MATHEMATICAL MODELS OF REPLACEMENT AND WEARING OF EQUIPMENT

2018. Т. 18, No 2. С. 121–130 12


Introduction
To ensure high competitiveness of a manufacturing enterprise it is necessary to provide timely upgrade and modernization of the current equipment. The main objectives of the theory of replacement of the equipment are to foresee the necessary expenses and to design the most effective strategy of replacements [1,2].
In this article the generalizing algorithm of the solution of a problem of replacement of the equipment is considered based on the analysis of the available models and algorithms [3][4][5]. Replacement processes are divided into two classes depending on the equipment type. The first class is connected

Deterministic models of replacement of equipment 1.Simple model
Assume that s i is expenses in the i period to purchase and operate the equipment. We will consider only such operational costs which change as service life increases. At the same time we will define interval n after which replacement will be made for the following reasons: -if the expenses in the following period are less than the average of the previous expenses, it isn't necessary to replace the equipment; -if the expenses in the following period exceed the average expenses, it is necessary to replace the equipment.
Thus, the following inequalities have to be fulfilled: (2) Example 1. Let the expenses associated with replacement and purchase of the equipment be presented in Table 1. In column 3 average expenses by the right part of formula (1) are calculated. We receive the decision which implies replacement of equipment in period 3 as   3  1  2  4  1  2  3   20  2 30,  30  3 26  (  (  .7  ) ) (3) The cost of money changes over time due to a possibility of investing it and receiving income. Therefore it is necessary to carry out calculations with a discounting coefficient. Let i be deposit percent in each period, discounting coefficient being r = 1/(1 + i/100). We replace average expenses into average discounted expenses in (1) Example 2. Let the expenses of purchase and replacement of equipment be similar to the previous example and i = 5 %. The average discounted expenses (r = 0,952) are given in the 3rd column of Table 2. The decision in this case is replacement of equipment in period 3 as well, since ratios (4)-(5) are fulfilled at n = 3. Both examples assumed that operational costs of the equipment increase over time.

Network model
Let us introduce a model of replacement of equipment using the language of network modelling. Let us assume that we need to devise a plan to rent the equipment for n years. We shall consider the event of i as the beginning of year i (i  n). Let the expenses value s ij include the rent and the expected maintenance costs of the equipment which at the beginning of year i is leased and replaced at the beginning of year j. Then we shall consider arch (i,j) the lease contract of the equipment since the beginning of year i until the beginning of year j. Thus we have received an oriented acyclic network. The plan of rent has to be made so that to minimize total expenses that, in the network language, corresponds to finding the shortest way in the network from event 1 to n + 1. Thus, to reduce the problem of renting the equipment to the network model we need to be able to calculate s ij coefficients.
Example 3. Let us assume that maintenance expenses of the equipment in years 1, 2, 3, 4 and 5 amount 1, 3, 6, 10 and 15 respectively. At the same time, let the rent amount 5, 9, 13, 16 and 19 respectively. Besides, every year rent payments increase by 1 i.e. if the equipment is leased at the beginning of the third year for 1, 2 or 3 years, for example, rent will amount 7,11 and 15 respectively. We will tabulate the calculated s ij coefficients in Table 3. In this example we are devising the plan of rent for 5 years, therefore we have introduced event 6 as the end of year 5. We shall now apply the algorithm of finding the shortest way to the received network (see [1]). Let y i be the minimum value of expenses which is reached upon transition from event i to final event n + 1. There are two shortest ways here (1-2-6) and (1-3-6). Thus, we have received two optimum strategies of rent: 1) to sign the contract for a year and to sign the contract for the remained 4 years at the beginning of the second year; 2) at first to sign the contract for two years and to sign the contract for the remained 3 years at the beginning of the third year.
At both options the minimum expenses amount 33. Example 4. It is necessary to define a plan of replacement of a machine tool for the next seven years (it will not be required any longer) if the following is known: a) p t is the cost of a new model in year t, b) v k is the salvage cost of the machine tool after k years of operation, c) r k is operational costs of the machine tool in year k of its operation. As well as in the previous example, we shall define events as beginnings of each year. We shall imply arch (i, j) as the decision to replace the machine tool vat the beginning of year j bought at the beginning of year i, then it will be operated for (j -i) years and the expenses associated with this decision are possible to calculate by the following formula: Let the given data be in Table 4. Let's calculate s ij by formula (6) and put the results in Table 5. Table 5 Values of s ij coefficients Calculating the expenses in column 8 we consider the fact that the new machine tool is not purchased any more at the beginning of year 8, there is no need for it under the terms of our example.
To find the optimum plan of replacements we shall use the algorithm of finding the shortest way again. 8  We have received the shortest way (1-3-8) which corresponds to the following optimum strategy: the machine tool bought at the beginning of year 1 needs to be replaced at the beginning of year 3 and operate it until the end of the considered period (the remained 5 years), at the same time the expenses will be minimum and equal 383 (without the cost of the machine tool purchased in the first year).

Algorithm of dynamic programming
It is possible to use algorithms of dynamic programming when modelling a problem of replacement of equipment (see [12][13][14][15]).
Let value s ij be the sum of expenses and expected maintenance costs of the equipment which is purchased at the beginning of year i, minus the salvage cost of the equipment as at the beginning of year j.
Let y i be the value of expenses which is reached at the strategy of the replacement providing the minimum of these expenses in intervals i,i+1,…,n if the new equipment is bought in year i.
Then finding the optimum strategy comes down to calculating y 1 and the corresponding strategy of the first step. At the same time the following recurrence equations are used: It is obvious that finding the optimum strategy in a multistep process using equation (7) coincides greatly with the algorithm of finding the shortest way in an acyclic network.
The problem of replacement of equipment by means of dynamic programming can be simulated in a bit different way.
Let the expenses corresponding with the strategy of replacement include two components: c im is costs of replacement of equipment of age m in interval i minus salvage cost; r im is operational costs of the equipment of age m in interval i. Let y i (m) be the strategy at which the minimum expenses on intervals i, i + 1, …, n is reached, if the age of the equipment at the beginning of interval i is m years.
At keeping the equipment in interval i an optimal solution is where y n+1 (m) = 0 for all m.
Let M be a possible operating life of the equipment. We are devising a plan for n years, therefore the beginning of (n + 1) period is the end of the planned period.
To find an optimal solution у 1 (m 0 ) is calculated, where m 0 is the age of the equipment as at the beginning of the period. If at the beginning some equipment is absent, the decision on replacement at i = 1 is its purchase.
Example 5. Let us devise a plan of replacement of the equipment for five years purchased at the beginning of year 1. The predicted expenses are given in Tables 6 and 7.  Empty cells in the tables mean that there is no equipment at the beginning of the planned period therefore there are no expenses, for example, in the third year there is no 4-year equipment and there is no 5-year-old equipment in any year due to which column 5 is absent in Table 7.
As the new equipment is purchased at the beginning of year 1 under the condition of the example, у 1 (0) = с 11 +r 11 + у 2 (1) = 100 + 20 + 200 = 320. We have received the optimal strategy which consists in the following: in the 3rd year we replace the equipment purchased at the beginning of year 1 and we use it until the end of the 5th year.

Stochastic models of replacement of the equipment 2.1. Stochastic model with a limited planned period
Above we have examined some deterministic tasks of replacement of equipment, where index m reflected the duration of normally operated equipment. We shall further consider a stochastic version of problem of reconditioning where breakage of the device before the planned replacement term is allowed (in this case it is replaced in the timepoint following breakage).
We shall set p j as probability of breakage of the equipment at moment j of its operation (j < m); r j is operational costs of the properly functioning equipment in interval j of its use; g j is additional expenses caused by premature breakage of the equipment in interval j. Let initial expenses r 1 include the initial cost of the equipment and the out-of-order devices are completely depreciated (for example, burned-down electric motors, lamps, etc.).
The optimal strategy will consist in minimization of mathematical expectation of the expenses which consider: -average expenses on reconditioning at the breakage of the equipment before planned moment m; -average expenses on reconditioning in cases of breakage of the equipment before planned moment m; -the expected operational costs during the period from the reconditioning till the next moment of reconditioning.
Thus, generalizing relations (7) and (8) where the first sum is mathematical expectation of expenses on premature replacement, the addend is product of minimum expenses from period i + m onwards and probabilities of normal operation of the equipment till this period. The third compound reflects the operational costs calculated by the following formula: These expenses are made up of the expenses of year 1 of use of the equipment plus the expenses of year 2 multiplied by probability of its non-breakage in year 1 plus the expenses of year 3 multiplied by probability of its non-breakage in the first two years, etc. till interval m plus mathematical expectation of expenses on repair at premature breakage till interval m.

Stochastic model with unlimited planned period
Now we shall consider a stochastic problem of replacement of the equipment at an unlimited planned period.
Let us assume that in this case the optimal strategy is time constant (i.e. replacement is made constantly at time interval m). Then the formula to define the optimal strategy becomes significantly simpler: where Е m is mathematical expectation of replacement age of the equipment The equation in the braces in (11) defines the expected expenses for one interval of the planned period which coincides with the technique to define the optimal strategy examined by us in example 1.
Example 6. The given data for the first 5 years of the unlimited planned period are presented in Table 8.