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Choosing the suitable demand distribution during lead-time is an important issue in inventory models. Much research has explored the advantage of following a distributional assumption different from the normality. The Birnbaum-Saunders (BS) distribution is a probabilistic model that has its genesis in engineering but is also being widely applied to other fields including business, industry, and management. We conduct numeric experiments using the R statistical software to assess the adequacy of the BS distribution against the normal and gamma distributions in light of the traditional lot size-reorder point inventory model, known as (

Inventory management permeates decision-making in countless firms. The topic has been extensively studied in academic and corporate spheres, for example, Braglia et al. [

According to Wanke [

The importance attached by firms to inventory management can be attributed to the following: first and foremost, it is the need to ensure that products, given the competitive pressure exercised by markets, are always supplied to customers at the least possible cost; see Eaves [

The inventory management models are frequently classified in two types: pull and push. On the one hand, according to Ballou and Burnetas [

According to Silver et al. [

Demand uncertainties directly affect the operation of the physical system of logistics. Moreover, to be closer to reality, single or multiple period inventory models must take into account that demand is occurring in a random fashion, which is explained by several factors. Thus, DPUT is taken to be a random variable (RV). Furthermore, during LT, due to the mentioned randomness, the corresponding demand (LTD) is also a RV; therefore, the behavior of DPUT and LTD must be described by statistical distributions; see Johnson et al. [

A unimodal, two-parameter probability model with positive support and asymmetry to the right that is receiving considerable attention is the Birnbaum-Saunders (BS) distribution; see Birnbaum and Saunders [

Our main objective is to explore the use of the BS distribution in inventory management. Differently from previous studies that exclusively considered the effects of one given distribution on inventory decision-making, we also analyze its adequacy in light of different operating characteristics and costs. Specifically, we assess how the BS, gamma, and normal LTD distributions interact with relevant product characteristics and affect the optimal EOQ and SS inventory indicators in terms of the optimization of the TC function. We minimize this function using stochastic programming, a technique where constraints and/or objective function of the problem to be optimized contain RVs that can follow any distribution; see Shapiro et al. [

Section

In this section, we discuss general aspects of inventory management models and demand statistical distributions used for the methodology presented in Section

The

The expected TC of the inventory is given by

Notice that it is necessary to specify the LTD distribution to determine the SS given in (

Silver [

In this section, we introduce our simulation scenario and formulate the stochastic programming used to optimize the expected TC associated with the

Assume BS, gamma, and normal distributions for the LTD. Then, fix values for the parameters of these distributions by considering values for means and SDs of DPUT and LT generated from uniform distributions. Now, generate holding, ordering, and shortage costs also from uniform distributions. This allows us to establish the expected TC to be minimized. Ten thousand (10000) different simulated scenarios of means and SDs for DPUT and LT as well as holding, ordering, and shortage costs are generated using an

Range of the mentioned uniformly distributed indicator for simulations.

Indicator | Minimum | Maximum |
---|---|---|

DPUT mean (units/day) | 80 | 120 |

DPUT SD (units/day) | 3 | 30 |

LT mean (in days) | 1 | 5 |

LT SD (in days) | 0.50 | 2.00 |

Holding cost ($/unit/day) | 0.00 | 0.68 |

Ordering cost ($/order) | 17 | 60 |

Shortage cost ($/shortage) | 0 | 100 |

Once the values for the inventory indicators are defined and the distributional assumptions (BS, gamma, and normal) for the LTD established, stochastic programming is performed on the expected TC function given in (

The problem of stochastic programming formulated in (

DE is a member of the family of genetic algorithms, which mimic the process of natural selection in an evolutionary manner; see Holland [

The DE algorithm has also been used to optimize problems that arise in inventory management, such as joint replenishment, replenishment coordination, and inventory location-allocation; see Qu et al. [

In what follows, we first discuss the DE algorithm used in our research work; see Storn and Price [

Let

Calculations in our simulation study were performed with the aid of the

… allows the user to pass additional arguments to the function

The return value of the function

The underlying idea of performing a sensitivity analysis on the testing variables related to product, demand, and operational characteristics is to discriminate between groups where the three distributional assumptions led to minimal TCs. Table

Summary of the simulations for the mentioned distribution and management indicator.

Indicator | Distribution | ||
---|---|---|---|

Gamma | Normal | BS | |

Sum of TCs (in $) | 205097.63 | 209175.49 | 202134.31 |

Sum of EOQs | 767449.00 | 764816.00 | 762843.00 |

Sum of SSs | 41166.66 | 53425.51 | 38103.58 |

Sum of average inventory level | 424891.16 | 435833.51 | 419525.08 |

Table

Number and percentage of times that the indicated distribution yielded minimal TC for the simulations.

Distribution | Number | % | Average CV of the LTD |
---|---|---|---|

BS | 8160 | 81.60 | 0.56 |

Normal | 1780 | 17.80 | 0.34 |

Gamma |
60 | 0.60 | 0.44 |

This paper has assessed how different demand distributions during lead-time interact with relevant characteristics of the product. We have considered the choice of the optimal inventory policy in terms of total costs of the inventory management. Differently from most previous studies on the topic that exclusively explored the effects of one single distributional assumption, this paper also analyzed its adequacy in light of some inventory management key elements for decision-making, such as cost, demand, and lead-time. It was shown that the Birnbaum-Saunders distribution outperformed the normal and gamma assumptions with respect to demand uncertainty during the lead-time. The contributions of this paper are on both the academic and the practical sides. Departing from what was found in previous studies, the obtained results provide a guidance on the selection of the most appropriate distributional assumption for the demand during the lead-time. Specifically, while the Birnbaum-Saunders distribution is more adequate to handle demand during lead-time with high coefficients of variation, the gamma and the normal assumptions should be restricted to well-behaved patterns. This paper also can present a practical contribution by means of numerical analyses conducted with the aid of a computational code developed for such purposes in the R statistical software.

With respect to future research endeavors, one possible extension of this work could be to analyze the results presented here in terms of inventory management in decentralized systems. The basic idea would be to assess, via numerical studies, the extent to which the decision to pool inventories is affected by different distributional assumptions and by their levels of skewness and kurtosis. In addition, in practical case studies, the usage of diagnostic statistical methods to detect influential data of demand or LT can be studied for this type of inventory models. In that case, demand distributions that provide parameter estimators to be robust to atypical data of demand must be considered; see, for example, Paula et al. [

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors thank the editors and referees for their constructive comments on an earlier version of this paper which resulted in this improved version. This research was supported by COPPEAD, Capes, and CNPq from Brazil and FONDECYT 1120879 from Chile.