Price discount facility in an EOQ model for deteriorating items with stock‐dependent demand and partial backlogging

• Published date: 01 July 2019
• Date: 01 July 2019
• Author: Gobinda Chandra Panda,Md. Al‐Amin Khan,Ali Akbar Shaikh,Ioannis Konstantaras
• DOI: http://doi.org/10.1111/itor.12632

Intl. Trans. in Op. Res. 26 (2019) 1365–1395

DOI: 10.1111/itor.12632

INTERNATIONAL

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IN OPERATIONAL

RESEARCH

Price discount facility in an EOQ model for deteriorating items

with stock-dependent demand and partial

backlogging

Ali Akbar Shaikha, Md. Al-Amin Khanb, Gobinda Chandra Pandacand

Ioannis Konstantarasd

aDepartment of Mathematics, University of Burdwan, Burdwan 713104, India

bDepartment of Mathematics, JahangirnagarUniversity, Savar, Dhaka 1342, Bangladesh

cDepartment of Mathematics, Mahavir Institute of Engineering and Technology, Bhubaneswar, Odisha 752054, India

dDepartment of Business Administration, School of Business Administration, University of Macedonia, 156 Egnatia Str.,

Thessaloniki 54636, Greece

E-mail: aliashaikh@math.buruniv.ac.in[Shaikh]; alaminkhan@juniv.edu [Khan]; gobinda1900@gmail.com [Panda];

ikonst@uom.gr [Konstantaras]

Received 12 October 2018; receivedin revised form 4 January 2019; accepted 8 January 2019

Abstract

All-units discount facilities are one of the attractive featuresin the competitive business situation. Due to the

globalization of the marketing policy, all-units discount facilities play an important role in the competitive

business. Typical economic order quantity (EOQ) models are cloistered by considering as constant not

only the purchase cost (irrespective of the order size of the product) but also the carrying cost during the

entire cycle period. However, the unit purchase cost has an antagonistic relationship with the order size,

and the carrying cost has a commensurate relationship with the storage time-period of the product, that

is, the higher the order size, the lower the unit purchase cost, and the longer the storage time-period, the

greater carrying cost per unit. Also deterioration is another imperative issue in inventory analysis as it has

a huge impact on profit or cost of the inventory system. Considering all of the above-mentioned factors,

we study two different inventory models, namely (a) inventory model for zero-ending case and (b) inventory

model for shortages case. The demand for both models is considered as price and stock dependent, whereas

shortages are partially backlogged at a rate with the length of the waiting time to the arrivals of the next lot.

The existence and uniqueness of the optimal solution for both models are examined theoretically and the

solution procedures are discussed along with two proposedalgorithms for minimizing the total cost. Finally,

we perform sensitivity analyses for both models and make a fruitful conclusion regarding the proposed

work.

Keywords:EOQ; deterioration; all-units quantity discount; stock-dependent demand; linearly time-varying carrying cost;

partial backlogged shortages

C

2019 The Authors.

International Transactionsin Operational Research C

2019 International Federation ofOperational Research Societies

Published by John Wiley & Sons Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main St, Malden, MA02148,

USA.

1366 A. A. Shaikh et al. / Intl. Trans. in Op. Res. 26 (2019) 1365–1395

1. Introduction

Nowadays, proper regulation and control of inventory (stock of products or goods or items pro-

duced or available in a business organization) is a big challenge for every business organization.

Proper management of inventoryin business satisfies the market behavior well, and business people

in general. One of the most important market behaviors is customer demand, which is directly

controlled by efficient and effective management of inventory. Until now, many inventory control

models have been developed assuming different types of demand along with various other aspects

related to inventory control, and therefore a suitable solution procedure is followed to maximize

profits or minimize costs.

This paper is developed considering different realistic assumptions, such as demand depends

on selling price and availability of adequate stock in the system. This type of demand is gen-

erally suitable for different types of industrial sectors of the economy, for instance food (pro-

cessed foods and raw foods), garment (clothes and dress materials), automobile (parts and fin-

ished products), electronic equipments, etc. Generally, in the food sector, the demand of raw food

items, such as vegetables, fruits, fishes, meat, and eggs, and of different processed foods in ho-

tels and restaurants are completely dependent on price. In this case, two situations may arise:

(a) if we increase the selling price of the items, then customers would move to other places

to satisfy their demand, on the other hand, (b) if we decrease the selling price, then demand

would rise for the concerned (specific) seller. However, the problem in decreasing in selling price

sometimes leads to stock-out problems due to the sudden rise in demand, and consequently, un-

availability of adequate stock in the system. Similar cases will also arise almost in every other

sector.

Another realistic assumption considered in this paper is that the holding cost depends on time

and increases linearly with it. When a new product (such as vegetables, fruits, etc.) reaches a seller,

it remains very fresh for some time and therefore the holding cost during this period remains very

small. But the freshness after the initial period gradually declines with time. Consequently, the

holding cost is an increasing function of time to preserve freshness. Thus, to keep the food products

fresh for a long period in a warehouse requires increasing holding cost, which depends on time.

Similarly, products from other sectors also need better warehousing facility from time to time to

meet this rising demand of the customers, which calls fortimely increasing of holding cost. Another

most important realistic assumption is purchase cost. Since demand depends on selling price, it

directly affects the order size of the customers, which creates scope for providing discounts and

other promotional offers to attractan increasing number of customers, and hence demand. Because

of less selling price, demand will be high and larger number of orders will be placed by customers,

and in this situation, all-units discount offer will be provided to the customer according to size of

the order.

Deterioration is another important issue in the inventory analysis. It has wide impact in the

inventory analysis. So, we cannot ignore this issue in the present study. Here, we assume that

the deterioration rate is constant and formulate two problems: one for zero-ending inventory and

another is partial backlogged shortages with the length of the waiting time of the customers. In the

next section, we describe the existing works that have been reported by several researchers and the

main contribution of the proposed paper.

C

2019 The Authors.

International Transactionsin Operational Research C

2019 International Federation ofOperational Research Societies