A Logistical Approach to Managing the Resources of Multi Nomenclature Spare Parts of a Corporate Car Service in Conditions of Risk and Uncertainty of Demand

The task of determining the optimal sizes of spare parts for an auto-service enterprise based on the maximum profit criterion for a discrete distribution of demand is formulated as a problem of quadratic programming with linear constraints. To calculate the probabilistic measure of the distribution of the values of the demand vector components, an approximation is used of the empirical distribution function of the demand components by hyper-Erlanger distribution functions, and the subsequent calculation of the corresponding distribution densities.


Introduction
In recent years, the concept of logistics has been developed and is being used as an important approach to inventory management [1][2][3][4][5][6][7][8]. Logistics is aimed at reducing costs, increasing reliability, reducing risks through harmonization and mutual systemic adjustment of plans and actions of the supply, production and marketing segments of the enterprise.
Current transformations in the transport sector of the republic are characterized by changes both in the size of the fleet of serviced rolling stock and in the structure of the management of motor transport enterprises (ATP). Unlike the conditions of the planned economy, when the demand for ATP transport services exceeded the capabilities of the motor transport service enterprises and it was possible to realize these opportunities regardless of the ATP composition used, this situation changed radically with the transition to the buyer's market. The task of economical and successful implementation of the capabilities of service centers in a competitive market in the transport services market is becoming one of the main tasks. Necessary conditions for its solution are the rapid reaction of enterprises to changing needs, reducing costs for the production of transport services and improving their quality and reliability. The most common model of applied logistics theory is the model of the optimal or economical order size (EOQ) for the replenishment period [7,9,10]. A review of the EOQ models and their bibliography is given in [11]. The problem of uncertainty and classification of types of uncertainty in supply chains are considered in [12].
In practice, there are often situations where the data on the prehistory of the supply of spare parts is either not sufficiently representative or inaccessible. Then, for inventory management, the demand is modeled on the basis of expert assessments, which contain more subjectivity than chance. In such cases, the inventory management problem is formulated as an optimization problem under fuzzy information [13][14][15][16][17]. In some works (see, for example, [18,19]) single-period (single-period) control problems of singleand multi-item reserves are solved using the strategy of minimal average and conditional Risk or neutral risk.
In this paper we use the method of neutral risk [19] for the one-time task of managing multinomenclature stocks, in which demand is described by a discrete probability distribution. For constructing a discrete probability distribution, we use the approximation of the empirical distribution function of actual demand in the previous period of replenishment of spare parts of an auto service plant with the help of the hyper-relang function of distribution in the Levy metric. The accuracy of the hypererlang approximation of arbitrary distributions in various metrics is estimated in [20].
It is shown that for an arbitrary empirical distribution function and a given degree of accuracy of the approximation δ, one can select the step of discretizing the values of the random demand of the i nomenclature of spare parts, in which the entire observation interval is divided into subintervals and the hyper-Erlanger distribution function (weighted sum of Erlanger distributions) corresponding to this partitioning, Approximates the empirical distribution function with a given accuracy δ in the Levy metric. The geometric meaning of this metric is that its value is equal to the side of the maximum square inscribed between the graphs of the empirical distribution function and the approximating distribution function . This approach is universal and can be used to approximate the empirical distribution functions of arbitrary random variables.
The probability of the appearance of the values of a discrete random variable = is equal to the values of the probability density function at a point = , where it is assumed that R u α will be a regularizing operator in the sense of Tikhonov only for The problem of optimal allocation of a multi nomenclature order ( ) based on the maximum profit criterion formulated in the paper is reduced to solving the problem of quadratic programming. In particular, with the proportional dependence of the amount of costs on the purchase of the ordered product of the type i on the value / i i A x ( i A -the need for the ordered product in the period under consideration), maximizing the expected average value of profit leads simultaneously to minimizing the total cost of fulfilling the order for this product, i.e. To the logistic model EOQ.

Formulation of the Optimization Problem and Its Solution
Let us formulate a one-period multinomenclature problem of managing a single-period multi-item for an auto service enterprise, taking into account two types of costs: 1) Costs for fulfilling orders: a i > 0, equal to the sum of costs for the purchase of the ordered product of type i, dollar; 2) Storage costs: the average number of units i, which will have to be stored in the warehouse, when ordering the size x i (pieces) is x i / 2 (pieces).
The amount of storage costs should be proportional to the number of stored product units and storage time = ⁄ , where D i is the fuzzy demand for the product i. Then the expected value of storage costs will be • ℎ • 2 ⁄ , where h i is the storage cost per unit of product i and Here, Cr is the credibility measure defined in [14]. When determining the optimal size of an order x i , using the maximum profit criterion, the expected value of profit is usually used as the objective function. In the case of the single-item inventory management task i, the objective function has the form Where: p i is the purchase price of product order unit i, dollar.
The expected value of the fuzzy profit & , is denoted by .& , /. Using the properties of the operation E, we obtain Thus, for a one-period one-nomenclature problem, the optimization problem will be written in the form where The solution of problem (4) is * = : +; As an approximate (whole) solution of the problem (4), we assume Where integ is the integer part of a number. For * = C D,:DEF • G ⁄ , where C D,:DEF -the cost of one order, dollar; A -the demand for the ordered product during the given period, pcs., The maximization of the expected value .& , / leads simultaneously to minimization of the total costs for the ordered product.
For the multinomenclature problem, we assume that there is no connection between any two standard terminals. Under this condition, the profit function is written in the form Where = , … , E , = , … , E are vectors from n components. Under the criterion of neutral risk, the multinomenclature problem of inventory management will be written down as an optimization problem Where the condition x≥0 means ≥ 0 = = 1, … , > . Suppose that the components of the vector D are mutually independent fuzzy quantities in the sense of the definition of [21], then their joint possible distribution is represented in the form As an approximate solution of the problem we take the vector < * = . < * , < * , … , < E * / where < * = => ?@ 5 ( ℎ 7

The Case of Discrete Distributions of the Demand of Multi Nomenclature Products
In [19] the one-period multiproduct inventory control problem for discrete and continuous distributions for some fuzzy variables . We will consider only the case of discrete variables possibility distributions , to which it is easy to make a discrete probability distribution. As will be shown in the next section, from which it can directly receive already discrete, piecewise constant distribution function, which has the same form with arbitrary empirical distribution function can be approximated by [20] (Continuous) Hyper distribution function (the sum of a finite number of Erlanger distribution functions) a probability distribution for a discrete random variable sequence under consideration (in this case, the demand D), the corresponding discrete sequence STI observation times. Suppose that in the model (4) the demand D has the following distribution possibilities where ≥ ≥ ⋯ ≥ ≥ ⋯ is the ordered series of discrete values of the quantity D taken with the probabilistic (or probability) measure > 0, and As was shown in [19], under these conditions the expected value of E [1/D] will be equal to where the weights ^ are determined by the formula ^ = _max`̃M `̃− max`̃M b `̃c + _sup`̃h `̃− sup`̃h [ `̃c (14) for any i ≥ 1; % = 0 The expected values of E (1/D i ) for the multi nomenclature problem are determined in a similar way:

Hypererlang Approximation of Arbitrary Distributions
Let X be a nonnegative random variable (abbreviated to rv) with an arbitrary distribution function (abbreviated df) F X .
We are given an arbitrary number δ> 0. We divide the semi axis .0, ∞ into half-open intervals k l = _ m − 1 , m /, m ≥ 1, and choose a natural number N such that We choose the points l ∈ k l , m = 1, … , − 1 and o ≥ − 1 . Let p l = lim ,→ls , m = 1, … , − 1 We define a piecewise constant f.r. F X according to the following rule: We note that by rule (18)  For comparison d.f. F X (x) and F (x), we use the Levi metric [20]: v , = =>wxy > 0: − y − y ≤ + y + y{ ∀ ∈ ℜ (20) The meaning of the Levi metric is very transparent -this is the side of the maximal square inscribed between the graphs of the d.
Where ( = p , ( l = @ l − p lb , 1 < m ≤ − 1, ( o = 1 − p ob , @ ,~ is the distribution, degenerate At the point l , i.e. @ ,~ • ,I,~= l . We approximate each of the degenerate distributions @ ,~ (x) by the Erlang distribution E€~. The Erlanger distribution is defined as follows [20]. Let = 1 be a sequence of independent identically distributed random variables (abbreviated as nos) having an exponential distribution with a unit mean: < = 1 − ? b, We fix the number τ> 0 (for example, τ = 1) and determine for each n≥1 a random variable with the Erlanger distribution of order n: where: † = > ‡ ⁄ . It is well known that • E ‚ E→$ ‰‰Š ‡ with probability 1, or, equivalently, Where @ ‚ is the distribution degenerate at the point τ. The limiting relation (25) is a consequence of equality (23) and the law of large numbers.

Conclusion
In conditions of competition in the market of motor transport services, ensuring maximum profit is one of the main tasks of managing multinational stocks of auto service enterprises using the logistics concept. Approximation of empirical distribution functions of the demand vector components allows us to calculate the corresponding density distribution of the values of the components of the demand vector and reduce the problem of determining the optimal stock size to the quadratic problem of conditional optimization.