An Integrated Inventory-Routing and Working Capital Requirement Model for a Two-Echelon Supply Chain

The most recent literature review by Cuda et al. [11] show the rise of two-echelon distribution systems in various practical scenarios. A supply chain can be viewed as consisting of various stages, with transportation occurring between every consecutive pair of stages, each representing a level within the distribution network, commonly known as an echelon. In recent years, a significant amount of research papers has been published that concentrate on issues related to different variants two-echelon inventory routing problems [12].

For instance, Shirzadi et al. [13] propose an innovative model to optimize the performance of fresh agri-food distribution, integrating traditional costs and modern environmental considerations. Farias et al. [14] introduce a mathematical formulation and a branch-and-cut algorithm to solve a variant of the problem, accounting for indirect deliveries and routing decisions at both levels. Sanghikian et al. [15] employ a hybrid metaheuristic to solve a maritime routing and inventory management problem, integrating a linear mathematical model into their approach.

Guimarães et al. [16] investigate the implementation of the Vendor-Managed Inventory paradigm, wherein a vendor oversees both their own inventory and that of their customers. The research focuses on a system comprising a supplier, multiple warehouses, and a group of customers, with centralized decision-making at the supplier level. It considers various delivery scenarios, including direct deliveries and deliveries through intermediate locations. Mohamed et al. [17] present the two-echelon stochastic multi-period capacitated location-routing problem (2E-SM-CLRP) dealing with uncertain and time-varying demand and cost. This problem represents a hierarchical decision-making scenario with a temporal hierarchy between design and transportation decisions. The research suggests a logic-based Benders decomposition method to tackle this model, aiming to meet the demand for a solution approach that offers high-quality design solutions along with accurate evaluation, even if it requires more time to run. Finally, the authors present an exact method for a stochastic multi-period location-routing problem, providing valuable insights and solutions for this complex and evolving area of research.

Schenekemberg et al. [18] present a metaheuristic that integrates the branch-and-cut algorithm to address a modified version of the IRP, which includes fleet planning in a two-echelon logistics system. The challenge lies in navigating an intricate supply chain arrangement, with plants positioned centrally and tasked with overseeing inventory management and routing for both material acquisition and final product distribution. Simultaneously, they are responsible for making tactical and operational decisions regarding the fleet. The authors' analysis underscores that expenses associated with rentals, cleaning services, and vehicle returns constitute a substantial share of overall logistics costs. Rohmer et al. [19] introduce a perishable product two-echelon inventory-routing problem, where items are conveyed from a supplier to an intermediary warehouse, where potential storage takes place before being further distributed to customer locations using smaller vehicles. Incurred holding costs are associated with storing these products at the depot. The authors formulate the problem as a mixed integer linear program, resolving it through an adaptive large neighborhood search metaheuristic. The objective is to minimize the total transportation and holding costs.

Integrating Working Capital Requirement into the supply chain is a critical component of managing business operations. WCR, which encompasses the funds required to maintain appropriate inventory levels, manage accounts receivable, and manage supplier debts, plays an essential role in the financial and operational stability of a company. Several studies highlight this importance.

Bian et al. [20] addresse the two-level uncapacitated lot-sizing problem while considering the financing cost associated with the WCR. The lot-sizing problem involves determining the quantities to produce or order for multiple periods to minimize total costs while meeting capacity and demand constraints. Enqvist et al. [9] investigate how working capital management affects firm profitability across different business cycles, using evidence from Finland. Working capital management involves managing a company's short-term assets and liabilities to ensure it maintains sufficient liquidity to meet its operational needs.

Pfohl and Gomm [21] explore the concept of supply chain finance, which involves optimizing financial flows within supply chains. They explore the methods and tools available for streamlining financial processes along the supply chain, focusing on working capital management, reducing financial risks, and improving collaboration among supply chain partners. Singhania et al. [22] examined the relationship between working capital management and profitability, specifically focusing on Indian manufacturing companies. By analyzing empirical evidence, the article aims to shed light on how different aspects of working capital management, such as inventory management, accounts receivable, and accounts payable, impact the profitability of manufacturing firms in India.

From this literature review, it's clear that the financial dimension wasn't considered in the two echelons inventory routing problems, despite the fact that managing financial flows constitutes a vital aspect of supply chain management.

Based on the literature review, the originality of this study can be summarized as follows:

1) The majority of studies on IRPs are based on a distribution structure with a single echelon [20, 23]. However, this study focuses on a two-echelon IRP.

2) Our problem diverges from the majority of existing two-echelon problems in two primary aspects:

• Firstly, we account for inventory levels at the warehouses rather than at the customer locations, which differs from the approach taken in many other papers [3, 14, 24].
• Secondly, while the classic version of the Inventory Routing Problem focuses on a system with a single supplier [19], this study delves into a system with multiple suppliers, warehouses, and customers.

3) We are the first to present a WCR model tailored specifically to the 2E-IRP context. This enables us to accurately gauge the WCR throughout the distribution network.

In this section, we provide a formalized representation of the mathematical model addressing the synchronization of logistical and financial flows. Our model aims to integrate economic, operational, and environmental aspects, thereby achieving an optimized and balanced solution.

4.1 Model notation

This section shows the notation used in setting the 2E-IRP-WCR model.

4.1.1 Sets and indices

|D|: the number of elements in set D

S: set of suppliers

s: index for suppliers (s=1, 2, …, |S|)

W: set of warehouses

w: index for warehouse (w=1, 2, …, |W|)

C: set of customers

c: index for customers (c=1, 2, …, |C|)

4.1.2 Parameters

Dem $_c$ : demand of customer $\mathrm{c}$

Appro $_w$ : quantity supplied by warehouse $\mathrm{w}$

$\mathrm{Cap}_w$ : maximum capacity of warehouse $\mathrm{w}$

$\mathrm{Cp}_s$ : maximum capacity of supplier $\mathrm{s}$

$\mathrm{DSW}_{s, w}$ : distance between supplier $\mathrm{s}$ and warehouse $\mathrm{w}$

$\mathrm{DWC}_{w, c}$ : distance between warehouse $\mathrm{w}$ and customer $\mathrm{c}$

$I O_w$ : initial inventory of warehouse w

$C S_s$ : unit purchase cost from supplier $\mathrm{s}$

$C P_w:$ unit inventory cost in warehouse w

$C T$ : unit transportation cost

$r_c$ : credit period of customer $\mathrm{c}$

$L_s:$ credit period of supplier $\mathrm{s}$

$P_w:$ product inventory period in warehouse $\mathrm{w}$

$\beta$ : interest rate for financing WCR per period

$C a_w:$ turnover of warehouse $\mathrm{w}$

$M$ : a big number

$W C R D$ : working capital requirement in number of days

VAT: Value Added Tax

4.1.3 Decision variables

Continuous variables

$Z_{s, w}:$ quantity delivered from supplier $\mathrm{s}$ to warehouse $\mathrm{w}$

$Y_{w, c}$ : quantity delivered from warehouse $\mathrm{w}$ to customer $\mathrm{c}$

$I_w:$ inventory level in warehouse $\mathrm{w}$

Binary variables

$B_{s, w}$ : binary variable that takes the value 1 if the warehouse w is delivered by the supplier s, otherwise, it takes the value 0

$X_{w, c}$ : binary variable that takes the value 1 if the customer $\mathrm{c}$ is delivered by the warehouse w, otherwise, it takes the value 0

4.2 The mathematical model

This section delves into the innovative 2E-IRP-WCR mathematical model proposed and each model equation.

4.2.1 Objective function

Eq. (1) gives the total cost function that needs to be minimized. It comprises four terms. These cost terms are, in order, transportation costs, costs of purchasing, inventory holding costs, and financial costs of WCR.

$\begin{gathered}\text { Min total cost }=\sum_{s, w}\left(D S W_{s, w} \times B_{s, w} \times C T\right)+ \\ \sum_{w, c}\left(\mathrm{DWC}_{w, c} \times X_{w, c} \times C T\right)+\sum_{s, w}\left(C S_s \times\right. \\ \left.Z_{s, w}\right)+\sum_w\left(C P_w \times I_w \times P_w\right)+\sum_w \frac{C a_w}{365} \times W C R D \times \beta\end{gathered}$     (1)

where,

$\begin{aligned} W C R D=\sum_w \sum_c & \left(\frac{C a_w \times(1+V A T)}{C a_w} \times r_c\right) \\ & +\sum_w \sum_s\left(\frac{Z_{s, w} \times C S_s}{C a_w} \times P_w\right) \\ & -\sum_w \sum_s\left(\frac{Z_{s, w} \times C S_s(1+V A T)}{C a_w}\right. \\ & \left.\times L_s\right)\end{aligned}$     (2)

4.2.2 Constraints

• Demand Satisfaction Constraint

Eq. (3) ensures that for each client c, the total quantity delivered from warehouses w meets or exceeds the demand.

$\sum_w Y_{w, c} \geq \operatorname{Dem}_c \forall \mathrm{c} € \mathrm{C}$     (3)

• Supply Commitment Constraint

Eq. (4) expresses the supply commitment constraint. For each warehouse w, the total quantity supplied from supplier s should match the committed supply.

$\sum_w Z_{s, w}=\operatorname{Appro}_w \forall \mathrm{s} € \mathrm{~S}$    (4)

• Inventory Balance Constraint

Eq. (5) maintains a balance between warehouse and supplier inventory.

$I O_w+\operatorname{Appro}_w \geq \sum_c Y_{w, c} \forall \mathrm{w} € \mathrm{~W}$     (5)

• Warehouse Capacity Constraint

Eq. (6) ensures that the total delivery from warehouse w to customer c does not exceed the warehouse capacity.

$I_w \leq \mathrm{Cap}_w \forall \mathrm{w} € \mathrm{~W}$    (6)

$I_w$ is the dynamics of inventory updates. The constraint, given by Eq. (7), ensures that the inventory of warehouse w is updated considering initial stock, supplies from suppliers s, and customer delivery.

$I_w=I O_w+\sum_s Z_{s, w}-\sum_c Y_{w, c} \forall \mathrm{w} € \mathrm{~W}$    (7)

• Supplier Capacity Constraint

Eq. (8) ensures that the total delivery from supplier s to warehouse w does not exceed the supplier capacity.

$\sum_w Z_{s, w} \leq \mathrm{Cp}_s \forall \mathrm{s} € \mathrm{~S}$     (8)

• Binary Variable Logical Constraint

To ensure a logical linkage between binary variables $Z_{s, w}$  and $B_{s, w}$, the constraint (9) restricts $Z_{s, w}$  to be zero if $B_{s, w}$  is zero.

$Z_{s, w} \leq M \times B_{s, w}$     (9)

To ensure a logical linkage between binary variables $Y_{w, c}$ and $X_{w, c}$, the constraint (10) restricts $Y_{w, c}$ to be zero if $X_{w, c}$ is zero

$Y_{w, c} \leq M \times X_{w, c}$    (10)