The effect of advance payment with discount facility on supply decisions of deteriorating products whose demand is both price and stock dependent

• Author: Leopoldo Eduardo Cárdenas‐Barrón,Md. Al‐Amin Khan,Ioannis Konstantaras,Ali Akbar Shaikh,Gobinda Chandra Panda
• DOI: http://doi.org/10.1111/itor.12733
• Date: 01 May 2020
• Published date: 01 May 2020

Intl. Trans. in Op. Res. 27 (2020) 1343–1367

DOI: 10.1111/itor.12733

INTERNATIONAL

TRANSACTIONS

IN OPERATIONAL

RESEARCH

The effect of advance payment with discount facility on supply

decisions of deteriorating products whose demand is both price

and stock dependent

Md. Al-Amin Khana, Ali Akbar Shaikhb, Gobinda Chandra Pandac,

Ioannis Konstantarasdand Leopoldo Eduardo C´

ardenas-Barr´

one

aDepartment of Mathematics, JahangirnagarUniversity, Dhaka, Bangladesh

bDepartment of Mathematics, University of Burdwan, West Bengal,India

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

dDepartment of Business Administration, School of Business Administration, University of Macedonia, Thessaloniki,

Greece

eDepartment of Industrial and Systems Engineering, School of Engineering and Sciences, Tecnol´

ogico de Monterrey,

Monterrey, M´

exico

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

ikonst@uom.gr [Konstantaras]; lecarden@tec.mx [C´

ardenas-Barr´

on]

Received 9 May2019; received in revised form 3 September 2019; accepted 20 September 2019

Abstract

Generally,in the business world, it is observed that suppliers give different kinds of benefits to retailers due to

advance payment. One of the popular benefits is instant cash discount due to advance payment. If a retailer

pays off his total purchase cost before receiving the products, then he receives a certain percentage of cash

discount instantly. However, if the retailer pays off a certain fraction of the total purchasing cost, then price

discount is given only at the time of receiving the products while paying the remaining amount of the total

purchasing cost. Using this concept, this paper formulates, under both cases of advance payment (full or

partial), an inventory model for deteriorating products where shortages are allowed and demand function

is considered as price and stock dependent. The closed-form solutions for each case are presented and two

numerical examples are solved. In addition, a sensitivity analysis is also performed to show the effects of

advance payment with discount facility.

Keywords:pricing; deterioration; advance payment; discount; price- and stock-dependent demand

1. Introduction

In the present-day competitivemarket dynamics, it is very challenging for businessmenand sellers to

attract and convert potential customers into actual buyers. In this end, to catch the attention of the

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.

1344 M.A.-A. Khan et al. / Intl. Trans. in Op. Res. 27 (2020) 1343–1367

customers, vendors embark on different penetrations and marketing strategies as market conditions

never remain stable for long periods. In order to run their business successfully, vendors offer

different types of discounts, namely, quantity discounts, trade discounts, promotional discounts,

seasonal discounts, cash discounts, and prepayment facilities. Among these, quantity and trade

discounts are most common in practice. In recent times, provisions for prepayment facilities have

been in vogue and are offered by vendors to attract their potential buyers in prevailing market

situations. The proposed model is applicable in every sector related with the deteriorating items

such as shopping mall, wholesale market, retail business, online shopping, among others.

At present, the concept of prepayment has caught the attention of a lot of researchers and aca-

demicians. Recently, Taleizadeh (2014a) presented an economic order quantity model by assuming

that the vendor requests to his buyer to prepay a fraction of the purchasing cost in equal size in

multiple installments. The remaining payment is made at the time of receiving the lot. Taleizadeh’s

(2014a) model considers deteriorating items under constant demand where shortagesare completely

backlogged. In another paper, Taleizadeh (2014b) improved his previous work assuming that short-

ages are partially backlogged. But in reality, the buyer most of the time wants to avoid this type of

situation because, in the case of prepayment, he has to bear some additional cost on the amount

that has been prepaid. In such cases, to create a center of attention of buyers, vendors can offer a

discount based on the amount of prepayment. Also in both the aforementioned studies (Taleizadeh

2014a, 2014b), demand rate of the product is considered as constant, although, in practice, it is

influenced by the selling price as well as the stock level of the product.

In recent years, inventory models for deterioratingitems have been widely investigatedby eminent

researchers/academicians.For instance, Chung and C´

ardenas-Barr´

on (2013) analyzed an inventory

model for a deterioratingproduct under stock-dependent demand and two-level trade credit system.

Yanget al. (2013) introduced retailer’s optimal ordering policy when suppliers offer a cash discount

or delay in payment facility. Sarkar and Sarkar (2013) developed an economic order quantity

(EOQ) inventory model for infinite replenishment rate with stock-dependent demand and time-

varying deterioration. Shah and C´

ardenas-Barr´

on (2015) presented a retailer’s credit policy for

deteriorating items with order-linked credit period and cash discount. Most recently, Teng et al.

(2016) studied an inventory model fordeteriorating items with seasonal demand where deterioration

rate gradually increases as the expiration date approaches. Wu et al. (2016) formulated an inventory

model with maximum lifetime of the product under downstream partial trade credit and credit-risk

customers. Khan et al. (2019b) considered an inventory model in all units discount environment for

a maximum lifetime–related deteriorating product.

In the typical economic order quantity model, the demand is taken as a constant and known,

but in real life, it is in fact a variable depending on the selling price and the availability of stock at

a particular point in time. Therefore, the demand rate must be considered as a function of either

the selling price or the stock level or both. Sana and Chaudhuri (2004) constructed an economic

order quantity inventory model by assuming that the demand rate depends on the stock level

as well as advertising expenditure. Balkhi and Tadj (2008) studied an economic order quantity

model for items with time-varying demand and deterioration. Yang et al. (2010) examined an

inventory model for deteriorating items in which demand rate depends on item availability with

partial backlogging under inflation. To maximize the total profit, using preservation technology for

a deteriorating product, Lee and Dye (2012) builtan economic order quantity inventory model with

partial backlogging and stock-dependent demand. Shaikh et al. (2017) formulated an inventory

C

2019 The Authors.

International Transactionsin Operational Research C

2019 International Federation ofOperational Research Societies