Classifications

• • G—PHYSICS
  • G06—COMPUTING OR CALCULATING; COUNTING
  • G06Q—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES; SYSTEMS OR METHODS SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES, NOT OTHERWISE PROVIDED FOR
  • G06Q10/00—Administration; Management
  • G06Q10/06—Resources, workflows, human or project management; Enterprise or organisation planning; Enterprise or organisation modelling
• • G—PHYSICS
  • G06—COMPUTING OR CALCULATING; COUNTING
  • G06Q—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES; SYSTEMS OR METHODS SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES, NOT OTHERWISE PROVIDED FOR
  • G06Q10/00—Administration; Management
  • G06Q10/06—Resources, workflows, human or project management; Enterprise or organisation planning; Enterprise or organisation modelling
  • G06Q10/063—Operations research, analysis or management
  • G06Q10/0631—Resource planning, allocation, distributing or scheduling for enterprises or organisations
  • G06Q10/06315—Needs-based resource requirements planning or analysis
• • G—PHYSICS
  • G06—COMPUTING OR CALCULATING; COUNTING
  • G06Q—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES; SYSTEMS OR METHODS SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES, NOT OTHERWISE PROVIDED FOR
  • G06Q10/00—Administration; Management
  • G06Q10/06—Resources, workflows, human or project management; Enterprise or organisation planning; Enterprise or organisation modelling
  • G06Q10/063—Operations research, analysis or management
  • G06Q10/0631—Resource planning, allocation, distributing or scheduling for enterprises or organisations
  • G06Q10/06316—Sequencing of tasks or work
• • G—PHYSICS
  • G06—COMPUTING OR CALCULATING; COUNTING
  • G06Q—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES; SYSTEMS OR METHODS SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES, NOT OTHERWISE PROVIDED FOR
  • G06Q10/00—Administration; Management
  • G06Q10/06—Resources, workflows, human or project management; Enterprise or organisation planning; Enterprise or organisation modelling
  • G06Q10/063—Operations research, analysis or management
  • G06Q10/0637—Strategic management or analysis, e.g. setting a goal or target of an organisation; Planning actions based on goals; Analysis or evaluation of effectiveness of goals
  • G06Q10/06375—Prediction of business process outcome or impact based on a proposed change

Definitions

• the present invention relates to a method for optimizing transport planning and, in particular, to a method for optimizing the storage ranges in a transport network having a plurality of distribution levels by using a computer-assisted optimization algorithm, in particular for quickly determining schedules for shipping and production.
• the individual allocation of a resource has a "return", ie the cost or profit for this allocation.
• the problem is thus to allocate resources in such a way that all restrictions are taken into account and at the same time the return is maximized, ie the disadvantages, e.g. B. costs, minimized or the advantages, z. B. Profits can be maximized.
• linear programming model One method of presenting such arbitration problems is known as the linear programming model.
• Such a model consists of a series of linear relationships, which are presented in a matrix form and quantitatively represent the relationships between allocations, restrictions and the results of the optimization process. In the linear relationships, the sum of constant coefficients multiplied by unknown allocation values is shown.
• Such modeling of linear programming is achieved in a multidimensional space, with multidimensional vectors providing a multidimensional figure or a polytope, each facet on an area thereof being limited by equations that define relationships between allocated resources in the process.
• One example of a solution to the linear programming problem is the simplex algorithm, which was developed by George Dantzig in 1947.
• a solution alternative is the Karmarkar algorithm.
• Network flow algorithms have many uses for planning problems in industry. You can for example, in the case of problems with regard to assignment, transport, minimum cost flow, shortest route and largest flow through a network.
• the optimization problem in this case is to reduce the transport costs to a minimum, on condition that all requirements in the distribution centers are met.
• the minimum cost flow problem is only the transport problem with intermediate nodes in the network.
• the sheets can have the smallest and largest capacity.
• the problem with the shortest path is a graph with positively weighted arcs.
• the optimization problem is to find the shortest path between two given nodes in the transport network.
• the problem of the greatest flow through a network is similar to the transportation problem, except that the arches between the nodes have limited transportation capacity but no transportation costs.
• the optimization problem is to get a large flow through the network without violating the transport capacity.
• the object of the present invention is to provide a new algorithm for maximizing the range of coverage profiles in the distribution problem that arises with planning systems in industrial production.
• Several algorithms are proposed that are applicable to this problem. The decision regarding which algorithm is appropriate for a given distribution problem depends on the size of the problem and the maximum acceptable CPU time for the calculation.
• the present invention proposes a new formulation of the network flow problem, which takes into account different types of transport, time and date restrictions, priority priorities and defined production flows. The algorithmic function is then applied to this formulation.
• the task of the algorithm is to select the free variables of the area profile formulation in such a way that first the area of coverage profiles is maximized and then the transport costs are minimized.
• the proposed algorithm can use any minimum cost flow algorithm as a basic building block.
• the present invention relates to a planning system for optimizing the flow of goods and goods in complicated distribution networks using a new, efficient algorithm.
• the new algorithms to be applied to this wording are introduced.
• the distribution algorithm described here is mainly intended to react to disruptions in the production process or to short-term changes in the distribution process.
• the system is well suited for products that are produced in large quantities and for which the demand fluctuates greatly.
• the optimization process outlined here tries to reduce transport costs to a minimum and avoid bottlenecks by planning for the future (push distribution). If there are still bottlenecks, the aim is to react optimally (fair share distribution) by setting priorities for the needs.
• the basic structure of a goods distribution system to which the present invention can be applied consists of three elements: the production facilities, the central warehouses and the distribution centers. Brief description of the drawings:
• Figure 1 is a graphical representation of a simple distribution network.
• Figure 2 is a graphical representation of a complicated distribution network.
• Figure 3 is a graphical representation of the mapping problem in the form of a two-sided graph.
• Figure 4 is a graphical representation of a network with balanced distribution.
• Figure 5 is a graphical representation of production and demand nodes with different possible assignments.
• Figure 6 shows a production facility with supply and demand.
• Figure 7 is a graphical representation of a distribution network with multiple modes of transport.
• Figure 8 is a graphical representation of an optimized distribution network with several modes of transport.
• Figure 9 is a flow diagram of both possible and optimized transport routes.
• Figure 10 shows another example of possible and optimized transport routes.
• Figure 11 shows the influence of impermissible periods on the possible transport options in a distribution network.
• Figure 12 is a graphical representation of a simple distribution network with time values on the sheets.
• Figure 13 is a graphical representation of the production and demand profiles of a distribution network.
• FIG. 14 shows a graphic representation of a global area profile for a possible assignment.
• FIG. 15 shows a graphic representation for a global area profile for a further possible assignment.
• Figure 16 shows an example of a monotonous optimal transport plan.
• Figure 17 is the search for the maximum monotonous area profile.
• FIG. 1 shows the simplest case of a distribution network.
• goods can be produced using production facilities that are available in limited quantities. These goods are then temporarily stored (eg in warehouses) 104.
• warehouses e.g., warehouses
• intermediate storage facilities or distribution centers 106 are required.
• a larger transport volume can be achieved with the intermediate storage facilities (intermediate nodes) or the central warehouses 104.
• central warehouses 104 leads to longer delivery times, but also to lower transport costs.
• a distribution center can also receive a direct delivery if necessary.
• This particular representation only shows a schematic structure.
• network structures can reach much larger dimensions with a large number of plants, warehouses and distribution centers. As shown in FIG. 2, these network structures comprise a number of manufacturing plants 202, central warehouses 204 and distribution centers 206.
• the task of the distribution problem can be summarized as follows.
• the sources are systems for which a planned production profile is available.
• the sinks are the distribution centers for which a demand profile (sales orders or a forecast ornamental need).
• the transport network consists of sheets with transport costs and time values between the intermediate nodes.
• the present invention searches for the optimal allocation of the spatially and temporally distributed production to the likewise spatially and temporally distributed need, so that the storage range of cover profiles is maximized.
• two cases are distinguished. In the "push" case, the number of products available exceeds demand and the products are distributed such that the area of coverage of each increases uniformly.
• the number of products available is below the requirement, and the products are distributed in such a way that more important customers are given priority and customers with the same priority are treated similarly.
• the invention prevents serving a customer need with lower priority before serving a customer need with higher priority.
• the minimum of the range of cover profiles is applied to all distribution centers.
• each priority class is treated sequentially and coverage is maximized to serve these high priority needs. If various types of transport, transport routes and sources of supply are also available, the entire distribution plan should continue to be optimized so that the total costs are as low as possible.
• the transport problem or the assignment problem plays a central role in the approach to solving the distribution problem described above.
• the underlying mapping problem is defined as follows: in a two-sided graphical representation with an equal number of left and right nodes, each node on the left should be assigned to a node on the right be ordered so that each node has exactly one partner. The resulting connection ("arch") between two nodes is associated with certain costs. The resulting total cost should be minimal.
• FIG. 3 describes the assignment problem in the form of a graphic representation. Arrow 302 represents a possible assignment of a source 304 on the left to a sink 306 on the right. The costs of assignments are entered as values 308 of the arcs.
• the classic transport problem differs from the assignment problem in that a node on the left side (now called “producer”) can be assigned to any number of nodes on the right side (now called “consumer”).
• the problem to be solved is how to find a suitable distribution of materials between the nodes so that the so-called inventory area is maximized. This means maximizing the smallest inventory across all nodes and periods.
• the solution with the lowest transport costs should be selected from the possible solutions (solution solution).
• the storage area of a warehouse is defined as the number of consecutive periods (including the current period) for which the current stock is sufficient. A range of 1 thus means that the stock exactly covers the needs of a single period, namely the current period.
• FIG. Figure 4 shows an example with two plants 402, 404 and two centers 406, 408. Normally there will be considerably more distribution centers than production plants.
• a production plant i has a production rate p (t) in a period t.
• a distribution center j there is a customer demand of O j (t).
• the initial inventory of all facilities and centers is zero. If the area is set for 1 period (that is, the stock of a distribution center in a certain period only has to be sufficient for this period), the following situation arises:
• the needs of a distribution center j in a period t can be covered by all productions that can be delivered to the distribution center in good time (i.e. not later than period t). To meet this need, the production of all systems i could therefore be used in all periods t ' ⁇ tT ⁇ j (where ⁇ j is the transport time).
• the transport times are set uniformly to one period.
• Transport costs Ci j arise when a material unit is transported.
• FIG. 5 shows that Si in our network 'situation, in the extended window in three consecutive periods (which in the network computation, the additional variable of time is introduced). The number of periods considered is called the horizon.
• each node 502 now represents a daily production or daily requirement.
• the arrows 504 no longer correspond directly to the transport routes, but instead correspond to the logical assignment between a specific production unit and a specific demand unit.
• everyone Arch can therefore be clearly identified by the location of the source and the destination (i and j) and by the start and end time (ti and t j ).
• the assignment of a certain amount of material from a node on the left to a node on the right is called the flow on the arch. This flow leads to costs in the amount of the corresponding transport costs d j in the distribution network.
• the transport network is referred to as the optimization network from now on, and the optimizer is referred to as the transport optimizer.
• the nodes in the distribution network are referred to as the distribution network nodes, and the nodes in the optimization network are referred to as optimization network nodes if both are referred to.
• the nodes in the optimization network are divided into sources and sinks.
• the target inventory L 2 (t) in a period t for a given area R (t) results from the orders of the following periods (the security inventory ⁇ ii (t) is not yet included):
• the demand 2 (t) depends on the difference of the target stock L 1 (t) - (t-1), which in turn depends on the area R (t)) and the sales order o 2 (t)).
• the demand d (t) results in:
• this node changes to an offer node with a positive offer:
• Si (t) - (o ⁇ (t) - tr ⁇ (t) + L x (t) - L ⁇ (t-1)).
• a distribution center can become a supply node (instead of a demand node) at a certain time t.
• a distribution center i at a specific point in time t can be assigned to any need of another node with regard to the transport time.
• the transportation time and transportation costs for the future needs of the same distribution center i are set to zero.
• the initial inventory of distribution centers can be modeled by defined deliveries tr (0).
• FIG. 6 shows a small network with a production facility 602 and two distribution centers 604, 606. In the distribution center 1, for example, there is an oversupply due to a very large initial inventory in period 1. This can now be assigned to the needs of period 2 608 or the needs of distribution center 2.
• FIG. Figure 7 illustrates a distribution network with several possible means of transportation, such as by truck 702 or by plane 704. Different arches correspond to different means of transportation in this case.
• the routes are always marked with the means of transport used and with the time 706 of the transport (in periods).
• each arch correlates with a maximum transport time. For example, only one means of transport can be selected for the assignment of an offer from period 1 to a requirement in period 2: no more than one period needed for transportation. It also makes sense to choose the best option, not just a feasible one.
• FIG. 8 The connection with the transport problem is shown in FIG. 8, in which an optimization network is shown in which there are several types of transport. The projection of transport costs is expanded by the different types of transport used.
• FIG. 9 shows an example with the shortest transport time between two nodes, the given network is shown above, and the associated transitive envelope below.
• the first flow diagram 902 shows the different transport routes and times. It takes three periods to get from node 904 to node 906. However, it only takes two periods to get from node 904 through node 908 to node 906. With this in mind, flowchart 910 is created that shows the minimum times required to move between the nodes.
• the number of the maximum permitted modes of transport in the network is specified.
• the modes of transport correspond to different speed classes (for example normal and express). If a direct connection is to be defined between two nodes, the costs and the duration are specified for each type of transport (ie speed, class, etc.).
• the optimizer then calculates the best way for each class separately, as described in the last section. This data is then used in the optimization network. In the case of routes with multiple segments, the modes of transport are not mixed together. If only one transport option is to be permitted on a route, the same data is used for all transport types.
• FIG. 10 shows an example. The drawing shows the network entered above and the network created from it by the optimizer below. Path 1002 shows a manufacturing center 1004, a central warehouse 1006 and a distribution center 1008.
• impermissible arches Every transport connection and every distribution node can be provided with a list of impermissible periods. Inadmissible periods are time windows such as public holidays, on which the transport is at a standstill or on which no goods can be accepted or issued. It is recalled from the classic case described above, in which each node in the optimization network stands for a starting point i, a starting time t 2 , a destination j and a specific target time t 3 of a specific material assignment. In order to take public holidays into account, each sheet is pushed into the future until t ⁇ falls on a permissible period for the delivery of goods at node i (that is, a common time delta is added to both the start time and the finish time).
• FIG. 11 shows an optimizing network with one
• the aim of the present invention is to supply the distribution centers of the demand nodes with the available material in such a way that the range profile of the inventory is maximized across all warehouses.
• the task is therefore to find a suitable assignment from the supply nodes to the demand nodes. So far, it has been shown how an assignment to a given area profile is determined which fulfills the above-mentioned conditions (minimum transport costs, the holidays, etc. are taken into account). This is done by mapping a transport problem and solving it using an efficient transport algorithm, for example the MODI algorithm, which can serve as the basis for the improved algorithm proposed by the present invention.
• the problem of maximizing the area of coverage is defined below.
• the range profile of each individual distribution center has to be described.
• the task according to the invention consists in balancing the ranges between the warehouses as much as possible at a given point in time, it is sufficient to speak only of a global range profile, which is then valid for each distribution center as the lower limit.
• the global profile R (t) is the minimum for the local profile R ⁇ (t) over the node I.
• the maximum area profile is the profile that is maximum in each period. This is not the case. The maximum range in a given period can differ significantly from the range in another previous period.
• FIG. 12 shows an example of a network with two production plants and two distribution centers. The corresponding transport times are entered on the sheets.
• FIG. 13 the associated production and demand profiles are shown.
• the example is based on the simple model of a distribution network and does not contain any security stock, initial stock or fixed access. All requirement classes have the same priority and there is only one type of transport.
• This simple model is always used for the following consideration of the range profile. In a certain period, the maximum possible global range is the range that both distribution centers can reach simultaneously in this period. In period 2, the range has a value of three. This value is only achieved if the production of the first period of plant 1202 (40 units) is allocated to the distribution centers in equal parts.
• Plant 1202 has no influence on the area in the second period because the material from plant 1202 cannot be delivered before the third period. With this assignment, a range of one automatically results for the distribution center 1 in the fourth period, since this distribution center is only reached by the plant 2 in the fifth period and because the initial production of the production plant 1 is already "used up".
• FIG. 14 shows the maximum global range for this first assignment. The minimum inventory of both Centers 1206, 1208 at the beginning of each period are shown in parentheses. There is always an infinite range if the inventory is sufficient for at least all subsequent periods, since a requirement of zero units is assumed for periods beyond the horizon.
• R 1 > R 2 : t in ⁇ R l (t) ⁇ R (t) ⁇ R 2 (t) ⁇ > min ⁇ R 2 (t) / R 1 (t) ⁇ R 2 (t) ⁇
• R mm is defined as the precision or granularity with which this area is to be determined, and H stands for the horizon.
• the optimizer must always optimize the data of the entire horizon.
• monotonous range profiles are first considered.
• this approach is improved to non-monotonous areas of range profiles. Since the minimum is always at the beginning of a monotonically increasing function, the profile that is the maximum among the monotonically increasing profiles is the first (with the smallest t) that achieves a greater value than the other profiles.
• a transportation plan that is calculated based on the maximum monotonous area profile is very good, but unfortunately it is not always optimal. An example shows this.
• the table in FIG. 16 shows a possible situation in a distribution network with one plant and two distribution centers. The transportation time is evenly a period.
• the maximum monotonous range profile R "1602 has a value of 1 in each period (with the exception of the last period, since the value 0 is assumed for the need of periods beyond the horizon), since no larger range is possible in period 5. Die The table shows the plan that results when the offer in the optimization network is assigned to the demand and when the delivery begins immediately.With Z M 1604, for example, the production of 30 units in the first period becomes the first three needs of the distribution center 1206 (see Fig. 12) If all 3 assigned deliveries begin immediately, ie in this first production period, then a total transport of 30 units takes place to the distribution center 1206.
• the transport plan Z M 1604 is sufficient for the maximum monotonous Range profile from; however, it forms an actual global range (that is, the minimum of the actual range of the distribution centers) of R ' M , 1606, which is smaller than the maximum possible range R 1608 (with the assignment Z).
• Plan Z M 1604 is very good for the maximum monotonous range (it is monotonously optimal), but is not completely optimal.
• An optimal transport plan for the first period is as follows: The assignment of offers from the first period should not have a negative effect on the area in all periods. The following assumption can significantly reduce the effort of such a search according to this plan. In order to obtain the monotonous optimal transport plan for the first period, it is sufficient to limit the search for the maximum area profile to those profiles that are not constant only in the initial area, i.e. with t ⁇ max 1 / J ⁇ T 1J ⁇ T 2J transport time between nodes i and j. You only have to increase the range profile up to the period of the maximum transport time; in all subsequent periods the profile can have a constant value.
• the optimization network delivers a maximum range profile? _ ⁇ , which is at least as large as in each period the maximum profile i? max across the entire horizon:
• the short horizon After the short horizon has been defined, it should be shortened to one period.
• the resulting optimization network (which only makes sense if it is possible to carry out the transport in one period) is very small, and the arc flows can be calculated very quickly.
• the maximum range in period 1 is determined by a binary search.
• the second period In the next step the second period is added and an attempt is made to find a maximum area profile over both periods.
• the range profile from the first step which so far only applies to the first period, is retained, and the second period is initialized with the value of the previous period (in this case, the first). If a solution is found for this profile, the range of the second period can again be maximized using a binary search, etc. If for a short one
• FIG. 17 shows an example according to the method according to the invention.
• Step 4 1702 shows a step backwards.
• the range in period (time interval) 4 cannot be kept at the value from period 3 and is therefore reduced to a value of 1.25. All previous periods with a larger area are reduced to this area. Then you start to enlarge the area of the affected period again.
• the value found for the range in the last period of the horizon just considered, i.e. R fH ⁇ ) represents an upper limit for the range that can be achieved in this period. Since the maximum monotonous profile also tolerates transportation plans that make it impossible to achieve a general maximum profile (for example, in this section), these limits can be used to visualize possible errors in the calculated monotonous profile and possibly react to them interactively.
• R ⁇ ax is the maximum range that can occur in a period
• R. is the accuracy (granularity) with which this range can be determined
• T ⁇ ax is the maximum transport time between two nodes (for the slowest transport mode)
• H tolal is the total horizon (for the
• Total running time including the running time for the transport optimizer.
• a combination is particularly useful when an insufficient supply is not just a short-term problem, that is, when production is generally below demand. In this situation, delays would continue to accumulate without reducing demand.
• Distribution network node i in period t is additionally marked with its priority P.
• the unimportant priority classes i.e. orders in these Classes
• the integer part LR (t) J specifies the number of priority classes whose orders are completely fulfilled.
• Part R (t) - LR (t) J is the part that is fulfilled by the next priority class.
• 4 priority classes are defined:
• Forecast Additional demand that is only forecast but has not yet been ordered by a customer
• the definition of a maximum range profile according to the invention guarantees that no need with a higher priority is satisfied to the disadvantage of a need with a lower priority.
• this problem is solved by introducing additional sheets with reduced transport time.
• the delivery with a delay time ⁇ corresponds to the reduction of the corresponding transport time by ⁇ .
• the expression 2 C max guarantees that by changing, ie by exchanging the assignment s 1 (t 1 ) -> d -, (t-,) and s 1 - (t 1 -) -> d -, - (t -
• R (t) J specifies the number of priority classes whose orders to be fully met.
• the rational part R (t) - LR (t) J is the part that is fulfilled by the next priority class.
• Cost transportation cost + delay penalty cost
• R (t) range of all distribution centers to
• Priority class P for distribution center j at time t m t) safety stock in distribution center j at time t
• Material unit with transport type D between nodes i and jr D transport time for the transport of a
• step 4 Various alternatives to step 4 according to the invention are proposed in the following section.
• the overall problem is first discussed, i.e. the efficient construction of a maximum range of range profiles.
• This algorithm can be significantly accelerated by restricting it to a monotonous range of range profiles, described below.
• the following is a fast algorithm for monotonous profiles to maximize the range profile.
• the actual profile is iteratively increased by 1 whenever this is possible for each time step that spans the entire planning horizon several times.
• R (t): R (t) + l; ⁇ R enlarge ⁇ Generate min_cost_flow problem F (R) for profile R (t)
• R (t): R (t) + ⁇ ; ⁇ R enlarge ⁇ Generate min_cost_flow problem F (R) for profile R (t)
• Step 4 algorithm - monotonic range of
• R (t + l) R (t); Generate min_cost_flow problem F (R) for profile R (t)
• the production should be shipped to at the start time t 0 , i.e. the shipping planning y lt0 J at time t 0 should be expandable to an optimal planning, however the shipping planning for the following time steps may not be optimal .
• they are corrected by shifting the planurig window.
• the range profiles are restricted to the following type: Evenly up to the maximum
• Step 4 algorithm - rolling planning scheme
• the maximum transport time is specified, ie t max
• R max : R max (t ma ) ⁇ change last constant part of R ⁇ Determine min_cost_flow problem F (R) for the profile R (t)
• step size ⁇ of the possible values for the area of the cover profile R (t) must be limited. If the
• Step size ⁇ is small, many iterations are required until the maximum profile is found.
• the following sections describe how the search for maximum monotonous profiles can be accelerated by a binary search process.
• step 3 If the whole profile is not set, go back to step 1 (and solve the remaining part).
• step 2 the time steps t must be selected for which
• T planning_end-planning__start (number of time steps in the planning window)
• T * 0 (R m> where 0 (R tn ) is the temporal complexity of the algorithm for the maximum monotonous range of the coverage profile
• the algorithmic complexity increases compared to monotonous profiles advantageously by at most a factor T, which makes it possible to even calculate extensive range profiles with conventional computer systems.

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