for the property attribute "location" is captured as the difference between the first attribute value assigned to the first object (Oi) and the second attribute value assigned to the second object (Oi) for the location. Table 12 shows the assigned attribute value relation for the property attribute "location" of object-2 tuples of the object set AsymmZ13.

[0308] P10001 - where

[0309] 01.03.2026 Table 12: Single-attribute value relation rule for the property attribute location of object-2 tuples ((Oi, Oj) formed from the object set AsymmZI 3 of 13 locations.

[0310]

[0311] Determining Different Representation Arrangements for Object-n-Tuples The following section determines different arrangement representations for the object sets SymmZ13 and AsymmZI3. The arrangement representation number is the product of a maximum of the first representation factor, which is four; a maximum of the second representation factor, which is equal to the number of objects (Z); a maximum of the third representation factor, which is equal to the number of objects (Z); and a maximum of the fourth representation factor, which is equal to the number of objects (Z) minus one. Each arrangement representation of object-n-tuples is associated with an object-n-tuple position index (Oti), where the object-n-tuple position index (Oti) takes integer values ​​in a range from one to the number of representation positions.

[0312] The maximum of the first representation factor corresponds to the number of different design regulations to be observed for the individual arrangement- P10001 -wo

[0313] March 1, 2026. Representations. The maximum of the first representation factor is four. Corresponding construction rules could be, for example: constructed from left to right and measured in the direction of construction (lk-ikg), constructed from left to right and measured against the direction of construction (lk-ekg), constructed from right to left and measured in the direction of construction (rk-ikg), constructed from right to left and measured against the direction of construction (rk-ekg).

[0314] The maximum of the second representation factor corresponds to the number of object n-tuples, where for the second representation factor one of the Z objects (Oj) is arranged to the left and right of the break edge of the arrangement representation.

[0315] The maximum of the third representation factor corresponds to the number of possible positions of a break edge in the different arrangement representations. The different positions of the break edges are denoted by offset = 0, offset = 1, offset = 2, offset = 3, offset = 4, ... , offset = 12.

[0316] Determining an arrangement-representation rule for object-n-tuples

[0317] Determining an arrangement-representation rule for object-n-tuples is carried out by way of example as described in connection with Figs. 1 to 20.

[0318] Determining a target-object arrangement

[0319] From the arrangement-representation rules for object-2 tuples of the object sets SymmZ13 and AsymmZ13, the target object arrangement of each of the 13 locations is finally determined. The target object arrangement for the 13 locations with a minimal sum of the attribute value relations for distance is, for the object set SymmZ13: A, C, I, B, M, E, F, H, K, L, J, G, D. In the target object arrangement (A, C, I, B, M, E, F, H, K, L, J, G, D), the sum of the attribute value relations of the locations is minimal and amounts to 138.

[0320] And the target object arrangement for the 13 locations with a minimal sum of attribute value relations for the distance is, for the object set AsymmZ13: A,B,C,G,D,E,K,M,J,L,I,F,H. With the target object arrangement (A, B, C, G, D, E, K, M, J, L, I, F, H), the sum of the attribute value relations of the locations is minimal and amounts to 200.

[0321] P10001 - where

[0322] March 1, 2026. The number of procedural steps required to determine the target object arrangement depends on the arrangement representation rule. The greater the frequency of object n-tuples that are part of the desired target object arrangement compared to the frequency of object n-tuples that are not part of the desired target object arrangement, the fewer steps are needed to determine the target arrangement. To a first approximation, the number of procedural steps scales with the time required to determine the target arrangement.

[0323] For example, if the target arrangement is determined using computer-aided methods, the number of procedural steps for determining the target arrangement from the arrangement representation rule for object-2 tuples is 72 times smaller than the number of procedural steps for determining the target arrangement from the arrangement representation rule for object-3 tuples. Table 10 also compares the times used to determine the target arrangement from the arrangement representation rule for all four design rules combined, as well as the times used for the four separate design rules: constructed from left to right and measured in the direction of construction (Ik-Ig), constructed from left to right and measured against the direction of construction (Ik-rg), constructed from right to left and measured against the direction of construction (rk-rg), and constructed from right to left and measured in the direction of construction (rk-lg).

[0324] Table 13: Time in s, which was determined to determine the target arrangement of the objects of the object set SymmZ13: A,C,I,B,M,E,F,H,K,L,J,G,D from the arrangement-representation rule for various construction rules.

[0325]

[0326] Determining the exact target arrangement using computer-implemented prior art methods scales with O(N!) and takes 21863 s. Determining the exact target arrangement using the method underlying the invention is in

[0327] P10001 - where

[0328] 01.03.2026 polynomial time < 3 s for the arrangement-representation rule for object-2 tuples and <200 s for the arrangement-representation rule for object-3 tuples.

[0329] For example, if the target arrangement is determined using computer-aided methods, the number of procedural steps for determining the target arrangement from the arrangement representation rule for object-2 tuples is 72 times smaller than the number of procedural steps for determining the target arrangement from the arrangement representation rule for object-3 tuples. Table 10 also compares the times used to determine the target arrangement from the arrangement representation rule for all four design rules combined, as well as the times used for the four separate design rules: constructed from left to right and measured in the direction of construction (Ik-Ig), constructed from left to right and measured against the direction of construction (Ik-rg), constructed from right to left and measured against the direction of construction (rk-rg), and constructed from right to left and measured in the direction of construction (rk-lg).

[0330] Table 14: Time in s, which was determined to determine the target arrangement of the objects of the object set AsymmZI 3: A, B,C,G, D, E,K,M, J, L, I, F, H from the arrangement-representation rule for various construction rules.

[0331]

[0332] Determining the exact target arrangement using state-of-the-art computer-implemented methods scales with O(N!) and takes 21863 s. Using the method for determining a primary target object arrangement, it is now possible to provide the exact determination of the target arrangement in polynomial time < 4 s for the arrangement-representation rule for object-2 tuples and < 40 s for the arrangement-representation rule for object-3 tuples.

[0333] P10001 - where

[0334] 01.03.2026 Preferably, the procedure for determining a primary target object arrangement additionally includes the following step:

[0335] a) Specifying an individual attribute value relation rule with respect to the attribute value relations and an overall attribute value relation rule with respect to the attribute value relation sums.

[0336] Preferably, the individual attribute-value-relation rule states that the relevant attribute-value relations should have a minimum, a maximum, or a reference value and / or should come as close as possible to it. Furthermore, preferably, the overall attribute-value-relation rule states that the relevant attribute-value-relation sums should have a minimum, a maximum, or a reference value and / or should come as close as possible to it.

[0337] Furthermore, the method for determining a primary target object arrangement preferably additionally includes the step:

[0338] b) Determining a respective attribute value assigned to the property attribute for, in particular, each, object.

[0339] Furthermore, the method for determining a primary target object arrangement preferably additionally includes the step:

[0340] c) Determining an object-n-tuple number from different object-n-tuples (Oi, ... Ok), wherein the object-n-tuples (Oi, ... Ok) each have a tuple content number (n) of at least two objects, wherein the objects of an object-n-tuple (Oi, ... Ok) are each distinct from one another, wherein the object-n-tuple number corresponds to a product of a number of factors corresponding to the tuple content number (n), wherein a first factor corresponds to the object number (Z), wherein further factors each correspond to a difference between the object number (Z) as the minuend and an integer starting with the value one and increasing by one from factor to factor as the subtrahend, wherein in the last factor the subtrahend corresponds to the tuple content number (n) decreased by one, wherein each object-n-tuple (Oi, ... Ok) has a number of objects corresponding to the tuple content number (n). has object positions, where for, in particular, each, object-n-tuple (Oi, ... OK) each object position has an object position index (j).

[0341] P10001 - where

[0342] is assigned to 01.03.2026, where the object position index (j) takes integer values ​​in a range from one to tuple content count (n).

[0343] Preferably, several object-n-tuples (Oi, ... Ok) are formed from several different objects (Oi). Here, i of object (Oi) is not equal to k of object (Ok).

[0344] Preferably, the number of object n-tuples corresponds to the maximum number of possible distinct object n-tuples (Oi, ... Ok). The number of object n-tuples is determined in particular according to the following formula, where Z is the number of objects (Z) and n is the number of tuple contents (n):

[0345]

[0346] Preferably, a single object-n-tuple (Oi, ... Ok) is assigned a number of attribute value relations, where this number corresponds to the tuple content count (n) minus one. For an object-2-tuple, where the tuple content count (n) is two, the aforementioned number is therefore one, and for an object-3-tuple, where the tuple content count (n) is three, the aforementioned number is therefore two.

[0347] Furthermore, the method for determining a primary target object arrangement preferably additionally includes the step:

[0348] d) Determining a number of relation sums of attribute value relation sums corresponding to the number of object n-tuples for, in particular, each object n-tuple (Oi, ... Ok), wherein the attribute value relation sums each correspond to a sum of the attribute value relations assigned to a single object n-tuple (Oi, ... Ok), wherein the attribute value relations assigned to a single object n-tuple (Oi, ... Ok) are determined pairwise between a first object (Oi) of the object n-tuple (Oi, ... Ok) and a second object (Oi) of the object n-tuple (Oi, ... Ok) taking into account the first attribute value assigned to the first object (Oi) and the second attribute value assigned to the second object (Oi) as well as the single attribute value relation rule, wherein the individual object n-tuple (Oi, ...OK) assigned attribute value relations for all possible pairs of first objects and second objects, each consisting of two objects of the object-n-tuple.

[0349] P10001 - where

[0350] 01.03.2026(Oi, ... Ok) result, of which the object position indices (j) are adjacent and of which the first object (Oi) has a lower object position index (j) than the second object (Oi).

[0351] Furthermore, the method for determining a primary target object arrangement preferably additionally includes the step:

[0352] e) Determining a number of arrangement representations corresponding to a plurality of arrangement representations, wherein the number of arrangement representations is a product of a maximum of the first representation factor, which is four, a maximum of the second representation factor, which is equal to the number of objects (Z), a maximum of the third representation factor, which is equal to the number of objects (Z), and a maximum of the fourth representation factor, which is equal to the number of objects (Z) less one, wherein each arrangement representation has a number of representation positions, which is equal to the number of objects (Z) increased by 1, wherein, in particular, each arrangement representation is assigned an object-n-tuple position index (Oti) to each representation position, wherein the object-n-tuple position index (Oti) takes integer values ​​in a range from one to the number of representation positions.wherein each representation position is assigned an object-n-tuple (Oi, ... Ok), wherein the representation position assigned to the first object-n-tuple (Oi, ... Ok) is assigned a first object-n-tuple position index (Oti) which corresponds to the second object-n-tuple position index (Oti) assigned to the representation position assigned to the second object-n-tuple (Oi, ... Ok) minus one, wherein in pairs of a respective first object-n-tuple (Oi, ... Ok) and a respective second object-n-tuple (Oi, ... Ok) that are assigned to immediately consecutive representation positions, the object position(s) of the second object-n-tuple (Oi, ... Ok) with a second object position index (j) or with second object position indices (j) are assigned the same objects that are assigned to the object position(s) of the first object-n-tuple (Oi, ... Ok). Object-n-tuples (Oi,... OK) are assigned to a first object position index (j) or to first object position indices (j), where the second object position index (j) is relative to the first object position index (j),

[0353] P10001 - where

[0354] 01.03.2026 or the second object position indices (j) are each reduced by one compared to the respective first object position indices (j), whereby representation positions whose object n-tuple position index (Oti) is at least two are each assigned an object n-tuple (Oi, ... Ok) whose attribute value relation sum best satisfies the overall attribute value relation rule, taking into account the object(s) already predetermined by the immediately preceding representation position and assigned to the respective object position(s).

[0355] Preferably, the object-n-tuple position index (Oti) takes on integer values ​​in a range from one to the number of representation positions. Object-n-tuple position indexes (Oti) with the values ​​Oti = 1 and Oti = the number of representation positions are also available.

[0356] The maximum of the first representation factor corresponds to the number of different design rules to be observed for the individual arrangement representations. Preferably, the maximum of the first representation factor is four. Corresponding design rules can be, for example: constructed from left to right and measured in the direction of construction (lk-ikg), constructed from left to right and measured against the direction of construction (lk-ekg), constructed from right to left and measured in the direction of construction (rk-ikg), constructed from right to left and measured against the direction of construction (rk-ekg).

[0357] According to the construction rule from left to right, the object-n-tuples (Oi, ... Ok) of an arrangement representation are determined such that an object-n-tuple (Oi, ... Ok) following in the sense of the construction has an object-n-tuple position index (Oti) that is one higher than the object-n-tuple position index (Oti) of the object-n-tuple (Oi, ... Ok) immediately preceding the following object-n-tuple (Oi, ... Ok).

[0358] According to the construction rule from right to left, the object-n-tuples (Oi, ... Ok) of an arrangement representation are determined such that an object-n-tuple (Oi, ... Ok) following a subsequent object-n-tuple in the sense of the construction has an object-n-tuple position index (Oti) that is different from the object-n-tuple position index P10001 -wo

[0359] 01.03.2026(Oti) of the object-n-tuple (Oi, ... Ok) immediately preceding the following object-n-tuple (Oi, ... Ok) has an object-n-tuple position index (Oti) reduced by one.

[0360] According to the construction rule, constructed from left to right and measured in the direction of construction (lk-ikg) as well as constructed from right to left and measured against the direction of construction (rk-ekg), the attribute value relation sums are determined in such a way that the object position index (j) assigned to the respective first object (Oi) is reduced by one compared to the object position index (j) assigned to the respective second object (Oi).

[0361] According to the construction rule, constructed from left to right and measured against the direction of construction (lk-ekg) as well as constructed from right to left and measured in the direction of construction (rk-ikg), the attribute value relation sums are determined in such a way that the object position index (j) assigned to the respective first object (Oi) is increased by one compared to the object position index (j) assigned to the respective second object (Oi).

[0362] The second representation factor addresses the number of objects (Z).

[0363] Preferably, each array representation includes a breakline. The breakline is the position between two immediately adjacent representation positions of an array representation, each of which is assigned the same object.

[0364] The maximum of the third representation factor addresses the number of possible positions of a break edge in the various arrangement representations, which preferably corresponds to the number of objects (Z). Preferably, the object (Oi) to the left of the break edge corresponds to the object (Oi) to the right of the break edge. The maximum of the fourth representation factor addresses the number of possible second objects (Oi) of the object arrangement that remain after the selection or specification of the first object (Oi). Therefore, the maximum of the fourth representation factor corresponds to the number of objects (Z) minus one.

[0365] Preferred practice is to assign one object-n-tuple (Oi, ... Ok) to each representation position whose object-n-tuple position index (Oti) is at least two, P10001 -wo

[0366] March 1, 2026, whose attribute value relation sum, taking into account the object(s) already predetermined by the immediately preceding representation position and assigned to the object position(s) in question, most closely satisfies the overall attribute value relation rule. The condition that the attribute value relation sum most closely satisfies the overall attribute value relation rule means that the attribute value relation sum in question comes closest to the overall attribute value relation rule, i.e., for example, becomes minimal, maximal, or is closest to a reference value.

[0367] Furthermore, the method for determining a primary target object arrangement preferably additionally includes the step:

[0368] f) Determining the arrangement-representation rule, taking the arrangement representations into account, such that the frequency of occurrence of similar object-n-tuples (Oi, ... Ok) in the arrangement representations is determined, and the attribute value relation sums assigned to the respective similar object-n-tuples (Oi, ... Ok) are assigned to this frequency. Preferably, the object position index (j) takes on integer values ​​in a range from one to the number of tuple contents (n). The object position index (j) with the values ​​j = 1 and j = n are also available. Preferably, each object position index (j) is uniquely assigned to an object index (i).

[0369] Preferably, several tuple content counts (n) are taken into account, whereby an arrangement-representation rule is determined for each of the several tuple content counts (n).

[0370] Preferably, the method is used to determine a primary target object arrangement for controlling and / or regulating a production machine.

[0371] Furthermore, the method is preferably used to determine a primary target object arrangement for controlling and / or regulating a robot, in particular an industrial robot, household robot and / or cobot.

[0372] The method for determining a primary target object arrangement for planning routes for robots, especially in warehouses or logistics centers, or in the design of telecommunications networks P10001 -wo

[0373] 01.03.2026 used to ensure optimal signal quality and to minimize costs.

[0374] Alternatively or additionally, the procedure for determining a primary target object arrangement can be applied to objects of different dimensions in various N-dimensional spaces. An example is rectangular, colored sheets of paper from a color fan deck in a color catalog. The color (corresponding to sensory perception) of the paper sheets is measured with a colorimeter and systematically ordered in the RAL Design color system according to hue (H), lightness (L), and chroma (C). The width and height of the paper sheets are measured with a ruler with an accuracy of 0.5 mm. The thickness of the paper sheets is measured with a micrometer screw gauge with an accuracy of 0.01 mm. The weight is measured with a paper scale with a resolution of 0.01 g.

[0375] Another example is astronomical objects in space. Another example is vehicles in a fleet. Another example is machines in a machine hall. Another example is droplets in a cloud. Another example is the atoms of a molecule.

[0376] An example of determining the property attribute "position" of an object relative to the Earth's surface, e.g., of near-Earth objects, is trigonometric parallax measurement using high-powered telescopes. An example of determining the property attribute "position" of an object relative to a reference point on the Earth's surface, e.g., of vehicles in a fleet, is the Global Positioning System (GPS). Here, the position of an object is determined with an accuracy of up to 1 meter. An example of determining the property attribute "position" of an object relative to the position of a laser scanner, e.g., of machines in a machine hall or of droplets in a cloud, is laser scanning. Here, the object's position is determined with an accuracy of up to 25 pm. An example of determining the property attribute "position" of an object, e.g.,Electron beam microscopy, with a resolution of 0.045 nm, is used to measure the size of atoms in a molecule relative to a reference molecule.

[0377] P10001 - where

[0378] March 1, 2026. An example of the mechanical determination of the property attribute of velocity of an object in relation to its position, e.g., a vehicle on the Earth's surface, is the speedometer with a deviation of 1%. An example of the optical determination of the property attribute of velocity of an object, e.g., droplets in a cloud, is optical position filter measurement technology, laser Doppler anemometry, or particle image velocimetry.

[0379] An example of determining the property attribute "force" between objects, such as the gravitational force between astronomical objects in space, is measuring the gravitational field above each of the astronomical objects using two satellites and a microwave rangefinder. An example of determining the property attribute "force" between objects, such as the intermolecular forces between droplets in a cloud, is measuring surface tension, capillary forces, and adhesive and cohesive forces. An example of determining the property attribute "force" between objects, such as the intermolecular forces between atoms of a molecule, is measuring electrostatic forces due to hydrogen bonds, ionic bonds, dipole-dipole bonds, van der Waals forces, and hydrophobic interactions.

[0380] When modeling the extension of an object arrangement by adding further objects, including the property attributes assigned to these additional objects and the relationships between identical property attributes between the additional objects and the objects already present before the extension, the problem often arises that existing object arrangements must first be captured in a computer model for the purpose of describing the extension. This is particularly relevant when adding an astronomical object to the astronomical objects in space, adding a droplet to the droplets in a cloud, or adding an atom to the atoms of a molecule. This effort is immense and, with an increasing number of objects in the arrangement, cannot be solved in polynomial time.

[0381] When planning extensions to an object arrangement with additional objects, including the property attributes assigned to these additional objects and relationships between identical property attributes between the additional objects,

[0382] P10001 - where

[0383] 01.03.2026 With regard to objects and objects already existing before the expansion, the problem often arises that existing object arrangements for the purpose of describing and planning the expansion, e.g. when adding a vehicle to the vehicle fleet, when adding machines to an engine room or when adding a rectangular, colored sheet of paper to a color fan, or the like, the object arrangement with the property attributes and relationships between identical property attributes between different objects must first be recorded in a computer model.

[0384] Further aspects of the invention are discussed below.

[0385] A fifth aspect of the invention relates to a control device of a computer system, comprising a control device processor and a control device memory, wherein the control device is configured to perform a method as described above using the computer system, i.e., a method according to the first, second, third, and / or fourth aspect of the invention. The control device processor can comprise at least one microprocessor and / or at least one microcontroller and / or at least one FPGA (Field Programmable Gate Array) and / or at least one DSP (Digital Signal Processor). In particular, a CPU (Central Processing Unit), a GPU (Graphics Processing Unit), or an NPU (Neural Processing Unit) can be used as the microprocessor.Furthermore, the control device processor can include program code configured to execute a method according to the invention when executed by the control device processor. The program code can be stored in the control device memory. The control device processor can, for example, be based on at least one circuit board and / or on at least one SoC (System on Chip).

[0386] A sixth aspect of the invention relates to a computer system, comprising a control device according to the fifth aspect of the invention and several computing devices, each comprising a computing device processor and coupled to the control device for data transmission.

[0387] A seventh aspect of the invention relates to a computer-readable storage medium comprising program code which, when executed by a computer system

[0388] P10001 - where

[0389] On March 1, 2026, according to the sixth aspect of the invention, this prompts the user to carry out a method as described above, i.e., a method according to the first, second, third, and / or fourth aspect of the invention. The storage medium can be provided at least partially as a non-volatile data storage medium (for example, as flash memory and / or as an SSD - solid state drive) and / or at least partially as a volatile data storage medium (for example, as RAM - random access memory). The storage medium can be located within the computer system. However, the storage medium can also be operated, for example, as an app store server and / or cloud server on the internet. The computer system can provide a processor circuit with, for example, at least one microprocessor. Such a processor circuit is formed, in particular, by the control device or its control device processor.The program code can be provided as binary code, assembly code, source code in a programming language (e.g., C), or a program script (e.g., Python). Alternatively, the computer-readable storage medium can be implemented as a signal containing computer-readable data, such as a time-varying voltage signal or a radio signal.

[0390] An eighth aspect of the invention relates to a computer program product, comprising a computer-readable storage medium according to the seventh aspect of the invention, which is in particular non-transferable, and instructions stored on the computer-readable storage medium which, when executed by a control device according to the fifth aspect of the invention, cause the control device to carry out a method as described above, i.e., a method according to the first, second, third and / or fourth aspect of the invention, using a computer system according to the sixth aspect of the invention.

[0391] The invention also includes combinations of the features of the embodiments or aspects described above. The invention therefore also includes implementations that each exhibit a combination of the features of several of the described embodiments or aspects, provided that the

[0392] P10001 - where

[0393] 01.03.2026 The forms of execution or aspects were not described as mutually exclusive.

[0394] The following are exemplary embodiments of the invention with reference to the accompanying drawing. The drawing shows:

[0395] Fig. 1: a computer system,

[0396] Fig. 2: Input data, each related to an object arrangement and extraction data,

[0397] Fig. 3: a relations matrix,

[0398] Fig. 4: an overall results matrix,

[0399] Fig. 5: a relations-sum matrix,

[0400] Fig. 6: Optimization data according to the first aspect of the invention, Fig. 7: multiple object n-tuples,

[0401] Fig. 8-21: various arrangement representations according to the third aspect of the invention,

[0402] Figs. 22-31: various arrangement-representation rules according to the third aspect of the invention,

[0403] Figs. 32-51: various arrangement-representation rules according to the fourth aspect of the invention,

[0404] Fig. 52: a threshold matrix,

[0405] Fig. 53: several levels and level vectors according to the second aspect of the invention, and

[0406] Fig. 54: Optimization data according to the second aspect of the invention.

[0407] The exemplary embodiments described below are preferred embodiments of the invention. In these exemplary embodiments, the described components each represent individual features of the invention that can be considered independently and that further develop the invention independently. Therefore, the disclosure is intended to include combinations of features of the embodiments other than those explicitly described. Furthermore, the described P10001-wo

[0408] 01.03.2026 The embodiments can also be supplemented by further features of the invention already described above.

[0409] Fig. 1 shows a computer system 100. The computer system 100 comprises a control device 110. The control device 110 includes a control device processor 112. The control device also includes a control device memory 111. The control device processor 112 is communicatively coupled to the control device memory 111, which is indicated by a dashed line in Fig. 1. The computer system 100 also comprises several computing devices 120. By way of example, the computer system 100 comprises four computing devices 120 as shown in Fig. 1. Each computing device 120 has a computing device processor 122. Also by way of example, each computing device 120 has a computing device memory 121. Furthermore, by way of example, the respective computing device processor 122 of a computing device 120 is communicatively coupled with its respective computing device memory 121, as shown accordingly in Fig.The control device 110 is communicatively coupled to each of the computing devices 120, as shown by a respective dashed line in Fig. 1. The computing devices 120 are not (directly) communicatively coupled to each other.

[0410] Fig. 2 shows input data 205, each of which is related to an object arrangement 230 and extraction data 210. By way of example, the object arrangement 230 comprises five objects 231, 232, 233, 234, and 235: a first object 231, a second object 232, a third object 233, a fourth object 234, and a fifth object 235. The input data 205 are provided in the control device memory 111. The input data 205 relate to the objects 231, 232, 233, 234, and 235, which is represented in Fig. 2 by a dashed double arrow. The objects 231, 232, 233, 234, and 235 are associated with an optimization problem. The extraction data 210 are extracted from the input data 205, as indicated by a corresponding arrow in Fig. 2. For this purpose, the control device processor 112 accesses the control device memory 111 (see Fig. 1).The following extraction data 210 will be extracted:.

[0411] P10001 - where

[0412] On March 1, 2026, an object count 217, several attribute values ​​211, 212, 213, 214, 215 corresponding to a property attribute of the objects 231, 232, 233, 234, 235, and an optimization goal 218 corresponding to the optimization problem. By way of example, a first attribute value 211, a second attribute value 212, a third attribute value 213, a fourth attribute value 214, and a fifth attribute value 215 are extracted. The extracted extraction data 210 are stored in the control device memory 111 (see Fig. 1).

[0413] Fig. 3 shows a relation matrix 220. The relation matrix 220 is provided by the control device processor 112 and stored in the control device memory 111 (see Fig. 1). The relation matrix 220 comprises a plurality of relation cells 221. In at least some of the relation cells 221, an attribute value relation between the attribute values ​​211, 212, 213, 214, 215 of two objects 231, 232, 233, 234, 235, corresponding to the respective relation cell 221, is stored. The relation matrix 220 can also be referred to as an arrangement representation (see below for Figs. 8 to 21). By way of example, the relation cells 221 are each assigned to a relation row 222 from a plurality of relation rows 222 corresponding to the number of objects 217, and to a relation column 223 from a plurality of relation columns 223 corresponding to the number of objects 217. In the figure shown.The exemplary relation matrix 220 shown in Figure 3 depicts six relation columns 223 and three relation rows 222. The relation matrix 220 can contain fewer relation columns 223 and / or fewer relation rows 222. Furthermore, the relation matrix 220 can contain more relation columns 223 and / or more relation rows 222, which is indicated in Figure 3 by three dots at the bottom and right ends of the relation matrix 220, respectively.

[0414] Example relation matrices are shown in Tables 2 and 3 for object-2 tuples and in Tables 4 to 8 for object-3 tuples.

[0415] Fig. 4 shows a total result matrix 250. The total result matrix 250 is provided by the control device processor 112 in the control device memory 111 (see Fig. 1). The total result matrix 250 comprises a plurality of total result cells 251. Initially, each total result cell 251 contains an empty value. The total result cells 251 are subsequently populated.

[0416] P10001 - where

[0417] The matrix 250, fully populated as of March 1, 2026, is completed according to the procedure(s) described above. The completed overall result matrix 250 can also be referred to as an arrangement-representation rule (see Figures 22 to 51 below). By way of example, the overall result matrix 250 comprises Z overall result rows 252. Furthermore, by way of example, the overall result matrix 250 comprises Z overall result columns 253. In the exemplary overall result matrix 250 shown in Figure 4, six overall result columns 253 and three overall result rows 252 are shown, purely by way of example. According to the preceding example, the overall result matrix 250 comprises exactly as many overall result columns 253 as it comprises overall result rows 252, which, however, is not fully shown in Fig. 4 for the sake of clarity. The overall result matrix 250 may contain fewer overall result columns 253 and / or fewer overall result rows 252 than shown in Fig. 4.Furthermore, the overall result matrix 250 can include more overall result columns 253 and / or more overall result rows 252 than shown in Fig. 4, which is represented in Fig. 4 by three points at the bottom and right ends of the overall result matrix 250, respectively.

[0418] Fig. 5 shows a relation sum matrix 260. For illustrative purposes, the relation sum matrix 260 is provided by the control device processor 112 and stored in the control device memory 111 (see Fig. 1). For illustrative purposes, the relation sum matrix 260 comprises a plurality of relation sum cells 261. Furthermore, for illustrative purposes, each of the relation sum cells 261 stores an attribute value relation sum. Specifically, as many attribute value relation sums are stored in each of the relation sum cells 261 as there are object n-tuples 301, 302, 303 (see Fig. 7 below). For illustrative purposes, the relation sum cells 261 of each relation sum row 262 are comprised of a plurality of

[0419] The relation-sum rows 262 and a relation-sum column 263 are assigned to a plurality of relation-sum columns 263. In particular, the relation-sum matrix 260 has Z relation-sum rows 262 and / or Z relation-sum columns 263. In the exemplary relation-sum matrix 260 shown in Fig. 5, six relation-sum columns 263 and three relation-sum rows 262 are shown purely by way of example. The relation-sum- P10001 -wo

[0420] 01.03.2026 Matrix 260 can contain fewer relation-sum columns 263 and / or fewer relation-sum rows 262. Furthermore, the relation-sum matrix 260 can contain more relation-sum columns 263 and / or more relation-sum rows 262, which is represented in Fig. 5 by three points at the bottom and right ends of the relation-sum matrix 260, respectively.

[0421] Fig. 6 shows optimization data 240 according to the first aspect of the invention. The optimization data 240 are provided by the control device processor 112 and stored in the control device memory 111. The optimization data 240 are assigned to several representation factors 242, 243, 244, 245, which are represented accordingly in Fig. 6 by dashed boxes. Each of the flat dashed boxes symbolizes one of the representation factors 242, 243, 244, 245. The optimization data 240 comprise several sub-data 241. The sub-data 241 each correspond to one of several optimization subproblems of the optimization problem. Three sub-data 241 are shown in Fig. 6 by way of example.Furthermore, the use of four representation factors 242, 243, 244, and 245 is exemplified: a first representation factor 242, a second representation factor 243, a third representation factor 244, and a fourth representation factor 245. By way of example, the first representation factor 242 corresponds to a formation rule. Such a first representation factor can, in particular, be one of the formation rules lk-ikg, lk-ekg, rk-ikg, and rk-ekg. By way of example, the second representation factor 243 corresponds to a start object. The third representation factor 244 corresponds, by way of example, to an offset image. The fourth representation factor 245 corresponds, by way of example, to a first object-n-tuple of an arrangement representation.

[0422] Purely by way of example, a maximum of the first representation factor is 242, which is four. Furthermore, by way of example, a maximum of the second representation factor corresponds to the number of objects 243, which is 217. Further by way of example, a maximum of the third representation factor corresponds to the number of objects 244, which is 217. Yet another by way of example, a maximum of the fourth representation factor corresponds to the number of objects 217 minus one. Arrangement representations can be determined.

[0423] P10001 - where

[0424] March 1, 2026, as exemplified in Figures 8 to 21 discussed below. Purely by way of example, a corresponding number of arrangement representations are determined for each arrangement representation number. The arrangement representation number is further exemplified as the product of the maximum of the first representation factor 242, the maximum of the second representation factor 243, the maximum of the third representation factor 244, and the maximum of the fourth representation factor 245.

[0425] By way of example, the partial data 241 are each represented as a vector, as shown in Fig. 6. Each coordinate of such a vector corresponds to a specific value of the respective representation factor 242, 243, 244, 245. Furthermore, by way of example, the coordinate relating to the first representation factor 242 in the vector of the left partial data 241 corresponds to the value “Ik-ikg”, the coordinate of the vector of the middle partial data 241 also corresponds to the value “Ik-ikg”, and the coordinate of the vector of the right partial data 241 corresponds to the value “rk-ikg”. As a further example, the coordinate relating to the second representation factor 243 of the vector of the left sub-data 241 corresponds to the value “5”, the vector of the middle sub-data 241 to the value “4” and the vector of the right sub-data 241 also to the value “4”.As a further example, the coordinate relating to the third representation factor 244 of the vector of left-hand data 241 corresponds to the value "5", the vector of middle data 241 to the value "4", and the vector of right-hand data 241 also to the value "4". As a further example, the coordinate relating to the fourth representation factor 245 of the vector of left-hand data 241 corresponds to the value "4", the vector of middle data 241 to the value "3", and the vector of right-hand data 241 also to the value "3".

[0426] Fig. 7 shows several object-n-tuples 301, 302, 303, namely, purely by way of example, the three object-n-tuples 301, 302, 303, the first object-n-tuple 301, the second object-n-tuple 302, and the third object-n-tuple 303. Furthermore, by way of example, each of the object-n-tuples 301, 302, 303 comprises three objects 231, 232, 233, 234, 235. As a further example, the first object-n-tuple 301 comprises the first object 231, the second object 232, and the third object 233. As a further example, the second object-n-tuple 302 comprises the fifth object 235, the second object 232, and the third object 233.

[0427] P10001 - where

[0428] 01.03.2026To give a further example, the third object-n-tuple 303 includes the fifth object 235, the fourth object 234 and the first object 231.

[0429] In the methods discussed below for determining an arrangement-representation rule and methods for determining a primary-target object arrangement, Figures 22 to 51 show various arrangement-representation rules, each with an associated primary-target object arrangement. The respective arrangement-representation rule is shown in the y-region or along the y-axis of the respective figure, and the primary-target object arrangement is labeled "correct" in the figure's legend. Each figure shows two primary-target object arrangements as examples, with Z "correct" object n-tuples 301, 302, 303, one of which has an object position index j -> j + 1 and the other an object position index j <- j + 1. The object n-tuples 301, 302, 303, which are also labeled "incorrect," do not form a primary-target object arrangement.

[0430] The following section discusses the method for determining an arrangement-representation rule according to the third aspect of the invention with reference to Figures 8 to 31. Here, the aforementioned method is explained in more detail for five color cards (Z = 5) with the property attribute "color value" and the relations for object-2 tuples assigned to the property attribute values. Each arrangement representation has objects from i = 1 to Z. Each object (Oi) is assigned a specific object index (i). In Figures 8 to 31, the color card with object index 1 is designated with the letter A, the color card with object index 2 with the letter B, the color card with object index 3 with the letter C, the color card with object index 4 with the letter D, and the color card with object index 5 with the letter E. Each of Figures 8 to 21 represents a total of Z-1 different arrangement representations.These four different arrangement representations of the five color cards A, B, C, D, and E are distinguished from one another by a solid line, a dashed line, a dotted line, and a dash-dotted line. These Z-1 arrangement representations are described by the fourth representation factor, where the fourth representation factor is the number of possible second objects (Oi) of the arrangement representation that are available after selection or specification.

[0431] P10001 - where

[0432] The first object (Oi) remains addressed as of March 1, 2026. Therefore, the fourth representation factor corresponds to the number of objects (Z) reduced by one; in the case of the 5 color cards, this is a factor of four.

[0433] Figures 8 to 11 show four different construction instructions.

[0434] Fig. 8 shows Z - 1 arrangement representations (solid, dashed, dotted and dash-dotted line from the fourth representation factor) of 5 color cards A, B, C, D, E for the construction instruction constructed from left to right and measured in the construction direction (lk-ikg) with fixed object to the left and right of the break edge (A) and with fixed position of the break edge (offset zero).

[0435] Fig. 9 shows Z - 1 arrangement representations (solid, dashed, dotted and dash-dotted line from the fourth representation factor) of 5 color cards A, B, C, D, E for the construction instruction constructed from left to right and measured against the direction of construction (lk-ekg) with a fixed object to the left and right of the break edge (A) and with the break edge in a fixed position (offset zero).

[0436] Fig. 10 shows Z - 1 arrangement representations (solid, dashed, dotted and dash-dotted line from the fourth representation factor) of 5 color cards A, B, C, D, E for the construction instruction constructed from right to left and measured in the construction direction (rk-ikg) with fixed object to the left and right of the break edge (A) and with fixed position of the break edge (offset zero).

[0437] Fig. 11 shows Z - 1 arrangement representations (solid, dashed, dotted and dash-dotted line from the fourth representation factor) of 5 color cards A, B, C, D, E for the construction instruction constructed from right to left and measured against the direction of construction (rk-ekg) with a fixed object to the left and right of the break edge (A) and with the break edge in a fixed position (offset zero).

[0438] The following section describes in more detail how to determine different arrangement representations using Figures 5 to 9 for 5 color cards (Z=5) with the property attribute "color value" and the relations assigned to the property attribute values.

[0439] P10001 - where

[0440] March 1, 2026: The object-2 tuple for the second representation factor, which represents the number of objects (Z), is explained. The object located to the left and right of the break line can be color card A, B, C, D, or E.

[0441] Fig. 12 shows Z - 1 arrangement representations (solid, dashed, dotted and dash-dotted line from the fourth representation factor) of 5 color cards A, B, C, D, E for the second representation factor, where the object which lies to the left and right at the break edge (offset zero) is object A, is constructed for the construction rule from left to right and measured in the construction direction (lk-ikg).

[0442] Fig. 13 shows Z - 1 arrangement representations (solid, dashed, dotted and dash-dotted line from the fourth representation factor) of 5 color cards A, B, C, D, E for the second representation factor, where the object which lies to the left and right at the break edge (offset zero) is object B, is constructed for the construction rule from left to right and measured in the construction direction (lk-ikg).

[0443] Fig. 14 shows Z - 1 arrangement representations (solid, dashed, dotted and dash-dotted line from the fourth representation factor) of 5 color cards A, B, C, D, E for the second representation factor, where the object which lies to the left and right at the break edge (offset zero), object C, is constructed for the construction rule from left to right and measured in the construction direction (lk-ikg).

[0444] Fig. 15 shows Z - 1 arrangement representations (solid, dashed, dotted and dash-dotted line from the fourth representation factor) of 5 color cards A, B, C, D, E for the second representation factor, where the object which lies to the left and right at the break edge (offset zero) is object D, is constructed for the construction rule from left to right and measured in the construction direction (lk-ikg).

[0445] Fig. 16 shows Z-1 arrangement representations (solid, dashed, dotted, and dash-dotted lines from the fourth representation factor) of 5 color cards A, B, C, D, E for the second representation factor, where the object located to the left and right of the breakline (offset zero) is object E for the

[0446] P10001 - where

[0447] 01.03.2026 Construction specification constructed from left to right and measured in construction direction (lk-ikg).

[0448] Figures 17 to 21 below illustrate how to determine different arrangement representations for 5 color charts (Z=5) with the property attribute "color value" and the relations assigned to the property attribute values ​​for object-2 tuples for the third representation factor, which represents the number of possible positions of a break edge in the different arrangement representations. The different positions of the break edges are labeled with offset 0, offset 1, offset 2, offset 3, and offset 4.

[0449] Fig. 17 shows Z - 1 arrangement representations (solid, dashed, dotted and dash-dotted line from the fourth representation factor) of 5 color cards A, B, C, D, E for the third representation factor, where the position of the break edge is offset 0, with object A to the left and right of the break edge (offset 0) and constructed for the construction rule from left to right and measured in the construction direction (lk-ikg).

[0450] Fig. 18 shows Z - 1 arrangement representations (solid, dashed, dotted and dash-dotted line from the fourth representation factor) of 5 color cards A, B, C, D, E for the third representation factor, where the position of the break edge is offset 1, with object A to the left and right of the break edge (offset 0) and constructed for the construction rule from left to right and measured in the construction direction (lk-ikg).

[0451] Fig. 19 shows Z - 1 arrangement representations (solid, dashed, dotted and dash-dotted line from the fourth representation factor) of 5 color cards A, B, C, D, E for the third representation factor, where the position of the break edge is offset 2, with object A to the left and right of the break edge (offset 0) and constructed for the construction rule from left to right and measured in the construction direction (lk-ikg).

[0452] Fig. 20 shows Z-1 arrangement representations (solid, dashed, dotted and dash-dotted line from the fourth representation factor) of 5 color cards A, B, C, D, E for the third representation factor, where the position of the break edge is offset 3, with object A to the left and right of the break edge.

[0453] P10001 - where

[0454] 01.03.2026(offset 0) and designed for the construction specification from left to right and measured in the direction of construction (lk-ikg).

[0455] Fig. 21 shows Z - 1 arrangement representations (solid, dashed, dotted and dash-dotted line from the fourth representation factor) of 5 color cards A, B, C, D, E for the third representation factor, where the position of the break edge is offset 4, with object A to the left and right of the break edge (offset 0) and constructed for the construction rule from left to right and measured in the construction direction (lk-ikg).

[0456] Figures 22 to 31 show various arrangement-representation rules derived from the different arrangement representations such that the occurrence of similar object-n-tuples 301, 302, 303 (Oi, ... Ok) in the various arrangement representations is determined, and this frequency is assigned to the attribute value relation sums associated with the respective similar object-n-tuples 301, 302, 303 (Oi, ... Ok). The determination of the arrangement-representation rule for object-2 tuples and for object-3 tuples of 5 color cards (Z=5) is illustrated below.We create an arrangement representation rule for all arrangement representations for all four construction rules combined and for the four separate construction rules: constructed from left to right and measured in the direction of construction (lk-ikg), constructed from left to right and measured against the direction of construction (lk-ekg), constructed from right to left and measured in the direction of construction (rk-ikg), constructed from right to left and measured against the direction of construction (rk-ekg).The object-n-tuples 301, 302, 303, which are an object-n-tuple 301, 302, 303 of the desired target object arrangement of object-n-tuples 301, 302, 303, where the target object arrangement comprises a number of object-n-tuples 301, 302, 303 corresponding to the number of objects (Z), are marked with a right-pointing triangle (object position index: j -> j + 1) or with a left-pointing triangle (object position index: j <- j + 1). The object n-tuples 301, 302, 303, which are not object n-tuples 301, 302, 303 of the desired target object arrangement of object n-tuples 301, 302, 303, are marked with a circle.

[0457] P10001 - where

[0458] Figure 22 (March 1, 2026) shows the arrangement-representation rule for all four construction rules combined from object-2 tuples of 5 color cards. The frequency of object-2 tuples that are part of the desired target object arrangement of object-2 tuples is higher than that of object-2 tuples with attribute-value relation sums in the same value range. Preferably, at least three, preferably at least five, and more preferably at least 10, and in particular at least 15, different value ranges, especially those directly adjacent to one another, are provided across the entire spectrum of all possible attribute-value relation sums. Figure 23 shows the arrangement-representation rule for the separate construction rules: constructed from left to right and measured in the construction direction (lk-ikg), from object-2 tuples of 5 color cards.The frequency of object-2 tuples that are object-2 tuples of the desired target object arrangement of object-2 tuples is higher for smaller attribute value relation sums than the frequency of object-2 tuples with attribute value relation sums of the same value range.

[0459] Fig. 24 shows the arrangement-representation rule for the separate construction rules: constructed from left to right and measured against the construction direction (lk-ekg), of object-2 tuples from 5 color cards. The frequency of the object-2 tuples that are an object-2 tuple of the desired target object arrangement of object-2 tuples has a higher frequency for smaller attribute-value relation sums than the object-2 tuples with attribute-value relation sums in the same value range.

[0460] Fig. 25 shows the arrangement-representation rule for the separate construction rules: constructed from right to left and measured in the construction direction (rk-ikg), of object-2 tuples from 5 color cards. The frequency of the object-2 tuples that are an object-2 tuple of the desired target object arrangement of object-2 tuples has a higher frequency for smaller attribute-value relation sums than the object-2 tuples with attribute-value relation sums in the same value range.

[0461] Fig. 26 shows the arrangement-representation rule for the separate construction rules: constructed from right to left and contrary to P10001 -wo

[0462] 01.03.2026 Construction direction measured (rk-ekg) of object-2 tuples from 5 color charts. The frequency of object-2 tuples that are part of the desired target object arrangement of object-2 tuples is higher for smaller attribute value relation sums than for object-2 tuples with attribute value relation sums in the same value range.

[0463] Fig. 27 shows the combined arrangement-representation rule for all four construction rules of object-3 tuples from 5 color cards. The frequency of the object-3 tuples that are part of the desired target object arrangement of object-3 tuples is all higher than that of the object-3 tuples with attribute value relation sums in the same value range.

[0464] Fig. 28 shows the arrangement-representation rule for the separate construction rules: constructed from left to right and measured in the construction direction (lk-ikg), of object-3 tuples from 5 color cards. The frequency of object-n tuples 301, 302, 303, which are object-3 tuples of the desired target object arrangement of object-3 tuples, has a higher frequency for smaller attribute-value relation sums than the object-3 tuples with attribute-value relation sums in the same value range.

[0465] Fig. 29 shows the arrangement-representation rule for the separate construction rules: constructed from left to right and measured against the construction direction (lk-ekg), of object-3 tuples from 5 color cards. The frequency of the object-3 tuples that are an object-3 tuple of the desired target object arrangement of object-3 tuples has a higher frequency for smaller attribute-value relation sums than the object-3 tuples with attribute-value relation sums in the same value range.

[0466] Fig. 30 shows the arrangement-representation rule for the separate construction rules: constructed from right to left and measured in the construction direction (rk-ikg), of object-3 tuples from 5 color cards. The frequency of the object-3 tuples that are an object-3 tuple of the desired target object arrangement of object-3 tuples has a higher frequency for smaller attribute-value relation sums than the object-3 tuples with attribute-value relation sums in the same value range.

[0467] P10001 - where

[0468] Figure 31, dated March 1, 2026, shows the arrangement-representation rule for the separate construction rules: constructed from right to left and measured against the direction of construction (rk-ekg), of object-3 tuples from 5 color cards. The frequency of the object-3 tuples that are object-n tuple 301, 302, 303 of the desired target object arrangement of object-3 tuples is higher for smaller attribute-value relation sums than the object-3 tuples with attribute-value relation sums in the same value range.

[0469] The method for determining a primary target object arrangement according to the fourth aspect of the invention is discussed below with reference to Figures 32 to 51. The respective arrangement representation rule (as a result of an exemplary method for determining an arrangement representation rule) is shown in the y-region or along the y-axis of the respective figure, and the primary target object arrangement (as a result of an exemplary method for determining a primary target object arrangement) is shown in the legend of the respective figure with the designation "correct". One primary target object arrangement from left to right (j -> j + 1) comprises Z "correct" object n-tuples 301, 302, 303 with object position index j -> j + 1, and the other primary target object arrangement from right to left (j <- j + 1) comprises Z "correct" object n-tuples 301, 302, 303 with object position index j <- j + 1 in the relevant figure.The relevant figure(s) show Z “correct” object n-tuples 301, 302, 303 with an object position index j -> j + 1 and Z “correct” object n-tuples 301, 302, 303 with an object position index j <- j + 1.

[0470] Preferably, an arrangement-representation rule is determined from the various arrangement representations, taking into account the arrangement representations, in such a way that the occurrence of similar object-n-tuples 301, 302, 303 (Oi, ... Ok) in the various arrangement representations is determined and this frequency is assigned to the attribute value relation sums assigned to the respective similar object-n-tuples 301, 302, 303 (Oi, ... Ok).

[0471] The following determines the arrangement-representation rule for object-2 tuples and object-3 tuples of the object sets SymZ13 and AsymZ13, respectively. We create an arrangement-representation rule for all

[0472] P10001 - where

[0473] 01.03.2026 Arrangement representations for all four design rules combined and for the four separate design rules: constructed from left to right and measured in the direction of construction (Ik-Ig), constructed from left to right and measured against the direction of construction (Ik-rg), constructed from right to left and measured in the direction of construction (rk-rg), constructed from right to left and measured against the direction of construction (rk-lg).

[0474] As a purely exemplary example, a primary target object arrangement from left to right (j -> j + 1) comprises Z "correct" object n-tuples 301, 302, 303 with object position index j -> j + 1, and another primary target object arrangement from right to left (j <- j + 1) comprises Z "correct" object n-tuples 301, 302, 303 with object position index j <- j + 1. Furthermore, as an example, the two primary target object arrangements are determined using the arrangement representation rules.If, for example, the arrangement-representation rules for at least one of the object-n-tuples 301, 302, 303 differ for at least two of the four construction rules (Ik-Ig, Ik-rg, rk-rg, rk-lg), then the two arrangement-representation rules with the construction rules Ik-Ig and rk-rg are used to determine the primary target object arrangement with object position index j -> j + 1, and then the two arrangement-representation rules with the construction rules Ik-rg and rk-lg are used to determine the primary target object arrangement with object position index j <-j + 1.If, as an example, the arrangement-representation rules for all object-n-tuples 301, 302, 303 are the same for all four construction rules with respect to one of the object-n-tuples 301, 302, 303, then the combined arrangement-representation rule is used to determine the primary-target object arrangement with object position index j -> j + 1 and to determine the primary-target object arrangement with object position index j <- j + 1.

[0475] The object n-tuples 301, 302, 303, which are object n-tuples 301, 302, 303 of the desired target object arrangement of object n-tuples 301, 302, 303, where the target object arrangement comprises a number of object n-tuples 301, 302, 303 corresponding to the number of objects (Z), are represented by a right-pointing triangle (object position index: j -> j + 1) or by a left-pointing triangle.

[0476] P10001 - where

[0477] 01.03.2026 (Object Position Index: j <- j + 1) is marked. The object n-tuples 301, 302, 303, which are not object n-tuples 301, 302, 303 of the sought target object arrangement of object n-tuples 301, 302, 303, are marked with a circle.

[0478] Fig. 32 shows the combined arrangement-representation rule for all four construction rules of object-2 tuples of the object set SymmZ13 from 13 locations. The frequency of the object-2 tuples that are part of the desired target object arrangement of object-2 tuples is all higher than that of the object-2 tuples with comparable attribute value relation sums.

[0479] Fig. 33 shows the arrangement-representation rule for the separate construction rules: constructed from left to right and measured in the construction direction (Ik-Ig), of object-2 tuples of the object set SymmZ13 from 13 locations. The frequency of the object-2 tuples that are an object-2 tuple of the desired target object arrangement of object-2 tuples has a higher frequency for small- to medium attribute-value relation sums than the object-2 tuples with comparable attribute-value relation sums.

[0480] Fig. 34 shows the arrangement-representation rule for the separate construction rules: constructed from left to right and measured against the construction direction (Ik-rg), of object-2 tuples of the object set SymmZ13 from 13 locations. The frequency of the object-2 tuples that are an object-2 tuple of the desired target object arrangement of object-2 tuples has a higher frequency for smaller attribute-value-relation sums than the object-2 tuples with comparable attribute-value-relation sums.

[0481] Fig. 35 shows the arrangement-representation rule for the separate construction rules: constructed from right to left and measured in the construction direction (rk-rg), of object-2 tuples of the object set SymmZ13 from 13 locations. The frequency of the object-2 tuples that are an object-2 tuple of the desired target object arrangement of object-2 tuples has a higher frequency for smaller attribute-value relation sums than the object-2 tuples with comparable attribute-value relation sums.

[0482] Fig. 36 shows the arrangement representation rule for the separate construction rules: constructed from right to left and contrary to P10001 -wo

[0483] 01.03.2026 Construction direction measured (rk-lg), of object-2-tuples of object set SymmZ13 from 13 locations. The frequency of the object-2-tuples that are an object-2-tuple of the sought-after target object arrangement of object-2-tuples has a higher frequency for smaller attribute value relation sums than the object-2-tuples with comparable attribute value relation sums.

[0484] Fig. 37 shows the combined arrangement-representation rule for all four construction rules of object-3 tuples of the object set SymmZ13 from 13 locations. The frequency of the object-3 tuples that are part of the desired target object arrangement of object-3 tuples is all higher than that of the object-3 tuples with comparable attribute value relation sums.

[0485] Fig. 38 shows the arrangement-representation rule for the separate construction rules: constructed from left to right and measured in the construction direction (Ik-Ig), of object-3 tuples of the object set SymmZ13 from 13 locations. The frequency of object-n tuples 301, 302, 303, which are object-3 tuples of the desired target object arrangement of object-3 tuples, has a higher frequency for smaller attribute-value-relation sums than the object-3 tuples with comparable attribute-value-relation sums.

[0486] Fig. 39 shows the arrangement-representation rule for the separate construction rules: constructed from left to right and measured against the construction direction (Ik-rg), of object-3 tuples of the object set SymmZ13 from 13 locations. The frequency of the object-3 tuples that are an object-3 tuple of the desired target object arrangement of object-3 tuples has a higher frequency for smaller attribute-value-relation sums than the object-3 tuples with comparable attribute-value-relation sums.

[0487] Fig. 40 shows the arrangement-representation rule for the separate construction rules: constructed from right to left and measured against the construction direction (rk-lg), of object-3 tuples of the object set SymmZ13 from 13 locations. The frequency of the object-3 tuples that are an object-3 tuple of the desired target object arrangement of object-3 tuples has a higher frequency for very small attribute-value-relation sums than the object-3 tuples with comparable attribute-value-relation sums.

[0488] P10001 - where

[0489] Figure 41, dated March 1, 2026, shows the arrangement-representation rule for the separate construction rules: constructed from right to left and measured in the construction direction (rk-rg), of object-3 tuples of the object set SymmZ13 from 13 locations. The frequency of the object-3 tuples that are object-n tuple 301, 302, 303 of the desired target object arrangement of object-3 tuples is higher for smaller attribute-value-relation sums than the object-3 tuples with comparable attribute-value-relation sums.

[0490] Fig. 42 shows the combined arrangement-representation rule for all four construction rules of object-2 tuples of the object set AsymmZ13 from 13 locations. The frequency of the object-2 tuples that are part of the desired target object arrangement is all higher than that of the object-2 tuples with comparable attribute-value relation sums. The attribute-value relation sums of the target object arrangement are significantly smaller for the correct "direction" (object position index: j->j+1) than the attribute-value relation sums of the target object arrangement for the reverse "direction" (object position index: j<-j+1).

[0491] Fig. 43 shows the arrangement-representation rule for the separate construction rules: constructed from left to right and measured in the construction direction (Ik-Ig), of object-2 tuples of the object set AsymmZ13 from 13 locations. The frequency of the object-2 tuples that are object-2 tuples of the desired target object arrangement of object-2 tuples has a higher frequency for smaller attribute-value relation sums than the object-2 tuples with comparable attribute-value relation sums. The frequencies of the objects of the target object arrangement are significantly higher for the correct "direction" (object-position index: j -> j+1) than the frequencies of the objects of the target object arrangement for the reverse "direction" (object-position index: j<-j+1).Figure 44 shows the arrangement-representation rule for the separate construction rules: constructed from left to right and measured against the construction direction (Ik-rg), of object-2 tuples of the object set AsymmZ13 from 13 locations. The frequency of the object-2 tuples that are an object-2 tuple of the desired target object arrangement of object-2 tuples has for.

[0492] P10001 - where

[0493] 01.03.2026 Smaller attribute value relation sums have a higher frequency than object-2 tuples with comparable attribute value relation sums.

[0494] Fig. 45 shows the arrangement-representation rule for the separate construction rules: constructed from right to left and measured against the construction direction (rk-lg), of object-2 tuples of the object set AsymmZ13 from 13 locations. The frequency of the object-2 tuples that are an object-2 tuple of the desired target object arrangement of object-2 tuples has a higher frequency for small-large attribute-value relation sums than the object-2 tuples with comparable attribute-value relation sums.

[0495] Fig. 46 shows the arrangement-representation rule for the separate construction rules: constructed from right to left and measured in the construction direction (rk-rg), of object-2 tuples of the object set AsymmZ13 from 13 locations. The frequency of the object-2 tuples that are an object-2 tuple of the desired target object arrangement of object-2 tuples has a higher frequency for smaller attribute-value-relation sums than the object-2 tuples with comparable attribute-value-relation sums.

[0496] Fig. 47 shows the combined arrangement-representation rule for all four construction rules of object-3 tuples of the object set AsymmZ13 from 13 locations. The frequency of the object-3 tuples that are part of the desired target object arrangement of object-3 tuples is all higher than that of the object-3 tuples with comparable attribute value relation sums.

[0497] Fig. 48 shows the arrangement-representation rule for the separate construction rules: constructed from left to right and measured in the construction direction (Ik-Ig), of object-3 tuples of the object set AsymmZ13 from 13 locations. The frequency of object-n tuples 301, 302, 303, which are object-3 tuples of the desired target object arrangement of object-3 tuples, has a higher frequency for smaller attribute-value-relation sums than the object-3 tuples with comparable attribute-value-relation sums.

[0498] Fig. 49 shows the arrangement-representation rule for the separate construction rules: constructed from left to right and measured against the direction of construction (Ik-rg), of object-3 tuples of object set P10001 -wo

[0499] 01.03.2026AsymmZ13 of 13 locations. The frequency of object-3 tuples that are part of the target object arrangement of object-3 tuples is lower than the frequency of some object-3 tuples that do not belong to the target object arrangement.

[0500] Fig. 50 shows the arrangement-representation rule for the separate construction rules: constructed from right to left and measured against the construction direction (rk-lg), of object-3 tuples of the object set AsymmZ13 from 13 locations. The frequency of the object-3 tuples that are part of the desired target object arrangement of object-3 tuples is lower than the frequency of some object-3 tuples that do not belong to the desired target object arrangement.

[0501] Fig. 51 shows the arrangement-representation rule for the separate construction rules: constructed from right to left and measured in the construction direction (rk-rg), of object-3 tuples of the object set AsymmZ13 from 13 locations. The frequency of an object-3 tuple, which is an object-n tuple 301, 302, 303 of the desired target object arrangement of object-3 tuples, is the highest and has a small attribute-value relation sum.

[0502] Fig. 52 shows a threshold matrix 270. The threshold matrix 270 is provided by the control device processor 112 and stored in the control device memory 111 (see Fig. 1). The threshold matrix 270 comprises a plurality of threshold cells 271. The threshold matrix 270 can also be referred to as a comparison value matrix. By way of example, the threshold cells 271 are each assigned to a threshold row 272 of several threshold rows 272 and to a threshold column 273 of several threshold columns 273. In the exemplary threshold matrix 270 shown in Fig. 52, six threshold columns 273 and three threshold rows 272 are shown, purely by way of example. The threshold matrix 270 can contain fewer threshold columns 273 and / or fewer threshold rows 272. Furthermore, the threshold matrix 270 can contain more threshold columns 273 and / or more threshold rows 272, as shown in Fig.52 is represented accordingly by three points at the lower and right ends of the threshold matrix 270, respectively.

[0503] P10001 - where

[0504] Figure 53, dated March 1, 2026, shows several levels: 281, 282, 283, 284, 285, and 286. Each of these levels has an index, or level index, ranging from 1 to Z. By way of example, the following levels are shown: a first level 281 (index = 1), a second level 282 (index = 2), a third level 283 (index = 3), a fourth level 284 (index = 4), a fifth level 285 (index = 5), and level Z 286 (index = Z). If Z could be six, for example, then no further levels would be provided between the fifth level 285 and level Z 286. Z can also be greater than six, in which case at least one further level would be provided between the fifth level 285 and level Z 286. Z can also be less than six, for example, three. Then the third level 283 corresponds to level Z 286, and neither the fourth level 284 nor the fifth level 286, nor any further levels after level Z 286, are provided.

[0505] Furthermore, by way of example, a level vector is filled step by step for each level 281, 282, 283, 284, 285, 286, or as a result of each level 281, 282, 283, 284, 285, 286. The level vector is shown to the right of the boxes labeled with the reference symbols 281, 282, 283, 284, 285, 286. As a further example, a first coordinate of the level vector is filled in the first level 281, a second coordinate of the level vector in the second level 282, and so on, until a coordinate Z of the level vector is filled in level Z 286. As a further example, it is provided that each coordinate of the level vector is filled with a corresponding object n-tuple 301, 302, 303. The resulting object-n-tuple 301, 302, 303 represents, in particular, a segment.In other words, the object-n-tuple in question becomes a segment when the level vector has been appropriately populated with this object-n-tuple 301, 302, 303, that is, when this object-n-tuple 301, 302, 303 has been appropriately placed in the level vector. For example, in the first level 281, a segment, which is labeled Segmentl in Fig. 53, is selected as an object-n-tuple 301, 302, 303 from a set of object-n-tuples 301, 302, 303 that contains a number of object-n-tuples 301, 302, 303 equal to the number of object-n-tuples (see above). Furthermore, for example, the first level 281 is considered the level preceding the second level 282. The second level, 282, is then the preceding level for the third level, 283. Similarly, level Z-1 is the preceding level for level Z 286. Considering, for example, the second level, P10001-wo...

[0506] If 01.03.2026282 is the current level, the segment from the aforementioned set of object-n-tuples 301, 302, 303 is selected in the current level. This segment, selected in the current level (i.e., the second level 282), is referred to as Segment2. If level Z 286 is the current level, the segment formed in this level Z 286 is referred to as SegmentZ.

[0507] Furthermore, by way of example, it is provided that a respective computing device 120 checks at least one, preferably several, and more preferably each, test condition of levels 281, 282, 283, 284, 285, 286. Several test conditions can be checked within one test step. It is further by way of example that if at least one, preferably several, and more preferably each, of the following test conditions is / are met, the level value is increased by means of the respective computing device 120. Simultaneously, by way of example, the object-n-tuple that meets the respective test condition(s) is entered as a segment for the respective level 281, 282, 283, 284, 285, 286 into the respective coordinate of the level vector. Increasing the level value means that a current level 281, 282, 283, 284, 285, 286 is left behind and a subsequent level 281, 282, 283, 284, 285, 286 is started.For example, the first level 281 can be exited and the second level 282 started as the subsequent level. By way of example, it is intended that after a checked test condition is not met, the checked object-n-tuple 301, 302, 303 in the current level 281, 282, 283, 284, 285, 286 is discarded and the next object-n-tuple 301, 302, 303 is checked. Furthermore, by way of example, it is provided that if no further object-n-tuple 301, 302, 303 to be checked is available in the current level 281, 282, 283, 284, 285, 286, a jump is made back to a previous level, in particular to the immediately preceding level 281, 282, 283, 284, 285, 286. By way of further example, it is also provided that, in the level 281, 282, 283, 284, 285, 286 to which the jump was made, a check is performed to see if any further object-n-tuples 301, 302, 303 to be checked are available.This process continues, by way of example, until level 281, 282, 283, 284, 285, 286 is reached, which immediately follows level 281, 282, 283, 284, 285, 286, which is fixed.

[0508] P10001 - where

[0509] March 1, 2026: Segment with the highest level index. The test conditions are discussed below.

[0510] A first check condition is whether the object-n-tuple 301, 302, 303 of the previous level shows overlap with the object-n-tuple 301, 302, 303 of the current level. Overlap preferably means that the n - 1 last objects 231 , 232, 233, 234, 235 of the object-n-tuple 301 , 302, 303 of the preceding level 281 , 282, 283, 284, 285, 286 correspond to the n - 1 first objects 231, 232, 233, 234, 235 of the object-n-tuple 301 , 302, 303 of the current level 281 , 282, 283, 284, 285, 286.

[0511] A second check condition is whether the last object of the object-n-tuple 301, 302, 303 has not yet been used in the object-n-tuples 301, 302, 303 of the preceding levels.

[0512] A third test condition is whether the content of threshold cell 271 of the object-n-tuple 301, 302, 303 of the current level satisfies a predefined threshold condition (see above) in comparison to the content of threshold cell 271 of the object-n-tuple 301, 302, 303 of the preceding level 281, 282, 283, 284, 285, 286 and, if applicable, in comparison to the content of a threshold cell 271 of the object-n-tuple 301, 302, 303 of at least one further preceding level 281, 282, 283, 284, 285, 286.

[0513] A fourth test condition is whether the summed differences between the respective attribute value relation sums of pairs of consecutive segments or object n-tuples 301, 302, 303 from the first level 281 up to the current level 281, 282, 283, 284, 285, 286 better satisfy the optimization goal 218 than a current best value stored in the control device memory 111. For example, if the optimization goal 218 is configured as a minimum, the fourth test condition is fulfilled if the aforementioned summed differences are less than or equal to the current best value.

[0514] As an example, the object-n-tuple 301, 302, 303 of the arrangement representation from the respective level 281, 282, 283, 284, 285, 286 that satisfies the aforementioned test conditions is used as the respective segment of each level 281, 282, 283, 284, 285, 286. If such an object-n-tuple 301, 302, 303 has been determined, the current level 281, 282, 283, 284, 285, 286 is closed and P10001 -wo

[0515] On March 1, 2026, the subsequent level 281, 282, 283, 284, 285, 286 was started, i.e., a corresponding object-n-tuple of the arrangement representation of the subsequent level 281, 282, 283, 284, 285, 286 was determined. Preferably, the levels 281, 282, 283, 284, 285, 286 are nested according to their index in the sense of a loop construction. The first level 281 forms an outermost loop, the second level 282 forms a first inner loop, and so on, until level Z 286 forms an innermost loop. The cross-level computation of an optimization subproblem can also be referred to as recursive computation. For example, it is provided that as long as a subsequent level 281, 282, 283, 284, 285, 286 is started, subsequent levels 281, 282, 283, 284, 285, 286 are present.Furthermore, by way of example, it is provided that if the current level 281, 282, 283, 284, 285, 286 is a last available level 281, 282, 283, 284, 285, 286, i.e., level Z 286, and the object-n-tuple 301, 302, 302 to be checked in the last level fulfills the check condition(s), the determined level vector is used as the optimal arrangement. By way of example, the result is then a completely filled level vector (cf. the level vector shown in Fig. 53 for level Z 286), which represents an optimal arrangement of object-n-tuples 301, 302, 303.

[0516] By way of example, it is provided that the summed differences between the respective attribute value relation sums of pairs of consecutive segments or object n-tuples 301, 302, 303 from the first level 281 to level Z 286 are compared with the current best value. If a global condition is met, namely that the aforementioned summed differences better meet the optimization goal 218 than the current best value, the current best value is overwritten with the aforementioned summed differences and stored accordingly in the control device memory 111. In particular, the control device 110 transmits the then current best value directly to at least some, in particular each, of the computing devices 120, specifically to store the then current best value in the respective computing device memory 121 and thus, if necessary, overwrite a current best value already stored there.Furthermore, by way of example, it is provided that the sum of the aforementioned amounts is to be calculated.

[0517] P10001 - where

[0518] March 1, 2026: The level vector corresponding to the differences is also stored in the control device memory 111. Furthermore, it is provided that if the summed differences of the aforementioned type assigned to other level vectors satisfy the optimization goal 218 in the same way as the current best value, these level vectors are also stored in the control device memory 111. Advantageously, it is provided that while the then current best value is transmitted to at least one of the computing devices 120, the level vector corresponding to this then current best value is not.

[0519] Fig. 54 shows optimization data 290 according to the second aspect of the invention. This optimization data 290 can also be referred to as further optimization data 290. By way of example, the optimization data 290 are provided by the control device processor 112 and stored in the control device memory 111. Furthermore, by way of example, the optimization data 290 are assigned to several levels 281, 282, 283, 284, 285, 286, which is shown accordingly in Fig. 54 with dashed boxes. Each of the flat, dashed boxes symbolizes a level 281, 282, 283, 284, 285, 286. For further example, the optimization data 290 comprises several sub-data 291. These sub-data 291 can also be referred to as further sub-data 291. Each of the sub-data 291 corresponds to one of several optimization sub-problems of the optimization problem.These optimization subproblems can also be referred to as further optimization subproblems. Furthermore, this optimization problem can also be referred to as a further optimization problem. By way of example, three partial data sets 291 are shown in Fig. 54. Also by way of example, the use of several levels 281, 282, 283, 284, 285, 286 is provided, namely a first level 281, a second level 282, a third level 283, a fourth level 284, a fifth level 285, and a level Z 286. As a further example, the partial data sets 291, as shown in Fig. 54, are each represented as a level vector. Each coordinate of each level vector corresponds to a fixed segment. If no segment is fixed, the relevant coordinates of the level vector(s) each contain the value 1 (see, for example, Fig. 54: the coordinates of the level vectors corresponding to level Z 286).

[0520] P10001 -wo

[0521] 01.03.2026

Claims

1. Patent claims 1. Computer-implemented method for the efficient computation of an optimization problem using a computer system (100) comprising a control device (110) and several computing devices (120) coupled to the control device (110) for data transmission, comprising the steps of: - Providing input data (205) in a control device memory (111) of the control device (110), wherein the input data (205) relate to a plurality of objects (231, 232, 233, 234, 235) that are related to the optimization problem, - Accessing the control device memory (111) using a control device processor (112) of the control device (110) to extract the following extraction data (210) from the input data (205): a number of objects (217), several attribute values ​​(211, 212, 213, 214, 215) corresponding to a property attribute of the objects (231, 232, 233, 234, 235), an optimization goal (218) corresponding to the optimization problem, and storing the extraction data (210) in the control device memory (111), - Providing a relation matrix (220) stored in the control device memory (111) by means of the control device processor (112), wherein the relation matrix (220) comprises a plurality of relation cells (221), wherein at least some of the relation cells (221) each contain an attribute value relation between the attribute values ​​(211, 212, 213, 214, 215) of two objects (231, 232, 233, 234, 235) corresponding to the relation cell (221) in question, - Providing optimization data (240) stored in the control device memory (111) by means of the control device processor (112), wherein the optimization data (240) are assigned to several representation factors (242, 243, 244, 245), at least one of which depends on the number of objects (217), wherein the optimization data (240) comprise several sub-data (241) each corresponding to one of several optimization sub-problems of the optimization problem, P10001 - where 01.03.2026- Providing, by means of the control device processor (112), a total result matrix (250) comprising a plurality of total result cells (251) in the control device memory (111), wherein the total result cells (251) each have an initially empty content, where, with respect to each of the sub-data (241), the following steps are further carried out once: i. Assigning the partial data (241) by means of the control device processor (112) to one of the computing devices (120) for which at least one of the following conditions is met: no partial data (241) are assigned to the computing device (120), an optimization partial result of an optimization subproblem corresponding to the partial data (241) assigned to the computing device (120) is or is stored in the control device memory (111), ii. In response to the fact that partial data (241) have been assigned to the computing device (120), computation according to a computational rule that is the same for all optimization subproblems of the optimization subproblem corresponding to the assigned partial data (241), taking into account these partial data (241) and the relation matrix (220), and by means of a computing device processor (122) of the computing device (120) in question, in order to obtain a respective optimization sub-result, and iii. Transmitting the optimization partial result to the control device (110) by means of the relevant computing device processor (122) and storing the optimization partial result in the control device memory (111) by additively adding the optimization partial result to the respective content of the relevant total result cell(s) (251), wherein, in response to the fact that no more partial data (241) can be assigned to one of the computing devices (120), the contents of at least some of the total result cells (251) form a total result.

2. Method according to claim 1 , characterized by the fact that P10001 - where 01.03.2026 at least some, in particular each, of the computing devices (120) each have a computing device memory (121), wherein the partial data (241) assigned to one of the computing devices (120) are stored or are stored in its computing device memory (121).

3. Method according to claim 2, characterized by that furthermore the relations matrix (220) and / or the optimization data (240) are stored in at least some, in particular each, of the computing device memory (121).

4. Method according to any one of the preceding claims, characterized by the fact that at least some, in particular each, of which partial data (241) are exclusively assigned to a single computing device (120).

5. Method according to any one of the preceding claims, characterized by the fact that at least some, in particular each, of the computing devices (120) are not coupled to another computing device (120) for data transmission.

6. Method according to any one of the preceding claims, characterized by the fact that at least one, in particular each, of which steps i. to iii. are carried out in parallel with respect to at least some, in particular each, of the computing devices (120).

7. Method according to any of the preceding claims, characterized by the fact that An object-n-tuple number of different object-n-tuples (301, 302, 303) is determined, wherein the object-n-tuples (301, 302, 303) each have a tuple content number (n) of at least two objects (231, 232, 233, 234, 235), wherein the objects (231, 232, 233, 234, 235) of an object-n-tuple (301, 302, 303) are each distinct from each other, wherein the object-n- P10001 -wo 01.03.2026 The number of tuples corresponds to a product of a number of factors corresponding to the number of tuple contents (n), wherein a first factor corresponds to the number of objects (217), wherein further factors each correspond to a difference between the number of objects (217) as the minuend and an integer starting with the value one and increasing by one from factor to factor as the subtrahend, wherein in the last factor the subtrahend corresponds to the number of tuple contents (n) less one, wherein each object-n tuple (301, 302, 303) has a number of object positions corresponding to the number of tuple contents (n), wherein for at least some, in particular each, of the object-n tuple (301, 302, 303) each object position is assigned an object position index (j), wherein the object position index (j) takes integer values ​​in assumes a range from one to the number of tuple contents (n).

8. Method according to claim 7, characterized by the fact thatA relation sum matrix (260) stored in the control device memory (111) is provided by means of the control device processor (112), wherein the relation sum matrix (260) comprises a plurality of relation sum cells (261), wherein in at least some, in particular each, of the relation sum cells (261) a relation sum number of attribute value relation sums corresponding to the number of object n tuples is stored, wherein the attribute value relation sums are determined for at least some, in particular each, of the object n tuples (301, 302, 303), wherein the attribute value relation sums each correspond to a sum of attribute value relations assigned to a single object n tuple (301, 302, 303), wherein the attribute value relation sums assigned to a single object n tuple (301, 302, 303) assigned attribute value relations each pairwise between a first object (231 , 232, 233, 234, 235) of the object-n-tuple (301, 302,303) and a second object (231 , 232, 233, 234, 235) of the object-n-tuple (301 , 302, 303) taking into account the relations matrix (220) and the optimization goal, whereby the attribute value relations assigned to the individual object-n-tuple (301, 302, 303) are determined for at least some, in particular each, of the possible pairs of first objects (231 , 232, 233, 234, 235) and second objects (231 , 303) P10001 - where 01.03.2026232, 233, 234, 235) are determined, which result from two objects (231 , 232, 233, 234, 235) of the object-n-tuple (301, 302, 303), of which the object position indices (j) are adjacent and of which the first object (231 , 232, 233, 234, 235) has a lower object position index (j) than the second object (231, 232, 233, 234, 235).

9. Method according to claim 8, characterized by the fact that at least some, in particular each, of the sub-data (241) are configured as arrangement representations, wherein the arrangement representations are determined such that at least some, in particular each, of the arrangement representations have a number of representation positions that corresponds to the number of objects (217) increased by one, wherein for at least some, in particular each, of the arrangement representations, each representation position is assigned an object-n-tuple position index (Oti), wherein the object-n-tuple position index (Oti) takes on integer values ​​in a range from one to the number of representation positions, wherein each representation position is assigned an object-n-tuple (301, 302, 303), and wherein the representation position assigned to a first object-n-tuple (301, 302, 303) is assigned a first object-n-tuple position index (Oti). is, which is one of the second object-n-tuple (301 , 302,The second object-n-tuple position index (Oti) assigned to the representation position of the first object-n-tuple (301, 302, 303) is reduced by one, where, in pairs of a respective first object-n-tuple (301, 302, 303) and a respective second object-n-tuple (301, 302, 303) that are assigned to immediately consecutive representation positions, the object position(s) of the second object-n-tuple (301, 302, 303) with one or more second object position index(s) (j) are assigned the same objects (231, 232, 233, 234, 235) that are assigned to the object position(s) of the first object-n-tuple (301, 302, 303) with one or more first object position index(s) (j). are assigned, wherein the second object position index(s) (j) is / are decreased by one compared to the first object position index(s) (j), where representation positions whose object n-tuple position index, P10001 - where 01.03.2026(Oti) is at least two, each of which is assigned an object-n-tuple (301, 302, 303), whose attribute value relation sum, taking into account the object(s) already predetermined by the immediately preceding representation position and assigned to the object position(s) in question (231, 232, 233, 234, 235), best satisfies the optimization goal (218).

10. Procedure after step 9, characterized by the fact that a corresponding number of arrangement representations is determined, where the number of arrangement representations is a product of a maximum of the first representation factor (242), which is four, a maximum of the second representation factor (243), which corresponds to the number of objects (217), a maximum of the third representation factor (244), which corresponds to the number of objects (217), and a maximum of the fourth representation factor (245), which corresponds to the number of objects (217) less one.

11. Method according to any of the preceding claims, characterized by the fact that The procedure further includes the following step: - By means of the control device processor (112) a threshold matrix (270) comprising a plurality of threshold cells (271) in the control device memory (111), wherein the threshold cells (271) each have an initially empty content, where a threshold value is further extracted from the input data (205) as extraction data (210), where, in response to the fact that the overall result has been formed, the following steps are also carried out with respect to at least some, in particular each, of the overall result cells (251) whose respective contents form the overall result: P10001 - where 01.03.2026 - Comparison of the content of the total result cell (251) with the threshold value, - in response to the fact that the content of the overall result cell (251) exceeds the threshold, setting the content of the threshold cell (271) corresponding to the overall result cell (251) to a positive value, wherein, in response to the fact that all contents of the relevant overall result cells (251) have been matched with the threshold, a primary target object arrangement is determined using the threshold matrix (270) and further optimization data (290), wherein the further optimization data (290) comprise several further sub-data (291) each corresponding to one of several further optimization sub-problems of a further optimization problem.

12. Control device (110) of a computer system (100), comprising a control device processor (112) and a control device memory, wherein the control device (110) is configured to perform a method according to one of the preceding claims using the computer system (100).

13. Computer system (100), comprising a control device (110) according to claim 12 and several computing devices (120) each having a computing device processor (122) coupled to the control device (110) for data transmission.

14. Computer-readable storage medium comprising program code which, when executed by a computer system according to claim 13, causes the computer system to perform a method according to any one of claims 1 to 11.

15. Computer program product, comprising a computer-readable storage medium according to claim 14, which in particular is non-transferable, and instructions stored on the computer-readable storage medium which, when executed by a control device (110) according to claim 12, cause the control device (110) to carry out a method according to one of claims 1 to 11 using a computer system (100) according to claim 13. P10001 - where March 1, 2026

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