EP1088283A1

EP1088283A1 - Method and system for maximising the range of stock management requirement profiles

Info

Publication number: EP1088283A1
Application number: EP99936457A
Authority: EP
Inventors: Heinrich Braun
Original assignee: SAP SE
Current assignee: SAP SE
Priority date: 1998-06-19
Filing date: 1999-06-18
Publication date: 2001-04-04
Also published as: WO1999067729A9; CA2332995A1; US6341266B1; WO1999067729A1; IL139850A0; AU5155199A; IL139850A; AU764495B2; JP2002519265A

Abstract

The invention relates to a method and system for carrying out stock management in a multilevel distribution chain by applying optimisation algorithms to a range profile built on a distribution network formulation and on different elements which are invoiced in the network. The optimisation problem is defined in a formal way and algorithms for maximising stock autonomy by means of requirement profiles are proposed. Said optimisation process allows transport costs to be reduced to a minimum and bottlenecks to be avoided in stocks. But should bottleneck occur, the system allocates priorities according to need and provides an optimum solution for monotonous product distribution. The inventive method comprises the following steps: initialisation of stock autonomy profile with a starting value; calculation of required entries to satisfy profile restrictions; creation of the most cost-effective flow for said needs by means of a minimal autonomy of the requirement profile until a solution is found.

Description

Process and system for maximizing the range of coverage profiles in inventory management

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.

General state of the art:

In many industries related to the manufacture and distribution of goods, there is a need for the allocation of inventory and transportation resources. These allocation decisions are usually subject to restrictions in terms of equipment, time, costs, storage capacity and other parameters that influence the result of a distribution process, in particular a goods distribution process. As an example that is of particular interest here, there is a need to optimize the distribution and inventory sizes of a supply chain that has multiple manufacturing facilities, with multiple distribution centers strategically located near customers. Resource allocation decisions are usually restricted. On the one hand, resources are always limited by their overall availability, and the usefulness of a certain resource in a certain application can also be limited. For example, the traffic management capacity of each individual link in a telecommunications system is limited, while the total traffic offered to the communication system is also limited. 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.

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. There are other optimization algorithms that can be used to solve minimum cost flow network problems. Network flow problems represent a special case of linear programming.

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.

In the case of assignment problems, there is a two-sided (bipartite) graph that includes a number of productive nodes and consuming nodes (for example, workers and tasks). The sheets between the workers and the tasks are weighted with costs for assigning the worker to the task. The optimization problem in this scenario is to assign a task to each worker in such a way that the total cost is reduced to a minimum. When it comes to transportation problems, there is a two-sided graph, where the productive and consuming nodes are factories and distribution centers. Every factory plant produces units, and every distribution center has a need for these units. The sheets between factories and distribution centers are weighted by the transportation costs of the units. 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. In addition, 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. However, in the prior art, network flow problems are limited to networks with nodes and arches in between, and the flow is charged with a linear penalty due to transportation costs between them. There is a need for a network flow solution where minimizing transportation costs is a lower priority and where maximizing the range of coverage profiles in the consumption nodes is of paramount importance. As in the prior art, this problem can neither be formulated as a linear optimization problem nor as a network flow problem.

Brief description of the invention:

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 The proposed algorithm can use any minimum cost flow algorithm as a basic building block.

Detailed description of the invention:

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. First, the wording of the planning system is discussed, then the new algorithms to be applied to this wording are introduced. When setting up the scenario in which the algorithm is to be used, it is assumed that the production volume to be distributed for the period under consideration has already been determined and that demand data are also known. 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:

The invention is explained in more detail below with reference to exemplary embodiments shown in figures. Show it:

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. In a production facility 102, goods can be produced using production facilities that are available in limited quantities. These goods are then temporarily stored (eg in warehouses) 104. In order to guarantee short goods transport routes to customers and to guarantee fast and economical delivery, 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. The use of 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. In practice, 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. There are sources, sinks and a transport network in the typical supply chain. 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. As mentioned above, 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. In the "fair share" case, 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. In the "push" case, the minimum of the range of cover profiles is applied to all distribution centers. In the "fair share" case, 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. In addition, 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.

The development of the problem to which the proposed algorithms should be applied will now be described.

THE CLASSIC CASE

In the simplest case, consider a simple two-way network with production facilities (PA) on one side and distribution centers (distribution center) on the other Page. In this classic case, deliveries are always made from the systems to the distribution centers and not in the opposite direction. A PA can supply any distribution center. 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. In 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). In this first example, 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. In this new scenario, 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.

As you can see, this is a classic, simplified transportation problem as it is known from Operations Research. As such, it can be solved using a standard Operations Research approach (for example, the MODI algorithm as is known in the art). Since the actual optimizer works at the level of the transport problem, 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.

This particular case of a distribution network is easier to deal with, since the production and needs of the optimizing network correspond exactly to those in the distribution network for a single period. If the area is enlarged over several time periods, this is no longer the case. The exact transfer of these values from the distribution network to the optimizing network is described in the next section. FROM THE DISTRIBUTION PROBLEM TO THE TRANSPORT PROBLEM IN THE CLASSIC CASE.

In the previous section, the distribution problem for a specified area was projected from a period onto the transport problem. The goal now is to generalize the projection for every possible area. Since the values of the sinks in the optimizing network already deviate from the requirements in the distribution network by a range of more than one period and since the source values can differ from the production values in a distribution network, a further distinction is made with regard to the data of the optimizing network made specifically from the original data in the distribution network. Furthermore, there is always talk of supply and demand in relation to the optimizing network - the terms production and order always refer to the values in the distribution network. The supply in a period t of a node i in the optimizing network is defined as s (t) and the demand is defined as d _∑ (t).

Definition of the area of coverage

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 πii (t) is not yet included):

t + [R (t) JL ₁ (t) = Σ o _Σ (τ) + (R (t) - LR (t) Jo ₁ (t + fR (t) 1) τ = t + l Alternative definition / generalization of the area of coverage

The general statement about the area of coverage (storage range) can alternatively be written in such a way that it contains a maximization of the multiplier of the current inventory:

Li (t) = θi (t) + R (t) * security inventory or generally for each monotonically increasing function f with t) = f (R (t)).

For the sake of clarity, only the first-mentioned definition of the area of coverage is considered below.

In the case of a distribution center, 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:

Distribution centers: d _x (t) = o ₁ (t) + L (t) - L ₁ (t -1)

[The conditions: d _λ (t) ≥0 <=> R (t) ≥ R (t -l) -1 should be fulfilled t. ] The offer s (t) of a system corresponds to real production in this simple model. A plant normally has no personal needs, which means that it has no target area that must be reached. The offer of the system s (t) results in:

s (t) = p _x (t)

Now the transport problem as shown in the previous section can be defined. In FIG. 5 only p _λ (t) has to be replaced by s ₂ (t) and o (t) by d _∑ (t). FROM THE DISTRIBUTION PROBLEM TO THE TRANSPORT PROBLEM IN THE GENERAL CASE.

The models for projecting the transport problem shown so far are limited to essential values. A model is now created that takes security stocks and orders with multiple priority classes into account:

t + R (t)

L, (t) = m _] (t + R (t)) + ∑ ∑ of (τ) τ = t + l P≤P forecast

(Generalization for non-integer R (t) see above)

Note: If transit loads tr (t) (specified deliveries) are also taken into account, the need is reduced to:

d _λ (t) = o (t) - tr _x (t) + L _x (t) - L (t -1).

Whenever the transit freight is large enough, this need can possibly become negative, i.e. this node changes to an offer node with a positive offer:

_Si (t) = - (o _λ (t) - tr _λ (t) + L _x (t) - L _λ (t-1)).

This means that whenever the specified delivery tr (t) is large enough, a distribution center can become a supply node (instead of a demand node) at a certain time t. In this case, such an offer from 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. In particular, the transportation time and transportation costs for the future needs of the same distribution center i are set to zero. At the beginning of the planning horizon t = 0, 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.

DIFFERENT TRANSPORT OPTIONS

The previous section was a very simple one

Network with only one means of transport between two nodes is taken into account. The change that leads to the process of finding a solution when different transportation methods are available is now examined. Different transport methods do not necessarily mean different means of transport, it can also refer to alternatives along the way (for example in the case of truck transports). The longer way has to be worth it, meaning that it only makes sense if it results in lower transport costs.

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). In the representation of the optimizing network, 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. One can then formulate the basic assumption that if a slower mode of transport is chosen, it is also the cheaper one. In principle, this additional assumption therefore only changes the costs of the assignment in the corresponding transport problem. 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.

TRANSPORTATION BY INTERIM STORAGE DEVICES.

In the following, the case is first considered with only one possibility of transport for each direct route between two nodes. The change that occurs when several alternatives are permitted for such a route is then taken into account.

INTERMEDIATE STORAGE FACILITIES IN THE EVENT OF POSSIBILITY OF TRANSPORTATION PER LINE. So far, only a two-tier distribution network has been considered. The consumers (in this case the distribution centers) are supplied directly by the manufacturing facilities. Distribution networks often also use central interim storage facilities. By introducing these facilities, the transport routes to the distribution centers can now be partially bundled together. Deliveries can also be bundled and thus possibly made cheaper. The central warehouse functions like a transit station to support delivery optimization. For this reason, the Production of the systems can be assigned directly to the distribution centers. The sum of the transport time to the central warehouse and from there to the distribution center can be used for the duration of the overall transport (via the intermediate storage facility). The transport times may already include storage and retrieval times. Since the method according to the invention is intended to maximize the area of the cover profiles of the warehouses, deliveries as early as possible and therefore short transport times are advantageous. For this reason, the fastest route between two nodes is always preferred, even if it is not necessarily the direct route. In the graphical representation of the transport times, in order to obtain the shortest total transport duration between two arbitrarily chosen nodes, the transitive envelope of the graphical representation must be calculated. 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.

If the duration of the fastest route has been calculated in this way, then the associated costs are calculated. Analogous to the times, they are made up of the sum of the costs for each part of the route. If there are several quick ways, the cost of the most efficient and / or cheapest among them will be used. INTERMEDIATE STORAGE FACILITIES IN THE EVENT OF ALTERNATIVE POSSIBILITIES OF TRANSPORT FOR EVERY DISTANCE. As explained above, alternative arches in the distribution network can also be taken into account with the method according to the invention. When transporting via intermediate nodes, the question arises as to which arch should be used for the different segments. If all combinations are permitted, then there are many possibilities for transport between the end nodes if there are several intermediate nodes. On closer inspection, however, it turns out that not every combination of these arcs makes sense. For example, one would normally not use an express connection in a first segment and a slow connection in a second segment. For this reason, the following procedure was chosen to enter the connections:

First, 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. It shows a transport route 1010 by train, which requires 1 period, transport route 1012 by truck, which requires 3 periods, transport routes 1014, 1016 by plane, which each require 1 period. The arcs of the normal mode are shown with dots, the arcs of the express modes are shown with lines. There is only one real connection between nodes 1006 and 1008, namely transport by plane. Route 1018 shows the routes compressed by the optimizer between manufacturing center 1004, central warehouse 1006, and distribution center 1008.

PROHIBITED PERIODS

The implementation of impermissible arches is discussed below. 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 _λ 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). Thereafter, t _{j is} also postponed until the sum of the permissible transport periods between t and t _{3 is} sufficient for the corresponding transport, and until t ₃ falls on a permissible period for accepting goods at node j. FIG. 11 shows an optimizing network with one

Horizon from six periods 1102, 1104, 1106, 1108, 1110, 1112. The nodes are marked with the distribution network node number i and the associated period number t _± . The impermissible periods are shown hatched / gray. In period 2, goods acceptance in distribution network node 1 is prohibited. The delivery of goods is not permitted in node 2 in period 5. Transport is not permitted in period 3 1114. The described distribution of the impermissible periods is of course an extreme case. In most cases, the holiday will affect at least the areas that receive and issue goods at the same time.

THE SEARCH FOR THE MAXIMUM AREA PROFILE.

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. MAXIMIZING THE RANGE PROFILES IN THE EVENT OF OVERBUILDING An oversupply means that the planned customer needs can be fully met. Thus, the transport optimizer has found a solution with the minimal range profile of one period across all distribution centers and times. Now the range profile must be maximized. At first glance, the only option is to try out different profiles with the given options and gradually work towards the optimum, e.g. B. by a trial-and-error process. However, assumptions regarding the existing supply and demand situation that do not unduly restrict the general validity of the process can greatly accelerate the search. However, one has to accept the risk that one cannot always find an optimal solution. Before a more detailed discussion, the question should be answered what is meant by a maximum range profile.

HOW DOES A MAXIMUM RANGE PROFILE LOOK?

If you want to describe the range of all distribution centers in the network, the range profile of each individual distribution center has to be described. However, since 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. More precisely, the global profile R (t) is the minimum for the local profile R _λ (t) over the node I. The local profiles may sometimes be above the global profile; however, they can never be below, ie: R (t) = min {R _x (t)}. As before, we will only speak of the range profile (which affects all distribution centers). One would intuitively assume that 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. In 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.

Do you want the global reach in period 4 of

Raise 1 to 2, the following assignment is possible. Of the 40 units produced in period 1 by plant 1202, 30 units are assigned to distribution center 1206 and 10 units to distribution center 1208. Of the 40 units of the first period in plant 1204, at least 20 units are assigned to distribution center 1208. With a suitable distribution of the remaining production of system 1202 (for example to the same parts in both centers 1206, 1208 ^" ), the global range shown in FIG. 15 is achieved.

This assignment has the additional advantage that the minimum of the global range is 1 over all periods

2 has risen. This example shows that it is not always cheap, starting with the largest possible global

To build reach. This sometimes prevents the possibility of later balancing out a minimum of the global reach. This process of balancing the inventory is desirable in any case in order to maintain a good overall profile. The range profiles in FIGS. 14 and 15 must therefore be classified in such a way that the range profile in FIG. 15 turns out to be better than the profile in FIG. 14 when evaluated. It is not enough to only compare the global minimum of the profiles (as one might think), since the other periods, in which a range may be just slightly above the minimum, are not taken into account in a comparison. The following order is therefore defined for the global range profile: Global Area Profile Rating:

The following applies to two global area profiles R ¹ and R ² :

R ¹ > R ² : t => in {R ^l (t) \ R (t) ≠ R ² (t)}> min {R ² (t) / R ¹ (t) ≠ R ² (t)}

For the completeness of this classification it has to be shown that if two elements cannot be compared (the profiles have the same minimum at different points), there is always an element that is larger than the other two elements.

THE MONOTONE RANGE PROFILE

It is very time-consuming to determine the optimal range profile using the known methods (trial and error method) if, for example, the following method is used: starting with a minimum uniform range of 1 at any given time, the range is increased in period 1 around a small delta of range. If a solution is found, the range is also increased in period 2 and so on. After all periods have been dealt with, the process starts again at the first period. If the range cannot be increased in a certain period, this period is changed by the following iterations

Except for enlargements. The temporal complexity of this approach is limited by 0 (R _max / R _min- H) calls of the transport optimizer, which may take too much time, where R _{max is defined} as the optimal range that can occur in the period; in which

R _{mm is defined} as the precision or granularity with which this area is to be determined, and H stands for the horizon. In this case, the optimizer must always optimize the data of the entire horizon. For this reason, according to the invention, monotonous range profiles are first considered. In a second step, 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). With others

Words, the 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.

If the entire horizon H is not taken into _account when determining the optimization network, but only the periods t ≤ H _laιrz up to a certain point in time H _terz <H, that is, if for t ≤ H _gates only supply s (t) and demand d ( t) are taken into account, then 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:

Vt ≤ H ^: RZ '(t) ≥ R _w (t).

FAST AREA PROFILE SEARCH

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. 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

Horizon H _kun the range R (H _short) is taken from R (H _laιrz -l), that is, if no solution is found, the range in the period H _gate a

Period decreased. The resulting profile is always kept monotonous, which means that the range is set to R (t) = min {R (t), R (H _tor- } in all periods t <H _lmrz . At the latest now a solution must be possible, because yourself a range of ^R ( ^H k _rz ) ^{= R} ( ^H short - ¹ ) - ^{1 us de} range R fH ^ -1) in period H _{kurτ z} . Now try again

Increase the value of this and the following periods, starting with the first period in which a range was changed in this step. The considered horizon remains at H ^. After T _mΛX = max _y {T _j } has been determined with the short horizon, in the next step all those that follow

Periods treated as a single period. The horizon under consideration is extended to the total horizon and all subsequent periods are extended to the

Value R (H _{k rz} ) set. After this step, the process is complete and the maximum monotonous profile has been found. It rises to a point T _ιmx and then remains constant.

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. In each step, 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.

If Opt (H) is the runtime of the transport optimizer for the horizon H, then the runtime in the worst case is:

O dog iR ^ / R ^) ^■ r _max ^• Opt (H _{π x} )).

R _πax is the maximum range that can occur in a period, R. is the accuracy (granularity) with which this range can be determined, T _πax is the maximum transport time between two nodes (for the slowest transport mode) and H _tolal is the total _horizon (for the

Total running time including the running time for the transport optimizer).

In practical use, it turned out that a reset step is hardly necessary. If one assumes that bottlenecks are more likely to occur at the beginning and that the total production volume slightly exceeds the demand volume (push distribution), then in the last step (in which all periods after 7 ^ are included in the optimization network) there is none

_{Reset step} instead, and the number of calls to the transport _optimizer is only log (R _mΛX /? _Mx ) for the

Overall horizon. Then together with the effort for the initial area, the following runtime results:

O dog iR _^ / R ^) ^■ (T _^ - Opt (T _m ) + Opt (H _total ))) The method according to the invention quickly generates a monotonous optimal transport plan for the first period.

INSUFFICIENT OFFER

In the previous section it was assumed that the production was always sufficient to always cover all needs in good time. Of course, this is not always the case. If there is insufficient supply, the need situation must be relaxed in a suitable manner. This can be done in two ways:

1. The demand itself is reduced, which means that a few customer orders are only partially or not at all executed.

2. A delay in the fulfillment of the orders is permitted.

Both variants can also be combined. 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.

REDUCTION OF NEEDS

Let us now turn to the first case (none

Delay). In order to shorten the need in a suitable way, customer orders become different

Priorities assigned. An order of (t) des

Distribution network node i in period t is additionally marked with its priority P. The unimportant priority classes (i.e. orders in these Classes) are deleted until the material volume available is sufficient to fulfill the remaining orders in good time. The same approach according to the invention as mentioned above is mentioned. The integer part LR (t) J specifies the number of priority classes whose orders are completely fulfilled. The rational one

Part R (t) - LR (t) J is the part that is fulfilled by the next priority class. For example, 4 priority classes are defined:

1- P _high : customer needs with high priority.

2. nor _m i '• Customer needs with normal priority.

3- P _medr g: customer needs with low priority.

4. Forecast: Additional demand that is only forecast but has not yet been ordered by a customer

A coverage area R (t) = 2.7 would mean that at time t all requirements for customers with high and normal priority can be met. For customers with low priority, the degree of fulfillment is 70%.

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.

DELIVERY DELIVERY

In the above considerations, no delayed delivery was allowed. This means that whenever an order could not be delivered on time, it was not delivered at all. Example: If there is a high priority requirement for t = 1 and a low priority requirement for t = 2 and if the first offer arrives at t = 2, then the low priority need will get the whole offer, but the high priority need will get nothing.

According to the invention, this problem is solved by introducing additional sheets with reduced transport time. The delivery with a delay time Δ corresponds to the reduction of the corresponding transport time by Δ.

The cost of these additional sheets is defined as follows:

Penalty = Δ ² * 2C "

Δ: the delay time

C _max : maximum cost of the sheets without

Delay Time

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 - | ') by; t _x ) -> d-, - (t-, -) and Si' ti- ^■ > d ₁ (t ₁ a delay cannot be prevented.

If there are no delays in the interchanged assignments, then their costs are ≤ 2C _max <Δ 2 ^ + 2C _ma χ and thus cheaper than the assignments without a delay. For this reason, the optimizer will never produce a solution with a delay that could be prevented by an exchange. Similarly, the expression Δ ² guarantees that a long delay due to an exchange cannot be shortened.

DESCRIPTION OF THE PROJECTION OF THE DEPLOYMENT PROBLEM INTO THE DISTRIBUTION PROBLEM.

Furthermore, the representation of the distribution problems previously shown with regard to the transport problem is summarized again in a mathematical form.

Case 1: oversupply

PREVIOUS CALCULATION

First, the production and needs of

Optimization network can be calculated:

Distribution centers:

D _j (t) = ∑ o (t) - tr (t) + L _j (t-1) -L _j (t)} P

If D, (t)> 0, then d, (t) = D, (t) {demand node}

Otherwise s, (t) = -D, (t) {knot offer}

with a target inventory of distribution centers:

L _] (t lR (t) -LR (t))) - L (tiR (t) J) ^ (R (t) -LR (t) J ₎ -L ' _l (t, FR (t)) t + r

L _j '(t, r) = m t + r) + ∑ ∑of (τ) τ = t + l P

Manufacturing plants: s _l (0) = p, (0) + L ^{a ngl, ch}

Vt> 0: s, (t) = P, (t) THE OPTIMIZATION PROBLEM

Minimize:

Transportation costs

Under the following additional conditions: Limited offer: Vϊ. s _l (t _l ) = 2L _l y _ltJ

Restricted need: v / ■ ^•

CASE 2A: REDUCTION OF PRELIMINARY CALCULATION NEEDS

First, production and demand of the optimization network have to be calculated again:

Distribution centers: D t) = ∑o (t) tr t) + L _j (t-1) -L _j (t)}

If D, (t)> 0, then d, (t) = D, (t) {demand node} otherwise s, (t) = -D, (t) {offer node}

with a target stock of

Distribution centers:

L / ^ l-RftflRf ^ UR j + fR -Rf jL / t / R) t + r

Note: The integer part [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.

Manufacturing plants: s, (0) = p (0) + L ^a " ^{g ch} Vt s, (t) = P, (t)

THE OPTIMIZATION PROBLEM Minimize: Transport costs

Under the following additional conditions: Limited offer: Vz: s _l (t _l ) = ∑ _J y _l ιi, J>, if,

Limited need: \ jj: d (t,) = ∑jy _lt

., t. ' ^J '>

CASE 2B: DELAY WITH SUB-TOTAL CALCULATION

Again, the production and demand of the optimization network have to be calculated in the same way as above, only the optimization function changes by assigning the delay sheets with penalty costs. It is also possible to limit the time delays by introducing only these additional sheets with an acceptable delay. THE OPTIMIZATION PROBLEM

Minimize:

Cost = transportation cost + delay penalty cost

Transport costs =

Delay penalty costs:

, y " _lJtj

Under the following additional conditions: Limited offer: Vz. s _l (t _l ) = ∑y _u ',)>,

Restricted need: V: d _J (t _l ) = £ _l y ",

Maximum delay limit:

Vi, t,, j, t _j : y _{lt Jtj} defines <=> t, + T ₁ j * ^d -— - t _j <A _Jtj

VARIABLE AND CONSTANT Free decision variables: y _a = part of the supply from plant or plant i at time t that is assigned to the needs of distribution center j at time t _j

Bound Variables (with limited decision to maximize R (t)):

R (t) = range of all distribution centers to

Time t L t) = Desired inventory in distribution center j at time / (as a function of range R (t)) s, (t) = supply of the node (production plant) i at time tdt) - demand for nodes (distribution center) j at time t

Constants:

P, (t) = production rate in production plant i at a certain time t tr (t) = previously defined offer (in

Passage) in the distribution center or central warehouse j to the specific one

Time ti ^ ^nghch _{= t∑ι (0) =} initial stock in

Production plant z or distribution center i

P - demand priority P, Pe {Ph _0C h, - ■ ■, P forecast} = {1, 2, 3 _,. , .} In which <- ■ ■ <P predicts θ _j ^P (t) - order (= accumulated demand) for

Priority class P for distribution center j at time t m t) = safety stock in distribution center j at time t

D = mode of transport

D e {slowest,. . ., fastest}

C, _j ^D = transportation costs for transportation of a

Material unit with transport type D between nodes i and jr D transport time for the transport of a

Material unit with transport type D between nodes i and j

D s, fastest Fastest mode of transport: VD, i, j: rp D fastest _< -r D 1 V - ¹ V

L> slowest slowest mode of transport: VD, i, j: T, _j ^slowest slower (D) next slowest mode of transport after

Transport type D:

^ 3D ': [\ / ij: T, _J ^D ≤r _v ^D' <' _lJ ^{slower (DJ} J

A J>, Maximum permissible delay of one

From distribution center j at a certain time t _}

ALGORITHMS

After the problem has been formulated, the algorithm to be used is discussed. The basic idea of the algorithm is to incrementally mutate the range profile R (t) until the optimal profile is determined. The basic procedural steps of the algorithm include the following:

1. Initialize the range profile R (t) (e.g. R (t) = l).

2. Calculate the required needs ₇ (t) in the demand nodes to meet profile constraints.

3. Construct the cheapest flow for these needs using any minimal cost flow algorithm.

4. If no solution is found, reduce the size Range profile R (t), otherwise increase the range profile R (t). 5. Go back to step 2 if the optimal solution is not found.

Step 2 is the same for all algorithms as described in the sections above: d / t) = ∑offO- tr / + L / t-lH / t)} p where L _j (t) = (lR (t) - lR (t) J) -L (t (t) J + (R (t) -LR (t) J- L (t / R (t)) t + r

L / 't, r) = m t + r) - ∑ ∑of (τ) τ = t + l P

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.

A. MAXIMUM RANGE OF COVER PROFILE

To construct the maximum (integer) area of 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.

Step 4 algorithm

While Profile_enlarged do begin

{Cover the planning horizon} enlarged = false;

for all t (planning horizon) do begin R (t): = R (t) + l; {R enlarge} Generate min_cost_flow problem F (R) for profile R (t)

{= 2nd step (see above)} If min_cost_flow (F (R)) solvable then enlarged = true ise R (t): = R (t) -l {reset}; end; end;

The generalization to enlarge the profile by a smaller step size Δ <1 instead of 1 gives:

Algorithm from step 4 with step size Δ for profile R

While Profile_enlarged do begin {enlarged the planning horizon} enlarged = false;

for all t <planning horizon do begin

R (t): = R (t) + Δ; {R enlarge} Generate min_cost_flow problem F (R) for profile R (t)

{= 2nd step (see above)} if min_cost_flow (F (R)) solvable then enlarged = true e \ se R (t): = R (t) -Δ {reset}; end; end; B. MAXIMUM MONOTONE RANGE OF RANGE PROFILES

The basic idea is to solve the problem inductively:

• Initialize with an empty profile.

• Construct a solution for t time steps and improve this solution for t + 1 time steps.

• Iterate until t = planning horizon.

Step 4 algorithm

Initialize R (t): Vt: R (t) = 0 t = to; {Planning start time} While t <planning horizon do begin {induction step} R (t + l) = R (t);

Generate min_cost_flow problem F (R) for profile R (t)

{= 2. Step (see above)}

If min_cost_flow (F (R)) solvable

Then Begin {R (t + 1) increase} Repeat

R (t + l): = R (t + l) + l; Generate min_cost_flow problem F (R) for profile R (t) Until min_cost flow (F (R)) not solvable R (t + l): = R (t + l) -l {reset}

End Else {shrink R

Repeat

R ^mαx = R (t + l) t _R = min {t | R (t) = R ™ "- 1}; W> t _R : R (t ') = R "- 1;

Generate min_cost_flow problem F (R) for profile R (t) Until min_cost_flow (F (R)) solvable

Step 4 algorithm - monotonic range of

Range profiles with increment Δ

The generalization to enlarge the profile by a smaller step Δ <1 instead of 1 gives:

Initialize R (t): Vt: R (t) = 0 t = tO; (Planning_Start) While t <planning horizon do begin {induction step}

R (t + l) = R (t); Generate min_cost_flow problem F (R) for profile R (t)

{= 2nd step (see above)}

If min_cost_flow (F (R)) solvable

Then Begin {R (t + 1)} enlarge}

Repeat

R (t + l): = R (t + l) + Δ; Generate min_cost_flow problem F (R) for profile

R (t) Until min_cost_flow (F (R)) not solvable R (t + l): = R (t + l) - Δ {reset}

End Else {shrink R

Repeat

_R max. _{= R} ( _{t + 1 t} t _R = min {t '/ R (t')> R ^max - Δ} W≥ _R : Rft ^ R "- Δ;

Generate min_cost_flow problem F (R) for profile R (t)

Until min_cost_flow (F (R)) solvable

C. IMPROVING EFFICIENCY FOR ROLLING PLANNING

For the rolling planning scenario, it only has to be known where 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 . According to the invention, they are corrected by shifting the planurig window. In the following, the range profiles are restricted to the following type: Evenly up to the maximum

Transport time t _max and constant after t _max As long as this monotonous profile is insoluble, the last constant will be

Functional part reduced.

Step 4 algorithm - rolling planning scheme

The maximum transport time is specified, ie t _max

Find the maximum monotonous range profile R ^max (t) within the planning window [planning_start, planning_start + t _max ];

Initialize R (t): {constant continuation of R] Vt ^ RQ-T ^* ®

R ^max : = R ^max (t _ma ) {change last constant part of R} Determine min_cost_flow problem F (R) for the profile R (t)

While min_cost_f low (F (R)) not solvable do begin

t _R = min {t / R (t) ≥R ^max} -l Vt≥ _R: R (t) = R ^max -l;

Determine min_cost_flow problem F (R) for the profile R (t)

End;

GENERALIZATIONS

To construct non-integer capacity profiles, the step size Δ of the possible values for the area of the cover profile R (t) must be limited. If the

Step size Δ 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.

1. BINARY SEARCH FOR MAXIMUM MONOTONAL PROFILES In the algorithm for the maximum monotonous range of cover profiles with step size Δ two parts are taken into account: "R (t) increase" and "R (t) decrease". In both cases, the search for the optimal matching of the last constant part of the capacity profile is accelerated using a binary search. A simple extension of the algorithm according to the invention with a binary search process is given below. For reasons of clarity, alternative variants are omitted (for example, choosing the step size δ of the binary search with values learned through experience instead of the smallest possible value δ = Δ). ALGORITHM FOR STEP 4 - MONOTONE AREA OF COVER PROFILES WITH STEP WIDTH Δ (BINARY SEARCH)

Initialize R (t): Vt: R (t) = 0 t = to; {Planning start time} While t <planning horizon do begin {induction step} R (t + l) = R (t) \ Generate min_cost_flow problem F (R) for profile R (t) {= step 2}

If min_cost_flow (F (R)) solvable

Then Begin {enlarge (R + 1)}

Ö: = Δ;

Repeat {binary search- £ magnify}

R (t + 1): = R (t + l) + δ δ: = 2 * δ;

Generate min_cost_flow problem F (R) for profile R (t)

Until min_cost_flow (F (R)) not solvable Repeat {binary search- «!) Reduce}

R (t + 1): = R (t + l) + δ; δ: = δ / 2; Generate min_cost_flow problem F (R) for profile R (t) If min_cost_flow (F (R)) solvable Until

If min_cost_flow (F (R)) not solvable Then R (t + 1): = R (t + l) -δ

End

Else [decrease R} δ: = Δ;

Repeat {binary search- δ increase} R ^max : = R (t + l); δ: = 2 * δ;

Vt'≥t _R : R (t ') = R ^max -δ; Generate min_cost_f low problem F (R) for profile R (t)

Until min_cost_f low (F (R)) solvable

Repeat {binary search- δ reduce} R ^max : = R (t + l); δ: = δ / 2; Generate min_cost_flow problem F (R) for profile

R (t)

If min_cost_flow (F (R)) solvable

3. If the whole profile is not set, go back to step 1 (and solve the remaining part).

For step 2, the time steps t must be selected for which

Ff (t) <R ^yes (t + l) => R "(t) = R (t) _f ie the strongly monotonically increasing parts of the maximum monotonous range of the range profile can be specified. In addition, the end of the planning horizon can also be specified :

R ^m (planning_end) = R (planning_end) Therefore the planning window can be reduced by at least 1 (possibly by more, if we can also reduce certain fixed, strongly monotonically increasing parts of R ™). This ensures that the above loop is repeated at most I steps, where T = planning_end-planning__start (number of time steps in the planning window), ie the time complexity is limited by 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.

Claims

claims

1. A method for maximizing the coverage range of cover profiles, comprising the following steps: a. Providing a warehouse reach function for the cover profile; b. Initialize the warehouse range function with at least one start value parameter; c. Determine an optimal stock quantity to solve the stock range function for the

Coverage profile, calculating the following

Steps include: i. Applying a minimum cost flow algorithm to determine an optimal transport solution to transport the required inventory; ii. Determining whether a solution was found after applying the minimum cost flow algorithm for the coverage profile function area with the seed value; iii. incrementally decreasing the starting value if no solution was determined and repeating the calculation of a solution until an optimal solution was found for the minimum cost flow algorithm; d. incrementally increasing the starting value if one

Solution was determined, and repeating the

Computing a solution until computing a solution

Solution to an optimal solution for the

Minimum cost flow algorithm leads.

2. Method of Maximizing a Range of

Cover profiles according to claim 1, further comprising the following steps: a. Coupling to an online Transaction processing system; b. Reading transaction data from the online transaction processing system; and c. Enter the transaction data for the area of the coverage profile function.

3. Computer-readable medium with a plurality of instructions stored thereon, the plurality of instructions containing instructions which, when executed by a processor, cause the processor to carry out the following steps: a. Interfacing with an online transaction processing system; b. Reading transaction data from the online transaction processing system; c. Entering the transaction data into the function for an area of the coverage profile; d. Initializing the function for an area of the coverage profile with a start value; e. Calculating a required inventory quantity to fulfill the coverage range function for the coverage profile; f. Calculating an optimal solution, the calculation comprising the steps of: i. Applying a minimum cost flow algorithm to determine an optimal transport solution to transport the required inventory; ii. Determining whether a solution was found after applying the minimum cost flow algorithm for the coverage profile function area with the seed value; iii. incrementally decreasing the starting value if no solution was determined and repeating the calculation of a solution until an optimal one Solution for the minimum cost flow algorithm was determined; G. incrementally increasing the starting value when a solution has been determined and repeating the calculation of a solution until the calculation of a solution leads to an optimal solution for the minimum cost flow algorithm.

4. A system for maximizing the coverage range of cover profiles, comprising: a. Means for interfacing with an online transaction processing system; b. Means for reading transaction data from the online transaction processing system; c. Means for entering the transaction data in the

Storage range function for the cover profile; d. Means for initializing the storage range of the coverage profile function with a start value; e. Means for calculating a required inventory quantity to fulfill the function for a storage range of the cover profile; f. Means for determining an optimal solution, comprising: i. Means for applying a minimum cost flow algorithm to determine an optimal transport solution to transport the required inventory; ii. Means for determining if a solution was found after the minimum cost flow algorithm for the coverage profile function area was applied with the starting value; iii. Means for incrementally reducing the size of the

Start value if no solution was determined and repeat calculating a solution until an optimal solution for the

Minimum cost flow algorithm was determined; Means for incrementally increasing the starting value when a solution has been determined and repeating computing a solution until computing a solution yields an optimal solution for the minimum cost flow algorithm.