Distribution problem - Design of logistics systems

Distribution problem

Many of the linear programming problems encountered in solving planning problems can be reduced to a task, one of the many economic interpretations of which is as follows. There are t types of raw materials in quantities units. There are n production points in which any kind of this raw material can be processed into a finished product, and the manufacture of a unit of the finished product at j-m production point is a ij of raw materials i -th type. The quantities of the product units to be produced at each of the production points are given. Let be the quantity of the product manufactured at the j production point from the raw material i , and - the cost of manufacturing a unit of product in this way.

It is required to find such production plan (matrix ), which does not lead to overspending of raw materials, provides requirements for the output of the product at each point of production, and reduces the total cost of production to a minimum.

The requirement of no overspending of each kind of raw materials can be written in the form of inequalities

(4.13)

The rest of the conditions imposed on the unknowns have the same form as in the transport task: the quantity of the product manufactured at each point of production must be equal to the specified:

(4.14)

and all unknowns must be nonnegative:

(4.15)

The production cost, determined by the plan , is expressed by the formula

(4.16)

Thus, we arrive at the following mathematical formulation of the problem: find the minimum of the linear function (4.16) under the constraints (4.13) - (4.15) imposed on the unknowns.

This task is called distributive. Often it is also called - a task (lambda task) or a generalized transport task. The latter name reflects the fact that in the special case, when all the distribution task is reduced to transport. In terms of transportation, the distribution task can be formulated differently. The numbers denote fuel stocks concentrated in t departure points, with different fuel at different points. All this fuel or part of it must be delivered to consumers, and for each consumer the amount of heat (the number of calories) that must be obtained from the combustion of imported fuel is indicated.

The numbers represent the calorific value of fuel from the i-th point of departure, which is supposed to depend not only on the source of supply, but also on which consumer this fuel burns. The cost of transporting one unit of fuel from the i -th point of departure to j -th consumer will be denoted by and we will also assume known. Any combination of numbers , meaning the values ​​of traffic originating from the points of departure to consumers, is a task plan if it meets the requirements for providing calories for each consumer, ie

(4.17)

and if it does not imply an excess of fuel stocks available at departure points:

(4.18)

In this case, it should be considered non-negative:

(4.19)

The task is to find a transportation plan that minimizes the total transportation costs:

(4.20)

Thus, it is required to find the minimum of the linear function (4.20) under the conditions (4.17) - (4.19). This formulation is somewhat different from the one given earlier. However, if we change the variables and designate , the problem is reduced to finding the minimum of the linear function

(4.16 ')

under conditions

(4.13 ')

(4.14 ')

(4.15 ')

which, up to the notation, coincides with the previously formulated distribution problem.

At present a number of algorithms for solving the distribution problem are developed, most of which realize the idea of ​​a consistent improvement of the plan.

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