Open AccessArticle

Leveraging Blockchain and Consignment Contracts to Optimize Food Supply Chains Under Uncertainty

Department of Mathematics, Lovely Professional University, Phagwara 144411, Punjab, India

Department of Mathematics, College of Engineering and Technology, SRM Institute of Science and Technology, Kattankulathur 603203, Tamil Nadu, India

Authors to whom correspondence should be addressed.

Appl. Sci. 2024, 14(24), 11735; https://doi.org/10.3390/app142411735

Submission received: 19 October 2024 / Revised: 27 November 2024 / Accepted: 5 December 2024 / Published: 16 December 2024

(This article belongs to the Special Issue Emerging Methods and Tools for Production Systems’ Design and Management in the Mass Customization Era, 2nd Edition)

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Abstract

The occurrence of the fourth industrial revolution (Industry 4.0) has led many industries to the path of adopting new technologies. Such technologies include blockchain, artificial intelligence (AI), and the Internet of Things (IoT). Blockchain creates the opportunity to access data and information in a decentralized manner, resulting in increased customer satisfaction. This study develops a smart newsvendor model of the food industry with consignment contracts and blockchain technology. Under a consignment policy, the central division (manufacturer) can utilize the retailer’s warehouse for storage. The producer may also have the opportunity to share the holding cost with retailers without losing the ownership of products. The main contribution of this study is to analyze the profitability of the retailing and supply chain when the blockchain technology is implemented by the food industry. Moreover, a thorough investigation of profit and loss is conducted under a consignment contract when uncertain demand is encountered. This study mainly concerns perishable food items, and increasing volatility in market demand. Two cases of probabilistic uncertainty are considered, including uniform and normal distribution. The key investigations of this study are presented in terms of (a) the effect of adopting blockchain on market demand for the food industry, (b) analysis of company profitability for perishable food items and demand uncertainty, and (c) the effect of the consignment contract under blockchain technology in the food industry. Finally, this research develops an optimization tool to numerically analyze the effect of several factors of the blockchain technology on demand. Moreover, the optimal values of the design variables and the resulting maximum profitability provide valuable insights that support industry in formulating effective policies and making informed strategic decisions.

Keywords:

Industry 4.0; food industry; blockchain technology; newsvendor problem; consignment contract

1. Introduction

Blockchain is a decentralized digital ledger. With blockchain, the supply chain company can document production updates to a single shared ledger, which provides complete data visibility and a single source of truths. Because transactions are always time-stamped and up-to-date, companies can query a product’s status and location at any point in time. Blockchain is one of the digital technologies that Industry 4.0 advances with. For both small and large businesses, blockchain can be used to enhance security, privacy, and data openness [1]. Blockchain technology opens up a wide range of options for the manufacturing and supply chain industries. Blockchain is a technology that has attracted a lot of attention and can improve the environment for manufacturing and supply chains. For example, Pfizer, a major pharmaceutical business, is working with other industry participants to put in place a blockchain-based medication tracking and tracing platform (Abdallah and Nizamuddin [2]). This aims to stop counterfeiting, guarantee the safe delivery of essential medications, and improve supply chain efficiency and transparency (Zhang et al. [3]). Blockchain technology is often tested by large retail companies to trace the provenance and travel history of food products like mangoes and leafy greens (Shahid et al. [4]). This makes it possible for them to locate possible sources of contamination more quickly and effectively, enhancing food safety and lowering the possibility of returns. The benefits of blockchain are now very well understood in a variety of fields. Blockchain contributes to the development of the smart manufacturing era, which will make use of Industry 4.0’s modern technology and systems to improve various manufacturing processes and boost profitability as well as efficiency. When developing applications for smart manufacturing, this poses numerous difficulties in terms of security, trust, traceability, and dependability. In 2015, Volkswagen was found to have changed information about emissions in its automobiles, showing the potential for manipulation and lack of trust in the correctness of data in interconnected systems. An E. coli outbreak in romaine lettuce that occurred in 2018 highlighted the difficulty of tracking the source of contamination in intricate, worldwide food supply networks, preventing efficient response to ensure the health of consumers. These events emphasize the necessity of blockchain technology to enhance transparency.

As blockchain can play an important role in the food supply chain to increase safety and transparency, modeling to quantify the managerial decisions is essential. The food supply chain can be modeled by newsvendor modeling as fresh food products are mostly perishable, which results in a significantly low shelf life. In operations management and applied economics, the newsvendor model (also known as the single-period newsboy problem) is used to calculate ideal inventory levels. Fixed prices and erratic demand for a perishable good are its main characteristics. In operations management, the newsvendor model is a well-known and beautiful mathematical formulation. These models involve decision-makers who decide how much to buy or order in order to balance the costs of underbuying and overbuying. The newsvendor problem can be seen in a variety of business situations, including purchasing for a single selling season, making a final production run, establishing safety stocks, establishing target inventory levels, and deciding on capacity. Each of these scenarios has the same problem structure: a single policy parameter, such as the order quantity, random demand, and costs associated with unit overage and underage. Managers can be guided by a model that calculates the critical ratio and explicitly defines the costs of overbuying and underbuying.

However, apart from adopting blockchain and developing a model for quantification and strategic decision-making, companies should also concentrate on delivery policies and inventory management. In this regard, adopting new ideas on inventory management should be another important approach to increase profitability. The traditional supply chain concept has changed and advanced over time with changes in the environment and technological advancements. The constant advancement of technology and the rapid change in customer expectations have made supply chain management essential. Effective supply chain management that is consistent with the rapidly evolving environment and technology has become essential to the success and competitiveness of organizations. This paper will define the traditional supply chain, outline how it differs from modern supply chains, and outline the key phases it involves.

A consignment contract is a legal contract pertaining to the consignment of goods between the consignor and the consignee. Under a consignment contract, the consignor gives the consignee custody of their products with the intention of selling them. But until the products are sold, the consignor retains ownership of them.

The main components of a consignment contract are as follows:

1.: Parties Identification: The consignor, who is the owner of the goods, and the consignee, who is the party entrusted with selling the items, will be mentioned in the contract.
2.: Products Description: A complete description of the products being consigned, including their amount, quality, and any related specifications, should be included as part of the contract.
3.: Terms and Conditions: The agreement will list all of the consignment’s terms and conditions, including how long it will last, where the goods will be kept or shown, and any limitations on the consignee’s ability to change or modify the products.
4.: Pricing and Payment: The contract may outline the good’s pricing schedule as well as the consignor and consignee’s split of sales proceeds. It could also specify the conditions and frequency of payments.
5.: Responsibilities of the Consignee: The contract will include the consignee’s responsibilities, which can include marketing and promoting the items, keeping them in excellent condition, and giving the consignor regular information on sales and inventory.
6.: Insurance: The contract could include matters related to insurance coverage for loss, theft, or damage during the consignment period, depending on the nature of the materials.
7.: Return of Unsold Goods: The conditions of the agreement of returning unsold goods to the consignor, together with any related expenses, may be outlined in the agreement.
8.: Termination Clause: A clause explaining how to end the consignment arrangement and stating any costs or penalties that may apply should be included.

In order to prevent miscommunications or disagreements, it is crucial that both parties comprehend and concur with the conditions specified in the consignment contract. It is advisable to seek legal advice when drafting or reviewing such contracts to make sure that both parties’ interests are sufficiently protected and that the relevant laws are followed.

Blockchain technology (BT) is one tool that could improve future food systems policies, traceability, and the flow and success of these supply chains. Blockchain technology (BT) can increase customer confidence, manufacturing efficiency, and product speed—all things the food industry could use right now. By facilitating faster and less expensive product deliveries, increased supply chain transparency and traceability, upgrading real-time trading partner coordination, and significantly improving record-keeping by all parties involved, BT can substantially enhance global food supply chains. A blockchain is a tamper-proof digital ledger that records trade transactions between many trading participants. For this distributed and decentralized system of record-keeping, food supply networks are the ideal fit. A huge, limitless number of trading partners may be able to conduct transactions secretly, anonymously, and securely thanks to blockchain record keeping. These exchanges can take place without a central a mediator. By implementing greater performance, control, and safety of the system, trading partners can use food supply blockchains to safeguard their business operations and the supply chain. A blockchain, in simple terms, is a network of several computers that keeps a digital “record” for a specific event.

1.1. Research Questions

Q1.: Is adoption of blockchain technology important for the food industry?
Answer: This research investigates the answer in several ways. Blockchain can help the food supply chain in many aspects, such as transparency, food safety, quality control, certification, fraud reduction, smart contracts, and consumer trust. Blockchain can provide a smooth and transparent record of each stage of the food supply chain. Customers can track a transparent report of a food contamination outbreak, which ultimately increases customer safety. Blockchain can verify several certifications, such as fair-trade or organic labels, which can reduce the cases of fraud. Moreover, due to the original information about the origin of food and company practices, the customer trust level automatically becomes higher.
Q2.: How is a consignment contract helpful for the food industry supply chain?
Answer: The food industry has to deal with many issues, such as food safety, contamination, and deterioration. Perishable food items are one of the greatest concerns of a food production, packaging, and supply chain. In the food industry, the consignment contract can play a crucial role. Under this contract, the manufacturer does not use its warehouse or storage facility to stock the food product. On the other hand, retailers have the opportunity to store and sell the products without purchasing the entire lots. Transactions are only conducted for sold units. This reduces the financial risk of stocking perishable items that might not sell before their expiration dates.
Q3.: How can a company deal with uncertain demand for perishable food items? Is adopting blockchain really helpful in addressing extreme demand fluctuation?
Answer: In the food industry, companies have to pay extra attention to perishable food items. The decay of food items is one of the biggest problems for the industry as the lifetime of these products is limited. Thus, with the production of extra quantities or a fewer number of items, in either case, a company shall face huge losses. This research identifies the strategies to deal with demand uncertainty and provides recommendations to companies on how much to produce.
Adopting blockchain technology can potentially offer several benefits to the perishable food industry supply chain, especially in managing extreme demand fluctuations. The ways in which this technology can help deal with demand fluctuation are transparency, traceability, and real-time data sharing. This research investigates how food industry and associated retail channels can improve profitability in various demand patterns.

1.2. Novelty of the Study

In recent years, transparency and traceability in supply chain management have become highly demanding issues. In the food industry, in particular, customers pay attention to food safety and contamination. The supply of fresh food items is more challenging than that of packaged food due to the chances of deterioration. Moreover, companies must achieve their targeted profit to run their business smoothly. Therefore, this study aims to consider some fundamental issues regarding food safety, customer satisfaction, and profitability. The originality of this manuscript relates to the impact analysis of adopting blockchain technology in the food industry. Several aspects are investigated, like how blockchain and its related costs impact customer demand, how the industry’s profitability varies when a company switches traditional contracts with consignment contracts under a decentralized blockchain system, and how companies can deal with demand uncertainty due to perishable food products. The novelty of this research also includes consideration of constructive mathematical modeling and quantifying the profitability analysis of food industry supply chain management under the adoption of decentralized blockchain technology, which has not been investigated in previous studies. Moreover, this study investigates a thorough numerical analysis of profit and loss and the effect on demand for new technology adoption.

2. Literature Review

Blockchain technology is still in its infancy and can have potentially transformational and fundamental effects on the food supply chain (FSC) as described in [5,6]. Regarding the advantages of the technology, these papers reported that adopting blockchain may boost FSC collaboration partnerships, increase food traceability, maximize operational efficiencies, and support food trading operations. Three categories—technical, organizational, and regulatory barriers—cover the majority of the drawbacks of blockchain technology. The main obstacles preventing the mainstream adoption of the technology are concerns about blockchain scalability, security, and privacy. Keskin et al. [7] derived closed-form expressions for the retailer’s anticipated profit growth and food waste reduction brought about by blockchain adoption, which demonstrated the importance of blockchain-enabled freshness transparency. A case study for the dairy sector was proposed by Casino et al. [1]. However, they ignored the concept of consignment policy in their study. Carvalho et al. [8] described an extension of the newsvendor problem for important commodities or items with larger excess or shortage costs but to the same degree. The authors discuss several estimators of the ideal order quantity based on a random sample of demand while also assuming that the parameters of the demand distribution are unknown. Malik et al. [9] present the basic ideas of smart contracts and NFTs, discussing how they can transform the market by reducing these transaction costs. Shi et al. [10] investigated the commercial methods used by many product and service platforms. comparing the benefits of various blockchain capabilities for operations management. Zhong et al. [11] provide a distributionally sound method to address the newsvendor issue. The method offers a lower bound on the expected profit and only requires the first and second moments of the redemption rate and demand distributions

Blockchain technology is fundamentally transforming the food supply chain by facilitating unprecedented levels of transparency and traceability. This decentralized ledger framework meticulously records each phase of a food product’s trajectory, from agricultural production to consumption, thereby ensuring both immutability and security. Such enhanced visibility permits swift identification and containment of foodborne illness outbreaks, significantly mitigating public health risks. Furthermore, blockchain technology empowers consumers to make well-informed decisions by authenticating the provenance, quality, and sustainability practices associated with their food products. By optimizing operational processes and diminishing administrative burdens, blockchain also enhances efficiency and cost-effectiveness for enterprises. Ultimately, blockchain technology equips stakeholders throughout the food supply chain, promoting trust, accountability, and a safer, more sustainable food ecosystem. Li et al. [12] discussed developing knowledge and information on how to maximise the application of blockchain technology to fortify the food supply chain’s traceability system. Bosona and Gebresenbet [13] proposed investigating food conditions and the numerous challenges experienced by carriers when selling fresh food by integrating blockchain technology with Internet of Things (IoT) devices. Khan et al. [14] developed a distribution-free newsvendor model considering environmental impacts and shortages with price-dependent stochastic demand.

Blockchain technology presents a revolutionary approach for fostering sustainable supply chains by improving traceability, transparency, and accountability. Through the establishment of an immutable and decentralized ledger, blockchain meticulously documents the progression of products from their origin to the final consumer, thereby ensuring adherence to ethical and sustainable practices throughout the supply chain (Dey and Seok [15]). This augmented visibility is instrumental in identifying and thwarting fraudulent activities, counterfeit products, and unethical sourcing methods. Additionally, blockchain facilitates the deployment of smart contracts to automate sustainable initiatives, such as monitoring carbon emissions and upholding fair labor standards. By harnessing the capabilities of blockchain, enterprises can cultivate trust with consumers, mitigate risks, and advance a more sustainable and responsible global supply chain. Nayal et al. [16] studied how adoption of blockchain technology (BLCT) impacts sustainable supply chain performance (SSCP) as a means of communication. Zubaydi et al. [17] give an overview of the fundamental ideas of BCT and IoT, including their architecture, protocols, consensus algorithms, features, and integration issues. Kouhizadeh et al. [18] suggested a comprehensive and complete circular economy (CE) performance metric as a tool to operationalize and measure circularity. Jiang et al. [19] explained some of the new uses for blockchain technology in the waste management industry and provided clear future directions for study in the areas of blockchain, digital waste management, and the circular economy. Noori-daryan et al. [20] investigated producer and supplier prices, the quantity of consignments that were ordered by the producer and supplier, and the sellers’ replenishment time for various structures. Kumar et al. [21] presented a drone delivery system in which the clients and the delivery firm work together to divide the shipments and construct a unique framework for determining the drones’ optimal path of travel. Rosales et al. [22] produced analytical models with an empirical foundation to better understand how a surprise increase in shrinkage might affect the price of physician- and general-preferred items. Kar et al. [23] provided a fresh federal categorization of the cloud, edge, and fog and a thorough research roadmap for offloading across various federated scenarios.

The newsvendor price-setting problem through the use of a barter trade methodology was analyzed by [24]. A store managing a stochastic price-dependent demand requires not just one product for internal usage but also another one to be sold on the market. Dey et al. [25] presented FoodSQRBlock (Food Safety Quick Response Block), a blockchain-based system that converts food production data and uses QR codes to make them easily accessible, traceable, and verifiable by producers and consumers. Ghode et al. [26] presented a secure, decentralized food industry architecture based on blockchain technology in order to address privacy and security concerns and provide a thorough solution taxonomy. Thanasi-Boçe et al. [27] discussed the detrimental effects of counterfeiting on the luxury market and the possible use of blockchain technology in the global fight against luxury counterfeiting. Dong et al. [28] analyzed a supply chain distribution system (which consisted of a supplier and many retailers) and performed an experiment in order to determine the impact of retailers trading digital claims (tokens) on the capacity of the supplier in supply–demand mismatches. Alacam and Sencer [29] explored how blockchain technology might enhance cooperation in the trucking sector by doing away with brokers while keeping them in their roles as organisers and trustees. Dutta et al. [30] evaluated a number of industry sectors that can be effectively transformed with blockchain-based technologies through improved visibility and business process management, including manufacturing, shipping, automotive, aviation, finance, technology, energy, healthcare, agriculture and food, e-commerce, and education. Čenreỳ et al. [31] analyzed the electronic bill of lading technology of container shipping, and solved key problems, such as the loss of the bill of lading during the circulation of the paper bill of lading.

In the literature, several studies have been formulated to show the advantages of block chain technology and several studies have been developed for the food supply chain. However, recognition of the importance of blockchain and consignment policy in the food supply chain in relation to the newsboy problem is still very rare in the literature. In this study, we seek to fill this gap by employing several case studies, such as use of a traditional system with and without blockchain and of consignment policy with and without blockchain.

3. Problem Definition, Notations and Assumptions

This section clarifies the definition of the problem, and the assumptions used for formulating the model. The basic notation are provided in Notations Section.

3.1. Problem Definition

This study considers a food supply chain platform using a newsvendor model with an uncertain demand scenario. The food supply chain has frequently used blockchain technology. The application of blockchain in the food supply chain has numerous potential advantages, including ensuring the reliability of the product’s information. Blockchain also improves the food supply chain’s visibility. In this study, we utilize the blockchain technology in a food industry model. We developed four different models including the following:

1.: TS without BT
2.: TS with BT
3.: CP without BT
4.: CP with BT

TS: Traditional System; CP: Consignment Policy; BT: Blockchain Technology.

As blockchain technology provides a highly secure and private transaction in information and fund sharing, this would be beneficial as well as popular for parties that are associated in the supply chain. Moreover, uncertain demand in the market is another risk for business. This study thus employs a probabilistic approach to integrate the threat scenario related to demand uncertainty. The problem and the model are formed in this study in such a way that we can analyse the risk and profitability under a blockchain environment with demand uncertainty. We also examine under various cost constraints for running a blockchain platform, how market demand fluctuates under the influence of encryption and privacy concerns. We adopt a supply chain model for the food industry as the maintenance of safety and privacy is crucial in this sector. The models are compared based on two business models, namely, a traditional supply chain model and the consignment contract. We investigate how blockchain technology affects profitability in both the traditional and consignment contract. The benefit of implementing the consignment contract and blockchain in the food industry is investigated in this study.

Figure 1 shows how consumer behavior is affected by the adoption of blockchain technology. With a blockchain platform, consumers can be attracted by two main features, such as no hassle cost and less privacy concern. The hassle cost is the cost of the effort, time, frustration, or complexity that a person has to endure. In this case, the consumer has to pay for gathering information about the authenticity and possible contamination of the food item which is to be purchased. Also, with blockchain adoption, consumers will be less concerned about any transaction privacy regarding funds or information. However, a platform charge must be incurred at both party’s (retailer or manufacturer) ends.

3.2. Assumptions

1.: Under the make-to-order policy, a single-period supply chain model is taken into account. Without enough supplies on hand to meet client demand, the manufacturer must produce and distribute goods.
2.: Although no specific probability distribution is taken into account, customer demand is viewed as random. The distribution of demand is one of the probability distribution functions. Despite this, the customer’s demand needs to be compared to a known mean and standard deviation.
3.: In the traditional structure, the manufacturer just bears the manufacturing costs; the retailer bears the full cost of inventory carrying. The retailer pays the manufacturer a wholesale price for the order quantity.
4.: The consignment policy uses the Stackelberg (leader-follower) game method to simulate a consignment contract. Manufacturers and retailers assume the positions of leaders and followers, respectively. Under the MTO manufacturing approach, the manufacturer provides the retailer with a consignment contract.
5.: The manufacturer keeps ownership of the goods even after they are transferred to the store or warehouse of the retailer. As the follower, the retailer would prefer to sign the agreement knowing that it will receive a specified fee as well as a commission for each item sold.
6.: The consignment contract divides the holding cost into two components: financial and operational. The manufacturer carries the financial portion, whereas the retailer bears the operational portion of the cost.
7.: A consignment contract may have a positive or negative sign associated with the fixed charge. A positive fixed fee is charged for guidance from the manufacturer to the retailer and vice versa.

4. Formation of Demand Equation

We assume that consumer demand follows a uniform distribution. Assuming X as a random variable, the following expression can be written:

\begin{matrix} i . e ., X \sim U [μ - \frac{σ}{2}, μ + \frac{σ}{2}] \end{matrix}

where U stands for “uniform distribution” with

μ

as the mean demand and

σ

as the standard deviation. Moreover, this study incorporates randomness and variability in demand simultaneously. As an example, if D and x denote the variable and random component of the demand, then the consumer demand function can be written in two ways, with the additive expression (

D + x

) and with the multiplicative expression (

D x

). In this study, the product expression is considered to develop the model.

4.1. Demand Function Without Blockchain: Model WBC

Without adoption of a blockchain platform, the customer’s decisions to buy food items are influenced by factors such as the price, hassle cost, and safety concerns. The expenses related to finishing a transaction, such as the time and care required, or the price of completing paperwork, are known as hassle costs. They can also be used to describe the degree of aggravation or frustration someone feels when they are inconvenienced. Demand may also be affected by privacy concerns in the WBC model. As this is a centralized system, a third party is always monitoring all transactions and information sharing. In the food supply chain, the buyer will generally be curious about the previous reputation of the seller. Thus, if the seller’s information regarding the quality of food is already uploaded in the blockchain platform in a decentralized manner, the buyer can easily access this and gather the information about the previous reputation of the seller. Infornation gathering through a decentralized blockchain platform is not available in the case of WBC. Therefore, some amount of demand must be affected by this. Without blockchain, the total demand for the food is as follows:

\begin{matrix} D_{W B C} = (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) x \end{matrix}

(1)

4.2. Demand Function with Blockchain: Model BC

With blockchain, customers may encounter fewer privacy concerns as a decentralized system is used. Thus, we consider

0 < ν < 1

, which reflects that the demand affected by the privacy concern in the WBC model is much less than the amount of demand influenced by the BC model. Evidently, the hassle cost is nullified in the BC model. The demand function in BC is as follows:

\begin{matrix} D_{B C} = (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) x \end{matrix}

(2)

5. Mathematical Model and Methods

Figure 2 describes the theoretical process flow and methods. The process starts with collection of data and information from the food industry. Then, the newsvendor model is created to optimize the decision variables. The model is developed for both traditional and consignment contracts and in both models, two separate cases are considered, namely, with the blockchain model (Model BC) and without the blockchain model (Model WBC). To solve the proposed models, we use the Leibniz rule as we need to obtain the differentiation under the sign of integration. In calculus, differentiation under the integral sign is a method for evaluating specific integrals. Differentiation under the integral sign permits one to switch the sequence of differentiation and integration under relatively liberal requirements for the function being integrated. The following equation is valid with minimal assumptions in its most basic form, known as the Leibniz integral rule, which is differentiation under the integral sign:

\frac{d}{d x} \int_{a}^{b} f (x, t) d t = \int_{a}^{b} \frac{\partial}{\partial x} d t

The uniform distribution is used for the obtained model after applying the Leibniz rule to deal with demand uncertainty. Handling the uncertainty is crucial in food manufacturing and the supply chain industry due to perishable items. Finally, using optimization techniques, the managerial decisions are taken.

We examine a supply chain that includes a blockchain platform, a retailer, and a food manufacturer. Whether to use blockchain to store product data and implement quality certification is up to the manufacturer and retailer. Without the blockchain platform, the sequence of events in this food supply chain is as follows: The manufacturer acts as the game leader and offers the retailer W (wholesale price). After receiving W (wholesale price), the retailer, i.e., the buyer, determines the food’s retail price

P_{w b c}

There are four models which are based on TP without blockchain, TP with blockchain, CP without blockchain, and CP with blockchain. In this section, all the models are developed and solved.

5.1. Traditional System Without Blockchain

The retailer’s total profit for the traditional system in model BC is

\begin{matrix} ϕ_{T S}^{r} = \{\begin{matrix} (p_{W B C} D_{W B C} - w q) - h_{T S}^{r} (q - D_{W B C}); D_{W B C} \leq q \\ (p_{W B C} - w) q - s^{r} (D_{W B C} - q); D_{W B C} > q \end{matrix} \end{matrix}

The retailer keeps ownership of the product, purchases the entire lot at wholesale from the manufacturer, and sells it to the customer at a selling price. If

x \leq q

, then the retailer incurs a carrying cost of

h_{T S}^{r}

for each unsold unit. Conversely, if

x > q

, the retailer suffers a loss of goodwill quantified at

s^{r}

for each unit of stockout. The anticipated profit of the retailer can be articulated as

\begin{matrix} E (ϕ_{T S}^{r}) & = & \int_{0}^{q} p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) x - h_{T S}^{r} (q - (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) x) \\ - & W (q) f (x) d x + \int_{q}^{\infty} [p_{W B C} q - S^{r} ((D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) x) - q) - W q] f (x) d x \end{matrix}

The retailer procures q from the supplier. The initial integral represents the anticipated profit of the retailer, predicated on the scenario where the available stock exceeds consumer demand. The retailer meets the demand and incurs a holding cost,

h_{T S}^{r}

, for each unsold unit. The following integral demonstrates that the retailer faces a stock-out cost, represented as

s^{r}

, for each unit that cannot be satisfied when depleting their existing inventory. Within the traditional system, a retailer ascertains customer demand and subsequently conveys the order quantity to the manufacturer in an indirect manner. The retailer determines the order quantity based on customer demand.

\begin{matrix} X \sim U [μ - \frac{σ}{2}, μ + \frac{σ}{2}] \end{matrix}

See Appendix A for details of the calculation.

\begin{matrix} Q^{*} = (μ - \frac{σ}{2}) + [\frac{p_{W B C} - w + s^{r} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c})}{(p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{T S}^{r} + S^{r})}] σ \end{matrix}

(3)

Owing to the order quantity

(Q^{*})

, the manufacturer maximizes his estimated revenue for a set wholesale price per unit. The profit of the producer can be explained as

\begin{matrix} ϕ_{T S}^{m} = (w - c) Q^{*} \end{matrix}

(4)

The estimated profit of the producer and the seller in the TS without blockchain is given below. The proof can be found in the Appendix A.

\begin{matrix} E (ϕ_{T S}^{r}) & = & (p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{T S}^{r} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c})) μ \\ - & (p_{W B C} + h_{T S}^{r} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + s^{r}) \frac{1}{2} ((μ + {\frac{σ}{2}}^{2} - ((μ - \frac{σ}{2}) \\ + & \frac{p_{W B C} - A + s^{r} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c})}{(p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{T S}^{r} + S^{r})} σ)) \end{matrix}

(5)

\begin{matrix} E (ϕ_{T S}^{m}) & = & [\frac{(p_{W B C} (σ^{2} - 2 μ σ + 2 c) - 2 s^{r}) D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + (σ - 2 μ σ + 2 c) (h_{T S}^{r} - s^{r}) - 2 σ (p_{W B C} - 1)}{2 (p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{T S}^{r} - s^{r} - 1)} - c] \\ \times & [(μ - \frac{σ}{2}) + \frac{p_{W B C} - A + s^{r} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c})}{(p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{T S}^{r} + S^{r})} σ] \end{matrix}

(6)

where,

\begin{matrix} A = \frac{(p_{W B C} (σ^{2} - 2 μ σ + 2 c) - 2 s^{r}) D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + (σ - 2 μ σ + 2 c) (h_{T S}^{r} - s^{r}) - 2 σ (p_{W B C} - 1)}{2 (p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{T S}^{r} - s^{r} - 1)} \end{matrix}

(7)

5.2. Traditional System with Blockchain

The retailer’s total entire profit for the traditional system with blockchain is

\begin{matrix} ϕ_{T S}^{r} = \{\begin{matrix} (p_{B C} D_{B C} - w q) - h_{T S}^{r} (q - D_{B C}); D_{B C} \leq q \\ (p_{B C} - w) q - s^{r} (D_{B C} - q); D_{B C} > q \end{matrix} \end{matrix}

The retailer keeps ownership of the product, purchases the entire lot at wholesale from the manufacturer, and sells it to the customer at a selling price. If

x \leq q

, the retailer incurs a holding cost of

h_{T S}^{r}

for every unsold item. Conversely, when

x > q

, the retailer suffers a loss of reputation quantified as

s^{r}

for each unit that is out of stock. The projected profit for the retailer can be articulated as

\begin{matrix} E (ϕ_{T S}^{r}) & = & \int_{0}^{q} [p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) x \\ - & h_{T S}^{r} (q - (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) x - w q)] f (x) d x \\ + & \int_{q}^{\infty} [p_{B C} q - S^{r} ((D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) x - q) - w q] f (x) d x \end{matrix}

The retailer acquires q from the producer. The initial integral represents the retailer’s projected profit, assuming that the existing stock surpasses the consumer demand. The retailer meets the demand and incurs a holding cost, denoted as

h_{T S}^{r}

, for each unit that remains unsold. The subsequent integral illustrates that the retailer faces a stock-out expense when they deplete their current stock, quantified as

s^{r}

, for each unit that remains unfulfilled.

In the TS, a retailer receives the demand from the customer and gives the manufacturer the order quantity indirectly. The retailer determines the order quantity based on customer demand.

\begin{matrix} X \sim U [μ - \frac{σ}{2}, μ + \frac{σ}{2}] \end{matrix}

The detailed calculation can be found in the Appendix A.1.

\begin{matrix} Q^{*} = (μ - \frac{σ}{2}) + [\frac{p_{B C} - w + s^{r} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d})}{(p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + h_{T S}^{r} + S^{r})}] σ \end{matrix}

(8)

The wholesale price and the manufacturer’s profit can be written as

\begin{matrix} ϕ_{T S}^{m} = (w - c) Q^{*} \end{matrix}

(9)

\begin{matrix} w & = & [(p_{B C} (σ^{2} - 2 μ σ + 2 c) - 2 s^{r}) (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) \\ + & (σ - 2 μ σ + 2 c) (h_{T S}^{r} - s^{r}) - 2 σ (p_{B C} - 1)] \\ \times & \frac{1}{2 (p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + h_{T S}^{r} - s^{r} - 1)} \end{matrix}

(10)

The estimated profit of the players in the TS with blockchain are given below. The proof is given in the Appendix A.1.

\begin{matrix} E (ϕ_{T S}^{r}) & = & (p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + h_{T S}^{r} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d})) μ \\ - & (p_{B C} + h_{T S}^{r} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + s^{r}) \frac{1}{2} ((μ + {\frac{σ}{2}}^{2} - ((μ - \frac{σ}{2}) \\ + & \frac{p_{B C} - A + s^{r} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d})}{(p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - Γ a_{d}) + h_{T S}^{r} + S^{r})} σ)) \end{matrix}

(11)

\begin{matrix} E (ϕ_{T S}^{m}) & = & [\frac{(p_{B C} (σ^{2} - 2 μ σ + 2 c) - 2 s^{r}) D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + (σ - 2 μ σ + 2 c) (h_{T S}^{r} - s^{r}) - 2 σ (p_{B C} - 1)}{2 (p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + h_{T S}^{r} - s^{r} - 1)} - c] \\ \times & [(μ - \frac{σ}{2}) + \frac{p_{B C} - A + s^{r} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d})}{(p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d})} + h_{T S}^{r} + S^{r}) σ] \end{matrix}

(12)

where,

\begin{matrix} A = \frac{(p_{B C} (σ^{2} - 2 μ σ + 2 c) - 2 s^{r}) D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + (σ - 2 μ σ + 2 c) (h_{T S}^{r} - s^{r}) - 2 σ (p_{B C} - 1)}{2 (p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + h_{T S}^{r} - s^{r} - 1)} \end{matrix}

5.3. Consignment Contract Without Blockchain

As the leader in the supply chain, the manufacturer uses make-to-order production to optimize profit by offering the “retailer a consignment policy” that comprises a fixed charge and per-unit commission. The two firms’ decision-making is modeled by the CP as a Stackelberg (leader-follower) game in this way. In the capacity of a follower, the retailer accepts the conditions of the agreement in return for a predetermined price and a consignment commission that is determined on a unit basis. In order to optimize his profit, he also chooses how many units of the assigned goods to order. Fixed fees may be valued negatively or positively. Typically, a positive fixed cost is seen as the manufacturer’s “admission fee” for supplying their goods to the retailer for display on the shelves. If, however, the set fee is negative, it might be thought of as a royalty that the seller pays to the manufacturer.

In order to optimize its profit, the retailer first chooses to receive a fixed charge (

C_{f}

) and a per-unit commission (

ψ

) from the manufacturer.

\begin{matrix} ϕ_{C P}^{r} = \{\begin{matrix} ψ D_{W B C} - h_{C P}^{r} (q - D_{W B C}) + C_{f}; D_{W B C} \leq q \\ ψ q - s^{r} (D_{W B C} - q) + C_{f}; D_{W B C} > q \end{matrix} \end{matrix}

The producer opts for a consignment arrangement aimed at maximizing its profitability, which encompasses a predetermined fee

C_{f}

along with a commission per unit (

ψ

) that is contingent upon the order quantity placed by the retailer.

\begin{matrix} ϕ_{C P}^{m} = \{\begin{matrix} (p_{W B C} - ψ) D_{W B C} - c q - h_{C P}^{m} (q - D_{W B C}) - C_{f};) D_{W B C} \leq q \\ (p_{W B C} - ψ - c) q - s^{m} () D_{W B C} - q) - C_{f};) D_{W B C} > q \end{matrix} \end{matrix}

The combination of the profit of the entire supply chain is given below

\begin{matrix} ϕ_{C P}^{t} = \{\begin{matrix} p_{W B C} D_{W B C} - c q - h_{C P}^{t} (q - D_{W B C}); D_{W B C} \leq q \\ (p_{W B C} - c) q - s^{t} (D_{W B C} - q); D_{W B C} > q \end{matrix} \end{matrix}

The supply chain receives a holding cost in order to satisfy the demand,

h_{C P}^{t}

, on each unit unsold, where

h_{C P}^{t}

h_{C P}^{r}

h_{C P}^{m}

. When consumer demand exceeds supply and the existing inventory is sold out, the second equation calculates the supply chain profit, with each unfilled unit resulting in a stock-out cost

s^{t}

The estimated profit of the manufacturer, the retailer, and the supply chain, respectively, are determined as follows:

\begin{matrix} E (ϕ_{C P}^{r}) & = & \int_{0}^{q^{*}} (ψ (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) x - h_{C p}^{r} (q^{*} - (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) x f (x) d x \\ + & \int_{q^{*}}^{\infty} ψ q^{*} - s^{r} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) x - q^{*} f (x) d x + X \end{matrix}

(13)

\begin{matrix} E (ϕ_{C P}^{m}) & = & \int_{0}^{q^{*}} [(p_{W B C} - ψ) ((D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c})] - h_{C P}^{m} ((D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) \\ - & c q) f (x) d x + \int_{q^{*}}^{\infty} (p_{W B C} - ψ) q^{*} - s^{m} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) - c q f (x) d x - X \end{matrix}

(14)

\begin{matrix} E (ϕ_{C P}^{t}) & = & \int_{0}^{q} [(p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c})] - c q - h_{C P}^{t} (q - D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) f (x) d x \\ + & \int_{q}^{\infty} ((p_{W B C} - c) q - s^{t} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) x f (x) d x \end{matrix}

(15)

Retailers receive a commission of

ψ

per unit from manufacturers when their order quantity is equal to the supply chain optimal production amount. The proof is given in the Appendix A.2.

\begin{matrix} ψ = \frac{h_{C P}^{r} (p_{W B C} - c + s^{t} ((D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c})}{c + h_{C P}^{t}} - s^{r} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) \end{matrix}

(16)

Assuming that there is no goodwill loss to the producer and the store in the event of a stock-out (i.e.,

s^{r}

s^{t}

= 0),

ψ

can be written as

\begin{matrix} ψ = \frac{h_{C P}^{r} (p_{W B C} - c)}{c + h_{C P}^{t}} \end{matrix}

(17)

In arranging to optimize the anticipated income for the retailer, the producer must ensure beyond any doubt the takings after the person judiciousness imperative are met. The retailer will profit at least as much as they would under the traditional system thanks to the individual rationality restriction. Therefore, until the manufacturer guarantees that the store would make at least as much as they would under the standard method, no retailer will consent to employ a CP.

\begin{matrix} m a x E (ϕ_{C P}^{m}) \end{matrix}

\begin{matrix} E (ϕ_{C P}^{r}) \geq E (ϕ_{T S}^{r}) \end{matrix}

(18)

Based on this assumption, the settled expense (X) ensures that the retailer will be at least as well off as within the conventional framework (See Appendix B)

\begin{matrix} X & = & (p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{T S}^{r} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) μ \\ - & (p_{W B C} + h_{T S}^{r} p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c})) \frac{1}{2} (μ + \frac{σ^{2}}{2} - (μ - \frac{σ^{2}}{2})) + (\frac{p_{W B C} - W + s^{R} D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}}{p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c} + h_{T S}^{r} + s^{r}})) σ \\ - & ψ D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c} + h_{r}^{C P} D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c} μ \\ - & \frac{1}{2 σ} (ψ D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c} + h_{r}^{C P} - D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + s^{r} + (μ + \frac{σ^{2}}{2} - {(q^{*})}^{2}) - h_{r}^{C P} q^{*} \end{matrix}

(19)

5.4. Consignment Contract with Blockchain

In order to optimize its profit, the retailer first chooses an order quantity and receives a fixed charge (

C_{f}

) and a per-unit commission (

ψ

) from the manufacturer.

\begin{matrix} ϕ_{C P}^{r} = \{\begin{matrix} ψ D_{B C} - h_{C P}^{r} (q - D_{B C}) + C_{f}; D_{B C} \leq q \\ ψ q - s^{r} (D_{B C} - q) + C_{f}; D_{B C} > q \end{matrix} \end{matrix}

The manufacturer selects a consignment deal while optimizing its profit, consisting of a fixed fee

C_{f}

and a per-unit commission (

ψ

) based on the retailer’s order quantity.

\begin{matrix} ϕ_{C P}^{m} = \{\begin{matrix} (p_{B C} - ψ) D_{B C} - c q - h_{C P}^{m} (q - D_{B C}) - C_{f}; D_{B C} \leq q \\ (p_{B C} - ψ - c) q - s^{m} (D_{B C} - q) - C_{f}; D_{B C} > q \end{matrix} \end{matrix}

The combination of the profit of the entire supply chain is given below

\begin{matrix} ϕ_{C P}^{t} = \{\begin{matrix} p_{B C} D_{B C} - c q - h_{C P}^{t} (q - D_{B C}); D_{B C} \leq q \\ (p_{B C} - c) q - s^{t} (D_{B C} - q); D_{B C} > q \end{matrix} \end{matrix}

The supply chain incurs a holding cost to meet the demand, denoted as

h_{C P}^{t}

, for each unsold unit, where

h_{C P}^{t}

h_{C P}^{r}

h_{C P}^{m}

. The subsequent equation delineates the profit for the supply chain, which is applicable when consumer demand surpasses supply and the existing inventory is depleted, whereby each unfulfilled unit incurs a stock-out cost represented as

s^{t}

. The estimated profit of the retailer, the manufacturer, and the supply chain, respectively, are determined as follows:

\begin{matrix} E (ϕ_{C P}^{r}) & = & \int_{0}^{q^{*}} (ψ (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) x - h_{C P}^{r} (q^{*} - (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) x f (x) d x \\ + & \int_{q^{*}}^{\infty} ψ q^{*} - s^{r} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) x - q^{*} f (x) d x + C_{f} \end{matrix}

(20)

\begin{matrix} E (ϕ_{C P}^{m}) & = & \int_{0}^{q^{*}} [(p_{B C} - ψ) ((D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d})] - h_{C P}^{m} ((D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) \\ - & c q) f (x) d x + \int_{q^{*}}^{\infty} (p_{B C} - ψ) q^{*} - s^{m} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) - c q f (x) d x - C_{f} \end{matrix}

(21)

\begin{matrix} E (ϕ_{C P}^{t}) & = & \int_{0}^{q} [(p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d})] - c q - h_{C P}^{t} (q - D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) f (x) d x \\ + & \int_{q}^{\infty} ((p_{B C} - c) q - s^{t} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) x f (x) d x \end{matrix}

(22)

When the supply chain optimal production amount equals the retailer’s order quantity, the manufacturer pays the retailer a commission of

ψ

per unit for the products sold. The proof is given in the Section Proof of Consignment Policy with Blockchain.

\begin{matrix} ψ = \frac{h_{C P}^{r} (p_{B C} - c + s^{t} ((D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d})}{c + h_{C P}^{t}} - s^{r} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) \end{matrix}

(23)

Assuming that there is no goodwill loss to the producer and the store in the event of a stock-out (i.e.,

s^{r}

s^{t}

= 0),

ψ

can be written as

\begin{matrix} ψ = \frac{h_{C P}^{r} (p_{B C} - c)}{c + h_{C P}^{t}} \end{matrix}

(24)

According to the consignment contract, the manufacturer will maximize the profit in order to satisfy the same or equal profit to the retailer in adopting CP.

\begin{matrix} m a x E (ϕ_{C P}^{m}) \end{matrix}

\begin{matrix} E (ϕ_{C P}^{r}) \geq E (ϕ_{T S}^{r}) \end{matrix}

(25)

Based on this assumption, the fixed fee (X) ensures that the retailer will be at least as well off as in the traditional system (See Appendix C)

\begin{matrix} X & = & p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + h_{T S}^{r} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) μ \\ - & (p_{B C} + h_{T S}^{r} p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d})) \frac{1}{2} (μ + \frac{σ^{2}}{2} - (μ - \frac{σ^{2}}{2})) \\ + & (\frac{p_{B C} - W + s^{R} D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}}{p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d} + h_{T S}^{r} + s^{r})}) σ \\ - & ψ D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d} + h_{r}^{C P} D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d} μ \\ - & \frac{1}{2 σ} (ψ D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d} + h_{r}^{C P} - D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + s^{r} \\ + & (μ + \frac{σ^{2}}{2} - {(q^{*})}^{2}) - h_{r}^{C P} q^{*} \end{matrix}

6. Numerical Illustration

The consignment contract and traditional system were used to conduct numerical studies of the parties and supply chain performance. Within the domain, the random part of the market demand x was uniformly distributed with mean

μ

and standard deviation

σ

such that

x \sim [μ - \frac{σ}{2}, μ + \frac{σ}{2}]

to facilitate closed forms for decision-making and profit. The input parameters taken are as follows

μ = 200

;

σ = 6

;

p_{w b c} = 100

($/unit);

W = 60

($/unit);

s^{r} = 20

($/setup);

D_{0}^{w b c} = 600

;

η 1 = 3.81

;

η 2 = 3.81

;

α = 25

;

β = 0.3

;

h_{c} = 10

;

a_{d} = 150

;

h_{T S}^{r} = 3

($/unit);

ν 1 = ν 2 = 0.2

;

c = 10

;

γ 1 = γ 2 = 0.3

;

Ψ = 60

($/unit).

6.1. Analysis of Profit

The estimated profit of both firms and the supply chain on TS with and without blockchain and CP with and without blockchain are shown in Figure 3. It is observed that the total profit of the entire supply chain increases when blockchain technology was applied by the firm. Moreover, the profit is higher for the consignment contract rather than the traditional system. For the traditional policy, the optimal value of the order quantities are observed as 99.04 and 99.09 for WBC and BC, respectively, which turns out to be a profit of

3.58 \times 10^{9}

and

4.39 \times 10^{9}

. Furthermore, the consignment contract shows a better profitability with

3.98 \times 10^{9}

and

4.57 \times 10^{9}

for WBC and BC, respectively. Moreover, in all cases, the manufacturer receives higher profit than the retailer. The fixed fee in this case is calculated as

- 4.05 \times 10^{8}

. This shows a negative fixed fee, which implies that the retailer has to pay the fixed fee or royalty to the manufacturer, as the manufacturer confirms higher profit in CP as compared to TS.

6.2. Analysis of Standard Deviation on System Profitability with Managerial Insights

Figure 4, Figure 5 and Figure 6 showing how the standard deviation of the random component of demand affects the profitability. When the standard deviation increases, the profit of the supply chain decreases accordingly. This scenario is observed for both the retailer’s profit and the total system profit. Moreover, the following points can be drawn from the figures: (a) The profit of the retailer and the total system profit decrease in a quadratic manner. (b) The difference in the decrease rate between the retailer’s and the total system profit is lower when blockchain technology is adopted by the firm.

The above phenomena happen because a high standard deviation increases the uncertainty, which results in inaccurate prediction of market demand and consumer behavior. Thus, company managers must be cautious about the collection of appropriate demand information to decrease the difference between the average and actual demand data.

6.3. Analysis of Retail Price on System Profitability with Managerial Insights

The retail price of the product has a significant effect on consumer behavior. Many market analysis results reflect that when the retail price increases, demand decreases. Figure 7, Figure 8, Figure 9 and Figure 10 show how the retail price impacts on supply chain profitability. For both TS and CP, the profit of both retailer and manufacturer dwindles irrespective of the adoption or non-adoption of blockchain technology. Thorough observation shows that in TS and CP, the rate of decrease in profitability is non-linear and the rate is almost the same for both WBC and BC.

According to the model formulation, the demand decreases linearly when the retail price gets higher. However, the expected profit function becomes non-linear with retail price increase. Thus, a company manager should expect a deep decline in system profitability per unit increase in the selling price. This could lead to a much wiser decision of the owner as to whether to increase the price or by how much to increase it. Moreover, as the retailer’s profit is lower in all cases than the manufacturer, the retailer should be more sincere about increasing its profit margin.

6.4. The Analysis of Consumer Behavior and Profitability for Adopting Blockchain Technology with Managerial Insights

6.4.1. Impact of Negative Security Concern

Consumer demand is dependent on several factors which are distinct for blockchain adoption and non-adoption. Without blockchain technology, consumers incur a hassle cost, influenced by a negative security concern and privacy concern. However, the security concern also influences the customer’s mindset after adopting blockchain, but the proportion of demand influenced by security concern is much less in blockchain adoption. Figure 11 represents how negative security concern impacts customer demand after adopting blockchain technology. Moreover, Figure 12 and Figure 13 show that the total profit is also impacted by negative security concern. The profit of both the manufacturer and the retailer reduces if security concern increases. This happens as the negative impact of security concern may affect consumer interest in signing any purchase/consignment contract, which directly affects system profitability.

As the security concern increases in blockchain adoption, the decline in demand is very slow in this case. Thus, managers should follow the decision of blockchain adoption under the condition that the company must confirm with their consumers to provide security in terms of information or fund sharing. Furthermore, with failure to provide the proper security, customers might lose interest in dealing with the companies who adopt blockchain. In this scenario, the decline in demand would be greater in blockchain adoption.

6.4.2. Impact of Hassle Cost and Privacy Concern

According to the theoretical model, consumer demand is only affected by the hassle cost in non-blockchain adoption. The privacy concern is incurred in blockchain adoption only. Hassle costs refer to the non-financial impediments or inconveniences that individuals encounter during the acquisition of goods or services, including factors such as time expenditure, physical effort, or the intricacies associated with the purchasing procedure. Thus, increase in hassle cost can directly impact consumer behavior. This study shows a linear relationship between demand and the hassle cost. Figure 14 and Figure 15 indicate that, along with demand, the total system profit is also affected by this cost and a sharp decline in profitability is observed in both TS and CP. However, the manufacturer receives more profit than the retailer in both cases and the rate of decrease in the profit percentage is the same as the change in the hassle cost.

A similar scenario happens for the consumer’s privacy concern when the blockchain system is implemented by the company. Many research studies show that while providing a multitude of advantages, including transparency and security, blockchain technology concurrently engenders considerable privacy issues that may adversely affect clientele. Increased privacy concern increases public concern, which eventually decreases the demand rate. However, if the hassle cost and privacy concern are compared then both may affect consumer behavior equally. The influence of these two parameters depends on

β

γ

h_{c}

, and

a_{d}

. For the sake of convenience, if we consider the same values of

h_{c}

and

a_{d}

, then two scenarios can occur. If

β > γ

, then the demand as well as the profitability of the supply chain will be greater for blockchain adoption (Table 1).

7. Conclusions

This study highlights the differences between consignment and traditional policies in a manufacturer-retailer supply chain system. A consignment arrangement benefits the retailer by dividing the operational and financial costs for maintaining the inventory between the manufacturer and retailer. Blockchain provides an immutable record of food origin, processing, and transportation. Blockchain can improve transparency and trust within the supply chain, allowing consumers to verify the authenticity and safety of their food. With blockchain, detecting tampering or substituting ingredients is easier, potentially reducing food fraud incidents. Blockchain can streamline communication and data sharing between all parties involved in the consignment contract. This can lead to faster processing times, reduced paperwork, and lower administrative costs. Real-time tracking of goods through the supply chain can improve inventory management for both consignors and consignees. This research identifies strategies to deal with demand uncertainty and suggest how much a company should produce. The food industry has many challenges, including deterioration, contamination, and safety. Perishable food items are one of the greatest food production, packaging, and supply chain concerns. Implementing blockchain technology improves the supply chain for perishable foods in several ways, particularly in controlling sharp demand fluctuations. Transparency, traceability, and real-time data sharing are three ways by which this technology might support flexible demand. Due to the lack of information manipulation, blockchain is commonly employed in food supply chains for information tracing. Utilizing blockchain technology also improves consumer safety when purchasing fresh food. The current research takes motivation from actual blockchain-based practices in the real world to track down fresh food information. In the fundamental models, we create models to investigate various situations with and without blockchain. The investigation evaluates the estimated profits obtained under each system by analytically modeling both a traditional and a consignment system. The main contribution of this study is to analyze the profitability of the retailing and supply chain without and with blockchain technology implemented by the food industry. The study findings ensure that the application of the blockchain technology in the food industry is very beneficial to achieve optimized profit. The model provides a quantitative analysis which clearly defines the cause and effect of several factors on a company’s profitability. The optimization tool developed in this study will be helpful for industry to predict the exact amount of investments appropriate for blockchain technology. This tool will also be helpful for decision-making and policy development. Companies can decide whether or not to invest in blockchain technology based on the effect of other factors bearing on demand deviation.

A focus on the food industry represents one limitation of this study. Future research might apply the concepts to build the model for other sectors also. By considering environmental impacts, such as carbon emissions and waste generation, research may include new area to contribute to the goal of sustainable development. Implementation of an online-to-offline retailing strategy will be an interesting addition in this research direction [32]. By considering the concept of different game polices, such as Bertrand competition and the Nash equilibrium, one can extend this model further in future.

Author Contributions

Conceptualization, B.K.D., G.K. and A.M.; methodology, A.M.; software, A.M., B.K.D., G.K. and I.S.; validation, B.K.D., A.M., G.K. and I.S.; formal analysis, B.K.D., G.K. and A.M.; investigation, B.K.D. and A.M.; resources, B.K.D. and A.M.; data curation, B.K.D. and A.M.; writing—original draft preparation, I.S.; writing—review and editing, B.K.D., A.M., G.K. and I.S.; visualization, B.K.D., A.M., G.K. and I.S.; supervision, B.K.D., G.K. and A.M. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Data used in this study will be made available upon request to the corresponding authors.

Conflicts of Interest

The authors declare no conflicts of interest.

Abbreviations

q	order quantity (units) (decision variable)
$C_{f}$	fixed cost given by the manufacturer to the buyer ($) (decision variable)
$α$	amount of demand impacted by privacy concern (units)
$D_{0}^{w b c}$	demand without blockchain (units)
$D_{0}^{b c}$	demand with blockchain (units)
$p_{W B C}$	product retail price without blockchain ($/unit)
$p_{B C}$	product retail price with blockchain ($/unit)
$h_{T S}^{r}$	traditional system holding cost of retailer ($/unit/unit time)
$s^{r}$	goodwill loss per unit item for retailer ($/unit)
w	wholesale price ($/unit)
$μ$	mean of the buyer’s demand
$σ$	standard deviation of the buyer’s demand
$h_{c}$	hassle cost
$h_{C P}^{m}$	holding cost of the manufacturer under consignment policy ($/unit/unit time)
$a_{d}$	total amount of the demand impacted by negative concern
$α$	the amount of demand affected by the privacy concern
$h_{C P}^{r}$	holding cost of the retailer under consignment policy ($/unit/unit time)
$β$	sensitivity factor for hassle cost
$γ$	sensitivity factor for consumer’s privacy concern
$s^{m}$	goodwill loss per unit item for the manufacturer ($/unit)
$ν$	sensitivity factor for consumer’s negative security concern
$η_{1}$	sensitivity factor for retail price without blockchain adoption
$η_{2}$	sensitivity factor for retail price with blockchain adoption
$E (.)$	mathematical expectation
$ϕ_{T S}^{r}$	retailer’s profit for traditional system
C	manufacturing cost per unit item ($/unit)
$ϕ_{T S}^{m}$	manufacturer’s profit for traditional system
$ψ$	per unit commission for retailer in consignment policy ($/unit)

Appendix A. Proof of Traditional System Without Blockchain

\begin{matrix} E (ϕ_{T S}^{r}) & = & \int_{0}^{q} (p_{W B C} D_{W B C} - h_{T S}^{r} (q - D_{W B C}) - w (q)) f (x) d x + \int_{q}^{\infty} [p q - S^{r} (D_{W B C} - q) - w q] f (x) d x \\ E (ϕ_{T S}^{r}) & = & \int_{0}^{q} [p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) . x - h_{T S}^{r} (q - (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) . x) \\ - & W (q)] f (x) d x + \int_{q}^{\infty} [p_{W B C} q - S^{r} ((D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) . x) - q) - W q] f (x) d x \end{matrix}

Taking the derivative for Q in the estimated profit of the retailer, and setting it equal to zero

\begin{matrix} \frac{d E (ϕ_{T S}^{r})}{d q} = 0 \\ \frac{d}{d q} \int_{0}^{Q} p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c} . x) - h_{T S}^{r} (q - (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c} . x) - W (q) f (x) d x \\ + & \int_{q}^{\infty} [p_{W B C} q - s^{r} ((D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c} . x) - q) - W q] f (x) d x = 0 \\ \frac{d}{d q} \int_{0}^{Q} p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c} . x) - h_{T S}^{r} (q - (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c} . x) - W q) f (x) d x \\ + & \frac{d}{d q} \int_{q}^{\infty} [p_{W B C} q - s^{r} ((D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c} . x) - q) - W q] f (x) d x = 0 \\ \frac{d}{d q} [p_{W B C} ((D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) . x)] + h_{T S}^{r} (p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) . x)] \\ \int_{0}^{q} x f (x) d x - [h_{T S}^{r} q + W q] f (x) d x + \frac{d}{d q} [p_{W B C} + s^{r} ((D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) . x) - W) q] \\ [1 - F (q)] - s_{r} \int_{0}^{q} x f (x) d x = 0 \\ \frac{d}{d q} [p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{T S}^{r} ((D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) \int_{0}^{q} x f (x) d x - [(h_{T S}^{r} + W) q] F (Q)} \\ + & \frac{d}{d q} [p_{W B C} + s^{r} ((D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) - W) Q] [1 - F (Q)] - s^{r} \int_{0}^{q} x f (x) d x = 0 \end{matrix}

We can obtain the following equation, using the Leibniz rule.

\begin{matrix} [p_{W B C} ((D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{T S}^{r} (p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) Q f (Q) - (h_{T S}^{r} + W) F (Q) - \\ (h_{T S}^{r} + W) q f (q) + [p_{W B C} + s^{r} ((D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c} - W) [1 - F (Q)] \\ + & [p_{W B C} + s^{r} (((h_{T S}^{r} + W) F (Q) - (h_{T S}^{r} + W) q (- F (Q)) + s^{r} q f (Q) = 0 \end{matrix}

We reorganized and re-wrote the previous equation to produce the following equation:

\begin{matrix} [p_{W B C} - W + s^{r} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) - (p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{T S}^{r} + S^{r}) F (Q)] = 0 \\ (p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{T S}^{r} + S^{r}) F (Q) = [p_{W B C} - W + s^{r} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) \end{matrix}

\begin{matrix} F (Q^{*}) = \frac{p_{W B C} - W + s^{r} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c})}{(p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{T S}^{r} + S^{r})} \end{matrix}

Suppose that

\begin{matrix} X : U [μ - \frac{σ}{2}, μ + \frac{σ}{2}] \end{matrix}

\begin{matrix} \int_{μ - \frac{σ}{2}}^{Q^{*}} \frac{1}{σ} d x = \frac{p_{W B C} - W + s^{r} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c})}{(p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{T S}^{r} + S^{r})} \end{matrix}

\begin{matrix} Q^{*} = (μ - \frac{σ}{2}) + \frac{p_{W B C} - W + s^{r} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c})}{(p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{T S}^{r} + S^{r})} σ \end{matrix}

\begin{matrix} E (ϕ_{T S}^{m}) = (w - c) Q^{*} \end{matrix}

Put

Q^{*}

in Equation (16)

\begin{matrix} E (ϕ_{T S}^{m}) = (w - c) (μ - \frac{σ}{2}) \frac{p_{W B C} - W + s^{r} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c})}{(p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{T S}^{r} + S^{r})} σ \end{matrix}

\begin{matrix} \frac{d E (ϕ_{T S}^{m})}{d w} = 0 \end{matrix}

\begin{matrix} \frac{d}{d w} E (ϕ_{T S}^{m}) = (w - c) (μ - \frac{σ}{2}) \frac{p_{W B C} - W + s^{r} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c})}{(p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{T S}^{r} + S^{r})} σ = 0 \end{matrix}

\begin{matrix} W = \frac{(p_{W B C} (σ^{2} - 2 μ σ + 2 c) - 2 s^{r}) D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + (σ - 2 μ σ + 2 c) (h_{T S}^{r} - s^{r}) - 2 σ (p - 1)}{2 (p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{T S}^{r} - s^{r} - 1)} \end{matrix}

substituting the wholesale price (W) into the Equation (15)

\begin{matrix} Q^{*} = (μ - \frac{σ}{2}) + \frac{p_{W B C} - \frac{(p (σ^{2} - 2 μ σ + 2 c) - 2 s^{r}) D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + (σ - 2 μ σ + 2 c) (h_{T S}^{r} - s^{r}) - 2 σ (p_{W B C} - 1)}{2 (p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{T S}^{r} - s^{r} - 1)}}{(p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{T S}^{r} + S^{r})} σ \end{matrix}

\begin{matrix} + \frac{s^{r} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c})}{(p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{T S}^{r} + S^{r})} σ \end{matrix}

The partial integral theorem can be used to solve the retailer’s projected profit

\begin{matrix} E (ϕ_{T S}^{r}) = (p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{T S}^{r} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c})) μ \end{matrix}

\begin{matrix} - (p_{W B C} + h_{T S}^{r} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + s^{r}) \frac{1}{2} ((μ + {\frac{σ}{2}}^{2} - (Q^{*})) \end{matrix}

\begin{matrix} E (ϕ_{T S}^{m}) = (\frac{(p_{W B C} (σ^{2} - 2 μ σ + 2 c) - 2 s^{r}) D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + (σ - 2 μ σ + 2 c) (h_{T S}^{r} - s^{r}) - 2 σ (p_{W B C} - 1)}{2 (p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{T S}^{r} - s^{r} - 1)} - c) \end{matrix}

\begin{matrix} (μ - \frac{σ}{2}) + \frac{p_{W B C} - \frac{(p_{W B C} (σ^{2} - 2 μ σ + 2 c) - 2 s^{r}) D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + (σ - 2 μ σ + 2 c) (h_{T S}^{r} - s^{r}) - 2 σ (p_{W B C} - 1)}{2 (p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{T S}^{r} - s^{r} - 1)} + s^{r} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c})}{(p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{T S}^{r} + S^{r})} σ \end{matrix}

\begin{matrix} E (ϕ_{T S}^{r}) = (p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{T S}^{r} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c})) μ \end{matrix}

\begin{matrix} - (p_{W B C} + h_{T S}^{r} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + s^{r}) \frac{1}{2} ((μ + {\frac{σ}{2}}^{2} - ((μ - \frac{σ}{2}) \end{matrix}

\begin{matrix} + \frac{p_{W B C} - \frac{(p_{W B C} (σ^{2} - 2 μ σ + 2 c) - 2 s^{r}) D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + (σ - 2 μ σ + 2 c) (h_{T S}^{r} - s^{r}) - 2 σ (p_{W B C} - 1)}{2 (p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{T S}^{r} - s^{r} - 1)} + s^{r} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c})}{(p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{T S}^{r} + S^{r})} σ)) \end{matrix}

Appendix A.1. Proof of Traditional System with Blockchain

\begin{matrix} E (ϕ_{T S}^{r}) = \int_{0}^{q} (p_{B C} D_{B C} - h_{T S}^{r} (q - D_{B C}) - w (q)) f (x) d x + \int_{q}^{\infty} [p_{B C} q - S^{r} (D_{B C} - q) - w q] f (x) d x \end{matrix}

\begin{matrix} E (ϕ_{T S}^{r}) = \int_{0}^{q} p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d} . x) - h_{T S}^{r} (q - (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d} . x) - W (q)) f (x) d x \end{matrix}

\begin{matrix} + \int_{q}^{\infty} [p_{B C} q - S^{r} ((D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d} . x) - q) - W q] f (x) d x \end{matrix}

Calculating the derivative with regard to Q in the retailer’s expected profit and setting it to zero

\begin{matrix} \frac{d E (ϕ_{T S}^{r})}{d q} = 0 \end{matrix}

\begin{matrix} \frac{d}{d q} \int_{0}^{q} p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d} . x) - h_{T S}^{r} (q - (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d} . x) - W (q)) f (x) d x \end{matrix}

\begin{matrix} \frac{d}{d q} + \int_{q}^{\infty} [p_{B C} - S^{r} ((D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d} . x) - q) - W q] f (x) d x = 0 \end{matrix}

\begin{matrix} \frac{d}{d q} \int_{0}^{Q} (p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d} . x) - h_{T S}^{r} (q - (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d} . x) - W q) f (x) d x \end{matrix}

\begin{matrix} + \frac{d}{d q} \int_{q}^{\infty} [p_{B C} q - s^{r} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d} . x) - q) - W q] f (x) d x = 0 \end{matrix}

\begin{matrix} \frac{d}{d q} [p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) \end{matrix}

\begin{matrix} + h_{T S}^{r} (p_{B C} q - (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}] \int_{0}^{q} x f (x) d x - [h_{T S}^{r} Q + W Q] f (x) d x + \end{matrix}

\begin{matrix} \frac{d}{d q} [p_{B C} + s^{r} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) q] [1 - F (Q)] - s^{r} \int_{0}^{Q} x f (x) d x = 0 \end{matrix}

We can obtain the following equation using the Leibniz rule.

\begin{matrix} [p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d})) + h_{T S}^{r} ((p_{B C} q - (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}] q f (Q) - (h {T S}^{r} + W) F (Q) - \end{matrix}

\begin{matrix} (h {T S}^{r} + W) q f (q) + [p_{B C} + s^{r} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d} - W) [1 - F (Q)] \end{matrix}

\begin{matrix} + [p_{B C} + s^{r} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) - w) q (- F (Q)) + s^{r} q f (Q) = 0 \end{matrix}

We re-organized the previous equation to produce the following equation:

\begin{matrix} [p_{B C} - W + s^{r} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d} - (p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + h {T S}^{r} + S^{r}) F (Q)] = 0 \end{matrix}

\begin{matrix} (p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + h {T S}^{r} + S^{r}) F (Q) = (p_{B C} - W + s^{r} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) \end{matrix}

\begin{matrix} E (ϕ_{T S}^{m}) = (w - c) (μ - \frac{σ}{2}) \frac{p_{B C} - W + s^{r} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c})}{(p_{B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{T S}^{r} + S^{r})} σ \end{matrix}

\begin{matrix} F (Q^{*}) = \frac{(p_{B C} - W + s^{r} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d})}{(p_{B C} D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d} + h {T S}^{r} + S^{r})} \end{matrix}

Suppose that

\begin{matrix} X : U [μ - \frac{σ}{2}, μ + \frac{σ}{2}] \end{matrix}

\begin{matrix} \int_{μ - \frac{σ}{2}}^{Q^{*}} \frac{1}{σ} d x = \frac{(p_{B C} - W + s^{r} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d})}{(p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + h {T S}^{r} + S^{r})} \end{matrix}

\begin{matrix} Q^{*} = (μ - \frac{σ}{2}) + [\frac{(p_{B C} - W + s^{r} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d})}{(p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + h {T S}^{r} + S^{r}))}] σ \end{matrix}

\begin{matrix} E (ϕ_{T S}^{m}) = (w - c) Q^{*} \end{matrix}

Putting

Q^{*}

in the above equation, we obtain

\begin{matrix} E (ϕ_{T S}^{m}) = (w - c) (μ - \frac{σ}{2}) + [\frac{(p_{B C} - W + s^{r} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d})}{(p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + h {T S}^{r} + S^{r}))} σ \end{matrix}

Now, for the optimized value of the wholesale price, we apply the following equation:

\begin{matrix} \frac{d E (ϕ_{T S}^{m})}{d w} = 0 \end{matrix}

\begin{matrix} \frac{d}{d w} E (ϕ_{T S}^{m}) = (w - c) (μ - \frac{σ}{2}) + [\frac{(p_{B C} - W + s^{r} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d})}{(p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + h {T S}^{r} + S^{r})]} σ = 0 \end{matrix}

\begin{matrix} w = \frac{(p_{B C} (σ^{2} - 2 μ σ + 2 c) 2 s^{r}) D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d} + σ^{2} - 2 μ σ + 2 c) (h {T S}^{r} - s^{r} - 2 σ (p_{B C} - 1)}{2 (p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + h {T S}^{r} - s^{r} - 1)} \end{matrix}

substituting the wholesale price (w)

\begin{matrix} Q^{*} = (μ - \frac{σ}{2}) + \frac{p_{B C} - \frac{(p_{B C} (σ^{2} - 2 μ σ + 2 c) - 2 s^{r}) D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + (σ - 2 μ σ + 2 c) (h_{T S}^{r} - s^{r}) - 2 σ (p_{B C} - 1)}{2 (p (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + h_{T S}^{r} - s^{r} - 1)} + s^{r} (D_{0}^{B C} - η_{p_{B C}} - ν α - γ a_{d})}{(p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + h_{T S}^{r} + S^{r})} σ \end{matrix}

Calculating the derivative with regard to Q in the retailer’s expected profit and setting it to zero

\begin{matrix} E (ϕ_{T S}^{r}) = (p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + h_{T S}^{r} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d})) μ \end{matrix}

\begin{matrix} - (p_{B C} + h_{T S}^{r} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + s^{r}) \frac{1}{2} ((μ + {\frac{σ}{2}}^{2} - (Q^{*})) \end{matrix}

\begin{matrix} E (ϕ_{T S}^{m}) = (\frac{(p_{B C} (σ^{2} - 2 μ σ + 2 c) - 2 s^{r}) D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + (σ - 2 μ σ + 2 c) (h_{T S}^{r} - s^{r}) - 2 σ (p_{B C} - 1)}{2 (p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + h_{T S}^{r} - s^{r} - 1)} - c) \end{matrix}

\begin{matrix} (μ - \frac{σ}{2}) + \frac{p_{B C} - \frac{(p_{B C} (σ^{2} - 2 μ σ + 2 c) - 2 s^{r}) D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + (σ - 2 μ σ + 2 c) (h_{T S}^{r} - s^{r}) - 2 σ (p_{B C} - 1)}{2 (p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + h_{T S}^{r} - s^{r} - 1)} + s^{r} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d})}{(p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + h_{T S}^{r} + S^{r})} σ \end{matrix}

\begin{matrix} E (ϕ_{T S}^{r}) = (p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + h_{T S}^{r} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d})) μ \end{matrix}

\begin{matrix} - (p_{B C} + h_{T S}^{r} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + s^{r}) \frac{1}{2} ((μ + {\frac{σ}{2}}^{2} - ((μ - \frac{σ}{2}) \end{matrix}

\begin{matrix} + \frac{p_{B C} - \frac{(p_{B C} (σ^{2} - 2 μ σ + 2 c) - 2 s^{r}) D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + (σ - 2 μ σ + 2 c) (h_{T S}^{r} - s^{r}) - 2 σ (p_{B C} - 1)}{2 (p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + h_{T S}^{r} - s^{r} - 1)} + s^{r} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d})}{(p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - Γ a_{d}) + h_{T S}^{r} + S^{r})} σ)) \end{matrix}

Appendix A.2. Proof of Consignment Policy Without Blockchain

Given a particular consignment agreement that includes a fixed fee

C f

and a commission per unit

ψ

, the manufacturer maximizes its profit based on the order quantity.

\begin{matrix} E (ϕ_{C P}^{r}) = \int_{0}^{q^{*}} (ψ D_{W B C} - h_{T S}^{r} (q^{*} - D_{W B C}) f (x) d x + \int_{q^{*}}^{\infty} [ψ q^{*} - S^{r} (D_{W B C} - q^{*})] f (x) d x + C_{f} \end{matrix}

\begin{matrix} E (ϕ_{C P}^{m}) = \int_{0}^{q^{*}} ((p_{W B C} - ψ) D_{W B C} - h_{T S}^{r} (q - D_{W B C}) - c q) f (x) d x + \int_{q^{*}}^{\infty} [(p_{W B C} - ψ) q - S^{m} (D_{W B C} - q) - c q] f (x) d x - C_{f} \end{matrix}

\begin{matrix} E (ϕ_{C P}^{r}) = \int_{0}^{q^{*}} (ψ (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) x - h_{C P}^{r} (q^{*} - (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) x f (x) d x + \end{matrix}

\begin{matrix} \int_{q^{*}}^{\infty} ψ q^{*} - s^{r} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) x - q^{*} f (x) d x + C_{f} \end{matrix}

\begin{matrix} E (ϕ_{C P}^{m}) = \int_{0}^{q^{*}} [(p_{W B C} - ψ) ((D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c})] - h_{C P}^{m} ((D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) \end{matrix}

\begin{matrix} - c q) f (x) d x + \int_{q^{*}}^{\infty} (p_{W B C} - ψ) q^{*} - s^{m} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) - c q f (x) d x - X \end{matrix}

\begin{matrix} E (ϕ_{C P}^{t}) & = & \int_{0}^{q} ((p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}))) - c q - h_{C P}^{t} (q - [D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}]) f (x) d x \\ + & \int_{q}^{\infty} ((p_{W B C} - c) q - s^{t} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) x f (x) d x \end{matrix}

\begin{matrix} \frac{d E (ϕ_{C P}^{r})}{d q} = 0 \end{matrix}

\begin{matrix} \frac{d}{d q} \int_{0}^{q *} (ψ (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) x - h_{C P}^{r} (q * - (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) \end{matrix}

\begin{matrix} \int_{q^{*}}^{x} ψ q^{*} - s^{r} ((D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) x - q *) f (x) d x + X \end{matrix}

\begin{matrix} \frac{d}{d q} (ψ ((D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{C P}^{r} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) \end{matrix}

\begin{matrix} \int_{0}^{q *} x f (x) d x [h_{C P}^{r}] q * F (Q) + \frac{d}{d q} (ψ + s^{r} ((D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) q^{*} [1 - F (Q)] - s^{r} \int_{Q^{*}}^{\infty} x f (x) d x \end{matrix}

using the Leibniz rule, we have left

\begin{matrix} ψ + s^{r} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) - (ψ (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) \end{matrix}

\begin{matrix} + s^{*} + h_{C P}^{r} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) F (Q^{*}) \end{matrix}

\begin{matrix} F (Q^{*}) = \frac{ψ + s^{r} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}}{(ψ (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + s^{r} + h_{C P}^{r}} \end{matrix}

now, similarly, we have

\begin{matrix} \frac{d E (ϕ_{C P}^{t})}{d q} = 0 \end{matrix}

\begin{matrix} \frac{d}{d q} \int_{0}^{q} [(p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c})] - c q - h_{C P}^{t} (q - D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) f (x) d x \end{matrix}

\begin{matrix} + \int_{q}^{\infty} ((p_{W B C} - c) q - s^{t} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) x f (x) d x = 0 \end{matrix}

\begin{matrix} \frac{d}{d q} (p_{W B C} D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{C P}^{t} D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) \end{matrix}

\begin{matrix} \int_{0}^{q} x f (x) d x - h_{C P}^{t} q F (Q) + \frac{d}{d Q} [(p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c})] q [1 - F (Q)] - s^{t} \int_{q}^{i n f t y} x f (x) d x = 0 \end{matrix}

\begin{matrix} F (Q^{*}) = \frac{p_{W B C} - c + s^{t} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c})}{p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + s^{t} + h_{C P}^{t}} \end{matrix}

The retailer receives payment for products supplied when the optimal amount in the supply chain matches the quantity ordered by the retailer, less a commission of

ψ

per unit.

\begin{matrix} I f F (Q *) = F (Q_{t}^{*}), t h e n \end{matrix}

\begin{matrix} ψ = \frac{h_{C P}^{r} (p_{W B C} - c + s^{t} ((D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c})}{c + h_{C P}^{t}} - s^{r} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) \end{matrix}

Now, suppose that

\begin{matrix} X : U [μ - \frac{σ}{2}, μ + \frac{σ}{2}] \end{matrix}

\begin{matrix} Q^{*} = (μ - \frac{σ}{2}) + [\frac{ψ + s^{r} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c})}{ψ (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + s^{r} + h_{C P}^{r}}] σ \end{matrix}

\begin{matrix} Q_{t}^{*} = (μ - \frac{σ}{2}) + [\frac{p_{W B C} - c + s^{t} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c})}{ψ (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + s^{t} + h_{C P}^{t}}] σ \end{matrix}

\begin{matrix} Q^{*} = Q_{t}^{*} = (μ - \frac{σ}{2}) + [\frac{p_{W B C} - c + s^{t} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c})}{ψ (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + s^{t} + h_{C P}^{t}}] σ \end{matrix}

Appendix B. Estimated Profit

\begin{matrix} m a x E (ϕ_{C P}^{m}) \end{matrix}

\begin{matrix} E (ϕ_{C P}^{r}) \geq E (ϕ_{C P}^{m}) \end{matrix}

\begin{matrix} X & = & p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{T S}^{r} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) μ \\ - & (p_{W B C} + h_{T S}^{r} p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c})) \frac{1}{2} (μ + \frac{σ^{2}}{2} - (μ - \frac{σ^{2}}{2})) + (\frac{p_{W B C} - W + s^{R} D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}}{p_{W B C} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c} + h_{T S}^{r} + s^{r})}) σ \\ - & ψ D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c} + h_{r}^{C P} D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c} μ \\ - & \frac{1}{2 σ} (ψ D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c} + h_{r}^{C P} - D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + s^{r} + (μ + \frac{σ^{2}}{2} - {(q^{*})}^{2}) - h_{r}^{C P} q^{*} \end{matrix}

Solving the retailer’s expected profit using integral calculus.

\begin{matrix} E (ϕ_{C P}^{r}) & = & \int_{0}^{q^{*}} (ψ x - h_{T S}^{r} (q^{*} - D_{W B C}) f (x) d x + \int_{q^{*}}^{\infty} [ψ q^{*} - S^{r} (D_{W B C} - q^{*})] f (x) d x + C_{f} \\ E (ϕ_{C P}^{r}) & = & \int_{0}^{q^{*}} (ψ (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c} . x) - h_{C P}^{r} (q^{*} - (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c} . x f (x) d x \\ + & \int_{q^{*}}^{\infty} [ψ q^{*} - S^{r} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c} . x) - q^{*})] f (x) d x + C_{f} \end{matrix}

\begin{matrix} E (ϕ_{C P}^{r}) & = & (ψ (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{C P}^{r} (D_{0}^{W B C} - η_{1 p_{W B C}} - α - β h_{c})) (μ - \int_{q^{*}}^{{(μ + \frac{σ}{2})}^{x}} f (x) d x) - h_{C P^{r}}^{q^{*}} + h_{C P^{r}}^{q^{*}} \int_{q^{*}}^{(μ + \frac{σ}{2})} f (x) d x \\ + & ψ (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) q^{*} \int_{q^{*}}^{(} μ + \frac{σ}{2}) f (x) d x - s^{r} \int_{q^{*}}^{(} μ + \frac{σ}{2}) x f (x) d x E (ϕ_{C P}^{r}) \\ = & ψ (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{C P}^{r} (- (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) μ) \\ - & (ϕ (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{C P}^{r} (- (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + S^{r} \int_{q^{*}}^{{(μ + \frac{σ}{2})}^{x}} f (x) d x) - h_{C P^{r}}^{q^{*}} \\ + & h_{C P}^{r} \int_{q^{*}}^{μ + \frac{σ}{2}} f (x) d x + (ϕ + S^{r} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) q^{*} \int_{q^{*}}^{μ + \frac{σ}{2}} f (x) d x) + C_{f} \end{matrix}

\begin{matrix} E (ϕ_{C P}^{r}) = [(α (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{r}^{C P} (- (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) μ - \end{matrix}

\begin{matrix} \frac{1}{2 σ} (α (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{C P}^{r} (- (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + S^{r} [{(μ + \frac{σ}{2})}^{2} - {(Q^{*})}^{2}] - h_{r}^{C P} Q^{*} \end{matrix}

\begin{matrix} + \frac{1}{σ} h_{C P}^{r} q^{*} [(μ + \frac{σ}{2}) - (q^{*})] + \frac{1}{σ} q^{*} (α + S_{r} (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) (μ + \frac{σ}{2}) - (q^{*})] + C_{f} \end{matrix}

\begin{matrix} E (ϕ_{C P}^{r}) = [(ψ (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{C P}^{r} (- (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) μ - \end{matrix}

\begin{matrix} \frac{1}{2 σ} (ψ (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{C P}^{r} (- (D_{0}^{W B C} - η_{1} p_{W B C} - α - β h_{c} + S^{r} [{(μ + \frac{σ}{2})}^{2} - {(q^{*})}^{2}] - h_{C P}^{r} q^{*} + C_{f} \end{matrix}

\begin{matrix} E (ϕ_{C P}^{m}) = [p_{W B C} - ψ ((D_{W B C} - η_{1} p_{B C} - α - β h_{c})) + h_{C P}^{m} (D_{W B C} - η_{1} p_{W B C} - α - β h_{c})) μ - \end{matrix}

\begin{matrix} \frac{1}{2 σ} (p_{W B C} - ψ (D_{W B C} - η_{1} p_{W B C} - α - β h_{c}) + h_{C P}^{m} (D_{W B C} - η_{1} p_{W B C} - α - β h_{c}) + S^{m}) [(μ + \frac{σ}{2}) - {(q^{*})}^{2}] - (h_{C P}^{m} + c) q^{*} - C_{f} \end{matrix}

\begin{matrix} E (ϕ_{C P}^{t}) = [p_{W B C} + h_{C P}^{t} (D_{W B C} - η_{1} p_{W B C} - α - β h_{c})) μ - \end{matrix}

\begin{matrix} \frac{1}{2 σ} (p_{W B C} + h_{C P}^{t} (D_{W B C} - η_{1} p_{W B C} - α - β h_{c}) + S^{t}) [(μ + \frac{σ}{2}) - {(q^{*})}^{2}] - (h_{C P}^{t} - c) q^{*} \end{matrix}

Proof of Consignment Policy with Blockchain

\begin{matrix} E (ϕ_{C P}^{r}) = \int_{0}^{q^{*}} (ψ D_{B C} - h_{T S}^{r} (q^{*} - D_{B C}) f (x) d x + \int_{q^{*}}^{\infty} [ψ q^{*} - S^{r} (D_{B C} - q^{*})] f (x) d x + C_{f} \end{matrix}

\begin{matrix} E (ϕ_{C P}^{m}) = \int_{0}^{q^{*}} ((p_{B C} - ψ) D_{B C} - h_{T S}^{r} (q - D_{B C}) - c q) f (x) d x + \int_{q^{*}}^{\infty} [(p_{B C} - ψ) q - S^{m} (D_{B C} - q) - c q] f (x) d x - C_{f} \end{matrix}

\begin{matrix} E (ϕ_{C P}^{t}) = \int_{0}^{q^{*}} (ψ (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) x - h_{C P}^{r} (q^{*} - (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) x f (x) d x + \end{matrix}

\begin{matrix} \int_{q^{*}}^{\infty} ψ q^{*} - s^{r} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ h_{c}) x - q^{*} f (x) d x + C_{f} \end{matrix}

\begin{matrix} E (ϕ_{C P}^{t}) = \int_{0}^{q^{*}} [(p_{B C} - ψ) ((D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) x)] - h_{C P}^{m} (D_{0}^{B C} - η_{2} p_{B C} ν α - γ a_{d}) x \end{matrix}

\begin{matrix} - c q) f (x) d x + \int_{q^{*}}^{\infty} (p_{B C} - ψ) q^{*} - s^{m} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) x - c q f (x) d x - C_{f} \end{matrix}

\begin{matrix} E (ϕ_{C P}^{t}) = \int_{0}^{q} [(p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) x] - c q - h_{C P}^{t} (q - D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) x f (x) d x \end{matrix}

\begin{matrix} + \int_{q}^{\infty} ((p_{B C} - c) q - s^{t} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) x f (x) d x \end{matrix}

\begin{matrix} \frac{d E (ϕ_{C P}^{r})}{d q} = 0 \end{matrix}

\begin{matrix} \frac{d}{d q} \int_{0}^{q *} (ψ (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) x - h_{C P}^{r} (q * - (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) x \end{matrix}

\begin{matrix} \int_{q *}^{x} ψ q * - s^{r} ((D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) x - q *) f (x) d x + C_{f} \end{matrix}

\begin{matrix} \frac{d}{d q} (ψ ((D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + h_{C P}^{r} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) \end{matrix}

\begin{matrix} \int_{0}^{q *} x f (x) d x [h_{C P}^{r}] q * F (Q) + \frac{d}{d q} (ψ + s^{r} ((D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) q^{*} [1 - F (Q)] - s^{r} \int_{Q^{*}}^{\infty} x f (x) d x \end{matrix}

Using the Leibniz rule, we have left

\begin{matrix} ψ + s^{r} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) - (ψ (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) \end{matrix}

\begin{matrix} + s^{*} + h_{C P}^{r} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) F (Q^{*}) \end{matrix}

\begin{matrix} F (Q^{*}) = \frac{ψ + s^{r} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}}{(ψ (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + s^{r} + h_{C P}^{r}} \end{matrix}

Now, similarly, we have

\begin{matrix} \frac{d E (ϕ_{C P}^{t})}{d q} = 0 \end{matrix}

\begin{matrix} \frac{d}{d q} \int_{0}^{q} [(p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d})] - c q - h_{C P}^{t} (q - D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) f (x) d x \end{matrix}

\begin{matrix} + \int_{q}^{\infty} ((p_{B C} - c) q - s^{t} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) x f (x) d x = 0 \end{matrix}

\begin{matrix} \frac{d}{d q} (p_{B C} D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + h_{C P}^{t} D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) \end{matrix}

\begin{matrix} \int_{0}^{q} x f (x) d x - h_{C P}^{t} q F (Q) + \frac{d}{d Q} [(p_{B C} D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d})] q [1 - F (Q)] - s^{t} \int_{q}^{i n f t y} x f (x) d x = 0 \end{matrix}

\begin{matrix} F (Q^{*}) = \frac{p_{B C} - c + s^{t} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d})}{p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + s^{t} + h_{C P}^{t}} \end{matrix}

The retailer receives payment for products supplied when the optimal amount in the supply chain matches the quantity ordered by the retailer, less a commission of

ψ

per Unit.

\begin{matrix} if F (Q *) = F (Q_{t}^{*}), t h e n \end{matrix}

\begin{matrix} ψ = \frac{h_{C P}^{r} (p_{B C} - c + s^{t} ((D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d})}{c + h_{C P}^{t}} - s^{r} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) \end{matrix}

Now, suppose that

\begin{matrix} X : U [μ - \frac{σ}{2}, μ + \frac{σ}{2}] \end{matrix}

\begin{matrix} Q^{*} = (μ - \frac{σ}{2}) + [\frac{ψ + s^{r} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d})}{ψ (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + s^{r} + h_{C P}^{r}}] σ \end{matrix}

\begin{matrix} Q_{t}^{*} = (μ - \frac{σ}{2}) + [\frac{p_{B C} - c + s^{t} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d})}{ψ (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ h_{c}) + s^{t} + h_{C P}^{t}}] σ \end{matrix}

\begin{matrix} Q^{*} = Q_{t}^{*} = (μ - \frac{σ}{2}) + [\frac{p_{B C} - c + s^{t} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d})}{ψ (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + s^{t} + h_{C P}^{t}}] σ \end{matrix}

Appendix C. Estimated Profit

\begin{matrix} m a x E (ϕ_{C P}^{m}) \end{matrix}

\begin{matrix} E (ϕ_{C P}^{r}) \geq E (ϕ_{C P}^{m}) \end{matrix}

\begin{matrix} X = (p_{B C} (D_{0}^{B C} - η_{2} p_{W B C} - ν α - γ a_{d}) + h_{T S}^{r} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) μ \end{matrix}

\begin{matrix} - (p_{B C} + h_{T S}^{r} p (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d})) \\ \frac{1}{2} (μ + \frac{σ^{2}}{2} - (μ - \frac{σ^{2}}{2}) \\ + (\frac{p_{B C} - W + s^{R} D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}}{p_{B C} (D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d} + h_{T S}^{r} + s^{r}}) σ \\ - ψ D_{0}^{B C} - η_{2} p_{B C} - ν α - β a_{d} + h_{r}^{C P} D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d} μ \\ - \frac{1}{2 σ} (ψ D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d} + h_{r}^{C P} - D_{0}^{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + s^{r} + (μ + \frac{σ^{2}}{2} - {(q^{*})}^{2}) - h_{r}^{C P} q^{*} \end{matrix}

Solving the retailer’s expected profit equation using integral calculus.

\begin{matrix} E (ϕ_{C P}^{r}) = \int_{0}^{q^{*}} (ψ x - h_{T S}^{r} (q^{*} - x) f (x) d x + \int_{q^{*}}^{\infty} [ψ q^{*} - S^{r} (x - q^{*})] f (x) d x + X \end{matrix}

\begin{matrix} E (ϕ_{C P}^{r}) = \int_{0}^{q^{*}} (ψ (D_{B C} - η_{2} p_{B C} - ν α - γ a_{d} . x - h_{C P}^{r} (q^{*} - D_{B C} - η_{2} p_{B C} - ν α - γ a_{d} . x) f (x) d x \end{matrix}

\begin{matrix} + \int_{q^{*}}^{\infty} [ψ q^{*} - S^{r} (D_{B C} - η_{2} p_{B C} - ν α - γ a_{d} . x - q^{*})] f (x) d x + X \end{matrix}

\begin{matrix} E (ϕ_{C P}^{r}) = (\int_{0}^{q^{*}} (ψ (D_{B C} - η_{2} p_{B C} - ν α - γ a_{d} . x - h_{C P}^{r} (q^{*} - ((D_{B C} - η_{2} p_{B C} - ν α - γ a_{d} . x) f (x) d x \end{matrix}

\begin{matrix} + \int_{q^{*}}^{\infty} [ψ q^{*} - S^{r} [(D_{B C} - η_{2} p_{B C} - ν α - γ a_{d} . x) - q^{*})] f (x) d x + X \end{matrix}

\begin{matrix} E (ϕ_{C P}^{r}) & = & ψ (D_{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + h_{C P}^{r} (D_{B C} - η_{2} p_{B C} - ν α - γ a_{d}) (μ - \int_{q^{*}}^{{(μ + \frac{σ}{2})}^{x}} f (x) d x) - h_{{C P}^{r}}^{q^{*}} + h_{{C P}^{r}}^{q^{*}} \int_{q^{*}}^{μ + \frac{σ}{2}} f (x) d x \\ + & α (D_{B C} - η_{2} p_{B C} - ν α - γ a_{d}) q^{*} \int_{q^{*}}^{μ + \frac{σ}{2}} f (x) d x - s^{r} \int_{q^{*}}^{μ + \frac{σ}{2}} x f (x) d x + X \end{matrix}

\begin{matrix} E (ϕ_{C P}^{r}) & = & [(ψ (D_{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + h_{C P}^{r} (((D_{B C} - η_{2} p_{B C} - ν α - γ a_{d}) μ - (ψ ((D_{B C} - η_{2} p_{B C} - ν α - γ a_{d}) \\ + & h_{C P}^{r} (D_{B C} - η_{2} p_{B C} - ν α - γ a_{d})) + S^{r} \int_{q^{*}}^{{(μ + \frac{σ}{2})}^{x}} f (x) d x) - h_{{C P}^{r}}^{q^{*}} + h_{{C P}^{r}}^{q} \int_{q^{*}}^{μ + \frac{σ}{2}} f (x) d x \\ + & (ψ + S^{r} ((D_{B C} - η_{2} p_{B C} - ν α - γ a_{d}) q^{*} \int_{q^{*}}^{μ + \frac{σ}{2}} f (x) d x)) + X \\ E (ϕ_{C P}^{r}) & = & [(ψ (D_{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + h_{C P}^{r} - (D_{B C} - η_{2} p_{B C} - ν α - γ a_{d}) μ - \end{matrix}

\begin{matrix} \frac{1}{2 σ} (ψ (D_{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + h_{C P}^{r} - ((D_{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + S^{r} [{(μ + \frac{σ}{2})}^{2} - {(q^{*})}^{2}] - h_{C P}^{r} q^{*} \end{matrix}

\begin{matrix} + \frac{1}{σ} h_{C P}^{r} q^{*} [(μ + \frac{σ}{2}) - (q^{*})] + \frac{1}{σ} q^{*} (ψ + S^{r} (D_{B C} - η_{2} p_{B C} - ν α - γ a_{d})) (μ + \frac{σ}{2}) - (Q^{*})] + X \end{matrix}

\begin{matrix} E (ϕ_{C P}^{r}) & = & [ψ ((D_{B C} - η_{2} p_{B C} - ν α - γ a_{d})) + h_{C P}^{r} (D_{B C} - η_{2} p_{B C} - ν α - γ a_{d})) μ \\ - & \frac{1}{2 σ} (ψ (D_{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + h_{C P}^{r} (D_{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + S^{r}) [(μ + \frac{σ}{2}) - {(q^{*})}^{2}] - h_{C P}^{r} q^{*} + X \\ E (ϕ_{C P}^{m}) & = & [p_{B C} - ψ ((D_{B C} - η_{2} p_{B C} - ν α - γ a_{d})) + h_{C P}^{m} (D_{B C} - η_{2} p_{B C} - ν α - γ a_{d})) μ - \frac{1}{2 σ} (p_{B C} - ψ (D_{B C} - η_{2} p_{B C} - ν α - γ a_{d}) \\ + & h_{C P}^{m} (D_{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + S^{m}) [(μ + \frac{σ}{2}) - {(q^{*})}^{2}] - (h_{C P}^{m} + c) q^{*} - X \\ E (ϕ_{C P}^{r}) & = & [p_{B C} + h_{C P}^{t} (D_{B C} - η_{2} p_{B C} - ν α - γ a_{d})) μ - \frac{1}{2 σ} (p_{B C} + h_{C P}^{t} (D_{B C} - η_{2} p_{B C} - ν α - γ a_{d}) + S^{t}) [(μ + \frac{σ}{2}) - {(q^{*})}^{2}] - (h_{C P}^{t} - c) q^{*} \end{matrix}

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Figure 1. Consumer behavior under blockchain and no-blockchain adoption.

Figure 2. Model and methodology.

Figure 3. Analysis of profit of supply chain.

Figure 4. Effect of standard deviation on profit. (a) Graphical representation of standard deviation with profit for traditional system without blockchain. (b) Graphical representation of standard deviation with profit for traditional system with blockchain.

Figure 5. Graphical representation of standard deviation with profit for consignment policy without blockchain.

Figure 6. Graphical representation of standard deviation with profit for consignment policy with blockchain.

Figure 7. Graphical representation of profit with retail price for TS without blockchain.

Figure 8. Graphical representation of profit with retail price of the product for TS with blockchain.

Figure 9. Graphical representation of profit with retail price of the product for CP without blockchain.

Figure 10. Graphical representation of profit with retail price of the product for CP with blockchain.

Figure 11. Graphical representation of demand impacted by negative security concern.

Figure 12. Graphical representation of profit vs. the negative security concern for TS with blockchain.

Figure 13. Graphical representation of profit vs. the negative security concern for CP with blockchain.

Figure 14. Graphical representation of profit vs. hassle cost for TS without blockchain.

Figure 15. Graphical representation of profit vs. hassle cost for CP without blockchain.

Table 1. Sensitivity of blockchain parameters on profitability.

	$ν_{1} > ν_{2}$	$ν_{1} < ν_{2}$	$β > γ$	$β < γ$
Profit for WBC	↓	↑	↓	↑
Profit for BC	↑	↓	↑	↑

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Sharma, I.; Kaur, G.; Dey, B.K.; Majumder, A. Leveraging Blockchain and Consignment Contracts to Optimize Food Supply Chains Under Uncertainty. Appl. Sci. 2024, 14, 11735. https://doi.org/10.3390/app142411735

AMA Style

Sharma I, Kaur G, Dey BK, Majumder A. Leveraging Blockchain and Consignment Contracts to Optimize Food Supply Chains Under Uncertainty. Applied Sciences. 2024; 14(24):11735. https://doi.org/10.3390/app142411735

Chicago/Turabian Style

Sharma, Isha, Gurpreet Kaur, Bikash Koli Dey, and Arunava Majumder. 2024. "Leveraging Blockchain and Consignment Contracts to Optimize Food Supply Chains Under Uncertainty" Applied Sciences 14, no. 24: 11735. https://doi.org/10.3390/app142411735

APA Style

Sharma, I., Kaur, G., Dey, B. K., & Majumder, A. (2024). Leveraging Blockchain and Consignment Contracts to Optimize Food Supply Chains Under Uncertainty. Applied Sciences, 14(24), 11735. https://doi.org/10.3390/app142411735

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Leveraging Blockchain and Consignment Contracts to Optimize Food Supply Chains Under Uncertainty

Abstract

1. Introduction

1.1. Research Questions

1.2. Novelty of the Study

2. Literature Review

3. Problem Definition, Notations and Assumptions

3.1. Problem Definition

3.2. Assumptions

4. Formation of Demand Equation

4.1. Demand Function Without Blockchain: Model WBC

4.2. Demand Function with Blockchain: Model BC

5. Mathematical Model and Methods

5.1. Traditional System Without Blockchain

5.2. Traditional System with Blockchain

5.3. Consignment Contract Without Blockchain

5.4. Consignment Contract with Blockchain

6. Numerical Illustration

6.1. Analysis of Profit

6.2. Analysis of Standard Deviation on System Profitability with Managerial Insights

6.3. Analysis of Retail Price on System Profitability with Managerial Insights

6.4. The Analysis of Consumer Behavior and Profitability for Adopting Blockchain Technology with Managerial Insights

6.4.1. Impact of Negative Security Concern

6.4.2. Impact of Hassle Cost and Privacy Concern

7. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Conflicts of Interest

Abbreviations

Appendix A. Proof of Traditional System Without Blockchain

Appendix A.1. Proof of Traditional System with Blockchain

Appendix A.2. Proof of Consignment Policy Without Blockchain

Appendix B. Estimated Profit

Proof of Consignment Policy with Blockchain

Appendix C. Estimated Profit

References

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