Digital transformation and optimization models in the sphere of logistics a

. The efficiency, optimization, speed and time limits have always been of crucial importance for the logistics system, while saving of speed and time in the real-time mode are the key factors with transition to digital technologies and establishment of Industry 4.0 since they become the competitive advantages. The innovative use of technologies in such fields as data analysis, Internet of things and cloud calculations significantly change the logistical and transportation systems as a result of mating digital and existing supply chains becoming the catalyst of transition to “Logistika 4.0 ” . This work offers a model and method of shaping an optimal plan of fulfilling a complex of interrelated logistical operations for such changing conditions.The model is based on the presentation of optimization procedure as a non-linear task of discrete programming consisting in minimization of time of fulfilling the above complex of operations by a limited number of contractors partially interchangeable under conditions of limited budgeting. A model obtained thereat for establishing an optimal plan will belong to the class of nonpolynimially challenging tasks. In order to solve these tasks, a method has been suggested supported by a procedure of branches and boundaries. Thealgorithmisbasedon dichotomous branchingdiagram. Itsapplicationprovidesforreceivingboth quasi-optimalandoptimallogisticsplansfor the finite number of steps. Atthat, theassessmentofaccuracy is provided for quasi-optimal plans. The proposed model and method help solve a wide spectrum of practical tasks of logistical planning under conditions of digital transformation.


Introduction
The problems of minimizing time of fulfilling a complex of interrelated logistical operations having restrictions for the number of contractors, interchangeability and cost there of make a class of problems widespread in the practice of organization logistical management. Inparticular, theproblemsof industrial production logisticalplanning, development and implementation of innovative projects, planning scientific work in the research establishments and many othersbelong to this class [1][2][3]. The complexity of such like problems and considerable material and time expenses determined by planning miscalculations bring about the fact that the solution thereof shall be supported not only by the experience and intuition, but by impersonal scientific substantiations too. The applicable mathematical models can be used as tools for such substantiations. The development of structure of such a model and optimization method of the logistical plan is the purpose of this article.

Structure of model of logistical plans optimization
It is reasonable to reflect composition and interrelation of logistical operations in the form of a network when building models of logistical plan optimization: where, , i jnumbers of network nodes.
Every logistical operation in the network (1) is assigned with an arch   , i j connecting in and j -n nodes. Node 0 i  corresponds to an event of commencing fulfillment of complex of logistical operations presented by this network. Nodes 1, 2, , i m  correspond to the events corresponding to completion of all logistical operations described by the arches included into each of these nodes. The total number of logistical operations (arches) equalsN.
Thesequenceof logistical operations obeystherule: operation corresponding to an arch outgoing from any node can not be launched before finishing all operations corresponding to the arches included into this node. where, Let us present the financial expenditures related to engaging operators in fulfilling logistical operations in the form of vector components: where, ( , ) k с i j -cost of time unit of engaging k -noperator to fulfilling   , i j -n logistical operation.
Thelogisticalplanof fulfilling complex of logistical operations formallycan be presented by a set: where, Taking into account the adopted designations, the financial expenses ( , ) Y W i j for fulfilling logistical operation   , i j will be calculated according to formula in the course of implementing plan Y : Let us designate by means of L H a set of all network (1) routes connecting its initial and final vertexes.
Timeoffulfillingtheentirecomplexof logistical operations when implementing plan (5) will be equal to maximum duration L T of route Taking into account the adopted designations, the model of forming an optimal plan of fulfilling the complex of logistical operations will take the following form: Determine logistical plan of fulfilling the complex of operations where,   Y G t -set of operations being fulfilled at every moment of timet in case of implementing the logistical plan Y ; W *maximum permissible financial expenditures for fulfilling a complex of logistical operations (1). Inproblem (7) -(13) condition (8) formalizes a commitment of minimizing time of fulfilling a complex of logistical operations.
Limitation (9) reflectsaruleofintegrating logistical operations identifying that operations outgoing from any network node (1) can begin only aftercompleting all operations included into this node. Limitation (10) formalizesarequirementof allocation of an established number of operators for every logistical operation.
Limitation (11) reflects a natural planning condition consisting in that the number of operatorsinvited for fulfilling logistical operations simultaneously can not exceed a total number thereof.
Limitation (12) formalizes a requirement consisting in that the number and interchangeability of operatorsshall provide for a possibility of fulfilling every logistical operation of complex (1).
Limitation (13) denotes that total financial expenditures in the course of fulfilling a complex of logistical operations (1) can not exceed an established limiting level. Problem (7) -(13) belongs to the non-linear problems of distributing discrete on-uniform resources in the random network. It is NPa complex problem of discrete programming [4][5][6]. Precise methods of solving problems of such class have been offered for the first time in [7][8]. However, renewableresourcesonlyhave been taken into account in the models given consideration in these works. At the same time it is necessary to take into account the necessary non-renewable resources too in the course of planning actual logistical processes in economicalones. In model (7) -(13) the financial resources belong to this type. Relation (13) formallyreflectsthelimitationoftheseresources.
Model (7) -(13) obtained as a result of introducing this relation is a further generalization considered in model [8]. The precise optimization algorithms for model of (7) -(13) type are absent presently. At the same time the demands of practices of planning logistical processes determine a necessity of developing thereof.
For the existence of solution of problem (7) -(13), it is necessary and sufficient that the composition and interchangeability of operators and the established limiting level of Ω * of financial expenses provide for a possibility of fulfilling all logistical operations of complex (1).
In the formalized form the fulfillment of the first of these requirements consists in fulfilling limitation (12).
In order to check fulfillment of the second requirement, one can use the following set: Of financial expenses Ω q (i, j) for fulfilling logistical operations with the corresponding variants of engaging operators. The elements of set (14) will be determined by the relation: Taking into account (15) these condrequirement providing for existence of a solution of problem under consideration (7) -(13) is presented in the formalized form by the relation: It is reasonable to check practicability of conditions (12)  ,will be determined on the basis ofinterrelation: where,   Thus, S t corresponds to a sequence of time moments, in to which logistical operations included into the next tree branchare completed and the corresponding operators get released. Condition 1 0 S t  reflects a fact that all the variants of a logistical plan begin at the moment of time 0 t  . In the interests of implementing the adopteddichotomousdiagramofbranchinglet us introduce a set for consideration: where, In this case the order number q of element 1 q d  of set D characterizes both fulfilled logistical operation and a variant of appointing operators, and financial expenses.
Taking into account (21) the process of branching in the interests of compiling an optimal logistical plan (7)  specified permissible target function deviation from an optimal one (optimization accuracy).
Fulfilling condition (26) means that it is impossible to improve a record received earlier by more than 100µ% in the branch being considered and its continuation in the framework of a specified accuracy of optimization has no sense.
A procedure of searching solution comes to an end, if condition (26) is met for all the remaining branches. In case of an established method of going around a tree of variants the second return to the root vertex corresponds to such a situation. In this case the last record is the sought value of target function (8), while a permissible logistical plan corresponding to it-an optimal plan * Y .
In general the offered model and method of planning logistical processes provide for an automatic viewing and assessing all possible plan variants and receiving on this basis both accurate and approximate solutions of the planning problem. In this case repetitions are excluded in the course of viewing permissible variants. It significantly reduces the time of solving problems.

Conclusion
The offered model and method can be used in the systems of supporting decisions-making used in the practice of logistical management of organizations. They help formalize complexes of logistical operations different by the nature thereof. In this case the system approach to forming a plan of logistical process, reviewing it as a unified non-separable complex of interrelated logistical operations and the life support thereof aimed at attaining general final goals, make it possible to reduce material and time expenses of the participants of logistical process, reduce time limits of resources delivery and increase the quality of servicing.