An integrated vendor-buyer supply chain model with inflation induced demand under conditional delivery

Introduction

An integrated vendor-buyer supply chain represents a coordinated framework involving the vendor, retailer and customer, collectively striving to achieve operational efficiency and sustainability. To ensure long-term profitability and seamless functioning of business operations, vendors and buyers typically prefer to engage in sustained collaborative arrangements. The seminal work of Goyal (1976) [7] pioneered this domain by introducing the optimization of the total cost in an integrated supply chain environment. Banerjee (1986) [4] extended this perspective by proposing ordering policies that enhanced joint profitability for both parties. Subsequently, Goyal and Gunasekaran (1995) [14] advanced the framework by incorporating deteriorating items within the inventory system. Huang (2004) [12] introduced the concept of unreliability in a single-vendor, single-buyer production-inventory setting and formulated the corresponding optimal policy. Hoque (2009) [9] also investigated a similar production-inventory model with the objective of deriving viable system solutions. Further contributions include Tayal et al. (2016) [24] , who integrated credit-period considerations into the vendor-buyer inventory model, and Mishra & Talati (2017) [22] , who enriched the framework by incorporating both advertising frequency and quantity discount policies. Collaboration, whether within a supply chain structure or in broader inter-firm contexts, is widely acknowledged as a key enabler of enhanced performance outcomes. A considerable body of literature has explored integrated inventory models encompassing two or more supply chain partners. Foundational studies in this domain was carried by Banerjee (1986) [4] . Goyal and Gupta (1989) [15] provided an influential review of integrated inventory modeling efforts, while Aderohunmu et al. (1995) [2] examined Just-In-Time (JIT) applications within such systems. Lu et al. (1995) [20] extended the scope by addressing inventory models involving a single vendor and multiple buyers. Hill (1997) [8] and Viswanathan (1998) [28] examined replenishment policies under scenarios of equal and unequal lot sizing, respectively. Goyal and Nebebe (2000) proposed a modular cost-reduction inventory framework, later extended by Wu and Ouyang (2003) [30] , who explicitly accounted for shortages in the system. further refined the algorithm proposed by Wu and Ouyang, thereby contributing to its practical applicability. The following decade witnessed intensified research in integrated modeling. Hsu and Hsu (2012, 2013) [10,11] emphasized the significance of planned backorders, while Khanna et al. (2016) [19] extended their work by embedding trade-credit policies within the modeling framework. The relevance of collaboration has also been validated through industrial practices. In recent years, the focus has shifted toward sustainability-oriented integrated models. Gautam and Khanna (2018) [13] developed an integrated inventory model that addressed imperfect production processes, setup cost reduction, and carbon emissions. Wang et al. (2019) [29] analyzed supply chain systems in which suppliers provide environmental friendly products to sustainability-conscious consumers. Manna et al. (2021) [21] proposed a single-manufacturer; multi-retailer framework that incorporated a realistic pre-payment mechanism, whereas Rout et al. (2021) [23] addressed the issue of vehicle routing in sustainable integrated supply chains. In the contemporary global economy, inflation serves as a critical factor that diminishes the purchasing power of money. Many nations face persistent inflationary pressures that directly affect consumer demand patterns for various goods and services. Rising inflation not only reduces the real value of savings but also encourages greater expenditure on non-essential or luxury items, thereby altering market demand structures. Hence, overlooking the implications of inflation would present an incomplete and ethically questionable assessment of economic dynamics. Buzacott (1975) [5] first discussed EOQ model with inflationary effect subject to different pricing policies. Chang et al. discussed the effect of inflation on the inventory model for deteriorating items in consideration with partial backlogging. Other researchers like Jaggi et al. (2006) [17] , Chern et al. (2008) [6] , Jaggi et al. (2016) [18] , Thangam and Uthayakumar (2010) [25] contribute their valuable efforts in developing the inventory models under inflationary effect. Bag et al. (2017) [3] , Tripathy &Bag (2018) [26] , Tripathy & Behera (2019) [27] are some authors who contribute their valuable efforts in developing the inventory models. Despite these significant contributions, existing literature continues to exhibit notable gaps. Most studies have predominantly emphasized either cost optimization or sustainability in isolation without adequately addressing the simultaneous impact of multiple real-world factors such as product deterioration, environmental considerations, credit policies, and consumer-driven demand variability. Furthermore, only limited attempts have been made to integrate advanced coordination mechanisms that can balance profitability with sustainability imperatives. This gap necessitates the development of more comprehensive and realistic integrated vendor-buyer models that jointly account for economic, operational, and environmental dimensions. The present study aims to bridge this gap by proposing an improved framework that captures these complexities and offers practical insights for both academia and industry. The rest of the chapter is developed as follows. Notations and assumptions are placed in the next section. Mathematical formulation with solution procedure is established next. In the third section empirical investigation is carried out. In the fourth section sensitivity analysis is performed with respect to major parameters. The conclusion and future research scope is demonstrated in the last section.

 ∅ 𝟐: Total profit of the retailer per unit time

Assumptions

1. A single item inventory model is considered here. 2. The inventory system deals with deteriorating items that deteriorate at a constant rate. 3. The lead time is zero or negligible. 4. The shortage is not allowed. 5. The demand rate depends upon the time, inflation and credit period and represented by

is the quadratic rate of change of demand and 0   is a constant and N is constant rate of inflation. 6. If longer credit period is given to the buyer, default risk decreases and the rate of default risk for given credit period M is considered as

, where 𝛾> 0 is a constant

Mathematical Model For vendor

The mathematical model deals with the perishable products which deteriorate with a constant rate k. The buyer is offered a credit period with a quadratic demand. With permissible credit period the inventory level at any instant of time't' is represented by the differential equation

The solution of differential equation ( 3) with initial condition I (T) = 0 is given by (

At the beginning, when t = 0, the number of units in the inventory system is

The associated costs of the inventory system per cycle for the supplier are as follows. Sales revenue after default risk: SR = Selling price*Total demand

Ordering cost (OC) =

The total profit of the seller per unit time is given by

For Retailer

The retailer does not give credit period to the customer and simultaneously default risk will be absent. The mathematical model deals with the perishable products which deteriorate with a constant rate k. The inventory level at any instant of time 't' is represented by the differential equation

The solution of differential equation ( 3) with initial condition I (T) = 0 is given by

At the beginning, when t = 0, the number of units in the inventory system is

The associated costs of the inventory system per cycle for the supplier are as follows.

Sales revenue: SR = Selling price*Total demand

Ordering cost (OC) =

The total profit of the seller per unit time is given by

Total Profit of the Integrated Supply Chain model for both vendor and retailer is

The necessary conditions to maximize the total profit of the inventory system and to find the optimal values of credit period and production period are

For concavity of the profit function the following sufficient conditions should be satisfied.

By solving the pair of equations in equation (20), the optimal values of credit period M and cycle time T can be obtained. The solutions are determined with the help of MATHEMATICA-5.1 software and the concavity conditions are also checked.

Solution procedure  Step 1:

The parameters in inventory system are assigned with values.  Step2: The simultaneous equations ( 20) are solved with MATHEMATICA 5.1.  Step 3: The sufficiency conditions for concavity are tested.  Step 4: The total profit of the inventory system is found out by equation (19).

Empirical Investigation Numerical

Sensitivity Analysis

The effect of changes of parameters on the optimal solution are studied. Sensitivity Analysis is performed by changing the parameters by 25% and and taking one parameter at a time, keeping the remaining parameters at their original value.

Note: **indicates irrelevant results

Observations

Based on the sensitivity analysis the following managerial insights are obtained.

1. The total profit (TP) is highly sensitive with respect to the parameter a , b , c ,  and k . If the mentioned parameter values increases then the total profit increases rapily.

2. The total profit (TP) is moderately sensitive w.r.t the parameter m . It indicates the TP gradually decreases if the given parameter value change.

3. The TP remain static w.r.t the parameter s . it means that if the given parameter value is changed then it is hardly any effect in the TP of the inventory system.

4. The total profit (TP) is highly sensitive with respect to the parameter h and N . If the mentioned parameter values increase then the total profit decreases rapily.

Conclusion

A vendor-buyer integrated inventory model for deteriorating items subject to inflation induced demand has been developed in this paper. The cycle time and optimal credit period with consideration to default risk has been found out which maximizes the total profit of the system. In the real market world all the factors considered here are equally important to make decision on implementation of various marketing policies in the inventory system. Sensitivity analysis provides a thorough idea about the impact of several system parameters on the total profit of the system which can help to establish marketing strategies. It is advised to make less production period and allow more credit period for more demand to maximize the total profit. This model can also be extended by considering partial backlogging and shortage. Preservation technology can also be included to reduce deterioration.

References

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