An optimization approach for inventory costs in probabilistic inventory models: A case study Un enfoque de optimización para costos de inventario en modelos de inventario probabilísticos: Un caso de estudio

Inventories represent stocks of goods necessary for operations of sales or manufacturing in a company. These allow to the companies meet their sales levels, while representing an opportunity to the cost control and the decision-making. This paper presents an optimization approach to minimize the inventory costs in probabilistic inventory models of independent demand. The approach has been validated for set the policy optimal of inventories with probabilistic demand within a company that markets disposable products. The established policy aims to minimize the inventory costs by using the standard deviation of the historical data, the mean deviation of forecast errors and the mean deviation of the historical data. For the determination of the economic order quantities, three types of products were selected, taking historical sales data. Likewise, different forecasting methods were used, selecting the one that minimizes the mean squared error for the forecasted demand. The proposed methodology is practical and easy to use in companies where inventories have probabilistic and independent demand. Also, the proposed approach allowed optimize the costs related to holding costs, ordering costs and safety stock costs.


INTRODUCTION
Organizations dedicated to the marketing of products usually have some questions concerning the ordering of products necessary. These questions, are usually related to two specific situations. First, in case of shortage (i.e. insufficient products to supply the demand) that causes potential losses due to miss sales opportunities. Secondly, in case of excess inventories that greatly increases the holding and maintenance costs [1][2][3]. The goal is to balance these two situations having the right quantities of each product and at the same time, avoid the decrease of sales due to lack of goods [4,5]. To establish the optimal quantities of inventory of a product, the demand must be considered over a period of time (days, weeks, months, and years), however, when we obtain this information, the product can have a constant or variable demand; which would hinder the process of setting the optimal quantities need to order. In any case, the responsible of these decisions aims to create and implement techniques, methods and methodologies that allow good inventory management to accurately determine the quantities to be ordered at the right time [5][6][7].
In this paper, an optimization approach for minimizing inventory costs in probabilistic inventory models of independent demand is proposed according to the behavior of demand from each product and the inventory model that best fits the company. The standard deviation of the historical data, the mean deviation of forecast errors and the mean deviation of the historical data were taken into account to calculate the inventory total cost. Furthermore, several forecast techniques were also used to establish the future demand values. The study was carried out in a company specialized in the commercialization of disposable product located in the city of Sincelejo-Colombia. This approach has been validated on three different products: Styrofoam dishes, rolls of plastic sheeting and die-cut bags. This paper has been structured as follows: Section 2 shows a summary of some studies related to inventories topic. Section 3 presents the methodology used in this document to the implementation of the proposed optimization approach. In Section 4, the evaluated and compared forecasting methods to minimize the mean squared error are displayed. The Section 5, presents the measures of dispersion used to determine the security stock of the evaluated inventory policies. Section 6 shows the approach to determining the economic order quantities and the total cost of inventory. The results and analysis of the numerical experiments are presented in Section 7, followed by the conclusions in Section 8.

RELATED WORKS
Inventory models have been recently studied by different authors, who have generated and presented significant approaches. For example, In [8] it was proposed an optimal combination of reserves instances on demand, so that the demand is satisfied and the costs are reduced to the minimum. To achieve this goal, this study presented a stochastic model based on inventory theory and it was formulated as an inventory management problem. In [9] it was proposed a model for the pricing and the inventory control of non-instantaneous deteriorating items. They determined the optimal selling price and the optimal inventory control variables by study of the behavior of the deterioration of the items. In [10] it was developed an inventory control model when replenishment intervals are probabilistic and partial backordering. They considered that inventory at the beginning of the period is not equal to zero and it can be positive or negative amounts, therefore, the goal was to determine the optimum amount of replenish-up-to level in special sale offer. In [11] it was presented an integrated inventory model to find optimal solutions for lot size, setup cost, and the total number of deliveries from the supplier to the buyer in a single production run, so that the expected total cost is minimum. In [12] it was proposed reducing investment in safety stock by using the standard deviation of forecast errors instead of standard deviation of the historical demand. In [13] it was developed an economic order quantity (EOQ) model to determine the joint ordering policy for two products under completion and substitution conditions. In [14] it was developed a deterministic inventory model with ramp-type demand depending on price and time. The aim was to maximize the total profit per unit time. In their approach, they assumed that the cumulative holding cost is a nonlinear function of time and presented a procedure to determine the economic lot size, the optimal inventory cycle and the maximum profit. In [15] it was considered an inventory model under the classical EOQ framework. The author assumed that shortages of inventory affect the customers' demand and, to obtain the total cost, it was approximated the backlogged demand rate by using a piecewise constant function. Finally, in [16] it was presented a unified EOQ model with financial constraints and market tolerance. It considered the EOQ paradigm with partial backorders to study factors related to the financial crisis. In this sense, an exact unified model was developed to calculate the total average profit/cost, the optimal profit/cost and the decision variables.

METHODOLOGY
In this research were determined and compared the total costs in a probabilistic inventory model using three different measures of dispersion of the demand: the standard deviation of the historical data, the mean deviation of forecast errors and the mean deviation of the historical data. Prior to that, three forecasting techniques (moving average method, weighted moving average method and exponential smoothing model) were compared to identify the forecasting method with the lowest value for mean squared error. This ensured that the mean deviation of forecast errors used to calculate the total cost of inventory was the minimum. In the determination of the security stock levels was taken into account the standard deviation of the data for the three specific cases mentioned above. Moreover, the quantity of goods to request was fixed based on three components: the costs of holding, ordering and security stock.
As a study case to demonstrate the usefulness of the proposed approach, three kind of products were chosen from a company specialized in the commercialization of disposable product: Styrofoam dishes Nº 20, rolls of plastic sheeting (6 meters wide and 450 meters long per roll) and die-cut bags (0.305 meters wide and 0. 4064 meters long per bag) (see Figure 1). The historical monthly demand of the last four years, as well as the order cost, lead time and holding cost per product are shown in the Tables 1 and 2.
-Moving Average Method (MA): It is a technique to get an overall idea of the trends in a set of n data from the last n periods. It can be calculated for any period of future time and any subset of data [18]. The MA uses the average of the k + 1 most recent data (in our case, the k + 1 most recent historical data of demand) in the time as a forecast for the next period and it can be calculated as follows: Where x i is the actual value of the demand for the period i, {i = 1…n}. "Moving" indicates that, while a new observation is available, it replaces the oldest observation, and a new average is calculated. As a result, the average will change, as new observations emerge [18][19][20].

Weighted Moving Average Method (WMA):
It is a forecasting technique which assign a heavier weighting to the most recent data than the oldest data. It is based on the principle that recent data are more relevant. Each observation x i in the calculation receives a different weight b i , and the sum of the weights b i must be equal to 1 [17,20]. WMA also uses the k + 1 most recent historical data of demand to calculate the forecast for the next period, as follows: Where β n ≥ β n−1 ≥ … ≥ β n−k and It is a technique appropriate for forecasting data with no trend and is also suitable when there is little data available. In this model, the forecast for the next period F n + 1 is equal to a weighted average between the most recent observation x n and the most recent forecast F n [19,21], so that it can be calculated as follow: Where 0 ≤ θ ≤ 1 is the smoothing parameter.
These methods have been proposed according to the characteristics of the demand of the studied products (although this will depend, in other contexts, on the type of demand). The accuracy of each technique is tested by calculation of the mean squared error (MSE); which measures the average of the squares of the errors between the forecasted demand and its corresponding actual demand for t forecasted periods:

DETERMINATION OF THE SECURITY STOCK
It is well known that there are different types of inventory models to establish the optimal quantities, and the use of them depends on the type of demand of the product concerned [2,[22][23][24][25]. According to authors such as [2], the demand can have a probabilistic behavior when the coefficient of variation of the historical demand (CV) is equal or greater than 20%, that is: where; x is the average of the historical demand and σ s is its standard deviation. Probabilistic inventory models take into account that the critical period in one inventory cycle is related to the waiting period or lead time (L), because during this time there might be an inventory shortage. To avoid these eventualities, the ideal is to have an additional amount of goods as a security stock (ss). So, if the historical demand follow a standard normal probability distribution N (µ L , σ L ) during the lead time, then the security stock will depend on the critical value (Z a ) of the standard normal probability distribution N (0,1) for a significance of a. The security stock (ss) is set in such a way that the probability of shortages during L is at most a, this is: where x L is the actual demand during the period L. This is equivalent to P{z ≥ ss/σ L} ≤ a, where z (x L -µ I )/σ L . In this way, to obtain the minimum stock quantities it must be met P{z ≥ z a } ≤a, and therefore, ss ≥ σ L · z a [2,12].
An important part of the proposed approach is to determine the levels of security stock which get a minimum total cost and, at the same time, guarantee the levels of services desired by the company. In this sense, it's proposed to establish an inventory policy where the security stock will depend on three different ways of calculating the dispersion of the demand; which is subsequently adjusted to L periods, so that we can finally obtain the S L value.
• The standard deviation of the historical data: • The mean deviation of forecast errors (Best forecasting method): • The mean deviation of the historical data: In this way, the values of the dispersion are used to determine the security stock with a desired service levels. Authors such as [12] have shown that the mean deviation of forecast errors is a recommended method to products with probabilistic demand, however, this may not always be the case. In our approach, it's compared the total cost of inventory in each case, identifying the best policy for the company (see section 7).

ORDER QUANTITY AND TOTAL COST
The company targeted in this case study set its optimal quantities by calculating the average historical sales. At present, the organization makes a single monthly order for the quantities that it expects to sell in this period. Obviously, this policy does not take into account losses due to shortages or holding costs, being inefficient in terms of profitability. It's proposed to evaluate two inventory policies which take as input the demand dispersion in each case displayed section 5. The first, where a monthly order is made for each product and, the second, an annual policy where the quantities to be ordered are: Here, D is the expected annual demand, C o is the ordering cost per order and C h is the holding cost per unit per year. However, the approach to calculating the economic order-quantity must be adjusted to the context of the organization.
The annual policy uses an economic order-quantity model and assumes that the company have sufficient storage capacity to store its products [5], which was confirmed by company. Both policy use a service level of 95%, causing an expected annual inventory cost represented by the following equations: As already stated, the total cost of each policy is also compared for each approach in the calculation of the values of S L (see section 5). These results are shown in the next section.

RESULTS AND ANALYSIS
First of all, a descriptive summary for the historical data in Table 1 is done to checking the type of demand for each product, these results are shown in Table 3. The skewness and kurtosis values confirm that the demands of the products follow a standard normal distribution with parameters x,σ s 2 ( ) . Note that the CV value obtained for Styrofoam dishes Minimum values for the mean squared error (MSE) by using the MA, WMA and ES methods are shown in Table 4. Note that the MA and WMA methods were tested for values of K + 1 between 2 and 8, and 2 and 6, respectively (see Tables 5  and 6). The b i values used in the WMA method are also shown in Table 6. These experiments were compared with each other, selecting the K + 1 value that minimizes the average of the MSE for each of the products. In this case, the outputs confirm an optimal value K + 1 of 3 and 4 for Styrofoam dishes and Die-cut bags in the WMA method (see Demand vs forecast demand in Figures 2 and 4, respectively). By using the  algorithm Solver for Microsoft Excel, it was obtained an optimal value of q = 0.2 for the Rolls of plastic sheeting in the ES method (see Demand vs forecast demand in Figure 3).
The values for the forecasted demand, the absolute forecast error and the mean absolute error for the best forecasting methods are displayed in Table 8. Also, the outputs for σs, σ f and σ a by using eqs. (7), (8) and (9) are shown in Table 7. These values will allow us to calculate the total cost for each policy proposed in section 6 and represented by eqs. (11) and (12). In cases where it implies to use the deviation for the historical monthly data σ s and σ a with      the same way, in cases where it is necessary to use the deviation for forecast errors (σ f with σ L = (σ f / √30·√L)), the average value for the predicted monthly demand F t is used to calculate the Q m and D values, that is, Q m = F t and D = 12· F t .
The results of the policies are presented in Table 9 for a level of service of 95%, i.e., a = 5% (value defined by the company). In general, note how policy 2 always offers the lowest values for annual inventory costs, regardless of the type of product. It is observed that this reduction is greater for Styrofoam dishes and Rolls of plastic sheeting when the mean deviation of the monthly historical data (σ a ) is used. Also, taking into account that σ a is lower in these products. It is also noted that the biggest reduction in costs for Die-cut bags occurs when σ f is used, although for this product, the differences in costs between the two policies are minimal. This could mean that for the Die-cut bags the policy used would be indifferent.
On the other hand, it is evident the inventory cycles are reduced in policy 2 and, this reduction is considerably greater for Styrofoam dishes. This increases the number of orders per year, which has to be discussed with the company's supplier. It's also noted how the security stock (ss) is minimum for cases where the dispersion is smaller.
In conclusion, it is recommended to use an annual inventory policy for all products, which takes into consideration: •

CONCLUSIONS
Inventories represent an important part of any company due to its implications in the costs of capital invested, storage, maintenance and ordering. These allow to meet internal and external demand, generating high levels of satisfaction when they are well managed. In this research, an optimization approach for independent demand inventory was proposed to establish the best inventory policy in a company specialized in the commercialization of disposable product.
The approach compared three forecasting methods and three different ways of calculating the safety stock by calculating the standard deviation of the historical data, the mean deviation of forecast errors and the mean deviation of the historical data. The total annual cost for two inventory policies were compared using the demand data for three types of products. In the case study, the results confirm that an annual policy, in which the required quantities are defined by a model of economic order quantity and the lowest value of the dispersion is used, would reduce the current inventory costs, guaranteeing a level of service of 95%. The approach evaluated several methods to calculate the predicted demand, the dispersion of the demand (actual or forecasted) and the economic order quantities as factors that directly influence the inventory policy and the total cost. In this sense, we concluded that the mean deviation of forecast errors does not always yield the lowest values for demand dispersion. In conclusion, this study demonstrated that the proposed methodology is practical and easy to use in companies where inventories have probabilistic and independent demand. In future research, it is proposed to study the influence of the proposed approach on items under probabilistic stationary demand.