A Higher-Order Markov Model for a Hybrid Inventory System with Probabilistic Remanufacturing Demand

Ali Khaleel Dhaiban

doi:10.1515/eqc-2022-0050

Article

A Higher-Order Markov Model for a Hybrid Inventory System with Probabilistic Remanufacturing Demand

Ali Khaleel Dhaiban

Published/Copyright: July 25, 2023

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From the journal Stochastics and Quality Control Volume 38 Issue 2

Abstract

This study develops a higher-order Markov model (HOM) for an inventory system with remanufacturing, substitution, and lost sales. Defective and disposed items are other factors that are considered in addition to probabilistic demand for both manufacturing and remanufacturing items. One year is the warranty period for items manufactured, and items sold return from customers to the manufacturer in increasing cumulative percentages over the months of the year. To the best our knowledge, a higher-order Markov model has rarely been used in a hybrid inventory system. The challenge is how to determine the steady state of the system with the probable demand for manufacturing and remanufacturing. We propose a new search algorithm to select the best control strategy from several strategies, and then compare it with the two-phase local search algorithm. Each state deals with (12) a probabilistic demand (policy), so the system steady state is set to (22632) policies in total for each production plan. The results showed profit maximization using the new search algorithm compared with the two-phase local search algorithm. Also, an increase in defective and returned items over time, and therefore an increase in remanufactured items. But it does not satisfy all the demand, so manufacturing increases over time due to substitution. Substitution strategy leads to increase the expected average profit.

Keywords: Inventory System; Higher-Order Markov; Substitution; Two-Phase Local Search Algorithm; New Search Algorithm; Probabilistic Demand

MSC 2020: 90B05; 65C40

A The Two-Phase Local Search Algorithm (TPLS)

This algorithm includes a two-phase search: the first is greedy search (GS), while the neighborhood search (NS) is the second. GS seeks to get the best value of the model parameters by changing the value of one parameter (increasing or decreasing) without changing the value of other parameters to maximize EAP. Then, the same method is used with the second parameter and so on for the other parameters. The process stops with EAP maximization, which means that there is no improvement in the EAP value for the current iteration compared to the previous iteration (Ahiska, Gocer and King [2]). In our model, the first and second warehouse ( x t , y t ) represent the parameters. The second warehouse for the remanufactured items and can be controlled by the percentage of items disposed of. The model assumes the lower limit of percentage is 0.1. Figures 16 and 17 show the results of the greedy search.

$Figure 16 EAP as a function of ux {\mathrm{ux}} ( ε = 0.1 \varepsilon=0.1 ).$

Figure 16

EAP as a function of ux ( ε = 0.1 ).

$Figure 17 EAP as a function of y ( ux = 11 , 000 \mathrm{ux}=11{,}000 ).$

Figure 17

EAP as a function of y ( ux = 11 , 000 ).

By Figures 16 and 17, the best values that maximize EAP are

ux = 11 , 000 , ε = 0.1 .

The second phase is neighborhood search that seeks to maximize EAP by changing the value of both parameters at the same time. If there is an improvement in the EAP, it means that the algorithm is repeated with new values. The opposite case, a greedy search solution is the optimal solution. The number of g-par neighbors is four with two parameters, and 20 with three parameters, according to ( 2 g ) ⁢ 2 g and ( 3 g ) ⁢ 2 g , respectively:

( 2 2 ) ⁢ 2 2 = 2 ! 2 ! ⁢ ( 2 - 2 ) ! ⁢ 2 2 = 4 ,

( 3 2 ) ⁢ 2 2 = 3 ! 2 ! ⁢ ( 3 - 2 ) ! ⁢ 2 2 = 12 ,

( 3 3 ) ⁢ 2 3 = 3 ! 3 ! ⁢ ( 3 - 3 ) ! ⁢ 2 3 = 8 .

From Figure 18, the solution of the greedy search is the optimal solution compared to the other four 2-par neighbor’s solutions.

Figure 18

EAP of the neighborhood search.

Acknowledgements

The authors would like to sincerely thank the referees for their valuable comments that improved the manuscript.

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Received: 2022-11-25

Revised: 2023-05-27

Accepted: 2023-05-28

Published Online: 2023-07-25

Published in Print: 2023-12-01

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Articles in the same Issue

https://doi.org/10.1515/eqc-2022-0050

Keywords for this article

Inventory System; Higher-Order Markov; Substitution; Two-Phase Local Search Algorithm; New Search Algorithm; Probabilistic Demand