Storage Efficiency As a Managerial Optimization Problem

The paper derives efficient methods for solving basic
fundamental tasks of warehouse logistics in what amount and
when to resupply the stocks by means of the “case-based
reasoning” approach. The aim is to give a satisfactory
response, including its economic interpretation under
conditions of uncertainty. Therefore, the discrete case is
first studied in detail by means of the tools of microeconomic
analysis and accomplished results are then generalized by the
limiting transition to the continuous case. In both cases, the
discreet and continues one, the method relies on the value of
the newly established indicator of the storage efficiency,
which indicates that what inventory items are worth storing in
the case of the limited storage capacity. Its value depends on
the fragment of the probability distribution of the demand
during the delivery time.


Introduction
In practice, most sellers or manufacturers, who cannot manage without storing the retailed goods or components required in manufacturing, are looking for the right answer to the question -in what quantity and when to resupply their stock‖. The criteria of -the correct‖ answer here is the minimization of the total cost associated with it (Williams & Tokar, 2008). The parameters, on which the correct answer depends, are:  DD-the variable daily demand with the DD values showing the number of removed units of the considered inventory items from storage in one working day of the current year;  D-the variable annual demanded consumption of the considered inventory items with the values D = ΣDD i (summing up all the working days of the current year);  H-the variable part of the annual costs associated with the storage of one unit of the inventory item;  K-costs associated with one supply of the inventory item (ordering, transferring, and handling costs of one delivery that within given specific logistic terms do not depend on the number of X units of supplement delivery);  DT-the variable delivery time of the ordered additional supply (its values DT indicate the number of working days that pass from placing the order of the additional supply to its acceptance into stock);  L-the loss from each considered inventory item that would not satisfy the current demand due to an empty storehouse (the loss from customer dissatisfaction in the case of a retailer or the failure of production in the case of a manufacturer, e.g. the lost margin). If authors assume from the above unlimited time duration of business, it is customary to consider the possible variants within one accounting period, which is usually a calendar year. It is also suitable to split the initial question into two questions: -how much to reorder‖ and -when to reorder‖. If X is the answer to the question -how much to reorder‖, then the values of the variable D/X are the annual numbers of orders of additional supply. If it denotes by RL (reorder level) the level of the considered inventory item (RL ≥ 0), at which level the storehouse gives the order for an additional supply, then authors can answer the question -when to reorder‖ by pointing to the selected RL.
In the following, this paper proceeds from the standard textbook model (see Brealey, Myers, & Marcus, 2011) and precises the solution in terms of certainty. It gives reasons for the adaptation and generalization of this model on conditions of uncertainty and then by the -case-based reasoning‖ approach, it derives a general procedure for solving the task.

Model and the Solution of the Problem in the Case of Even Storage Removal in Terms of Certainty
Ideally, in the case of even storage removal where DD and DT are the constants (DD = dd and DT = dt), the dynamics of stocks (i.e. the course of the state of stock, depending on the time measured by the number of working days) can be approximated by the continuous model, which is the -saw‖ in Figure 1. This is exactly valid, for example, for the state of stock of even flowing fluid refilled to the initial level of X at the moment of depletion of the tank (Chen, Ray, & Song, 2006). The variable DDDT (the demand during the delivery time) is given by the product of DDDT = DD  DT, and therefore in this case, it has a constant value DDDT = dddt = dd  dt. For RL opt (i.e. RL minimizing the total costs associated with the storage of the inventory item and thus correctly answering the question -when to reorder‖), it applies that RL opt = DDDT independently of X ≥ DDDT.
It is obvious that the deviation of the RL value from the value DDDT on this or that side results in the increase of loss (costs) associated with the storage of the inventory item. Therefore RL opt = DDDT. If RL = DDDT, then the resulting dependence of the value of the total annual costs N(X) associated with the storage of the inventory item on the amount X of the additional supply can be described by: where H  X/2 is the annual costs of storing and K  D/X is the annual amount of ordering, transferring, and handling costs associated with the replenishment of stock. If it denotes EOQ (economic order of quantity) as the volume of the supplemental supply (i.e. the value of variable X), minimizing the expression (1) (Maddah & Jaber, 2008), then EOQ can be obtained by solving the system dN(X)/dX = H/2 -K  D/X 2 = 0, d 2 N(X)/dX 2 = 2  K  D/X 3 > 0 due to the variable X: By substituting EOQ for X in (1) and by the subsequent substitution of the expression (2  K  D/H) 1/2 for EOQ to the right side of the equation, the following can be successively got:

Adaptation of the Model to the Conditions of Uncertainty
In the examined ideal case ( Figure 1) when RL = DDDT, the last unit of the inventory item is contracted out at the time of arrival of a new supply (Shapiro & Wagner, 2009;Wokurka & O'Leary-Kelly, 2000). If RL > DDDT, then the decrease of its stock would regularly stop at the level SL = RL -DDDT. For the value SL, the teeth of the -saw‖ shown in Figure 1 (starting from the second tooth) would move upward. The level of SL is a safety stock, which in a case of need (an unexpected increase in the value of DT or an unexpected occurrence of dd i > dd within the time of delivery) acts as a buffer contained in RL. Its aim is to eliminate partially or completely the threat of customer dissatisfaction, due to the unexpected presence of DDDT above the usual level. In the described ideal case, where this is excluded, this buffer is not needed. It would only increase the storage cost by H  SL. Therefore, in Figure 1, SL = 0 is selected.
The practice sometimes substantially deviates from the examined ideal case. Both the daily demand (DD) and the length of delivery time (DT) may vary over time and their fluctuations make their product variable, thus the value of the variable DDDT. If values of DD and DT can, in this context, be regarded as independent random variables with known probability distributions of their values, then DDDT will be the random variable with probability distribution inferred from the distribution of variables DD and DT. Under certain assumptions, this paper can consider every -saw teeth‖ from the perspective of a long time horizon as the results of repeating of the same random process consisting of elementary random experiments that take place within individual working days. If a number of repetitions is large enough, so that the law of large numbers could be applied (Jiroušek & Kushmerick, 1997) In it (unlike Figure 1), the safety stock level SL is an essential part of RL used to eliminate the unavoidable overhangs of the variable values DDDT over E(DDDT). Therefore, it makes sense to consider SL only within the limits 0 ≤ SL ≤ (dddt max -E(DDDT)), where for RL applies E(DDDT) ≤ RL ≤ dddt max . This model can, in a certain sense, be regarded as a statistical average of all possible random realizations of a long-term process of -development inventory stock‖ with the additional supply in the amount of X opt units and with RL = E(DDDT) + SL. For determination X opt (the correct answer to the question -how much to reorder‖) of this model, it applies all that have already been mentioned in connection with the model in Figure 1. Much more difficult is finding RL opt , i.e., the correct answer to the question -when to reorder‖ (Wei & Chen, 2009). The following part of this paper deals with the solution to this problem.

General Solution of the Problem by the "Case based Reasoning" Approach
Both considered models here (i.e. the model in terms of certainty in Figure 1 and the model in terms of uncertainty in Figure 2) answer the question -how much to reorder‖ in the same way. In both cases, authors obtain X opt by the procedure described in Article 2. The certain annual consumption of D will be replaced by the expected value E(D) = E(Σdd i ) in terms of uncertainty. The same thing can be said of the total annual cost (1), induced by the -saw‖ component of the examined fluctuations of the stock. The only difference is that while in the first case it deals with the real costs, in the second case it deals with the expected cost (Chen et al., 2006).
The source of the substantial difference between the two cases is the existence of the expected annual cost E(NSL(RL)) of the safety stock that depends on the chosen RL level under conditions of uncertainty (Federgruen & Heching, 1999). By analogy with equation (1), valid in terms of certainty, at X = X opt for the expected total cost E(N(RL)), it can be written: (4) In the general case, E(NSL(RL)) has two components: the expected annual storage costs of the safety stock level in the amount H  SL = H  (RL -E(DDDT)) and the expected annual loss E(LLS(HD)) from a lack of stock. Therefore,

E(NSL(RL)) = H  SL+ E(LLS(RL))
(5) To minimize E(N(RL)), according to RL, means to minimize E(NSL(RL)), according to RL. In the following, it derives an algorithm to minimize E(NSL(RL)), according to RL in the discrete case and then it generalizes the idea of a solution to the continuous case.

Minimizing E(NSL(RL)) in the Discrete Case
By -the discrete case‖, it means the case when the unit of inventory is no more arbitrarily divisible (e.g. the washing machine in the warehouse store of washing machines, etc.). If assuming RL max = dddt max , then E(NSL(RL max )) = H  SL max = H  (dddt max -E(DDDT)). Because there is no shortage of stock, E(LLS(dddt max )) = 0. A gradual reduction of RL from RL max to RL -1 gradually decreases SL to SL -1 (see Figure 2). This leads

ΔE(LLS(RL)) = E(LLS(RL  1))  E(LLS(RL))
(6) The idea of the algorithm of minimization E(NSL(RL)) can be described as follows: As the first current value RL, it chooses the value equal to dddt max and then for each current RL, it counts according to (6) ΔE(LLS(RL)). If ΔE(LLS(RL)) ≥ H, then by the transition to RL -1, it is no better off. Thus RL opt = RL and the process ends. In the opposite case when ΔE(LLS(RL)) < H, the new current RL becomes RL -1 and the calculation according to (6) is repeated for the new current RL. The flowchart of the algorithm of this idea, in which the designation ΔE(RL) is an abbreviation for ΔE(LLS(RL)), is shown in Figure 3.
The algorithm elaborates from RL max to RL opt from above and RL opt is the one RL, to which the algorithm arrived by its recursive steps, and that as a result of (6) satisfies the condition ΔE(LLS(RL)) ≥ H ≥ ΔE(LLS(RL + 1)) (7)  (7), it obtains after adjustment:

HD = RLmax
Calculation where in both summations, it is summed over all i > RL. Sought RL opt is the RL, which satisfies the condition (8). This condition can be made more transparent by designation of the constant H  X opt /(L  E(D)) by the symbol Φ, by which it takes the shape of The constant Φ = H  X opt /(L  E(D)), sandwiched in (9) among the inequalities is the inverse value of the efficiency storage indicator.
This indicator is the product of the ratio of the unit loss from the lost margin to the annual cost of the unit storage (L/H) and the annual number of orders (E(D)/X opt ). It measures the loss of the dissatisfaction of the annual consumption of the inventory item with the annual costs of the storing of the optimum additional supply. The condition (9) in combination with Figure 2 then says that the higher this indicator is (and therefore Φ lower), the higher the safety stock level is worth holding. In the case of the limited storage capacity, it can be decided on the basis of the analysis of indicators, which are worth storing and which are not.

Minimization of E(NSL(RL)) in the Continuous Case
By the continuous case, it means that the unit of stock is arbitrarily divisible (as it is in the case of coal in the boiler room, the fluid reserve in the reservoir, and etc.). The transition from discrete to continuous case can be reached by the -from below‖ unbounded physical division of the original unit. In terms of mathematical analysis, it corresponds to the possibility of the limit transition from differential Δx = 1 in the discrete model to an arbitrarily small value of the differential dx → 0 in the continuous model. This is related to the transition from a finite number of elements in the set of values Ω = {0, 1, …, dddt max } of the discrete random variable DDDT to the interval (0, dddt max ) of countless values of dddt of the continuous variable DDDT.
The initial distribution p: Ω → (0, 1) of the discrete DDDT in the form {(i, p(i)): i  Ω}, where p(i) is the probability occurrence of dddt = i, passes in the case of the continuous DDDT in the continuous function f: (0, dddt max ) → (0, 1), in which for each x  (0, dddt max ) and for a sufficiently short interval dx  (0, pbdd max ), the product of f(x)dx gives the probability that x ≤ dddt ≤ x + dx. The function f is the density of probability of a continuous random variable DDDT. By assigning each value RL  (0, dddt max ) to its corresponding value F(RL) obtained by integrating the density of probability f(x) in the range from 0 to RL, i.e. by the rule F(RL) = ∫f(x)dx, this paper defines the distribution function F: (0, dddt max ) → (0, 1) of the continuous random variable DDDT with the extreme values of F(0) = 0 and F(dddt max ) =1. In the space among these extreme values, F(RL) value gives the probability that dddt < RL.
The discrete version of the continuous distribution function is the expression F(RL) = Σp(i), where it is summed over all i < RL. Taking into account the fact that, if summing over the entire field Ω, Σp(i) = 1 applies, then by substituting F(RL) to the equation (9), it can be obtained: The condition (11) can be achieved only when  = 1  F(RL) or F(RL) = 1  (which says the same thing) (12) It is obvious that condition (12) holds for both the discrete and continuous case. What it means in the continuous case specifically is apparent from Figure 4. It can be seen that the sought RL opt is placed under the intersection of the distribution function F(x) of a random variable DDDT with the level of y = 1 -Φ.
This model can be used as a default model when contracting the logistic optimization model in the context of statistical reliability analysis (Volf & Timková , 2014).

Conclusions
The paper focuses in detail on solving one of the fundamental problems of logistics-the questions -in what quantity‖ and -when‖ to resupply stock. While the literature gives a relatively accurate and satisfactory response in terms of certainty and uncertainty to the first question, the question -when‖ is often in terms of uncertainty answered more or less vaguely. The first part of the paper clarifies the standard definition of a model of the dynamics of the stock state with even storage removal in terms of certainty ( Figure 1) and draws conclusions from this, which further extends and generalizes on the conditions under uncertainty.
The reality reflecting general model of the dynamics of stock in terms of uncertainty (Figure 2), which can answer the question -when‖, is gradually built. For this purpose, this paper first studies the discrete case in detail, the result of which is described by the algorithm of the search of the optimum by means of the flow chart in Figure 3, alternatively by the condition (9), which when converted to the condition (11) by the limit transition is also valid for the continuous case.
To answer satisfactorily the question -when‖ requires calculating the probability distribution of the demand during the delivery time (DDDT) and the ascertainable value of, in this paper, newly established indicator of the storage efficiency. The indicator measures the loss of the dissatisfaction of the annual consumption of the inventory item and the annual storing costs of the optimum additional supply. The higher it is, the higher the safety stock level is worth holding. Its knowledge especially grows in importance in the case of the limited storage capacity indicating, which inventory items are worth storing and which are not. The frequently in practice occurring discreet case is presented within the solution of the model case study performing the economical procedure of the calculation of the DDDT built on the data of the analytical evidence of financial accounting. As shown, the correct result can be estimated from its several values, while the requirement on accuracy of these values is not high.