Managing slow-moving item: a zero-inflated truncated normal approach for modeling demand

Simple exponential smoothing / Simple exponential smoothing sum

This approach assume that LTD follows a normal statistical distribution and a fixed LT. The LTD mean and variance are calculated as follows:

(i) Considering the mean level of DPUT as μ_SES, and estimate using ${\displaystyle {\mu }_{SES}=\frac{1}{T+m}\sum _{t=T+1}^{T+m}\left(\gamma y_t+\left(1-\gamma \right){\mu }_{t-1_{SES}}\right),}$ where γ is a smoothing constant between 0 and 1, selected to minimize ${\sum }_{t=T+1}^{T+m}{\left(y_t-{\mu }_{SES}\right)}^2,t=T+1,\dots ,T+m.$ To initialize the smoothing, we can use the average of the first two demands ${\mu }_o=\frac{y_1+y_2}{2}.$

(ii) The DPUT variance with this approach can be calculated from: ${\displaystyle \text{Var}\left(Y\right)={\sigma }_{SES}^2=\frac{1}{T+m}\sum _{t=T+1}^{T+m}{\left(y_t-{\mu }_{SES}\right)}^2,\forall y.}$ The mean of K demands over the LT (μ_SSES) is given by ${\displaystyle {\mu }_{S_{SES}}={\mu }_K{\mu }_{SES},}$ and the variance of K demands over the lead time (${\sigma }_{SES}^2$) is calculated using one-step ahead forecast difference between the actual DPUT data and the mean lag, using the expression ${\sigma }_{SES}^2=\frac{1}{m}{\sum }_{t=T+1}^{T+m}{\left(y_t-{\mu }_{t-1_{SES}}\right)}^2.$ The variance of the LTD distribution (${\sigma }_{S_{SES}}^2$) can be estimated as: ${\displaystyle {\sigma }_{S_{SES}}^2={\sigma }_{SES}^2{\mu }_K.}$

Croston’s / Croston’s sum variant of Simple exponential smoothing method

Croston’s approach considers the DPUT mean using exponential smoothing that is separate from:

(i) the mean intervals of data conformed between non-null demands (here, the smoothed estimate is denoted by I_t) and (ii) the mean sizes of these intervals (here, the smoothed estimate is denoted by S_t).

In addition, q is the time interval since the last non-zero demand. Croston’s approach can be described as follows:

if Y = 0,  then

${\displaystyle S_t=S_{t-1}}$ ${\displaystyle I_t=I_{t-1}}$ (2)${\displaystyle q=q+1,}$

else

${\displaystyle S_t=\gamma Y_t+\left(1-\gamma \right)S_{t-1}}$ ${\displaystyle I_t=\gamma q+\left(1-\gamma \right)I_{t-1}}$ (3)${\displaystyle q=1.}$

The combination of the size and interval estimates from Eqs. (2) and (3), the DPUT mean can be expressed as: ${\displaystyle {\mu }_{CROST}=\frac{1}{T+m}\sum _{t=T+1}^{T+m}\left(\frac{S_t}{I_t}\right)}$ These estimates update whenever a demand non null realization occurs. When a demand occurs during the same review interval, Croston’s approach is identical to conventional exponential smoothing, where S_t = μ_tCROST.

To initialize Croston’s approach, we use the time until the first event and the size of the first event.

The DPUT variance when using this method can be expressed as: ${\displaystyle \text{Var}\left(Y\right)={\sigma }_{CROST}^2=\frac{1}{T+m}\sum _{t=T+1}^{T+m}{\left(y_t-{\mu }_{CROST}\right)}^2.}$ Croston’s method also considers LTD with a constant LT and normal statistical distribution. The mean is expressed as: ${\displaystyle {\mu }_{S_{CROST}}={\mu }_K{\mu }_{CROST},}$ and the variance is calculated as: ${\displaystyle {\sigma }_{S_{CROST}}^2={\sigma }_{CROST}^2{\mu }_K.}$

Zero inflated truncated normal / zero inflated truncated normal sum

In this model, we assumed that DPUT has a ZITNO distribution and LTD has a ZITNOsum distribution with a constant LT. We estimated the mean level of DPUT (μ_ZITNO) and its variance (${\sigma }_{ZITNO}^2$) using: ${\displaystyle \text{E}\left(Y\right)={\mu }_{ZITNO}=\left(1-\nu \right){\int }_{R_{y>0}}y\varphi \left(\right.\frac{y-{\mu }_{NO}}{{\sigma }_{NO}}\left)\right.\frac{1}{{\sigma }_{NO}}\text{d}y;\text{with}y>0,}$ and ${\displaystyle \text{Var}\left(Y\right)={\sigma }_{ZITNO}^2=\frac{1}{T+m}\sum _{t=T+1}^{T+m}{\left(y_t-{\mu }_{ZITNO}\right)}^2,}$ respectively, where μ_NO and σ_NO are mean parameters and the standard deviation (SD) of a normal distribution of subset y > 0. Note that Y forecasting length measures from T + 1 to T + m. On the other hand, the expected LTD value (μ_SZITNOsum) and its variance (${\sigma }_{S_{ZITNO}}^2$) under ZITNOsum distribution is calculated by: ${\displaystyle {\mu }_{S_{ZITNOsum}}={\mu }_K{\mu }_{ZITNO},}$ and ${\displaystyle {\sigma }_{S_{ZITNOsum}}^2={\sigma }_{ZITNO}^2{\mu }_K,}$ respectively.

Total cost of continuous review with the shortage inventory model

By carrying out the iterations shown in a previous subsection, we calculated the decision variables that minimize the total cost of continuous review with shortage inventory model TC = G(Q∗, r∗),  applied to Eq. (4) for each model in Tables 1 and 2. An inventory model is more effective when it results in a lower annual cost.

Expected shortage per cycle

We used a previously explained performance measurement to obtain the value of the expected amount of shortage per cycle (S(r∗)) given in Eq. (5), for each of the tested models in Tables 1 and 2. An inventory model is more effective when it results in a lower expected shortage per cycle.

Fill-rate

(Sobel, 2004) defined the fill-rate of a supply system as: “the fraction of demand that is met from on-hand inventory, understanding that the satisfaction of the demand is restricted to the amount purchased and available”. We calculated the Fill-rate for each statistical distribution shown in Tables 1 and 2 using Eq. (8): (8)${\displaystyle Fill-rate=\frac{{\int }_o^{Q\ast }yf\left(y\right)\text{d}y}{{\int }_0^{my}yf\left(y\right)\text{d}y},}$ where f(y) are the PDFs that are shown in Table 1, and my is the maximal DPUT for each of the evaluated distributions. An inventory model is more effective when its Fill-rate indicator value is closer to 1.

Kullback–Leibler divergence

To determine the quality of the proposed PDF and LTD approximation in Table 2, we used the Kullback–Leibler divergence (Cobb, 2004): (9)${\displaystyle \text{Kullback– Leibler}={\int }_{-\infty }^{\infty }log\left(\frac{f\left(x\right)}{\tilde{f}\left(x\right)}\right)f\left(x\right)\text{d}x,}$ where f(⋅) is the unknown true PDF and $\tilde{f}\left(\cdot \right)$ its approximation. To calculate the Kullback–Leibler divergence show in (9), we used the kernel estimate to establish the true PDF (Langseth et al., 2014). Using Eq. (1), we computed the sequence {s₁, …, s_n} of n LTD realizations (or data). According to the data, we defined the kernel estimate of the unknown PDF of LTD f_S(⋅) using: (10)${\displaystyle {\widehat{f}}_S\left(s\right)=\frac{1}{nh}\sum _{i=1}^{n}K\left(\frac{s_i-s}{h}\right),s\ge 0,}$ where K(⋅) is a kernel function satisfying ${\int }_0^{\infty }K\left(x\right)\text{d}s=1$, h a smoothing parameter and s the point at which the PDF is estimated. Once Eq. (10) has been calculated, the Kullback–Leibler divergence for each LTD distribution established in Table 2 with support [a, b] can be expressed as: (11)${\displaystyle {\text{Kullback– Leibler}}^{\prime }={\int }_a^{b}log\left(\frac{{\widehat{f}}_S\left(s\right)}{{\tilde{f}}_S\left(s\right)}\right){\widehat{f}}_S\left(s\right)\text{d}s,}$ where ${\widehat{f}}_S\left(s\right)$ is the kernel estimate and $\tilde{f}\left(s\right)$ are the approximations proposed in Table 2. We selected one of the approximations with a smaller Kullback–Leibler value than the one calculated using Eq. (11).

TOPSIS

This multi-criteria decision analysis analyzes the geometric distances between a chosen solution, the ideal solution, and the least suitable solution (Yoon & Hwang, 1995; Aye, Gupta & Wanke, 2018; De Andrade, Antunes & Wanke, 2020).