Page 50 - IJOCTA-15-2

An International Journal of Optimization and Control: Theories & Applications
ISSN: 2146-0957 eISSN: 2146-5703
Vol.15, No.2, pp.245-263 (2025)
https://doi.org/10.36922/ijocta.1676

RESEARCH ARTICLE

Multiple item economic lot sizing problem with inventory
dependent demand

¨
Duru Balpınarlı and Mehmet Onal *
˙
¨
Department of Industrial Engineering, Ozye˘gin University, C¸ekmek¨oy, Istanbul, T¨urkiye
duru.balpinarli@ozu.edu.tr, mehmet.onal@ozyegin.edu.tr

ARTICLE INFO ABSTRACT
Article History: We consider a multiple item Economic Lot Sizing problem where the demands
Received: September 5, 2024 for items depend on their stock quantities. The objective is to find a production
Accepted: February 5, 2025 plan such that the resulting stock levels (and hence demands) maximize total
Published Online: March 20, 2025 profit over a finite planning horizon. The single item version of this problem
Keywords: has been studied in the literature, and a polynomial time algorithm has been
Economic lot-sizing proposed when there are no bounds on production. It has also been proven
Inventory dependent demand that the single item version is NP-hard even when there are constant (i.e.,
time-invariant) finite capacities on production. We extend this single item
Lagrangian relaxation
model by considering multiple items and production capacities. We propose
Tabu search algorithm
a Lagrangian relaxation method to find an initial solution to the problem.
AMS Classification:
This solution is a hybrid solution obtained by combining two distinct solutions
90B05; 90C59
generated in the process of solving the Lagrangian dual problem. Starting with
this initial solution, we then implement a Tabu Search algorithm to find better
solutions. The performance of the proposed solution method is compared with
the performance of a standard commercial software that works on a mixed
integer programming formulation of the problem. We show that our solution
approach finds better solutions within a predetermined time limit in general.

1. Introduction demand, pointing out a popular product, reassur-
ing customers about high service standards, and
availability in the future.
In order to increase sales and revenues, retailers
may follow various strategies. For instance, intel- In this paper, we consider the production
ligent pricing decisions or advertising campaigns planning problem of a manufacturer who pro-
may increase retailers’ sales and revenues. It is
duces multiple items, where we assume that the
observed in many cases that increasing the vis-
ible stock levels may also increase customer de- demands of items depend on their stock quan-
tities. We model this problem as an Economic
mand (and hence revenues) for some product cat-
egories. This is sometimes called the “Stack them Lot Sizing (ELS) problem. Throughout this pa-
1
high, let them fly” phenomenon. shows empiri- per, we will refer to our problem as multiple item
2
cal evidence from magazine retailing, and pro- ELSIDD (Economic Lot Sizing problem with In-
vides evidence from automotive dealerships that ventory Dependent Demand), where we assume
3
high stock levels stimulate sales. proposes that that demand for each item is a non-decreasing
increasing the stock level of a product can increase and convex function of the available amount of
sales for a variety of reasons, such as indicating that particular item in the stocks. We also as-
freshness (for food products), stimulating hidden sume that there are finite production capacities.
*Corresponding Author
245