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Using this matrix it is possible to calculate at any set assortment vector At not only necessary gross production x (for what S matrix is used), but also necessary total expenses of the work xn+1, capital investments of xn+2, etc. providing release of these end products At.

(250 and 80 or 750 and 800), here they are distributed by types of end products: on production of the 1st branch 268 and on production of the 2nd branch 62; according to expense of capital investments make 1176 and 37

Let's supply with a stroke (x'ik, y’i, etc.) the data relating to the expired period, and the same letters, but without stroke – the similar data connected with the planned period. Balance equalities (1) have to be carried out both in expired, and in the planned period.

Thus, xi difference - yi makes the part of production of i-y of branch intended for intra production consumption. Let's believe further that the balance is formed not in natural, and in a cost section.

This system of two equations can be used for definition h1 and h2 at preset values u1 and u2, for use of influence on gross release of any changes in the range of the final product, etc.

Thus, having counted a matrix of full expenses of S, it is possible on formulas (7) – (11) to calculate gross release of each branch and cumulative gross release of all branches at any set assortment vector At.

Having solved this system, we will receive h1=8 and h2 = Therefore to make an end production unit of the 2nd branch, production needs to let out in the 1st branch h1 = This size call coefficient of full expenses and designate it through S1 Thus if a12=4 characterizes the costs of production of the 1st branch of production of a unit of production of the 2nd branch used directly in the 2nd branch (why they and a factor cost was called), S12 consider cumulative expenses of production of the 1st branch both direct (a12), and the indirect expenses realized through others (in this case through 1st) branches, but eventually end production units of the 2nd branch, necessary for ensuring release. These indirect expenses make S12-a12=8-4=4

Theorem. If exists though one non-negative vector x> 0, satisfying to an inequality ( - And) · x> 0 i.e. if the equation (6') has the non-negative decision x> 0, at least for one At> 0, it has for any At> 0 only non-negative decision.

i.e. the total number of work and the capital investments necessary for providing an assortment vector of end products At, are equal to scalar products of the corresponding additional lines of a matrix of S' a vector At.

(u1=1) we will find an expense of raw materials of I on unit end products of the 1st shop from expression 4S11 + 4S21 + 8S3 Therefore, we will receive the corresponding coefficients of full expenses of raw materials, fuel and work on each end production unit from work of a matrix:

Follows from a way of formation of a matrix of expenses that for the previous period equality ( - And) · is carried out x' = At' where a vector plan x' and an assortment vector At' are determined by the executed balance for the last period, thus At'> Thus, the equation (6') has one non-negative decision x> on the basis of the theorem we conclude that the admissible plan and a matrix ( always has the equation (6') - And) has the return matrix.

At the solution of the balance equations only the main part of a matrix is still used (a structural matrix And). However at calculation for the planned period of expenses of work or the capital investments necessary for release of this final product, additional lines take part.

Let's designate through xi gross production of i-y of branch for the planned period and through yi – the final product going for consumption, external for the considered system (means of production of other economic systems, consumption of the population, formation of stocks, etc.).

To let out only an end production unit of k-y of branch, it is necessary to let out in the 1st branch x1=S1k, in the 2nd x2=S2k, etc., in i-y of branch to let out xi=Sik and, at last, in n-y of branch to let out xn=Snk of units of production.

resource on the unit of production released by k-y branch. Having included these coefficients in a structural matrix (i.e. having added them in the form of additional lines), we will receive a rectangular matrix of coefficients of a factor cost: