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A joint inventory–finance model for coordinating a capitalconstrained supply chain with financing limitations
Financial Innovation volume 7, Article number: 6 (2021)
Abstract
Faced with economic recession, firms struggle to find ways to stay competitive and maintain market share. Effective coordination of the supply chain can solve this problem, but this may fail if existing capital constraints and financial flows are ignored. This study addresses the challenge by exploiting coordination through joint decisionmaking on the physical and financial flows of a capitalconstrained supply chain. We also consider the capitalconstrained member’s financing limitations that lead to lost sales. Two scenarios based on noncoordinated and coordinated structures are modeled in the form of biobjective optimization problems that simultaneously optimize system costs and service levels. The models are solved using the \(\varepsilon\)constraint method. The results indicate that the noncoordinated model cannot satisfy more than about 50% of the demand due to capital shortage and financing limitations, while the coordinated model can satisfy all of the demand via internal financing. Furthermore, the proposed coordination scheme leads to cost reduction for the members and the total system. To motivate all members to accept the proposed coordination scheme, a costsharing mechanism is applied to the coordination procedure. Finally, a sensitivity analysis concerning financial parameters is provided for validating the coordination model.
Introduction
Supply chain management involves coordination and cooperation among several business partners linked through flows of material, money, and information. These partners include suppliers of raw materials and basic components, manufacturers, wholesalers, distributors, transporters, retailers, banks, and financial institutes. Two key factors ensure the proper functioning of a supply chain: first is coordination between supply chain members, and second is integration of the said three flows. This study addresses both factors for enhancing supply chain profitability and effectiveness.
Supply chain coordination involves collaborative work—joint planning, joint product development, mutual information exchange, integrated information systems, cross coordination on several levels of the companies in the network, longterm cooperation, and fair sharing of risks and benefits (SkjoettLarsen 2000). In a supply chain with a decentralized structure, each member independently optimizes its own operational and financial decisions without considering their effects on other members. Hence, a decision that is optimal for a member may cause additional costs for other members, which leads to inefficiencies in the processes of supplying, producing, distributing, and delivering goods to customers, and consequently, poor service. However, in a coordinated supply chain, all members operate to optimize the whole supply chain, leading to an increase in profitability and customer satisfaction in the supply chain.
To have an effective supply chain, proper management of upstream financial flows is as important as that of downstream physical flows (Gupta and Dutta 2011). The supply chain management literature has focused mainly on material and information flows, and financial flows have been ignored despite its great importance and inevitable effects on the physical flows (Naimi Sadigh et al. 2013; Sadeghi Rad et al. 2018; Sadeghi Rad and Nahavandi 2018). This literature stream assumes that a firm can always finance its operations at an optimal level or borrow at a constant interest rate. When financial markets are efficient, that is, external funding is plentiful and relatively inexpensive, firms have sufficient funds for daily operations, and financial and operational decisions can be made separately (Modigliani and Miller 1958). In such cases, a downstream entity in the supply chain pays the inventory ordered from an upstream entity. Consequently, financial flows become an output of logistics decisions (Luo and Shang 2015). However, capital shortage becomes a bottleneck, especially for small and medium enterprises. Financial activities complement the physical flow of materials and ensure funding of logistics operations. Thus, integrating operational and financial decisions may ensure proper implementation of operations in practice.
This study addresses both of these important issues—supply chain coordination and the integration of operational and financial decisions.
We consider an original equipment manufacturer (OEM) that faces a capital shortage and financing limitations. Specifically, financial institutions limit the OEM’s financing amount for reasons such as low credit. Because of this limitation on external financing, the OEM cannot obtain as much financing as it requires to support its operations. Therefore, it cannot satisfy a portion of the demand and thus, face lost sales (lost sales refer to selling opportunities that a firm has lost because an item was out of stock). The OEM operates in a supply chain with multiple suppliers of raw materials and components (upstream members), and an auto manufacturer (downstream member). The financial problems of the OEM affect all members in the supply chain and may lead in the loss of a portion of the market share of the supply chain’s final products. Thus, coordination among all members to solve the OEM’s financial problems is prudent so as to improve the supply chain’s performance. To understand better the proposed coordination mechanism, two scenarios are formulated: a noncoordinated supply chain (or decentralized supply chain) and a coordinated supply chain. The advantages of the second scenario for all supply chain members are illustrated using an example from a supply chain in the automotive industry.
The automotive industry is one of the world's most important economic sectors in terms of revenue and employment, and any kind of disruption in the production and profitability of a key member can be detrimental to the whole industry. Therefore, it is necessary to investigate issues such as the financial challenges and coordination of operational and financial decisions in this industry. The OEMs in automotive supply chains commonly try to establish longterm relationships with their suppliers and/or customers. Proper coordination mechanisms that align the parties' objectives can help build such relationships. This study finds ways to build these relationships in light of an OEM’s financial constraints (capital shortage and financing limitations). Nevertheless, the proposed coordination mechanism can be adopted by any manufacturing business that faces such financial challenges.
This study looks for effective ways to solve the financial problems of a supply chain and improve its performance through coordination of all members in light of a difficult financial situation. To this end, this study proposes a biobjective nonlinear optimization model that minimizes systemwide costs (including operational and financial costs) and maximizes the satisfied demand, which is equal to the optimal customer service level.
In the supply chain coordination literature, it is a novel idea to consider financing limitations (which is a realworld problem for capitalconstrained firms) and the consequent lost sales. Moreover, we believe that our study is the first to address supply chain coordination together with financing issues for maximizing service level and minimizing operational and financing costs.
The rest of this paper is organized as follows. Section “Literature review” provides a brief review of the related literature. Section “Model description and solution approach” presents the proposed model, including the model’s assumptions and notations, and the mathematical formulations of scenarios 1 and 2. Section “Numerical analysis” discusses the numerical results, along with a costsharing mechanism and sensitivity analysis of the financial parameters. Section “Conclusion” presents the conclusion and suggestions for further research.
Literature review
Our study relates to two literature streams: supply chain coordination and integration of the physical and financial flows of a supply chain (also known as operations–finance interface).
Supply chain coordination (SCC)
Coordination mechanisms in supply chains are generally based on centralized and decentralized decisionmaking processes (Jaber and Osman 2006). In a centralized decisionmaking process, a unique decision maker manages the whole supply chain, with the main objectives of minimizing (maximizing) the total supply chain cost (profit). By contrast, in a decentralized decisionmaking process with multiple decisionmakers, each member focuses on its own objectives. Li and Wang (2007) conducted a review of coordination mechanisms of supply chain systems in a framework that is based on the supply chain’s decision structure (centralized or decentralized) and the nature of demand (deterministic or stochastic).
Abundant research exists regarding the coordination between supply chain members. Several review papers have focused on various aspects of supply chain coordination (Fugate et al. 2006; Li and Wang 2007; Arshinder et al. 2008, 2011; Govindan et al. 2013). Accordingly, various classifications of coordination mechanisms exist in the literature. Arshinder et al. (2008), in their review of the literature, grouped coordination mechanisms into four categories: supply chain contracts, information technology, information sharing, and joint decisionmaking. As major categories of coordination mechanisms, Sahin and Robinson (2002) proposed price, nonprice, buyback and returns policies, quantity flexibility, and allocation rules. Subsequently, Fugate et al. (2006) adapted their classification into three major categories: price, nonprice, and flow coordination mechanisms. Bernstein and Federgruen (2007) showed the types of coordination mechanisms that allow a decentralized supply chain to generate aggregated expected profits equal to the optimal profits in a centralized system, and how the parameters of these (perfect) coordination schemes can be determined.
One common coordination mechanism is coordinating contracts, which are formulated using game theory in a decentralized structure. This stream of SCC literature uses some incentives in the form of contracts to achieve coordination among supply chain members. Some of these incentives include buyback contracts (Padmanabhan and Png 1997; Hou et al. 2010; Shen and Zhang 2012; Tibrewala et al. 2018), revenue sharing contract (Giannoccaro and Pontrandolfo 2004; Cachon and Lariviere 2005), service levelbased contracts (Sieke et al. 2012), discount policy (Xu et al. 2018a; Nouri et al. 2018), credit period (delay in payment) (Ebrahimi et al. 2019), bilevel credit period (Johari et al. 2018). The readers may refer to Cachon (2003) for comprehensive information about the structure of coordinating contracts.
Some researchers have shown that coordination can be achieved by integrating lotsizing models (Goyal and Gupta 1989). Joint economic lot size models are especially useful planning tools in situations where companies have established longterm relationships with their suppliers or customers, which is common in the automotive industry (Glock 2012). Glock (2012) focused on coordinating inventory replenishment decisions between buyer and vendor and their impact on the performance of the supply chain. He reviewed joint economic lot size models aimed at minimizing total system costs. Some of the first studies on the coordination of inventory replenishment and production decisions between supply chain members include Goyal (1977), Goyal and Gupta (1989), and Goyal et al. (2003). Sarmah et al. (2006) classified buyer–vendor coordination models that use quantity discount as a coordination mechanism under a deterministic environment. In a recent study, AlDurgam et al. (2017) considered a singlevendor, singlemanufacturer joint economic lot size problem under demand uncertainty. They developed an integrated mathematical model that investigates the impact of a variable production rate on the system. In most of these works, coordination is achieved through joint consideration of the systemwide costs. This coordination scheme is based on a centralized decisionmaking process. Some other works that adopted coordination by joint consideration of systemwide costs include Chen and Chen (2005), Jaber and Osman (2006), Chiadamrong et al. (2007), Jayaraman and Pirkul (2001), Kim et al. (2005), and MoussawiHaidar et al. (2014). In this study, coordination among members is adopted through this coordination scheme.
Operations–finance interface
Supply chain management literature mostly focuses on the flow of goods and services without considering the effects of capital shortage and financial flows on operations management. Researchers in this field assume that there is always adequate working capital to supply the required inventory. However, operational decisions are constrained by limited working capital and often critically depend on external financing (Xu and Birge 2006). These constraints influence supply decisions. Nevertheless, the interaction of physical and financial flows has been ignored by many researchers in the field of operations management. The interface of finance, operations, and risk management is a relatively new research area, which deals with timely, complex, and boundaryspanning issues in a variety of settings from startups to global enterprises (Babich and Kouvelis 2018). Zhao and Huchzermeier (2015) identified three types of interdependence between operations management and corporate finance in their review of the operations–finance models: (i) financial constraints on operations, (ii) correlation between operational and financial risks, and (iii) alternative risk mitigation. Financial risks arise when there are uncertainties related to any form of financing, including credit, business, investment, and operational risks (Kou et al. 2014).
There is no specific framework for the works on the operation–finance interface in the literature. However, some classifications of existing works on supply chain finance (SCF) can be found in the review papers of Gelsomino et al. (2016) and Xu et al. (2018b).
To address the financial flows of the supply chain, some works in this field have focused on the management of accounts payables/receivables of a firm in the supply chain. Gupta et al. (1987) and Gupta and Dutta (2011) extended the application of scheduling theory to the financial decisions of the supply chain. Longinidis and Georgiadis (2011) proposed a mixed integer linear programming model that integrates financial issues with a supply chain design under stochastic demand. Jahangiri and Cecelja (2014) developed a stochastic model to maximize the profit of a firm and find the optimal strategy for paying its accounts payables. Luo and Shang (2015) studied a centralized twodivision supply chain and integrated inventory decisions and cash management in the mentioned supply chain. Miloudi et al. (2016) proposed a model to optimize the working capital of a firm in a threelevel supply chain by scheduling its accounts payables and receivables.
Some researchers have considered capital shortage and financing decisions. In this group, two financing methods have been proposed: (1) external financing and (2) internal financing. In the first method, the capitalconstrained member gets direct financing from financial institutions such as banks (Xu and Birge 2006; Guillén et al. 2007). We note that any kind of financing from a source outside the supply chain can be considered as external financing such as bank financing, peertopeer lending, and so on (Interested readers are referred to Wang et al. (2020) for further information regarding peertopeer lending and the associated credit ratings.). The second method deals with financing from other members of the supply chain through various options such as trade credit (Jaber and Osman 2006; Lee and Rhee 2011; Lee et al. 2018).
Trade credit refers to a seller’s shortterm loan to a buyer, allowing the buyer to delay payment of an invoice; this has been the largest source of working capital for a majority of businesstobusiness firms in the United States (Lee and Rhee 2011). Aggarwal and Jaggi (1995) developed a mathematical economic order quantity model for perishable goods using delayed payments. Teng (2009) established an economic order quantity (EOQ)based model for a retailer who receives full trade credit from its supplier and offers full or partial trade credit to its good or bad customers, respectively. Using delayed payments, MoussawiHaidar and Jaber (2013) developed a joint model for cash and inventory management for a retailer. Taleizadeh et al. (2016) proposed an economic production quantity model (with defective items and rework processes) that considers both upstream and downstream trade credit. Yang and Birge (2018) adopted a trade credit contract as a risksharing mechanism for a twolevel supply chain with limited working capital. Babich and Kouvelis (2018) discussed recent contributions and future directions on the interface of finance, operations, and risk management. Kouvelis and Zhao (2018) studied the impact of credit ratings on operational and financial decisions of a supply chain with a supplier and a retailer interacting via an early payment discount contract. Tang et al. (2018) compared two schemes—purchase order financing and buyer direct financing—for financing a capitalconstrained supplier. Shin et al. (2018) found it beneficial to choose a tradecreditfinancing policy in a centralized twoechelon supply chain model with imperfect quality items. They did not consider budget constraints in their study. Li et al. (2019) optimized dynamic credit term decisions, along with inventory production decisions in a twolevel supply chain. They ignored financial constraints in their study as well. Mashud et al. (2019) extended the EOQ inventory model for deteriorating items with pricedependent demand under a twolevel trade credit policy. In their study, budget constraints and external financing were not addressed. Krommyda et al. (2019) explored some effects of the global financial crisis on inventory management decisions within the general EOQ that has a partial backorder paradigm. In this study, an EOQ inventory policy is used to manage inventories in the whole supply chain. An EOQ inventory policy has many applications in realworld problems. (See e.g. Khalilpourazari and Pasandideh 2019; Khalilpourazari et al. 2019a, b, and Khalilpourazari et al. 2019b for more information about the application of an EOQ policy in different problems.) Pramanik and Maiti (2019) considered the time value of money and inflation in an inventory model with twolevel partial trade credit. They did not consider financial problems such as budget shortage and limitations in financing. Chakuu et al. (2020) conducted an empirical study to explore the conditions in which logistic service providers offer inventory financing as an SCF service. Yan et al. (2020a, b) examined two financing strategies—loan and investment—for the financing of a capitalconstrained supplier by a riskaverse retailer. In another study, Yan et al. (2020b) analyzed the pricing competition in a dualchannel supply chain consisting of one eretailer that provides financing to a capitalconstrained supplier. Huang et al. (2020) compared trade credit financing (TCF), credit guarantee financing, and buyback guarantee financing as three types of SCF modes. They provided managerial insights for the application of these financing modes.
Combining supply chain coordination and operations–finance interface
A few studies in the literature have considered the coordination of supply chain members together with the integration of financial and operational decisions. Jaber and Osman (2006) proposed a centralized model where players in a twolevel (supplier–retailer) supply chain coordinate their orders to minimize supply chain costs. In their model, permissible delay in payment was considered a decision variable and was adopted as a trade credit scenario to coordinate the order quantity between the two levels. They did not address budget constraints for members in their study. Dada and Hu (2008) applied a nonlinear loan schedule to coordinate the decisions of a profitmaximizing bank and a capitalconstrained retailer. Yin and Xu (2010) proposed a twolevel supply chain with delay in payments and inventoryinpawn financing policy to minimize supply chain costs by coordinating material and capital flows. Lee and Rhee (2011) studied trade credit from a supplier’s perspective and presented it as a tool for supply chain coordination. Chen and Wang (2012) studied the interactions between a capitalconstrained retailer and its supplier under TCF. They found that trade credit can create value in a capitalconstrained supply chain and partly achieve coordination. MoussawiHaidar et al. (2014) coordinated a threelevel supply chain consisting of a capitalconstrained supplier, a retailer, and a financial intermediary (bank) by considering operational and financial decisions. Kouvelis and Zhao (2016) addressed two coordinating contracts (revenue sharing and buyback contracts) in the presence of capital constraint and default risk. Feng et al. (2015) developed a revenuesharingandbuyback contract to coordinate a capitalconstrained supply chain. Yan et al. (2016) proposed a partial credit guarantee contract for SCF and analyzed coordination conditions for this contract. Xiao et al. (2017) designed a generalized revenue sharing contract to coordinate a supply chain with financial constraints. Johari et al. (2018) proposed a bilevel credit period (delay in payment) for a periodic review inventory system that coordinates the inventory and pricing decisions of a twolevel supply chain. Ebrahimi et al. (2019) used the delay in payment contract as an incentive to coordinate a supply chain with stochastic, promotionaleffortdependent demand. Devalkar and Krishnan (2019) examined the use of trade credit in decreasing moral hazard and adopting supply chain coordination in the presence of information asymmetry and financial friction. Ding and Wan (2020) studied supply chain coordination in the presence of capital constraints and production yield uncertainty. In contrast to our research, in their study, the upstream member (supplier) is capitalconstrained and needs financing from the downstream member (manufacturer).
Literature that simultaneously considers operational and financial decisions is novel, and many assumptions about the real world have not yet been considered. For example, some works have not addressed budget shortage and financing options. The primary purpose has only been to improve supply chain performance by integrating physical and financial flows. Meanwhile, other works that have considered budget constraints and financing decisions have ignored the limitation on the amount of borrowing. However, in realworld situations, some firms cannot borrow an unlimited amount of cash from financial institutions, thereby leading to lost sales due to insufficient working capital for supporting the operations.
In this study, we address both of these important issues: coordination of inventory replenishment and production decisions among all members, and the integration of operational and financial decisions. Specifically, a coordination scheme based on joint decisionmaking on inventory replenishment, production decisions, and financial decisions is applied to a capitalconstrained supply chain with financing limitations in the automotive industry. This coordination mechanism integrates the operational and financial decisions of all members and prevents lost sales. For a better understanding of the coordination model, two scenarios are formulated: i) a noncoordinated supply chain used as a basis for the coordinated model and ii) a coordinated supply chain. The advantages of the coordinated model for all members are illustrated by solving the model using data from a supply chain in the automotive industry.
Table 1 compares previous literature against the current paper, with the contributions of this study shown in bold and italic font. Our study, with some differences, is most related to Jaber and Osman (2006) and MoussawiHaidar et al. (2014), which adopted the centralized decisionmaking process as a scheme for coordination (joint consideration of systemwide costs) in their supply chain models. They considered both inventory and financial decisions in their coordination schemes. Jaber and Osman (2006) did not consider budget constraints and external financing in their study. They applied coordination and integration of operations and finance to improve supply chain profitability. MoussawiHaidar et al. (2014) addressed budget constraints, but they assumed that the firms could borrow an unlimited amount of cash. Moreover, internal financing in their study was limited to the trade credit granted by the supplier. We consider a threelevel supply chain with multiple suppliers and multiple products. The OEM is capital constrained and faces financing limitations, leading to lost sales. To solve these difficulties, in addition to external financing, the OEM can use (1) trade credit granted by the suppliers and (2) prepayment of the auto manufacturer where needed, as two sources of internal financing in the coordination scheme proposed here.
This work contributes to the existing literature in the following ways:

We consider financing limitations in addition to budget constraints, highlighting that there is a maximum limit on borrowing from financial institutions, which means limited financing capacity. Furthermore, we consider lost sales as a consequence of financing limitations. Previous related works used a capitalconstrained firm financed from an unlimited external source, and the shortage of financing was not considered.

In the current study, the prepayment of the downstream partner is suggested as an internal source of financing, in addition to TCF. This option leads to new decisionmaking regarding different financing options (loan and prepayment of the downstream partner), specifically, which option to choose and how much to finance from each source. As such, the loan amount is a decision variable in this study, whereas in previous works, the loan amount is equal to the production or inventory costs minus initial capital.

We consider the service level to the customers as a decision variable of the coordinated supply chain. Moreover, we formulate the problem as a biobjective optimization problem, where both the costs and the satisfied demand are optimized. To the best of our knowledge, considering this issue is novel in the related literature.

We model a realworld supply chain in the automotive industry that has three levels and multiple products, while most previous works have been limited to the relationships in a single buyer–single vendor with a single product scenario.
Model description and solution approach
In this section, the proposed model and solution approach are explained in detail based on two scenarios. First is the decentralized noncoordinated structure, in which each member tries to optimize its own objectives. We show that under this setting, the supply chain performance is not satisfactory, and lost sales are inevitable. To improve the supply chain performance, a second scenario is proposed in which all members operate in a way that optimizes the financial and operational decisions of the entire system by adopting a coordination scheme. In the end, we apply a costsharing mechanism for sharing the benefits of coordination among all members. This can be a motivation tool for all members to accept the proposed coordination scheme.
Assumptions and notations
We consider a threelevel multiproduct supply chain in the automotive industry. Figure 1 shows the structure of the mentioned supply chain. This supply chain includes several suppliers of raw materials and the components of automobile original parts, a capitalconstrained OEM that manufactures original parts of automobiles (powertrain components including engines, gearboxes, and axles), and an auto manufacturer.
In the said supply chain, the necessary components and materials (shown by index k) are shipped from outside sources to the suppliers (\(S_{1}, S_{2}, \ldots ,S_{K}\)). These components are purchased by the OEM to produce the original parts of automobiles (shown by index n) according to the deterministic demand of the auto manufacturer. Afterward, these original parts are shipped to the auto manufacturer for assembling and producing final products (shown by index j). All of these activities take place continuously during the planning horizon (T).
The auto manufacturer has a predetermined production capacity for each final product j, which can be considered as the deterministic demand (shown by \(D_{j}\)) in the specified planning horizon (T). The OEM manufactures original parts of the final products (shown by index n) according to the auto manufacturer’s demand. The demand for the original parts manufactured by the OEM (\(D_{n}\)) can be calculated simply according to \(D_{j}\), that is, \(D_{n} = \mathop \sum \limits_{j} m_{jn} D_{j}\), where \(m_{jn}\) represents the quantity of part n needed for producing each unit of product j. The ordering and replenishment of required components and materials by the OEM take place in multiple periods during the planning horizon. Note that each component k is purchased from one supplier (\(S_{k}\)).
The OEM faces liquidity constraints and limitations in shortterm financing, leading to lost sales and consequently, an unsatisfactory service level. In this study, we define service level as the satisfied demand of original parts divided by the total demand of original parts, which is shown by \(\alpha_{n} \,(\alpha_{n} = \frac{Satisfied\,demand\,of\,part\,n}{{Total\,demand\,of\,part\,n}})\). It is a decision variable for the proposed model.
The inventory policy adopted in the proposed problem is based on the EOQ inventory policy, with some different and additional assumptions. Table 2 provides a comparison between the classical EOQ assumptions and the assumptions of the proposed model.
The notations of the mathematical formulation are given in Table 3.
The mathematical formulation of the noncoordinated supply chain
In this section, we model the decentralized noncoordinated supply chain in which each member optimizes its own problem without considering other members' costs and benefits. As such, three models are discussed, namely, the OEM's, the suppliers', and the auto manufacturer's models.
The OEM's model
As previously mentioned the OEM is faced with capital shortage and financing limitations. Since the OEM does not have sufficient initial budget to carry out its operations and the amount of external financing is limited, it might not be able to satisfy all of the auto manufacturer's demand, which leads to lost sales. In this case, the OEM should decide on the fraction of satisfied demand for original parts (\(\alpha_{n}\)) while minimizing cost. To deal with this situation, we propose a biobjective model that simultaneously minimizes the operational–financial costs of the OEM (\({\Pi }_{OEM}\)) and maximizes satisfied demand of the original parts (\(\mathop \sum \limits_{n} \alpha_{n} D_{n}\)).
The first objective function includes inventory replenishment costs (ordering, purchasing, and inventory carrying costs), fixed and variable production costs, and the cost of capital for borrowing the required capital. Inventory carrying cost or holding cost is the opportunity cost of the capital tied up in the inventories plus other costs (storage, obsolescence, damage, deterioration, insurance, etc.). Therefore, in this study, the parameter \(h_{k}\) only represents the “cost of capital,” and \(s_{k}\) represents the “other costs,” which we refer to as the storage cost of component k for the OEM.
Lemma 1
The opportunity cost of the capital tied up in the inventories is calculated as follows:
Proof
See “Appendix 1”.
The OEM replenishes its inventory from suppliers in the shipments of size \(Q_{k}\) every \(T_{{OEM_{k} }} = Q_{k} /D_{k}\) units of time where \(D_{k} = \left( {\mathop \sum \limits_{n} N_{nk} \alpha_{n} D_{n} } \right)\). We model the OEM's problem as a biobjective optimization problem that minimizes the total costs of inventories, production, and finance per unit time, and maximizes the satisfied demand:
The first objective (2) is to minimize the OEM's total costs, which include ordering cost (first term), purchasing cost (second term), storage cost (third term), the opportunity cost of the capital tied up in inventories (fourth term), fixed and variable production costs (fifth term), and loan repayment at the end of the time horizon (sixth term). The first constraint (4) represents the limitation of external financing (debt). The second (5), demonstrates the OEM’s budget constraint, and the third constraint (6) implies that the amount of cash by the end of the time horizon should not be less than the lower limit (L) determined by the manager.
The suppliers' model
Each supplier supplies one component for the OEM. They replenish their inventory from outside vendors in the shipments of size \(\lambda_{k} Q_{k}\) every \(T_{{S_{k} }} = \lambda_{k} Q_{k} /D_{k}\) units of time where \(D_{k} = \left( {\mathop \sum \limits_{n} N_{nk} \alpha_{n} D_{n} } \right)\) and \(\lambda_{k} = 1, 2, \ldots\) and deliver \(Q_{k}\) to the OEM every \(T_{{OEM_{k} }} = Q_{k} /D_{k}\) units of time (Fig. 2). Note that if \(\lambda_{k} = 1\), the suppliers follow the lotforlot policy. Figure 2 shows the inventory levels of each supplier and the OEM. According to Fig. 2, the average of each supplier's inventory per cycle can be calculated as the area under the inventory level, that is, \(\left( {\lambda_{k}  1} \right)Q_{k} T_{{OEM_{k} }} + \left( {\lambda_{k}  2} \right)Q_{k} T_{{OEM_{k} }} + \cdots + Q_{k} T_{{OEM_{k} }} = Q_{k} T_{{OEM_{k} }} \mathop \sum \limits_{i = 1}^{{\lambda_{k} }} \left( {\lambda_{k}  i} \right)\), which is equal to \(\frac{{Q_{k}^{2} \lambda_{k} \left( {\lambda_{k}  1} \right)}}{{2D_{k} }}\).
Similar to the OEM's problem, two elements are considered for inventory holding cost—the storage cost and cost of capital. Since the suppliers pay immediately to their vendors, the amount of inventory that is included in the cost of capital is the same as the one included in the storage cost. Thus, we calculate the total inventory holding cost for supplier \(k\) in the planning horizon as \(\left( {s_{k}^{{\prime }} + h_{k}^{{\prime }} } \right)\frac{{Q_{k} \left( {\lambda_{k}  1} \right)}}{2}\) by dividing the holding cost per period, \((\left( {s_{k}^{{\prime }} + h_{k}^{{\prime }} } \right)\frac{{Q_{k}^{2} \lambda_{k} \left( {\lambda_{k}  1} \right)}}{{2D_{k} }})\), by \(T_{{s_{k} }}\).
It is supposed that the suppliers have sufficient budget. Thus, we just considered inventory replenishment costs for the suppliers such as procurement, ordering, and holding costs. The cost function of each supplier k in the planning horizon (1 year) is modeled as follows:
The auto manufacturer’s model
The auto manufacturer produces the final products of the supply chain. It supplies the required original parts from the OEM. It is assumed that the required parts are shipped every day from the OEM to the auto manufacturer to be included in the production line of automobiles. The auto manufacturer pays for these parts at the end of the planning horizon. The majority of the auto manufacturer's inventory costs are related to the procurement of original parts. Therefore, in this study, the auto manufacturer's cost function is modeled only as the sum of the procurement cost of parts from the OEM, since inventory holding and ordering costs are negligible as compared with the procurement cost.
(Note that AM refers to the auto manufacturer in the mathematical terms and tables in the rest of this paper.)
Supply chain coordination model
In the coordination scheme, the suppliers offer trade credit to the OEM as a financing source in exchange for an increase in the OEM's orders. Having delayed (extended) its payments, the OEM invests its cash, if any, until \(t_{k}\) and gains interest equal to \(\mathop \sum \limits_{k} w_{k} *Q_{k} *\left[ {\left( {1 + r_{OEM} } \right)^{{t_{k} }}  \left( {1 + \gamma_{k} } \right)^{{\left( {t_{k}  d_{k} } \right)^{ + } }} } \right]\) units of cash each period, where \(\left( x \right)^{ + } = \max \left( {x ,0} \right)\). Meanwhile, the auto manufacturer pays in advance a fraction of payables to help the OEM improve its service level by increasing its production rate. Consequently, the auto manufacturer incurs an opportunity cost by prepaying. The members' cost functions are rewritten based on the aforementioned assumptions (\({\text{TC}}_{.}^{c}\) refers to the cost function of the members and supply chain under the coordination scheme.). The OEM's cost function in the coordinated model becomes:
The last term in the objective function is the capital gain of the OEM from extending its payables. There are upper bounds on the OEM's payment time to the suppliers, which are considered as constraints for the coordination problem:
The budget constraint in (5) will change based on the coordination settings as:
The balance of the cash at the end of the time horizon in (6) is rewritten as follows:
In the coordination scheme, the members should decide on the due date of the interestfree period (\(d_{k}\)). According to this setting, as the OEM delays its payments, an opportunity cost is incurred by the suppliers, which is added to their cost function. The cost function for each supplier k is rewritten as:
By adding the opportunity cost due to prepayment to the auto manufacturer's objective function, it becomes:
where \(\beta\) is the prepayment rate that is a decision variable in the coordinated scenario.
According to the members' cost functions, we can write the coordinated nonlinear biobjective supply chain model as follows:
subject to (4), (10), (11), and (12).
All of the variables are assumed to be positive to decrease the problem’s complexity, that is, the variables of time, \(t_{k}\) and \(d_{k}\), and the coefficient \(\lambda_{k}\) are considered as positive continuous variables. Moreover, \(\lambda_{k}\) has a lower bound which is equal to 1.
Solution approach
To solve the biobjective optimization models in both scenarios, we use the εconstraint method by considering the cost function as the main objective and the other objective as a constraint of the problem. It is assumed that \(\alpha_{1} = \alpha_{2} = \cdots = \alpha_{n} = \alpha\) (here \(n = 3)\), that is, all of the n parts should be produced in equal quantities. In other words, the demand satisfaction level for all parts should be the same. Based on this assumption, maximizing the satisfied demand for original parts (\(\mathop \sum \limits_{n} \alpha D_{n}\)) is equal to the maximization of the service level for original parts (\(\alpha )\). As such, for simplicity, in the εconstraint method, we use \(\alpha \ge x \left( {where \,\varepsilon = 0.1, 0.2, \ldots ,1} \right)\) in the constraints of the models instead of \(\mathop \sum \limits_{n} \alpha_{n} D_{n} \ge y\) (where y is the lower bounds of the satisfied demand). In this method, the nondominated (Pareto) solutions are found according to different service levels. The best solution among these nondominated solutions is chosen according to the experts' opinions. In the optimal solution of the proposed model, it can be proved that \(t_{k}\) is less than or equal to \(d_{k}\). This reduces the problem’s complexity.
Lemma 2
\(t_{k}\) is less than or equal to \(d_{k}\) in the optimal solution of the proposed model.
Proof
See “Appendix 2”.
After simplifying the model according to the abovementioned lemma and assumptions, the models are solved by \(the \ \varepsilon\)constraint method in the general algebraic modeling system (GAMS).
Numerical analysis
In this section, we compare the two scenarios explained in the previous section. We show the effects of coordinating a budgetconstrained supply chain on the financial and operational decisions of the members. Furthermore, a costsharing mechanism is developed to share coordination benefits among all members. In the end, we provide a sensitivity analysis to show the sensitivity of the model variables and cost function to some critical financial parameter fluctuations.
Data
The models are solved using data from an OEM of an automotive supply chain in Iran (Table 4). This supply chain includes 10 suppliers of strategic components and materials for the OEM (Each component is supplied by only one supplier; thus, K = 10), which produces three kinds of original parts (powertrain components including engines, gearboxes, and axles) for the auto manufacturer (N = 3), and an auto manufacturer with 2 kinds of final products (J = 2; Fig. 1). The required data are collected from the company’s archives and analyzed under the supervision of the OEM’s supply chain manager. In solving the model, the initial budget (B) and minimum cash level at the OEM (L) are assumed to be zero.
Solution results
This section presents the results of solving the proposed models in Section “Model description and solution approach”. The results are reported in two tables in the form of nondominated solutions. Table 5(a) and (b) show the costs of the members and the total cost of the supply chain according to different service levels in both scenarios, and Table 6 presents the quantities of the problem’s decision variables.
The OEM is faced with capital shortage and financing limitations, that is, it cannot obtain more than M units of cash from external financial institutions such as banks (M is the maximum limit for external financing). As explained earlier, this limitation may have some consequences such as the inability to satisfy all of the auto manufacturer’s demand and losing part of the OEM’s market share.
According to the numerical results, in the noncoordinated scenario with this circumstance (Table 5(a)), the OEM is not able to satisfy more than about 50% of the demand, as the quantity of required external financing for 60% of the auto manufacturer's demand is 55,479,000 (Table 6), which is more than the maximum limit of external financing (M = 50,000,000). Therefore, the problem is infeasible for \(\alpha \ge 0.6\) (italics in Tables 5(a) and 6) for the noncoordinated structure in accordance with the financing limitations (For \(\alpha \ge 0.6\), the problem is solved by eliminating constraint (5), which is related to the financing limitations, to track the amount of required loan from financial institutions.). The proposed coordination mechanism is developed to deal with these kinds of difficulties. Table 5(b) shows that in the coordinated scenario, all of the demand can be satisfied despite the financing limitations because of internal financing. Figure 3 illustrates the differences between the costs of the OEM, auto manufacturer, and the supply chain for the two scenarios for different service levels.
The numerical results indicate that the proposed coordination mechanism leads to cost reduction for the whole supply chain as compared with the noncoordinated scenario. As seen in Table 5(b), the supply chain cost is reduced in the coordinated scenario, as compared with the noncoordinated one, for all service levels (According to the results, for \(\alpha > 0.5\), the problem is infeasible for the noncoordinated scenario, i.e., the costs become infinity.). Moreover, through coordination, the OEM can order in larger quantities and larger periods than in the noncoordinated case (Table 6). In the coordinated scheme, the OEM takes advantage of the prepayment option and avoids external financing for all service levels in this example. As explained later in Lemma 3 in section “Sensitivity analysis”, the reason is that in this example, the interest rate of external financing is larger than the opportunity cost of the auto manufacturer. Consequently, the auto manufacturer's loss due to the prepayment to the OEM is less than the interest paid to the external institution for the loan at the end of the time horizon. It is also observed that for service levels of 0.9 and 1, the OEM takes advantage of both options: trade credit and prepayment of the auto manufacturer.
Costsharing mechanism
As previously mentioned, the supply chain benefits from the proposed coordination scheme through cost reduction. In the process of coordination, it is normal that some of the members face increased costs and others, reduced costs. As seen in Fig. 3, through coordination, the OEM’s costs decrease but those of the auto manufacturer increase.
In such cases, to persuade members whose costs increased to accept the coordination scheme, the last step is sharing the gains and losses among all members of the supply chain so that all the parties benefit equally from the coordination. In this regard, we apply a costsharing mechanism to the coordination scheme, as adopted by MoussawiHaidar et al. (2014), wherein the gains and losses are shared according to the ratio of the individual cost per unit time with respect to the chain’s noncoordinated cost per unit time. Let \(C_{chain}\) and \(C_{chain}^{0}\) be the total unit cost of a coordinated supply chain and the total unit cost of a noncoordinated supply chain, respectively, and \(C_{x}^{0}\) the cost per unit time of member x of the noncoordinated supply chain. x is a member of set \(M_{x}\) = OEM, auto manufacturer, \(S_{1}\), \(S_{2}\), …, \(S_{K}\). We get the following ratios:
We can calculate the cost of each member by:
Table 7 shows the results of sharing the costs among members. As seen in Table 7, compared with the noncoordinated case, the costs of all members in the coordinated scenario are reduced for all service levels (Table 5(a)) after applying the costsharing mechanism.
Figure 4 shows the difference between the costs of the OEM, auto manufacturer, and supply chain in the noncoordinated and coordinated scenarios after applying the costsharing mechanism. A comparison between Fig. 3 and Fig. 4 indicates the fair distribution of gains and losses among supply chain members as a result of applying the costsharing mechanism.
Figure 5 shows the difference between the costs of the OEM and auto manufacturer before and after applying the costsharing mechanism in the coordinated scenario.
Sensitivity analysis
In this section, we analyze the sensitivity of the system’s costs and model variables to the fluctuations in both the operational and financial parameters. To avoid excessive calculations, the analysis is only performed for the case with \(\alpha = 1\).
Sensitivity analysis for the operational parameters
A sensitivity analysis regarding inventoryrelated parameters, \(A_{k}\), \(S_{k}\), \(A_{k}^{\prime}\), and \(S_{k}^{\prime}\), is provided. To this end, for simplicity, we define a coefficient denoted by \(\omega\) to make changes in the parameters. For instance, for parameter \(A_{k}\), if \(\omega = 2\), this means that for all k, \(A_{k}\) is multiplied by 2. The results are illustrated in Fig. 6. As expected, the results indicate that the impact of changes in \(A_{k}\) and \(S_{k}\) on the order quantities and the total cost is similar to the classical EOQ model (see Fig. 6a, b). Moreover, by decreasing \(A_{k}\) or increasing \(S_{k}\), the average amount of \({\uplambda }_{k}\) increases (see Fig. 6c) because of the reduction in the OEM’s order quantities (Fig. 6a).
Any changes (increase or decrease) in \(A_{k}^{{\prime }}\) has a direct effect on the order quantities at the OEM (see Fig. 6a). Furthermore, by increasing \(A_{k}^{\prime}\) or decreasing \(S_{k}^{{\prime }}\) the average amount of \({\uplambda }_{k}\) increases. In this case, the suppliers tend to order in larger quantities to support more than one order at the OEM every time and store the remaining inventory for the next orders. According to the results shown in Fig. 6a, the average order quantity at the OEM decreases by decreasing \(S_{k}^{{\prime }}\). This can be justified according to the model. As seen in Fig. 6c, d, by increasing \(S_{k}^{\prime}\) (for \(\omega < 1\)) the average amount of \({\uplambda }_{k}\) decreases until it equals 1 (the lotforlot policy). In this situation, if the average order quantity also decreases, the average amount of \({\uplambda }_{k} *{\text{Q}}_{k}\) will largely decrease and cause a large increase in the total cost. Therefore, to avoid this large loss, the OEM increases the order quantities. (The order quantities of the OEM and the total cost do not change by increasing \(S_{k}^{\prime }\) from its values in the main example, that is, where \(\omega = 1\) in the horizontal axis, because of the lotforlot policy of suppliers).
Sensitivity analysis for financial parameters
A sensitivity analysis is provided with respect to some important financial parameters namely, \(i, r_{OEM}, r_{AM} \ ,and\,r_{k}\). The results are presented in Tables 8, 9, 10 and 11. Note that the rates in the tables are daily rates. The analysis of the results is summarized in the form of some lemmas.
Lemma 3
When \(i < r_{AM}\), the OEM borrows from the external financial institution as much as possible, that is, the maximum debt limit (Table 8(a)). Conversely, when \(i \ge r_{AM}\), the OEM obtains financing from its internal source, the auto manufacturer’s prepayment (Table 8(b), (c) and (d)).
Proof
\(i < r_{AM}\) means that the interest rate of the loan is less than the opportunity cost of the auto manufacturer, such that the interest paid to the external institution for the loan at the end of the time horizon is less than the auto manufacturer's loss arising from its prepayment to the OEM. This is why the OEM uses the maximum limit of the loan in this situation. By contrast, when \(i > r_{AM}\), the OEM uses internal financing from the auto manufacturer (Table 8(c), (d)). For the equality mode, \(i = r_{AM}\), no difference is expected between external and internal financing, but by making the debt equal to zero, constraint (12) becomes less constrictive. Therefore, in this case, internal financing is preferred (Table 8(b)).□
Lemma 4
If \(r_{k} < r_{OEM}\) then the OEM delays its payments until the maximum limit (Table 9(c), (d)) is reached, and if \(r_{k} > r_{OEM}\) the OEM pays at the time of ordering (\(t_{k} = 0;\) Table 9(a)).
Proof
To prove this lemma, we consider \(Z^{{\prime }}\) (Eq. 25 in “Appendix 2”) because the rest of the cost function does not affect the process of proving. According to \(Z^{{\prime }}\), we get the following rule:
The smaller the \(Z^{{\prime }}\) the better is the objective function. Therefore, if \(r_{k} > r_{OEM}\), the greater the \(t_{k}\), the greater is the \(Z^{{\prime }}\) and consequently, the worse is the objective function. From a mathematical viewpoint, we prove the proposition below:
Proposition 1: If \(r_{k} > r_{OEM} > 0\) and assuming we have \(t_{k}^{{\prime }}\) such that \(t_{k}^{{\prime }} > t_{k} > 0\); then:
Proof of proposition 1
This inequality implies that under this circumstance, the lower the \(t_{k}\), the better is the objective function. To prove this, we write the inequality below:
The inequality above is true because \(r_{OEM} > 0\) and \(t_{k}^{{\prime }}  t_{k} > 0;\) consequently, \(\left( {1 + r_{OEM} } \right)^{{t_{k}^{{\prime }}  t_{k} }} > 1\); therefore, we reach:
If \(r_{k} > r_{OEM} > 0\), then \(\left( {1 + r_{k} } \right)^{{t_{k}^{{\prime }}  t_{k} }} > \left( {1 + r_{OEM} } \right)^{{t_{k}^{{\prime }}  t_{k} }}\); thus, we can substitute \(\left( {1 + r_{k} } \right)^{{t_{k}^{{\prime }}  t_{k} }}\) with \(\left( {1 + r_{OEM} } \right)^{{t_{k}^{\prime }  t_{k} }}\) in the inequality (22). Then we get:
which is equal to \(\left( {1 + r_{k} } \right)^{{t_{k}^{{\prime }} }}  \left( {1 + r_{OEM} } \right)^{{t_{k}^{{\prime }} }} > \left( {1 + r_{k} } \right)^{{t_{k} }}  \left( {1 + r_{OEM} } \right)^{{t_{k} }}\).
This is why the OEM should pay at the beginning of the period when \(r_{k} > r_{OEM}\).
Lemma 5
The lower the \(r_{AM}\), the lower is the total cost of the supply chain (Table 10).
Proof
Taking the derivative of the cost function with respect to \(r_{AM}\), we get \(T\beta \left( {\mathop \sum \limits_{n} p_{n} \alpha_{n} D_{n} } \right)\left( {1 + r_{AM} } \right)^{T  1}\), which is greater than or equal to zero. Thus, the total cost function is increasing in \(r_{AM}\).
Lemma 6
If \(r_{k} \ge r_{OEM}\) (Table 11(c), (d)), then the total cost of the supply chain becomes greater than the total cost in the case where \(r_{k} < r_{OEM}\) (Table 11(a), (b)), that is, \(Z_{{r_{k} \ge r_{OEM} }} > Z_{{r_{k} < r_{OEM} }}\).
Proof
Again, we consider only \(Z^{\prime }\) (Eq. 25 in “Appendix 2”) because the other parts do not have any effect on the proof of this theorem. According to \(Z^{\prime }\), it is obvious that the greater \(the \ r_{OEM}\) from \(r_{k}\), the lesser is the \(Z^{\prime }\), which means the lesser is the cost function of the supply chain.
Conclusion
In this study, we exploited two vital issues in supply chain management—coordination between supply chain members and integration of the supply chain’s physical and financial flows—to improve system performance and service level.
As previously mentioned, combining supply chain coordination and financial decisions is a relatively new research area and needs more attention and extensions to realworld assumptions. In this stream of literature, some important financial issues, such as budget constraints and different financing options, have not been addressed in many of the existing works. Moreover, the idea of considering financing limitations and lost sales as a consequence of financing limitations is novel in the supply chain coordination literature. Meanwhile, most of the previous works have been limited to the relationships in a single buyer–single vendor with a single product scenario. To fill these gaps in the literature, we proposed a coordination scheme that is based on joint decisionmaking and that coordinates the operational (inventory–production) and financial decisions of the members of a capitalconstrained supply chain (with financing limitations) in the automotive industry. To the best of our knowledge, this is the first work to address supply chain coordination together with financing issues for maximizing service level and minimizing operational–financial costs.
The proposed supply chain model consists of three levels and multiple products. Two scenarios were developed according to the different structures of the supply chain: i) noncoordinated supply chain and ii) coordinated supply chain. In the first scenario, the members optimize their own costs without considering the benefits and costs of others, but in the second one, all the members work in a way that optimizes the supply chain costs.
The models were solved using the εconstraint method, applying GAMS based on data from an automotive supply chain in Iran. The results indicate that in the noncoordinated scenario, the OEM cannot satisfy all of the auto manufacturer’s demand and faces lost sales. This happens because of the capital shortage and limitations in external financing. As observed in Table 5(a) (italics), the OEM is not able to satisfy more than about 50% of the auto manufacturer's demand in the noncoordinated scenario. However, in the coordinated scenario, all of the auto manufacturer's demand can be satisfied through the OEM’s internal financing.
The advantages of this coordination scheme are twofold: (1) By applying the proposed coordination scheme, the total cost of the supply chain decreases compared with the noncoordinated scenario; and (2) The scheme makes it possible to satisfy all of the auto manufacturer's demand despite the capital shortage and financing limitations of the OEM. In other words, by adopting this coordination scheme, the supply chain manager can finance the supply chain operations and still ensure the service level for the auto manufacturer.
In such coordination methods, some of the members take advantage of the coordination more than others, and some of them may even lose a fraction of their profit. To avoid this inequity and motivate the members that lose a portion of their profits because of the coordination, a costsharing mechanism was applied to the coordination scheme. The results demonstrate that after applying the costsharing mechanism, the costs of all members are lower than their costs in the noncoordinated scenario. Thus, the proposed coordination scheme is beneficial to all members of the supply chain. Finally, to validate the proposed coordination model, a sensitivity analysis for the financial parameters was conducted to measure the sensitivity of the model variables and cost function to the fluctuations of these parameters.
Our results provide the following managerial insights for manufacturers that face financial constraints such as insufficient capital and limitations on external financing. First, the supply chain manager should integrate the operational and financial decisions of the firm. Second, he/she should entice the upstream and downstream partners to adopt operational–financial coordination. Furthermore, some detailed managerial insights after adopting the proposed coordination scheme are listed below.

If the interest rate of the loan is more than the downstream partner’s rate of return (ROR), the manufacturer should finance all the required working capital from the downstream partner. Conversely, it should borrow from the external source as much as it needs until the maximum limit of the external financing is reached.

If supplier k’s ROR is less than the manufacturer’s ROR, supplier k should offer a credit period (interestfree period) to the manufacturer for its payment, and the manufacturer should delay its payment to supplier k until the maximum deadline. Conversely, it should pay at the time of ordering.
We end this paper with two remarks. First, we have modeled the deterministic state of the problem based on an EOQ policy. However, sometimes in the real world, firms face fluctuations in demand. Thus, this research can be extended to the case with stochastic demand, which calls for a new coordination scheme. Second, we used trade credit and prepayment of the auto manufacturer as two options for internal financing. Adopting other options to finance the capitalconstrained member is another avenue for further research.
Availability of data and materials
Data are included in this published article.
Abbreviations
 AM:

Auto manufacturer
 EOQ:

Economic order quantity
 GAMS:

General algebraic modeling system
 OEM:

Original equipment manufacturer
 ROR:

Rate of return
 SC:

Supply chain
 SCC:

Supply chain coordination
 SCF:

Supply chain finance
 TCF:

Trade credit financing
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FE helped to conceive and design the study, conducted the data collection and analysis and drafted the manuscript. NN helped to conceive and design the study and coordinated the research activities. FM helped to conceive and design the study. All authors read and approved the final manuscript.
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Appendices
Appendix 1
The proof of lemma 1
As mentioned in section “The mathematical formulation of the noncoordinated supply chain”, this paper has assumed that the total holding cost is a composite of two cost components, one associated with the opportunity cost of capital, while the other is the cost of stocking one unit of an item. Since the suppliers offer the OEM to settle his/her accounts by the time \(t_{k}\) (\(t_{k}\) is in terms of days) after the shipment receives, the OEM's holding cost will be different from the case that the OEM settles his/her accounts at the time of receiving the shipments. We have considered two cases: 1) \(t_{k} \le T_{{OEM_{k} }}\) and 2) \(t_{k} > T_{{OEM_{k} }}\). In case 1, the holding cost of the OEM with respect to the cost of capital (\(H\left( k \right)\)) is calculated as \(h_{k} \frac{{\left( {Q_{k}  t_{k} D_{k} } \right)\left( {T_{{OEM_{k} }}  t_{k} } \right)}}{2}\) according to Fig.
7. Where \(\frac{{\left( {Q_{k}  t_{k} D_{k} } \right)\left( {T_{{OEM_{k} }}  t_{k} } \right)}}{2}\), is the area of shaded triangle and \(h_{k}\) is the unit holding cost of component k for the OEM, representing the cost of capital, excluding the storage cost. We substitute \(T_{{OEM_{k} }} = Q_{k} /D_{k}\) in the above statement and reach \(H\left( k \right) = h_{k} \frac{{\left( {Q_{k}  t_{k} D_{k} } \right)^{2} }}{{2D_{k} }}\).
In case 2, since the OEM settles his/her account for component k after \(T_{{OEM_{k} }}\), i.e., \(t_{k} > T_{{OEM_{k} }}\), his/her inventory level of component k will be zero and subsequently \(H\left( k \right) = 0\). We have merged these two cases as bellows:
Appendix 2
The proof of lemma 2
The total system cost function (23) is as follows:
In the above cost function, we consider only the parts containing \(t_{k}\) and \(d_{k}\) as the other parts do not have any influence on our process of proving. We have shown this part by \(Z^{\prime}\):
By simplification of (Eq. 24) we have:
We can see that \(\left( {t_{k}  d_{k} } \right)^{ + }\) is omitted from the objective function. Therefore, we need to check these two variables only in the constraints (11) and (12).
If \(t_{k} > d_{k}\) then it exacerbates these two constraints, on the contrary, putting \(t_{k} \le d_{k}\) meliorates the two constraints. So in the optimal solution, we will have \(t_{k} \le d_{k}\).
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Emtehani, F., Nahavandi, N. & Rafiei, F.M. A joint inventory–finance model for coordinating a capitalconstrained supply chain with financing limitations. Financ Innov 7, 6 (2021). https://doi.org/10.1186/s4085402000223z
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DOI: https://doi.org/10.1186/s4085402000223z