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''network method of planning and control''

A method of control in the realization of a certain complex of operations (designs, programs, topics, etc.) on the basis of a [[Network model|network model]] of the complex; it is also known as PERT (see [[#References|[2]]]). Network planning allows one to raise significantly the quality of the planning and control in the realization of the complex of operations; in particular, it makes it possible to coordinate efficiently the activity from all sides (organizations) participating in the realization of the complex, to distinguish the most important problems, to assess the most appropriate times to realize designs, to correct plans for realization in good time, etc.

Network planning can be conditionally split into two phases: 1) the construction of a network model of the complex of operations; and 2) the use of this network model for planning and control in the realization of the complex of operations (see [[#References|[1]]]–[[#References|[3]]]). The construction of a network model of a complex reduces to the representation of the set of stages (events) and the natural order (generally speaking, special) of the operations of the complex as a specially oriented graph, as well as of representing the numerical information necessary (the time required to accomplish each operation, resources, etc.). Depending on the formulated aims, after constructing a network model one proceeds to analyze it to find the best preparations of a plan to achieve these aims. For example, if a network model is constructed in terms of a time orientation, that is, when the whole complex of operations is to be achieved in minimal time for given resources, this analysis reduces to finding the critical paths and determining the least time in which the realization of the complex is possible. This means the following. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066400/n0664001.png" /> be the structure of a network model of the complex, where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066400/n0664002.png" /></td> </tr></table>

are, respectively, the sets of events and operations, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066400/n0664003.png" /> is the time required to accomplish the operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066400/n0664004.png" />. One considers the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066400/n0664005.png" /> of all paths of the graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066400/n0664006.png" /> that are maximal with respect to inclusion. In general, there are many such paths in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066400/n0664007.png" />, and for the simple situation when there is one initial event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066400/n0664008.png" /> and one terminal one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066400/n0664009.png" />, all these paths begin at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066400/n06640010.png" /> and end at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066400/n06640011.png" />. Among the paths of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066400/n06640012.png" /> one can find a path with least length (the length of a path <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066400/n06640013.png" /> is the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066400/n06640014.png" />). A path <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066400/n06640015.png" /> with this property is called a critical path of the network planning and its length expresses the fact that the complex of operations cannot be realized in a time less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066400/n06640016.png" />. Therefore, the method of network planing is also called the critical path method (see [[#References|[1]]]–[[#References|[4]]]).

During the period of the realization of a complex of operations, network planning plays the role of a control mechanism that helps one to process information about the actual state of the operations at a given instant, and to forecast changes and necessary corrections of plans to accomplish the remaining operations.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> , ''Fundamentals in the development and application of systems of network planning and control'' , Moscow  (1974)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Kaufmann,   G. Desbazeille,   "La méthode du chemin critique" , Dunod  (1964)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> , ''Network planning and control'' , Moscow  (1967)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.A. Abramov,   M.P. Marinichev,   P.D. Polyakov,   "Network methods of planning and control" , Moscow  (1965)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> , ''Encyclopaedia of cybernetics'' , '''1–2''' , Kiev  (1974)  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  L.I. Lopatnikov,   "A short mathematical-economic dictionary" , Moscow  (1979)  (In Russian)</TD></TR></table>



====Comments====
For additional references see also [[Network|Network]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.A. Taha,   "Operations research" , Macmillan  (1982)</TD></TR></table>