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A multi-product multi-period inventory control problem under inflation and discount: a parameter-tuned particle swarm optimization algorithm

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Abstract

In this paper, a seasonal multi-product multi-period inventory control problem is modeled in which the inventory costs are obtained under inflation and all-unit discount policy. Furthermore, the products are delivered in boxes of known number of items, and in case of shortage, a fraction of demand is considered backorder and a fraction lost sale. Besides, the total storage space and total available budget are limited. The objective is to find the optimal number of boxes of the products in different periods to minimize the total inventory cost (including ordering, holding, shortage, and purchasing costs). Since the integer nonlinear model of the problem is hard to solve using exact methods, a particle swarm optimization (PSO) algorithm is proposed to find a near-optimal solution. Since there is no bench mark available in the literature to justify and validate the results, a genetic algorithm is presented as well. In order to compare the performances of the two algorithms in terms of the fitness function and the required CPU time, they are first tuned using the Taguchi approach, in which a metric called “smaller is better” is used to model the response variable. Then, some numerical examples are provided to demonstrate the application and to validate the results obtained. The results show that, while both algorithms have statistically similar performances, PSO tends to be the better algorithm in almost all problems.

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Author information

Authors and Affiliations

  1. Faculty of Engineering, University of Malaya, Kuala Lumpur, Malaysia

    Seyed Mohsen Mousavi

  2. Young Researchers and Elite Club, Qazvin Branch, Islamic Azad University, Qazvin, Iran

    Vahid Hajipour

  3. Department of Industrial Engineering, Sharif University of Technology, P.O. Box 11155-9414, Azadi Ave, Tehran, 1458889694, Iran

    Seyed Taghi Akhavan Niaki

  4. Department of Science and Technology, Universiti Kebangsaan Malaysia, 43600, Bangi, Selangor, Malaysia

    Najmeh Aalikar

Authors
  1. Seyed Mohsen Mousavi
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  2. Vahid Hajipour
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  3. Seyed Taghi Akhavan Niaki
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  4. Najmeh Aalikar
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Corresponding author

Correspondence to Seyed Taghi Akhavan Niaki.

Appendices

Appendix 1. The holding cost

The holding cost for product i between periods T i,j and T i,j (1 − L i,j + 1) + T O i,j L i,j + 1 is calculated as.

$$ \begin{array}{l}{h}_i{\displaystyle {\int}_{T_{i,j}}^a{I}_{i,j+1}(t) dt={\displaystyle {\int}_{T_{i,j}}^a\left({I}_{i,j}-{D}_{i,j}\left(t-a\right)\right){e}^{- ft} dt}}=\\ {}{h}_i{\displaystyle {\int}_{T_{i,j}}^a\left({I}_{i,j}+{D}_{i,j}a\right){e}^{- ft} dt-{\displaystyle {\int}_{T_{i,j}}^a{D}_{i,j}}}t{e}^{- ft} dt=\\ {}\frac{h_i}{f}\left({I}_{i,j}+{D}_{i,j}a\right)\left({e}^{-f{T}_{i,j}}-{e}^{- fa}\right)+\frac{h_i{D}_{i,j}}{f^2}\left({e}^{- fa}\left(1- fa\right)-{e}^{-f{T}_{i,j}}\left(1-f{T}_{i,j}\right)\right)=\\ {}\frac{h_i}{f^2}\left({e}^{-f{T}_{i,j}}\left(f\left({I}_{i,j}+{D}_{i,j}a\right)+{D}_{i,j}f{T}_{i,j}-1\right)\right)-{e}^{- fa}\left(f\left({I}_{i,j}+{D}_{i,j}a\right)+\left(1- fa\right){D}_{i,j}\Big)\right)=\\ {}\frac{h_i}{f^2}\left(f\left({I}_{i,j}+{D}_{i,j}a\right)\left({e}^{-f{T}_{i,j}}-{e}^{- fa}\right)-\left({e}^{-f{T}_{i,j}}\left(1-f{T}_{i,j}\right)+{e}^{- fa}\left(1- fa\right)\right){D}_{i,j}\right)\end{array} $$

where a = T i,j + 1(1 − L i,j + 1) + T O i,j L i,j + 1.

Appendix 2. The shortage cost

The shortage cost for product i between two periods T i,j O and T i,j+1 is obtained as.

$$ \begin{array}{l}{\displaystyle {\int}_{T_{i,j}^O}^{T_{i,j+1}}{I}_{i,j+1}(t){e}^{- ft} dt=}{\displaystyle {\int}_{T_{i,j}^O}^{T_{i,j+1}}{D}_{i,j}\left(t-{T}_{i,j}^O\right){e}^{- ft} dt=}{\displaystyle {\int}_{T_{i,j}^O}^{T_{i,j+1}}{D}_{i,j}t{e}^{- ft} dt+{\displaystyle {\int}_{T_{i,j}^O}^{T_{i,j+1}}{D}_{i,j}{T}_{i,j}^O{e}^{- ft} dt}=}\\ {}\left(\frac{e^{-f{T}_{i,j+1}}{D}_{i,j}}{f^2}\left(1-f{T}_{i,j+1}\right)-\frac{D_{i,j}}{f}{T}_{i,j}^O{e}^{-f{T}_{i,j+1}}\right)-\left(\frac{e^{-f{T}_{i,j}^O}{D}_{i,j}}{f^2}\left(1-f{T}_{i,j}^O\right)-\frac{D_{i,j}{T}_{i,j}^O}{f}{e}^{-f{T}_{i,j}^O}\right)=\\ {}\frac{D_{i,j}}{f}\left({e}^{-f{T}_{i,j+1}}\left(\frac{\left(1-f{T}_{i,j+1}\right)}{f}-{T}_{i,j}^O\right)-{e}^{-f{T}_{i,j}^O}\left(\frac{\left(1-f{T}_{i,j}^O\right)}{f}-{T}_{i,j}^O\right)\right)\end{array} $$

If \( {T}_{i,j+1}-{T}_{i,j}^O=\frac{b_{i,j+1}}{D_{i,j}} \) then \( {T}_{i,j}^O={T}_{i,j+1}-\frac{b_{i,j+1}}{D_{i,j}} \). Now, have.

$$ \begin{array}{l}{\displaystyle {\int}_{T_{i,j}^O}^{T_{i,j+1}}{I}_{i,j+1}(t){e}^{- ft} dt=}\frac{D_{i,j}}{f}\left({e}^{-f\left({T}_{i,j+1}\right)}\left(\frac{\left(1-f{T}_{i,j+1}\right)}{f}-\Big({T}_{i,j+1}-\frac{b_{i,j+1}}{D_{i,j}}\right)\right)-\\ {}\kern4.32em {e}^{-f\left({T}_{i,j+1}-\frac{b_{i,j+1}}{D_{i,j}}\right)}\left(\frac{\left(1-f\left({T}_{i,j+1}-\frac{b_{i,j+1}}{D_{i,j}}\right)\right)}{f}-\left({T}_{i,j+1}-\frac{b_{i,j+1}}{D_{i,j}}\right)\right)\end{array} $$

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Mousavi, S.M., Hajipour, V., Niaki, S.T.A. et al. A multi-product multi-period inventory control problem under inflation and discount: a parameter-tuned particle swarm optimization algorithm. Int J Adv Manuf Technol 70, 1739–1756 (2014). https://doi.org/10.1007/s00170-013-5378-y

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