Blockchain-Enhanced Inventory Management in Decentralized Supply Chains for Finite Planning Horizons

The disruptive impact of COVID-19 underscores the vulnerability of centralized supply chain systems. Disruptions originating from a single source can have far-reaching consequences, impacting the global supply of goods and services. Decentralized supply chain models are emerging as robust solutions during this critical period [1, 2]. Through a multi-participant approach, these models distribute decision-making and resources among various participants, ensuring that a small issue in one part does not affect the entire system. In the aftermath of this pandemic, the flexibility, robustness, and power of a decentralized supply chain have become apparent, eliminating dependency on a single source.

In today's global business environment, supply chains have become increasingly complex and dispersed, leading to a need for better information sharing, visibility, and coordination among multiple stakeholders. The literature on decentralized supply chain models addresses the challenges and opportunities associated with managing these complex supply chains. Several studies have highlighted the importance of decentralized supply chain models in improving operational efficiency, reducing costs, and enhancing customer satisfaction. One key advantage of decentralized supply chain models is the greater autonomy they provide to individual subsidiaries or locations within a business [3].

In parallel, the advent of blockchain technology has transformed various industries by revolutionizing the storage and verification of transactions and data. One particular area where blockchain has shown great potential in the realm of supply chain management [4]. As supply chains become more complex and globalized, there is an increasing demand for innovative solutions that can improve traceability, efficiency, and transparency.

This research paper delves into the transformative journey of integrating Blockchain Technology into our decentralized supply chain. We aim to analyze the potential impact of this technology, especially within the finite planning horizon, to effectively manage our supply chain during disruptions [5]. Like a well-crafted recipe, this paper blends theory, case study, and practical insights, showcasing how blockchain can become a powerhouse for our decentralized supply chain. Together, feedback on a journey to explorer this new variant supply chain management, where innovative solutions powered by blockchain pave the way for a more liberated, silent, and efficient future.

In addition, this study incorporates quantitative analysis by employing numerical examples extracted from a secondary dataset compiled from a diverse array of [6, 7]. Our analysis is firmly grounded in a robust foundation. Utilizing an optimal iterative method, we systematically investigate the model's performance under varying wholesale prices. The primary objective is to contribute valuable insights into the optimization potential of the model across a spectrum of scenarios. This study not only extends the current body of research findings but also expands the examination of the model's behavior in response to diverse conditions of wholesale prices.

In this research paper, each section explains how blockchain technology enhances decentralized supply chain inventory models. Detailed literature reviews are presented in Section 2, including the impact of blockchain. Section 3 defines symbols and variables within the theoretical framework. We establish a theoretical framework and develop a model in this section. Section 4 explains the mathematical model for a blockchain-based supply chain inventory system. Literature challenges are addressed by the model, as well as its explained approach. A practical case study and numerical examples illustrate the model in Section 5. A sensitivity analysis is provided in Section 6, which evaluates the model's robustness under varying conditions, enabling optimization. This section offers supply chain decision-makers practical recommendations based on theoretical and numerical findings. Section 7 summarizes key findings, managerial implications, and future research directions.

To develop the proposed model in this paper, we use the notations, assumptions and Boundary Conditions listed below.

3.1 Notations

3.2 Assumptions

• The planning horizon is defined by constant holding, ordering, & shortage costs.
• The supply chain consists of a single retailer and a single supplier, managing a specific item.
• The supplier maintains no inventory, given the instantaneous replenishment and infinite production capacity.
• Replenishment by suppliers occurs in a lot-by-lot fashion.
• The setup cost incurred by the supplier is higher than that incurred by the retailer.
• The retailer, being a rational actor, strategically selects products to maximize profitability.

3.3 Boundary conditions

• I_i+1(t_i+1) = 0 signifies that the inventory level becomes zero at the end of each cycle, indicating complete consumption of the replenished inventory.
• I_t+1(t_i) = Q_i+1 ensures that the inventory level is initialized with the order quantity Q_i+1 from the previous cycle at the beginning of each cycle.

The specified boundary conditions play a fundamental role in solving the associated differential equation, serving as essential constraints for our model. They serve to enhance the accuracy and reliability of our model by providing a well-defined framework, enabling precise simulation and analysis of the inventory system's dynamics within the finite planning horizon.

To optimize a process, the aim is to minimize a specific Eq. (6) while keeping the value of ‘n’ constant. By taking the first partial derivative of the equation, we can obtain Eq. (9), which provides the optimal values for the ordering intervals, represented by t_j. Once Eq. (9) is satisfied, it reveals these optimal values. By imposing certain constraints such as ‘t₀=0’ & ‘t_n=H’, the uniqueness of these optimal solutions is established and forms the basis for an efficient and effective model.

$\frac{\partial}{\partial t_j} \mathrm{TCR}\left(\mathrm{n}, \mathrm{t}_0, \mathrm{t}_1, \mathrm{t}_2, \ldots \ldots \ldots \ldots, \mathrm{t}_{\mathrm{n}}\right)=[\beta \mathrm{W}-\mu \mathrm{d}-(1-\mu) \mathrm{d} \alpha] \mathrm{t}_{\mathrm{j}}\left[e^{\theta\left(t_j-t_{j-1}\right)}-1\right]-\theta \int_{t_j}^{t_{j+1}}[\beta \mathrm{W}-\mu \mathrm{d}-(1-$ $\mu) \mathrm{d} \alpha]+e]^{\theta(t-\mathrm{tj})} \mathrm{dt}=0 .\mathrm{j}=1,2,3 \ldots, \mathrm{n}-1$.     (9)

Let n*, t₁*, t₂*, ..., t_n-1* represent the optimal solution for the Minimum TCR problem with parameters n, t, t₁, t₂, ..., t_n.

Theorem 1: Uniqueness and Optimality interval

Consider a fixed parameter, n, and let Eq. (9) represent the total cost of retailer (TCR) function, in the context of supply chain optimization.

Proof: Our objective is to establish that the optimal ordering interval for this system is the unique solution to Eq. (9). To substantiate this claim, we delve into the properties of the Hessian matrix associated with TCR.

In comprehending the intricate relationship between replenishment cycles and times, we formulate key expressions. Notably,

$\begin{gathered}\frac{\partial^2 \mathrm{TC}_r^{\mathrm{IND}}\left(n_1, \mathrm{t}_0, \mathrm{t}_1, \mathrm{t}_2, \ldots \ldots, \mathrm{t}_n\right)}{\partial t_j^2}=(-\mathrm{W} \beta+\mathrm{d} \alpha(1-\mu)+\mathrm{d} \mu(W+\left.\frac{\mathrm{hr}}{\theta}\right)\left(e^{\theta\left(t_j-\mathrm{t}_{\mathrm{j}-1}\right)}-e^{\theta\left(t_{j+1}-t_j\right)}+e^{\theta\left(t_j-t_{j-1}\right)} \theta \mathrm{t}_j+\right.\left.e^{\theta\left(t_{j+1}-t_j\right)} \theta \mathrm{t}_{1-j}\right) \\ \frac{\partial^2 \mathrm{TC}_r^{\mathrm{IND}}\left(n_1, \mathrm{t}_0, \mathrm{t}_1, \mathrm{t}_2, \ldots \ldots \ldots, \mathrm{t}_n\right)}{\partial t_j \partial t_{j-1}}=-e^{\theta\left(t_j-\mathrm{t}_{\mathrm{j}-1}\right)} \theta(-\mathrm{W} \beta+d \alpha(1-\mu)+d \mu\left(W+\frac{h_r}{\theta}\right) t_j \\ \frac{\partial^2 \mathrm{TC}_r^{\mathrm{IND}}\left(n_1, \mathrm{t}_0, \mathrm{t}_1, \mathrm{t}_2, \ldots \ldots \ldots, \mathrm{t}_{n_1}\right)}{\partial t_j \partial t_{j+1}}=-e^{\theta\left(t_{j+1}-\mathrm{t}_j\right)} \theta(-\mathrm{W} \beta+d \alpha-\mu)+\frac{d \mu)\left(W+\frac{h_r}{\theta}\right) t_{1+j}}{\partial t_j \partial t_k}=0 \\ \text { and } \frac{\partial^2 \mathrm{TC}_r^{{I N D}}\left(n_1, \mathrm{t}_0, \mathrm{t}_1, \mathrm{t}_2, \ldots \ldots, \mathrm{t}_{n_1}\right)}{\partial t_j \partial t_k}=0\end{gathered}$

Furthermore,

$\begin{aligned} & \frac{\partial^2 T C_r^{I N D}\left(n_1, t_0, t_1, t_2, \ldots \ldots ., t_{n_1}\right)}{\partial t_j^2}>\left|\frac{\partial^2 T C_r^{I N D}\left(n_1, t_0, t_1, t_2, \ldots \ldots \ldots \ldots, t_n\right)}{\partial t_i \partial t_{j-1}}\right|+\left|\frac{\partial^2 T C_r^{I N D}\left(n_1, t_0, t_1, t_2, \ldots \ldots ., t_n\right)}{\partial t_j \partial t_{j+1}}\right| \text { for all } j=1,2, \ldots \ldots, n_1-1 .\end{aligned}$

Due to its diagonal properties, a Hessian matrix containing positive diagonal elements must also be positive definite. As a result, Eq. (9) has a unique solution which is the optimal replenishment interval as well as a global minimum value. If TCR is minimum, it must have a positive definite Hessian matrix r for a fixed n.

Proposition:

$t_{j+1} e^{\theta \mathrm{T}_{j+1}}<t_j\left(e^{\theta \mathrm{T}_j}+1\right)+\frac{1}{\theta} e^{\theta T_j}$

Proof: According to [13]:

$\mathrm{f}\left(\mathrm{t}_{j+1}\right)-\mathrm{f}\left(\mathrm{t}_j\right)<\frac{f^{\prime}\left(t_j\right)}{f\left(t_j\right)} \int_{t_j}^{t_{j+1}} f(u) d u$

We let $\mathrm{f}(\mathrm{t})=(\beta \mathrm{w}-\mu \mathrm{d}-(1-\mu) \mathrm{d} \alpha) \mathrm{t} e^{\theta\left(t-t_j\right)}$

Equation simplifies to

$\begin{gathered}{[\beta \mathrm{w}-\mu \mathrm{d}-(1-\mu) \mathrm{d} \alpha] \mathrm{t}_{j+1} e^{\theta\left(t_{j+1}-t_j\right)}-[\beta \mathrm{w}-\mu \mathrm{d}-(1-\mu) \mathrm{d} \alpha] \mathrm{t}_{\mathrm{j}}} \\ \left.\quad<\left(\theta+\frac{1}{t_j}\right) \int_{t_j}^{t_{j+1}} \beta w-\mu d-(1-\mu) d \alpha\right) t e^{\theta\left(t-t_j\right)} d t\end{gathered}$

Using Eq. (9) given by

$\begin{gathered}{[\beta \mathrm{w}-\mu \mathrm{d}-(1-\mu) \mathrm{d} \alpha] \mathrm{t}_{j+1} e^{\theta\left(t_{j+1}-t_j\right)}-[\beta \mathrm{w}-\mu \mathrm{d}-(1-\mu) \mathrm{d} \alpha] \mathrm{t}_{\mathrm{j}}}<\left(\theta+\frac{1}{t_j}\right)[\beta w-\mu d-(1-\mu) d \alpha] t_j e^{\theta\left(t_j-t_{j-1}\right)} \\ t_{j+1} e^{\theta \mathrm{T}_{j+1}}-t_j<\left(\theta+\frac{1}{t_j}\right) \frac{t_j e^{\theta T_j}}{\theta} \\ t_{j+1} e^{\theta \mathrm{T}_{j+1}}<\left(\theta+\frac{1}{t_j}\right) \frac{t_j e^{\theta T_j}}{\theta}+t_j \\ t_{j+1} e^{\theta \mathrm{T}_{j+1}}<e^{\theta T_j}+\frac{1}{\theta} e^{\theta T_j}+t_j\end{gathered}$

This leads to the conclusive result:

$t_{j+1} e^{\theta \mathrm{T}_{j+1}}<t_j\left(e^{\theta \mathrm{T}_j}+1\right)+\frac{1}{\theta} e^{\theta T_j}$

Lemma:

The monotonicity of t_j (where j = 1, 2, …, n - 2) is evident concerning the parameter t_{n - 1}. This lemma establishes the consistent increase of t_j concerning the penultimate time point in the planning horizon.

Proof: The equation is simplified using the relationship T_n = H - t_n-1 is constant if t_n-1 is known, as per [15] say.

This implies that T_n and t_n-1 are inversely related.

To prove that rate of change of TCR’s increase with T_n for j= n-1, n-2,…,3,2,1.

For j = n - 1, after differentiating Eq. (9) w.r.t. Tn, we get,

$\begin{gathered}\mathrm{TCR}=[\beta \mathrm{w}-\mu \mathrm{d}-(1-\mu) \mathrm{d} \alpha]\left[(-1) e^{\theta \mathrm{T}_{n-1}}+(H-\right.\left.\left.T_n\right) e^{\theta \mathrm{T}_{n-1}} \frac{\mathrm{dT}_{n-1}}{\mathrm{dT}_n}-t_n e^{\theta \mathrm{T}_n} \cdot \theta+e^{\theta \mathrm{T}_n} \cdot \theta\right]=0 \\ t_n e^{\theta \mathrm{T}_n} \cdot \theta=e^{\theta \mathrm{T}_n} \cdot \theta-e^{\theta \mathrm{T}_{n-1}}+\left(H-T_n\right) e^{\theta \mathrm{T}_{n-1}} \frac{\mathrm{dT}_{n-1}}{\mathrm{dT}_n}\end{gathered}$

Using preposition and Eq. (9) we get

$\begin{gathered}e^{\theta \mathrm{T}_n} \cdot \theta-e^{\theta \mathrm{T}_{n-1}}+\left(H-T_n\right) e^{\theta \mathrm{T}_{n-1}} \frac{\mathrm{dT}_{n-1}}{\mathrm{dT}}>\theta \mathrm{t}_{n-1}\left(e^{\theta \mathrm{T}_{n-1}}+\right.\text { 1) }+e^{\theta \mathrm{T}_{n-1}} \frac{d T_{n-1}}{d T_n}>0\end{gathered}$

Inequality concludes that

$\frac{d T_{n-1}}{d T_n}>0$

Indicating an increase in t_n-1 as T_n increases. A generalization is made, suggestion that this relationship holds for j= n-1, n-2,…,3,2,1 and can be reasonably extended.

Expanding on the established formulations, we can apply the subsequent optimization procedure to uncover the most advantageous values and outcomes. The methodology involves an iterative optimization process, commencing with the practical decision to set 'n' to 2 as the initial point. When n = 1, we assign t₀ = 0 and t₁ = 4, and then initiate the Mathematica program to solve for optimality. For n = 2, corresponding to the number of replenishment cycles, where t₀ = 0 and t₂ = 4, the value of t₁ is determined using an iterative method in the Mathematica program.

5.1 Methodology

The retailer should determine the most efficient way to schedule orders.

1. In this case, we will assign n a value of 2.
2. Calculate the unique optimal ordering interval using the nonlinear Eq. (6) system for a fixed n.
3. Using Eq. (6), determine the total cost of Retailers (n).
4. If TCR (n) is less than TCR (n − 1), increase n by 1, then return to Step (ii). Then stop because the algorithm has reached an optimal ordering policy.
5. Based on Eqs. (7) and (8), calculate TCS and Q.

5.2 Numerical example

Using the assumptions in Section 3, this numerical example shows how changes in the key parameter value W affect the optimal results.

EXAMPLE- Given $\alpha=\beta=5, \mu=0.40, \mathrm{C}=\$ 12 /$ unit, $\mathrm{d}=$ 120/unit, $\mathrm{Ss}=\$ 120$ /setup, $\mathrm{Sr}=\$ 90 /$ order, $\mathrm{hr}=\$ 2 /$ unit/year, $\theta=0.75, \mathrm{H}=4$-year, $\mathrm{d}=120$ units/year, $\mathrm{W}=10,20,30$ units/year, respectively.

The results are shown in Tables 1-2 because of the algorithm and its corresponding expressions.

Table 1. Total cost for retailer when

Table 2. Replenishment time for TCR, TCS, and Q

Maintaining a consistent value for 'n' ensures uniformity throughout the analysis, thereby facilitating a fair comparison across diverse iterations of the model. The provided table delineates the retailer's total cost for varying values of 'W' (wholesale price) and 'n' (replenishment cycles). The steadfastness in 'n' simplifies the assessment of how different wholesale prices influence the total cost over an identical number of replenishment cycles.

11.png

(a) n = 4

12.png

(b) n = 2

13.png

(c) n = 2

Figure 1. Optimal values of total cost of retailer and replenishment cycles

21.png

(a) n = 4

22.png

(b) n = 2

Figure 2. Increased order of optimal values of the replenishment cycles

The iterative process involves adjusting 'n' based on the comparison of total costs (TCR), aiming for convergence toward the optimal ordering policy. This iterative refinement is pivotal for attaining the most effective solution, allowing researchers to converge toward the configuration that minimizes total retailer costs. The algorithm's termination condition, highlighted in the table, dictates halting when the TCR for 'n' is less than TCR for 'n-1'.

Maintaining a consistent value for 'n' ensures uniformity throughout the analysis, thereby facilitating a fair comparison across diverse iterations of the model. The provided table delineates the retailer's total cost for varying values of 'W' (wholesale price) and 'n' (replenishment cycles). The steadfastness in 'n' simplifies the assessment of how different wholesale prices influence the total cost over an identical number of replenishment cycles.

The iterative process involves adjusting 'n' based on the comparison of total costs (TCR), aiming for convergence toward the optimal ordering policy. This iterative refinement is pivotal for attaining the most effective solution, allowing researchers to converge toward the configuration that minimizes total retailer costs. The algorithm's termination condition, highlighted in the table, dictates halting when the TCR for 'n' is less than TCR for 'n-1'.

For instance, when 'W' is 10 and 'n' is 2, the retailer's total cost amounts to 36978. With an incremental increase in 'n' from 2 to 6, the total cost fluctuates, culminating in its nadir at 'n' equals 4, registering a cost of 36881. This illustrates how the iterative process systematically refines 'n' to pinpoint the optimal ordering policy within defined constraints.

To visually comprehend these fluctuations, refer to Figure 1(a), which illustrates how values oscillate and reach their lowest point at 'n' equals 4. The same insights are conveyed for different 'W' values in both Table 1 and the Figure 1(b) and (c).

In Table 2 the intricate details of optimal outcomes in scenarios where compensation is factored into each optimal case. It can be rigorously established that the inequality t_j+1 - t_j < t_j - t_j-1 holds true, where j=1, 2,…,n−1. Here, t_j denotes the j^th replenishment time, with t₀=0 and t_n−1= H, where H is a non-negative integer.

Table 2 meticulously tabulates values for t_j (replenishment time), Total Cost of Replenishment (TCR), Total Cost of Setup (TCS), and Order Quantity (Q). This tabulation distinctly delineates how diverse W (wholesale prices) correlate with specific t_j values, thereby accurately characterizing optimal ordering policies for our model. For instance, when W is set at 10, t₁ assumes a value of 1.105, t₂ registers as 1.961, t₃ stands at 2.704, and t₄ reaches 4. Additionally, TCR manifests as 36681, TCS as 34370, and Q as 2866.

This illustrative example underscores the dynamic interplay of optimal replenishment times, costs, and order quantities across distinct W values. It underscores the adaptive nature of our model, optimizing parameters and enabling judicious decision-making to foster cost-effectiveness in the supply chain. Complementing the tabulated values in Table 2, our analysis extends to scrutinizing t_j values, symbolizing replenishment time, elucidated through Figure 2(a) and (b).

In these graphical representations, the discernible convexity of the curve, influenced by t_j fluctuations, serves as a visual indicator of optimal cycle points and cost minimization. This visual inspection of t_j values enriches our understanding of the nuanced shifts in our model's performance, empowering strategic decision-making to optimize the intricacies of the supply chain.