# Multi-Source Backlogged Probabilistic Inventory Model for Crisp and Fuzzy Environment

### Abstract

This paper proposed a multi-item multi-source probabilistic periodic review inventory model under a varying holding cost constraint with zero lead time when: (1) the stock level decreases at a uniform rate over the cycle. (2) some costs are varying. (3) the demand is a random variable that follows some continuous distributions as (two-parameter exponential, Kumerswamy, Gamma, Beta, Rayleigh, Erlang distributions).

The objective function under a constraint is imposed here in crisp and fuzzy environment. The objective is to find the optimal maximum inventory level for a given review time that minimize the expected annual total cost. Furthermore, a comparison between given distributions is made to find the optimal distribution that achieves the model under considerations. Finally, a numerical example is applied.

### References

Abuo - El-Ata M., Fergany H. and El-Wakeel M., Probabilistic Multi-Item Inventory Model with Varying Order

Cost under Two Restrictions: A Geometric Programming Approach, International Journal of Production

Economics, 83 (2003) PP. 223-231.

Chiang C., A periodic review replenishment model with a refined delivery scenario, International Journal of

Production Economics, 118 (2009) PP. 253-259

Chiang C., Optimal ordering policies for periodic-review systems with a refined intra-cycle time scale,

Production, Manufacturing and Logistics, European Journal of Operational Research 177 (2007) PP. 872-881.

Chiang C., Optimal replenishment for a periodic review inventory system with two supply modes, European

Journal of Operational Research 149 (2003) PP. 229-244

Chiang C., Periodic Review Inventory Models with Stochastic Supplier's Visit Intervals, International Journal of

Production Economics, 115 (2008) PP. 433-438.

Chu K., Lung Y., Shing K. and Thomas N., Note on inventory model with a mixture of back orders and lost

sales Peter, European Journal of Operational Research 159 (2004) PP. 470-475.

Fabrycky W. J. and Banks Jerry. (1967). Procurement and Inventory Systems: THEORY AND ANALYSIS.

Newwork, Amsterdam, London.

Fergany H., Probabilistic Multi Item Inventory Model with Varying Mixture Shortage Cost Under Restrictions,

Springer Plus Journal of Springer Plus 5(1), (2016), PP. 1-13.

Fergany H., El-Hefnawy N. and Hollah O., Probabilistic Periodic Review Inventory Model Using

Lagrange Technique And Fuzzy Adaptive Particle Swarm Optimization, Journal of Mathematics and Statistics

(3) (2014) PP. 368-383.

Fergany H. and El-Wakeel M., Probabilistic Single - Item Inventory Problem with Varying Order Cost Under

Two Linear Constraints. Journal of the Egyptian Mathematical Society, 12(1), (2004) PP. 71-81.

Konstantaras I. and Papachristos S., Production, Manufacturing and Logistics Optimal policy and holding

cost stability regions in a periodic review inventory system with manufacturing and remanufacturing

options, European Journal of Operational Research 178 (2007) PP. 433-448.

Rong M., Mahapatra N. and Maiti M., A multi-objective wholesaler-retailers inventory-distribution model

with controllable lead-time based on probabilistic fuzzy set and triangular fuzzy number, Applied

Mathematical Modelling 32 (2008) PP. 2670-2685.

Sadjadi S., Ghazanfari M. and Yousefliâ€ši A., Fuzzy Pricing and Marketing Planning Model: A Possibilistic

Geometric Programming Approach, Expert System Applied 37 (2010) PP. 3392-3397.

Silver A. and Robb J., Some insights regarding the optimal reorder period in periodic review inventory

systems, International Journal of Production Economics 112 (2008) PP. 354-366

Yuyue S. and Hoong C., A periodic-review inventory model with application to the continuous-review

obsolescence problem, European Journal of Operational Research 159 (2004) PP.110-120.

*JOURNAL OF ADVANCES IN MATHEMATICS*,

*14*(2), 7729-7743. https://doi.org/10.24297/jam.v14i2.7365

Copyright (c) 2018 JOURNAL OF ADVANCES IN MATHEMATICS

This work is licensed under a Creative Commons Attribution 4.0 International License.

Authors retain the copyright of their manuscripts, and all Open Access articles are distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided that the original work is properly cited.