Scenario-Based Optimization of Supply Chain Performance under Demand Uncertainty

Abstract

This study presents a comprehensive supply chain performance optimization model that addresses the trade-off between supply chain cost and customer service level in the distribution network. The model incorporates both deterministic and scenario-based approaches, allowing for a more realistic representation of supply chain operations. The model is applied to a case company operating in the pharmaceutical supply chain in Ethiopia. The goal is to improve the company’s supply chain performance by optimizing various factors such as establishment costs, handling costs, transportation costs, and demand satisfaction. The study considers both financial measures (supply chain cost) and non-financial measures (customer service level) to evaluate the performance of the supply chain. The results of the study demonstrate the effectiveness of the proposed model in identifying the optimal trade-off between supply chain costs and customer service levels. By comparing the results of the model with the current situation of the case company, it is determined that the company can achieve significant cost reductions of up to 25.26% while still meeting customer demands. The model also takes into account the uncertainty in demand, providing more realistic recommendations for distribution center locations, transportation planning, and order fulfillment. The implications of using this optimization model include the potential for cost savings, improved decision-making, enhanced customer satisfaction, and, ultimately, a more successful supply chain. However, the study has some limitations, including a need for further research on other objectives and considerations, such as environmental impacts and disruptions, which could be addressed in future research directions.

1. Introduction

Supply chain management (SCM) involves managing a network of interconnected businesses, optimizing product flows, and reducing transportation costs [1,2]. It encompasses various activities, from material acquisition to the distribution of final products [1]. To ensure successful operations, it is crucial to measure supply chain performance and monitor financial indicators [3]. Distribution networks play a critical role in supply chain performance, as they directly impact costs and customer satisfaction [4]. As a result, companies need a well-designed distribution network to succeed. Designing an effective supply chain network involves determining facility locations and capacities, sourcing demand, and selecting transportation modes to ensure cost-efficient customer service [2,5,6,7]. A lack of an adequately designed supply chain network is a major problem for most companies, especially those in developing countries [8]. As a result, their supply chain performance is low, from different performance measurement perspectives [8].

The pharmaceutical industry relies heavily on supply chain management (SCM) to transport goods, services, and information from the source to the consumer. In the pharmaceutical industry, an efficient supply chain network is vital for improving performance and meeting market demand consistently [9,10]. A pharmaceutical supply chain typically consists of primary and secondary manufacturers, main and local distribution centers (DCs), and destination zones such as pharmacies and hospitals (see Figure 1) [9,11], where each level or echelon of the supply chain may comprise numerous facilities. The complexity of the supply chain emerges from the numerous echelons and facilities within each echelon [12,13,14]. Consequently, selecting appropriate performance measures becomes crucial for effective supply chain analysis, considering the inherent complexity of the pharmaceutical supply chain [4].

The pharmaceutical supply chain is susceptible to various risks, resulting in resource wastage and shortages of medication [16]. These shortages have a significant impact on a large portion of the global population, underscoring the importance of efficient healthcare supply chain management [17]. Developing countries such as Ethiopia face specific challenges related to the efficiency and reliability of their supply chains, leading to inadequate access to medicine [8].

The Ethiopian pharmaceutical supply chain suffers from poor medicine delivery performance, attributed to demand fluctuations, distribution problems, inventory problems, and organizational structure-related factors [18,19]. Furthermore, Ethiopia’s pharmaceutical supply chain faces constraints due to limited local manufacturing capabilities [20,21], leading to heavy reliance on foreign manufacturers and incurring significant foreign currency expenditures. Moreover, the distribution network’s poor design results in excessive costs and inefficient delivery to end users, as evidenced by increasing expenditures over the past 11 years [15,20,21] (see Figure 2).

To address these challenges, this study aims to develop an optimization model that incorporates supply chain costs, customer service levels, and demand uncertainties. By considering both financial and non-financial perspectives, the study seeks to optimize supply chain costs while improving customer service levels. The model is validated using data from the Ethiopian Pharmaceutical Supply Agency (EPSA), a major pharmaceutical distributor in Ethiopia. Hence, the study aims to achieve the following objectives:

• Identify relevant supply chain performance measures to be considered;
• Develop a supply chain performance optimization model;
• Implement the model in a real case study.

The remainder of the article is structured as follows: Section 2 discusses the study’s contribution, Section 3 provides a literature review, Section 4 discusses supply chain performance measures and metrics, Section 5 presents the optimization models, Section 6 covers the model’s implementation, and Section 7 concludes the article.

2. Our Contributions

The study makes the following key contributions to optimizing supply chain performance under demand uncertainty:

• Financial and non-financial measures of performance (supply chain cost and customer service level) are incorporated into the optimization model (Section 4);
• A bi-objective scenario-based optimization model is developed, considering uncertainty handling approaches (Section 5);
• The model is implemented using real data from a company in a developing country, demonstrating its effectiveness (Section 6).

3. Literature Review

This review of the literature encompasses a wide range of articles focusing on the optimization of supply chain networks. Mousazadeh et al. [11] addressed the pharmaceutical network design problem with a bi-objective model that aims to minimize total cost and unmet customer demand. Fattahi et al. [22] developed a supply chain network model that takes into consideration customer sensitivity to delivery lead times, which is crucial for meeting customer expectations. Tsiakis et al. [23] tackled the design of multi-commodity, multi-echelon supply chain networks, considering predetermined locations of production facilities and customer zones. Vanteddu and Nicholls [24] proposed linear programming-based models for supply chain network design and materials transformation, providing insights into efficient resource allocation and material flow optimization.

Georgiadis et al. [25] presented an optimization model for supply chain networks operating under time-varying demand uncertainty, enabling better decision-making in dynamic and uncertain environments. Manzini and Gebennini [26] focused on location–allocation problems in the context of supply chain network design, helping organizations determine the optimal placement of facilities to minimize costs and improve efficiency. Nasiri et al. [27] developed an integrated production and supply chain distribution network model considering demand uncertainty, allowing organizations to optimize production and distribution strategies to meet fluctuating demand patterns. Camacho-Vallejo et al. [28] proposed a bi-level mathematical model for optimizing distribution costs and operations costs in manufacturing plants, facilitating cost-effective decision-making in supply chain network design.

Ghahremani-Nahr and Ghaderi [29] presented a multi-objective model for a resilient lean supply chain network, incorporating economic, social, and environmental aspects. This model enables organizations to make strategic decisions that balance efficiency, sustainability, and resilience in the supply chain. Fragoso and Figueira [30] introduced a multi-objective model that considers economic, social, and environmental dimensions, providing a holistic framework for decision-making that accounts for multiple stakeholder interests. Goodarzian and Fakhrzad [31] suggested a multi-objective model for a citrus supply chain, considering economic and environmental objectives. This model helps optimize supply chain design to maximize profit while minimizing environmental impacts.

Yildiz et al. [32] formulated a dual-objective nonlinear program to optimize a reliable supply chain network, investigating the trade-off between cost and reliability. Margolis et al. [33] developed a multi-objective resilient supply chain network, considering the network cost and connectivity to enhance the resilience of the supply chain. Lotfi et al. [34] considered robustness, risk awareness, resiliency, and sustainability in a closed-loop supply chain network, providing insights into designing robust and sustainable supply chains. Taleizadeh et al. [35] proposed a bi-level mixed integer programming model that captures the resiliency of two supply chains, enabling organizations to make decisions that enhance the resilience of their supply networks.

Wang et al. [36] designed a supply chain network considering the competition, multiple periods, products, and layers, offering a comprehensive approach to supply chain optimization in a competitive environment. Mohammadi Bidhandi et al. [37] addressed uncertainty in a supply chain network optimization model, comparing the performance of different solution approaches to account for uncertainty in decision-making. Arabsheybani and Arshadi Khasmeh [38] incorporated uncertainty and resiliency in a mathematical model, allowing organizations to consider various sources of uncertainty and improve the robustness of their supply chains. Schildbach and Morari [39] proposed a scenario-based model predictive control framework for optimizing supply chain management, considering uncertainties and disruptions. Their article provides insights into scenario-based modeling techniques and includes practical case studies. While further validation and implementation insights would enhance the research, they offer valuable guidance for researchers and practitioners in advancing supply chain management strategies.

In the context of uncertain supply chain environments, Tan et al. [40] considered hybrid uncertainties and developed a model that accounts for both random and fuzzy factors affecting supply chain performance. This comprehensive approach allows organizations to make robust decisions in the face of ambiguity. Additionally, Chatzikontidou et al. [41] addressed the design of reliable supply chain networks under demand uncertainty, aiming to minimize costs while ensuring reliable product availability to meet customer demand, thus enhancing supply chain resilience and customer satisfaction.

Emergency supply chain networks have also received attention. Fazli-Khalaf et al. [42] designed emergency blood supply chain networks, optimizing the delivery of blood products during critical situations. Khalilpourazari and Hashemi Doulabi [43] proposed a model for emergency supply chain network design, focusing on effectively responding to emergency situations such as natural disasters or pandemics, considering factors that included demand surge and time sensitivity. Similarly, Ghasemi et al. [44] focused on relief chain design for earthquake victims, aiming to minimize response time and cost while maximizing the coverage and effectiveness of relief operations.

Some studies have focused on optimizing pharmaceutical supply chain networks. Goodarzian et al. [45] developed a model considering the unique characteristics of the pharmaceutical industry, such as shelf life and regulations. Delfani et al. [46] proposed a model considering demand uncertainty, production capacity, and transportation costs. These models contribute to efficient and responsive supply chain design. Uthayakumar and Priyan [47] provide valuable insights into optimizing supply chain and inventory management in the pharmaceutical industry, emphasizing efficient inventory practices and discussing optimization techniques. USAID [48] conducted a network analysis of the Ethiopian Pharmaceuticals Supply Agency, identifying challenges in distribution and proposing strategies for improvement. The article provides valuable insights into pharmaceutical supply chain management in Ethiopia, but more elaborate considerations of important parameters would enhance its practical applicability.

Lastly, environmental sustainability in pharmaceutical supply chains has been investigated. Ahlaqqach et al. [49] developed a closed-loop supply chain network model for pharmaceutical products that incorporates the recovery and recycling of packaging materials, promoting sustainability. Diaz et al. [50] conducted a simulation-based analysis of the global pharmaceutical supply chain, exploring inventory management, transportation, and demand variability to identify areas for improvement and enhance efficiency and resilience.

Based on a literature review, it is evident that significant efforts have been made to address supply chain network optimization problems. Numerous custom solution algorithms have been developed, rivaling large commercial software solutions. However, many of these studies lack practical applicability and primarily rely on hypothetical scenarios [24,28,30,31,32,36,46,49]. Some of them neglect real-world factors, such as supply chain uncertainties, and consider only limited parameters [28,42]. Additionally, only a few studies have been applied to actual case studies [41,43,44].

While some articles focus on single-objective optimization models [22,26,27,28,37], the current study develops a bi-objective supply chain optimization model that incorporates both financial and non-financial objectives. This study also addresses demand uncertainty using a scenario-based optimization approach. Moreover, it demonstrates the practical application of the model by implementing it in a pharmaceutical supply chain in the context of a developing country, utilizing real-world data from an example company.

4. Supply Chain Performance Measures and Metrics

Chae [51] discusses the significance of performance measurement factors in supply chain management, acting as vital feedback mechanisms that bridge planning and execution. These factors assist in identifying potential issues and areas for improvement. However, developing practical guidelines for performance measurement factors remains a challenge, due to various factors such as the lack of supply chain orientation and the complexity of capturing metrics across multiple companies [15,52]. Altiparmak et al. [53] emphasized the establishment of appropriate performance measures to evaluate system efficiency, compare alternatives, and guide the design of new systems. They categorized performance measures into qualitative and quantitative ones, addressing objectives related to cost and customer responsiveness.

On the other hand, Gunasekaran et al. [52] and Bhagwat and Sharma [54] proposed a comprehensive supply chain performance measurement framework encompassing strategic, tactical, and operational decision levels, as well as financial and non-financial measures. Their frameworks enable companies to evaluate financial performance while considering aspects such as customer satisfaction. These articles collectively underscore the significance of performance measurement and offer frameworks for optimizing supply chain networks.

For the supply chain performance optimization in this study, a supply chain performance measurement framework, as proposed by Gunasekaran et al. [52] and Bhagwat and Sharma [54], is utilized. Financial measures include fixed establishment costs at the strategic decision level, transportation costs at the tactical decision level, and material handling costs at the operational decision level. As a non-financial indicator of supply chain performance, the customer service level is considered at the strategic decision level.

5. Material and Methods

This study aims to optimize a supply chain network that can efficiently deliver goods and services from suppliers to customers. The network design involves strategic choices of facility locations, supplier selection, production allocation, and warehouse management [25]. The optimization model is formulated using deterministic and scenario-based methods for a three-tier supply chain structure. The model considers both cost reduction and customer satisfaction objectives. The scenario-based approach accounts for demand uncertainty and is also extended to incorporate variability in distribution patterns. This section describes the model formulation, which defines indexes, parameters, and variables, and presents the deterministic and scenario-based models. It also explains how to handle uncertain demands using a scenario-based approach.

5.1. Indexes, Parameters, and Variables

Notations used in the basic model formulation are described as index sets, parameters, and decision variables (continuous and binary), as shown in

Table 1. Notation; index sets, parameters, and decision variables.

Index Sets
Notation	Description
I	Set of products indexed by i;
J	Set of candidate locations for local distribution center indexed by j;
K	Set of customer zones indexed by k;
S	Set of scenarios indexed by s.
Parameters
$F_j$	Fixed cost of establishing local distribution center j;
$c_{ij}$	Per-unit cost of shipment of product i from main distribution center to local distribution center j;
$b_{ik}$	Per-unit cost of shipping of product i from main distribution center to customer zone k;
$a_{ik}$	Additional cost if a product i is delivered directly from main distribution center to customer zone k;
$C_{ij}^H$	Unit handling cost of product i at local distribution center j;
$g_{ijk}$	Per-unit cost of shipping product i from local distribution center j to customer zone k;
$d_{iks}$	Demand for product i from customer zone k in scenario s;
${\mathsf{\Omega }}_j^{max}$	The maximum storage capacity available at local distribution center j;
${\mathsf{\Omega }}_j^{min}$	The minimum storage capacity allowed at local distribution center j;
$f_j^{max}$	The maximum flow amount between main distribution and local distribution center j;
$L_{jk}^{max}$	The maximum flow amount between local distribution center j and customer zone k;
$V_k^{max}$	The maximum flow amount between the main distribution center and customer zone k;
${\rho }_s$	Probability of occurrence of scenario s.
Decision Variables
$f_{ij}^s$	Quantity of product i transported from main distribution center to local distribution center j in scenario s;
${\mathsf{\Omega }}_j$	Storage capacity of the local distribution center j;
$L_{ijk}^s$	Quantity of product i transported from local distribution center j to customer zone k in scenario s;
$V_{ik}^s$	Quantity of product i shipped from main distribution center to customer zone k in scenario s.
Binary Variables
$Y_j$	Is a binary variable, which is 1 if a facility j is to be located, and 0 otherwise;
$X_j$	Is a binary variable, showing the link between the main distribution center and local distribution center j and 1 if the link is created, and 0 otherwise;
$Z_{jk}$	Is a binary variable, showing the link between local distribution center j and customer zone k; it is 1 if link is to be created, and 0 otherwise;
$M_k$	Is a binary variable, showing a link between the main distribution center and customer zone; it is 1 if the link is to be created, and 0 otherwise.

.

5.2. Deterministic Model

This section presents a deterministic model. The model aims to minimize supply chain cost and maximize customer service level, as illustrated in Equations (1) and (2):

$$\begin{array}{cc}\hfill & {\mathrm{Min}}_{w1}=\sum _j{F}_j{Y}_j+\sum _{i,j}{C}_{ij}^H{f}_{ij}+\sum _{i,j}{c}_{ij}{f}_{ij}+\sum _{i,j,k}{g}_{ijk}{L}_{ijk}+\sum _{i,k}\left(b_{ik}+a_{ik}\right)V_{ik}\hfill \end{array}$$

(1)

$${\mathrm{Min}}_{w2}={\mathrm{Max}}_{i,k}\left\{d_{ik}-(\sum _{i,j,k}{L}_{ijk}+\sum _{i,k}{V}_{ik})\right\}$$

(2)

subject to the following constraints:

$$\begin{array}{cc}\hfill & X_j\le Y_j,\forall j\in J\hfill \end{array}$$

(3a)

$$\begin{array}{cc}\hfill & Z_{jk}\le Y_j,\forall j\in J,\forall k\in K\hfill \end{array}$$

(3b)

$$\begin{array}{cc}\hfill & \sum _j{Z}_{jk}+M_k=1,\forall k\in K\hfill \end{array}$$

(3c)

$$\begin{array}{cc}\hfill & \sum _k{L}_{ijk}\le f_{ij},\forall i\in I,\forall j\in J\hfill \end{array}$$

(3d)

$$\begin{array}{cc}\hfill & \sum _j{L}_{ijk}+\sum V_{ik}\ge d_{ik},\forall i\in I,\forall k\in K\hfill \end{array}$$

(3e)

$$\begin{array}{cc}\hfill & {\mathsf{\Omega }}_j^{min}{Y}_j\le {\mathsf{\Omega }}_j\le {\mathsf{\Omega }}_j^{max}{Y}_j,\forall j\in J\hfill \end{array}$$

(3f)

$$\begin{array}{cc}\hfill & \sum _i{f}_{ij}\le {\mathsf{\Omega }}_j{Y}_j,\forall j\in J\hfill \end{array}$$

(3g)

$$\begin{array}{cc}\hfill & f_{ij}\le f_j^{max}{X}_j,\forall i\in I,\forall j\in J\hfill \end{array}$$

(3h)

$$\begin{array}{cc}\hfill & L_{ijk}\le L_{jk}^{max}{Z}_{jk},\forall i\in I,\forall j\in J,\forall k\in K\hfill \end{array}$$

(3i)

$$\begin{array}{cc}\hfill & V_{ik}\le V_k^{max}{M}_k,\forall i\in I,\forall k\in K\hfill \end{array}$$

(3j)

$$\begin{array}{cc}\hfill & Y_j,X_j,Z_{jk},M_k=\{0,1\},\forall j\in J,\forall k\in K\hfill \end{array}$$

(3k)

$$\begin{array}{cc}\hfill & {\mathsf{\Omega }}_j,f_{ij},L_{ijk},V_{ik}\ge 0,\forall i\in I,\forall j\in J,\forall k\in K\hfill \end{array}$$

(3l)

The first objective in Equation (1), representing supply chain cost, can be divided into parts to illustrate different cost components. The first part represents the fixed establishment cost, the second part represents the material handling cost at the local distribution center, the third part represents the transportation cost from the main distribution center to the local distribution centers, and the fourth part represents the cost of transportation from the local distribution centers to the customer zones. The last component considers the cost of direct delivery from the main distribution center to the customer zones, which includes both transportation costs and additional expenses associated with direct delivery.

The second objective function, Equation (2), represents the maximization of customer service level in terms of minimizing the maximum unmet demand for each product i from the customer zones.

On the other hand, constraints (3a) and (3b) establish the connection between the main distribution centers and the local distribution centers, as well as between the local distribution centers and the customer zones, respectively. These constraints ensure that the links exist only if the corresponding local distribution centers are established. Constraint (3c) ensures that each customer zone is assigned to a single source, either from the main distribution center or from a local distribution center. Constraint (3d) maintains the balance of material flow to and from the local distribution centers. Constraint (3e) guarantees that the demand from the customer zones is fulfilled by ensuring that the combined delivery from the local and main distribution centers is at least equal to the demand. The capacity of each local distribution center j is restricted by constraint (3f). The capacity should fall within the predetermined minimum and maximum values. Constraint (3g) ensures that the amount of supply to local distribution center j does not exceed its capacity. Constraints (3h)–(3j) impose limits on the flow quantity within the distribution network, preventing it from exceeding the predetermined maximum values. Constraint (3k) restricts the binary variables to either 1 or 0, and constraint (3l) enforces the non-negativity of variables.

Lemma 1.

The constraint in Equation (3g) can be simplified. The overall optimization model is classified as a Mixed Integer Quadratically Constrained Programming (MIQCP) model due to the constraint’s right-hand side (RHS). However, with a minor simplification, the model can be transformed into a Mixed Integer Linear Programming (MILP) model. Equation (3f) specifies that the capacity of local distribution centers must fall within a predetermined range of minimum and maximum values:

$${\mathsf{\Omega }}_j^{min}{Y}_j\le {\mathsf{\Omega }}_j\le {\mathsf{\Omega }}_j^{max}{Y}_j,\forall j\in J$$

and in Equation (3g), the constraint could have been a quadratic:

$$\sum _i{f}_{ij}\le {\mathsf{\Omega }}_j{Y}_j,\forall j\in J$$

However, due to the restriction on the previous Equation (3f), the RHS of this constraint can be simplified to ${\mathsf{\Omega }}_j$.

Proof. 

Let us consider the constraint in Equation (3g): ${\sum }_i{f}_{ij}\le {\mathsf{\Omega }}_j{Y}_j$.

We know that $Y_j$ is a binary variable, which means it can only take the values 1 or 0. If $Y_j=1$, then the constraint becomes ${\sum }_i{f}_{ij}\le {\mathsf{\Omega }}_j$. In this case, the RHS of the constraint is simply ${\mathsf{\Omega }}_j$, and the constraint remains linear. If $Y_j=0$, then the constraint becomes ${\sum }_i{f}_{ij}\le 0$, in which case, the RHS of the constraint is 0. However, it is already restricted in Equation (3f) that, if $Y_j$ is zero, ${\mathsf{\Omega }}_j$ is also zero. Therefore, regardless of the value of $Y_j$, the RHS of the constraint remains the same, either ${\mathsf{\Omega }}_j$ or 0.

Hence, the constraint in Equation (3g) can be simplified to a linear equation (Equation (4)) and the overall model can be considered a Mixed Integer Linear Programming (MILP) model:

$$\sum _i{f}_{ij}\le {\mathsf{\Omega }}_j,\forall j\in J$$

(4)

□

5.3. Model Considering Demand Uncertainty

Supply chain networks experience demand fluctuations due to changing consumption patterns and product life cycles [25]. Steady-state models are inadequate for capturing long-term fluctuations and systematic demand variations. Handling uncertainty in supply chain networks presents challenges in modeling and solution selection [55]. Uncertainty mainly revolves around product demands, prices, costs, and resource availability [25]. Two common approaches to handling uncertainty are the scenario-based approach, which divides the planning horizon into stages and uses a small number of scenarios, and the probabilistic approach, which employs the stochastic programming approach [56]. Factors affecting supply chain network operation can be categorized as short-term fluctuations or long-term trends. Stochastic programs and recourse programs deal with uncertain data and enable decisions before and after uncertainty realization. In the two-stage stochastic optimization approach, which is one of the stochastic programming approaches, uncertain parameters are treated as random variables, and the objective function combines first-stage and expected second-stage performances.

Considering uncertainty is crucial for effective decision-making in supply chain networks, various approaches are available to optimize performance under uncertain conditions. In this study, a scenario-based optimization approach is employed. Demand is considered uncertain, and this uncertainty is captured by considering scenarios that can express the variability in demand. The optimization model is formulated in two stages: the model when demand is uncertain and the distribution pattern is fixed, and the model when both demand and distribution pattern are variable.

5.3.1. Model When Demand Is Uncertain and Distribution Pattern Is Fixed

In the case of a fixed distribution pattern, the interconnections between nodes in the distribution network remain constant under all scenarios of demand uncertainty, and the bi-objective scenario-based model is as follows:

$${\mathrm{Min}}_{w1}=\sum _j{F}_j{Y}_j+\sum _{s\in S}{\rho }_s\left(\sum _{i,j,s}\left(C_{ij}^H+c_{ij}\right)f_{ijs}+\sum _{i,j,k,s}{g}_{ijk}{L}_{ijks}+\sum _{i,k,s}\left(b_{ik}+a_{ik}\right)V_{iks}\right)$$

(5)

$${\mathrm{Min}}_{w2}={\max }_{i,k,s}\left\{d_{iks}-(\sum _{i,j,k,s}{L}_{ijks}+\sum _{i,k,s}{V}_{iks})\right\}$$

(6)

subject to the following constraints:

$$\begin{array}{cc}\hfill & X_j\le Y_j,\forall j\in J\hfill \end{array}$$

(7a)

$$\begin{array}{cc}\hfill & Z_{jk}\le Y_j\forall j\in J,\forall k\in K\hfill \end{array}$$

(7b)

$$\begin{array}{cc}\hfill & \sum _j{Z}_{jk}+M_k=1,\forall k\in K\hfill \end{array}$$

(7c)

$$\begin{array}{cc}\hfill & \sum _j{L}_{ijks}+\sum V_{iks}\ge d_{iks}\forall i\in I,\forall k\in K,\forall s\in S\hfill \end{array}$$

(7d)

$$\begin{array}{cc}\hfill & {\mathsf{\Omega }}_j^{min}{Y}_j\le {\mathsf{\Omega }}_j\le {\mathsf{\Omega }}_j^{max}{Y}_j,\forall j\in J\hfill \end{array}$$

(7e)

$$\begin{array}{cc}\hfill & \sum _i{f}_{ijs}\le {\mathsf{\Omega }}_j,\forall j\in J,\forall s\in S\hfill \end{array}$$

(7f)

$$\begin{array}{cc}\hfill & \sum _k{L}_{ijks}\le f_{ijs},\forall i\in I,\forall j\in J,\forall s\in S\hfill \end{array}$$

(7g)

$$\begin{array}{cc}\hfill & f_{ijs}\le f_j^{max}{X}_j,\forall i\in I,\forall j\in J,\forall s\in S\hfill \end{array}$$

(7h)

$$\begin{array}{cc}\hfill & L_{ijks}\le L_{jk}^{max}{Z}_{jk},\forall i\in I,\forall j\in J,\forall k\in K,\forall s\in S\hfill \end{array}$$

(7i)

$$\begin{array}{cc}\hfill & V_{iks}\le V_k^{max}{M}_k,\forall i\in I,\forall k\in K,\forall s\in S\hfill \end{array}$$

(7j)

$$\begin{array}{cc}\hfill & Y_j,X_j,Z_{jk},M_k\in \{0,1\},\forall j\in J,k\in K\hfill \end{array}$$

(7k)

$$\begin{array}{cc}\hfill & \sum _s{\rho }_s=1,\forall s\in S\hfill \end{array}$$

(7l)

$$\begin{array}{cc}\hfill & {\mathsf{\Omega }}_j,f_{ijs},L_{ijks},V_{iks},{\rho }_s\ge 0,\forall i\in I,\forall j\in J,\forall k\in K,\forall s\in S\hfill \end{array}$$

(7m)

5.3.2. Model When Both Demand and Distribution Pattern Vary

The previous model with a fixed distribution pattern is extended to consider a variable distribution pattern that can be adjusted based on the realization of uncertain parameters. In this case, the interconnections between nodes in the distribution network are not predetermined but depend on the specific values of the uncertain parameters.

The objective functions of the model remain similar to Equations (5) and (6), aiming to optimize the performance of the supply chain network. However, the binary variables representing the linkages between nodes now incorporate probability scenarios. The following changes are made to the previous constraints:

$$\begin{array}{cc}\hfill & X_{js}\le Y_j,\forall j\in J,s\in S\hfill \end{array}$$

(8a)

$$\begin{array}{cc}\hfill & Z_{jks}\le Y_j\forall j\in J,k\in K,s\in S\hfill \end{array}$$

(8b)

$$\begin{array}{cc}\hfill & \sum _j{Z}_{jks}+M_{ks}=1,\forall k\in K,s\in S\hfill \end{array}$$

(8c)

$$\begin{array}{cc}\hfill & f_{ijs}\le f_j^{max}{X}_{js},\forall i\in I,\forall j\in J,\forall s\in S\hfill \end{array}$$

(8d)

$$\begin{array}{cc}\hfill & L_{ijks}\le L_{jk}^{max}{Z}_{jks},\forall i\in I,\forall j\in J,\forall k\in K,\forall s\in S\hfill \end{array}$$

(8e)

$$\begin{array}{cc}\hfill & V_{iks}\le V_k^{max}{M}_{ks},\forall i\in I,\forall k\in K,\forall s\in S\hfill \end{array}$$

(8f)

By incorporating the variability in both demand and distribution pattern, this model allows for a more realistic representation of supply chain operations. It enables decision-makers to assess the robustness of their supply chain designs and identify strategies that can adapt to different scenarios. This approach helps companies improve their agility and responsiveness to changes in market conditions and uncertainties.

6. Implementation

6.1. Problem Description

In this study, the focus is on improving the distribution aspect of a representative company operating in the pharmaceutical supply chain in Ethiopia. The company faces challenges due to limited funding and ineffective management practices. It currently operates nineteen local distribution centers, including two branches in Addis Ababa (

Table 2. List of existing local distribution centers of the case company.

S.No.	Center Name	S.No.	Center Name	S.No.	Center Name	S.No.	Center Name	S.No.	Center Name
1	Mekelle	5	Gondar	9	Assosa	13	Negele Borena	17	Addis Ababa 1 *
2	Shire	6	Jimma	10	Gambella	14	Dire Dawa	18	Addis Ababa 2 *
3	Semera	7	Nekemte	11	Hawassa	15	Jigjiga	19	Adama
4	Bahir Dar	8	Dessie	12	Arba Minch	16	Qebri Dehar

* In this study, local distribution centers Addis Ababa 1 and 2 are merged to form one distribution center: Addis Ababa.

). To simplify the analysis, data from these similar branches in Addis Ababa are combined, treating them as from a single local distribution center. Additionally, the study considers a potential location called Qebri Dehar, although data availability for it is limited for the existing network.

The optimization model is applied to the company’s three-tier supply chain, which includes 1 main distribution center, 18 local distribution centers, and 916 customer zones (districts), as illustrated in Figure 3. The demand from healthcare facilities in the districts is aggregated into a single demand point. The geographical map of the locations of these demand points, along with the local distribution centers and the main distribution center, is shown in Figure 4. Geospatial data from [57,58] were used for this illustration.

Data analysis reveals that the case company’s 19 local distribution centers experience overlaps in delivery areas, potentially due to the increased number and unplanned locations of centers, as well as the poor distribution plan.

Table 3. List of local distribution centers with their respective customer zones.

Code	Local Distribution Centers (LDCs)	Number of Customer Zones Assigned
LDC1	Mekelle	58
LDC2	Shire	21
LDC3	Semera	35
LDC4	Bahir Dar	84
LDC5	Gondar	36
LDC6	Jimma	81
LDC7	Nekemte	83
LDC8	Dessie	67
LDC9	Assosa	28
LDC10	Gambella	13
LDC11	Hawassa	150
LDC12	Arba Minch	45
LDC13	Negele Borena	76
LDC14	Dire Dawa	114
LDC15	Jigjiga	66
LDC16	Qebri Dehar	-
LDC17	Addis Ababa	154
LDC18	Adama	72
Total number of customer zones		1029

Source: Obtained from analysis of the case company’s data.

shows that, despite having 916 ustomer zones, the company’s centers deliver to approximately 1029 customer zones, resulting in an overlap of 113 zones. This overlap, known as the cannibalization of existing coverage, is also highlighted in a report by the USAID Global Health Supply Chain Program—Procurement and Supply Management (GHSC—PSM) project team [48]. While overlap can be beneficial in urgent delivery scenarios, it leads to inefficient resource utilization in year-round deliveries, negatively impacting the economy.

The company distributes three types of products: pharmaceutical products, chemicals and reagents, and medical equipment. However, this analysis specifically focuses on pharmaceutical products. Out of the company’s extensive range of over 2000 types of pharmaceutical products, only the 47 products consistently supplied each month are considered in the analysis.

The study also examines the distribution cost breakdown of the case company, as shown in

Table 4. Existing cost components of the case company (in relative money units).

Cost Components	Cost
Supply cost from MDC to LDCs	1.03 × 10${}^8$
Delivery cost from LDCs to CZs	5.24 × 10${}^7$
Handling cost at LDCs	3.10 × 10${}^7$
Fixed establishment cost of LDCs	2.37 × 10${}^8$
Total cost	4.23 × 10${}^8$

. The costs are classified into four main components: supply cost from the main distribution center to local distribution centers, material handling costs at the local distribution centers, delivery costs from the local distribution centers to customer zones, and fixed establishment costs. The cost analyses focus on the selected products, and the same are used to optimize supply chain performance in subsequent chapters, enabling meaningful comparisons. The specific parameters used in the modeling process are further discussed below.

Main distribution center (MDC): The main distribution center in Addis Ababa serves as the central hub for the pharmaceutical distribution network. It supplies products to local distribution centers and, for this study, it is considered also to occasionally deliver directly to customer zones. Direct deliveries are assumed to incur an additional cost denoted as $a_{ik}$, which covers specific handling operations. The cost is assumed to be constant for all products and depends on the quantity being delivered directly.

Local distribution centers (LDCs): All products in the distribution network pass through local distribution centers for temporary storage and further distribution. From a pool of 18 possible locations, the centers are selected. Each center has a maximum capacity (${\mathsf{\Omega }}_j^{max}$) and a minimum capacity of zero (${\mathsf{\Omega }}_j^{min}=0$). The maximum capacities are estimated based on existing capacity and normalized with respect to overall demand (

Table 5. Maximum storage capacity and fixed establishment cost of each local distribution center j.

Local Distribution Center (j)	Maximum Capacity, ${\mathsf{\Omega }}_{\mathit{j}}^{\mathit{max}}$ (Kg)	Fixed Establishment Cost, ${\mathit{F}}_{\mathit{j}}$ (Millions) ${}^{\mathit{a}}$	Local Distribution Center (j)	Maximum Capacity, ${\mathsf{\Omega }}_{\mathit{j}}^{\mathit{max}}$ (Kg)	Fixed Establishment Cost, ${\mathit{F}}_{\mathit{j}}$ (Millions) ${}^{\mathit{a}}$
$j_1$	3000	14.5	$j_{10}$	3000	13.5
$j_2$	3000	9.5	$j_{11}$	3000	14
$j_3$	3000	13	$j_{12}$	3000	13.5
$j_4$	3000	15	$j_{13}$	3000	9
$j_5$	3000	11.5	$j_{14}$	3000	12
$j_6$	3000	12.25	$j_{15}$	3000	12
$j_7$	3000	12.5	$j_{16}$	3000	9.5
$j_8$	3000	14	$j_{17}$	3000	22
$j_9$	3000	12.5	$j_{18}$	3000	16.5

${}^a$ in relative money units.

). Fixed establishment costs ($F_j$) vary among centers (Table 5), while material handling costs ($C_{ij}^H$) are assumed to be the same for all locations but depend on the product type (

Table 6. Material handling cost for candidate local distribution center (in relative money units).

Product (i)	Per Kg Handling Cost at LDC, ${\mathit{C}}_{\mathit{ij}}^{\mathit{H}}$	Product (i)	Per Kg Handling Cost at LDC, ${\mathit{C}}_{\mathit{ij}}^{\mathit{H}}$	Product (i)	Per Kg Handling Cost at LDC, ${\mathit{C}}_{\mathit{ij}}^{\mathit{H}}$	Product (i)	Per Kg Handling Cost at LDC, ${\mathit{C}}_{\mathit{ij}}^{\mathit{H}}$
$i_1$	0.06250	$i_{13}$	0.04300	$i_{25}$	0.08480	$i_{37}$	0.03815
$i_2$	0.94710	$i_{14}$	0.87475	$i_{26}$	0.09080	$i_{38}$	0.29820
$i_3$	0.21475	$i_{15}$	1.82050	$i_{27}$	0.31305	$i_{39}$	0.44120
$i_4$	1.76530	$i_{16}$	0.06325	$i_{28}$	0.23130	$i_{40}$	0.31310
$i_5$	1.77405	$i_{17}$	0.05445	$i_{29}$	0.22000	$i_{41}$	0.68500
$i_6$	1.78000	$i_{18}$	0.40030	$i_{30}$	0.03990	$i_{42}$	0.97500
$i_7$	0.11310	$i_{19}$	0.16750	$i_{31}$	0.21355	$i_{43}$	0.68880
$i_8$	0.56095	$i_{20}$	0.14285	$i_{32}$	0.65600	$i_{44}$	0.97660
$i_9$	0.10720	$i_{21}$	0.42550	$i_{33}$	0.09550	$i_{45}$	0.06600
$i_{10}$	0.22465	$i_{22}$	0.42425	$i_{34}$	0.10350	$i_{46}$	0.50000
$i_{11}$	0.21415	$i_{23}$	0.34195	$i_{35}$	0.04685	$i_{47}$	0.37110
$i_{12}$	0.35315	$i_{24}$	0.15025	$i_{36}$	0.61250

). Material handling costs are estimated at 5% of the unit selling price of each item.

Transportation flows and related costs: To simplify the modeling process, a large value is assigned as the maximum flow amount for products transported between the main distribution center and local distribution centers ($f_j^{max}$) and between local distribution centers and customer zones ($L_{jk}^{max}$). However, the maximum flow from the main distribution center to each customer zone ($M_k^{max}$) is restricted to 1000 kg to avoid technical difficulties at the main distribution center. The transportation cost between nodes is assumed to be independent of the material type but dependent on the distance traveled and the amount of products transferred. The average transportation cost is calculated based on the case company’s data of 8.84 units per kilometer for a full-load truck. The actual travel distance can be determined using either the Google Maps API or OpenStreetMap, with OpenStreetMap chosen in this study to minimize costs. The transportation costs between nodes are provided as a piece-wise linear function of the distance traveled (refer to

Table 7. Unit transportation cost from main distribution center to local distribution center ($c_{ij}$) (in relative money units, per kg of product i).

LDC (j)	MDC	LDC (j)	MDC	LDC (j)	MDC	LDC (j)	MDC	LDC (j)	MDC	LDC (j)	MDC
$j_1$	7.87	$j_4$	4.79	$j_7$	3.11	$j_{10}$	6.92	$j_{13}$	5.98	$j_{16}$	10.01
$j_2$	9.36	$j_5$	6.50	$j_8$	4.01	$j_{11}$	2.86	$j_{14}$	4.61	$j_{17}$	0.04
$j_3$	5.79	$j_6$	3.51	$j_9$	6.62	$j_{12}$	4.51	$j_{15}$	6.23	$j_{18}$	0.98

for values from main distribution center to local distribution centers). The transportation cost data for other routes are not presented due to their large sizes.

Product demand: Monthly product demand data over a two-year period were collected. For the deterministic part of the model, the average demand value for each product i was normalized to make it compatible with the storage size of the local distribution centers. Unfortunately, the demand data could not be included in this manuscript because of their size, but they can be provided upon request.

Additional cost: The model allows direct delivery from the main distribution center to the customer zone but at an additional cost. This cost, referred to as $a_{ik}$ in the model, includes expenses related to operations at the warehouse due to direct delivery. To normalize this cost, it is estimated to be five units per unit of an item directly delivered. Two factors contribute to this cost: the additional work required and the company’s inability to serve every customer zone directly.

6.2. Solution Approach

The models were implemented on an AMD Ryzen 7 5800H with Radeon Graphics, with 3.20 GHz and 16 GB RAM, using Gurobipy 10.0.0 solver in a Python environment. Solving a bi-objective model involves managing trade-offs between competing objectives, which can be challenging. Fortunately, Gurobi software offers two approaches to address this challenge: combining objectives through weighted combinations and hierarchical/lexicographic prioritization. Both approaches of the Gurobi solver were used to optimize the model.

In the bi-objective model, the second objective function of customer service level, represented by minimizing the maximum unmet demand (Equation (2)), is non-linear. To ease the implementation, the model was simplified. To simplify the model, a new decision variable $\psi$ was introduced and the model was then defined as a function of this variable:

$${\mathrm{Min}}_{w2}=\psi$$

(9)

In addition to the constraints explained in the previous chapter in Equations (3a)–(3l), the constraints related to the new variable in Equations (10) and (11) were then incorporated in the bi-objective model:

$$\begin{array}{cc}\hfill & \psi =d_{i,k}-\left(\sum _j{L}_{i,j,k}+\sum V_{i,k}\right)\forall i\in I,\forall k\in K\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & \psi \ge 0\hfill \end{array}$$

(11)

Similarly, for the case of demand uncertainty, Equation (6) can be simplified in the same manner. In this case, the scenario subscript s was added to the new variable $\psi$ and the model was modified as follows:

$${\mathrm{Min}}_{w2}={\psi }_s$$

(12)

In addition to the constraints in the previous section, the following constraints (13) and (14) were incorporated:

$$\begin{array}{cc}\hfill & \psi =d_{i,k,s}-\left(\sum _j{L}_{i,j,k,s}+\sum V_{i,k,s}\right)\forall i\in I,\forall k\in K,\forall s\in S\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & {\psi }_s\ge 0\forall s\in S\hfill \end{array}$$

(14)

6.2.1. Solution for the Deterministic Model

To solve the model, both weighted and hierarchical approaches of the Gurobi solver were employed. In the weighted approach, different weights are assigned to each objective to obtain results;

Table 8. Computational statistics of the weighted approach for the deterministic model.

Weight *	Obj. 1	Obj. 2	Integ. Gap	CPU Time
Highest/Lowest	3.192619772315 × 10${}^8$	0.00	0.0088%	247.37 s
Equal weight	3.192619772315 × 10${}^8$	0.00	0.0088%	242.16 s
Lowest/Highest	3.192619772315 × 10${}^8$	0.00	0.0088%	242.16 s

* For the first objective/For the second objective.

summarizes the optimal values for different weight assignments. The results show that the optimal values of both objectives remain the same, regardless of the weight assignments.

In the hierarchical/lexicographic approach, each objective is given priority alternately. Firstly, the highest priority is given to the first objective function, then to the second objective function.

Table 9. Computational statistics of hierarchical approach for the deterministic model.

	First Case		Second Case
Parameters	Obj. 1	Obj. 2	Obj. 1	Obj. 2
Priority	Highest	Lowest	Lowest	Highest
Best value	3.1924761 × 10${}^8$	0.00	3.1926172 × 10${}^8$	0.00
Integrality Gap	0.0041%	0.0000%	0.0090%	0.0000%
CPU time (s)	130.34	130.58	164.44	101.63
Nodes explored	159	0	113	1
Variables	818,853 continuous and 17,440 binary

displays the results of both cases. When the first objective function is given the highest priority, a slightly better optimal value is achieved compared to other cases. Compared to the weighted approach, giving priority to the first objective in the hierarchical approach leads to an improvement of about 1.43684383 × 10${}^4$ unit cost values. Additionally, when compared to the optimal value in the hierarchical approach with the second objective function given priority, the difference is 1.41073696 × 10${}^4$ unit cost value.

In all the approaches, 10 local distribution centers (LDCs) are selected and their locations are determined to be similar, namely, LDC${}_4$, LDC${}_5$, LDC${}_6$, LDC${}_7$, LDC${}_8$, LDC${}_{11}$, LDC${}_{13}$, LDC${}_{14}$, LDC${}_{17}$, and LDC${}_{18}$. There are some variations in the allocation of customer zones to these locations in different solution approaches. Moreover, some of the customer zones are assigned to the main distribution center for the direct delivery of products. Since all resulting values of the cases cannot be presented in this report due to their sizes, for clarity, the customer allocation/distribution link for the case having a better optimal value (i.e., when priority is given to the first objective in a hierarchical solution approach) is presented. The summary of the distribution link is shown in

Table 10. Allocation of customer zones to local DC j for the deterministic model.

Local Distribution Center (j)	Number of Customer Zones Allocated
LDC04 ≡ Bahir Dar	66
LDC05 ≡ Gondar	72
LDC06 ≡ Jimma	114
LDC07 ≡ Nekemte	124
LDC08 ≡ Dessie	103
LDC11 ≡ Hawassa	81
LDC13 ≡ Negele Borena	82
LDC14 ≡ Dire Dawa	143
LDC17 ≡ Addis Ababa	40
LDC18 ≡ Adama	58
MDC * ≡ Addis Ababa	33
Total	916

* Direct delivery from the main distribution center to customer zones.

.

The geographical location of selected local distribution centers was also determined using QGIS v3.28.0 software, as shown in Figure 5.

The comparison of the cost components obtained by the deterministic model and the existing distribution cost of the case company are shown in

Table 11. Cost component data for existing case and the deterministic model (in relative money units).

Cost Components	Model Value	Existing Value
Supply cost from MDC to LDCs	1.02 × 10${}^8$	1.03 × 10${}^8$
Delivery cost from LDCs to CZs	4.67 × 10${}^7$	5.24 × 10${}^7$
Direct delivery cost from MDC to CZs	3.52 × 10${}^5$	-
Handling cost at LDCs	3.10 × 10${}^7$	3.10 × 10${}^7$
Fixed establishment cost of LDCs	1.39 × 10${}^8$	2.37 × 10${}^8$
Total cost	3.19 × 10${}^8$	4.23 × 10${}^8$

and in Figure 6. From Figure 6, we can understand that, compared with the existing costs associated with the delivery of pharmaceutical products, the model results in a much better estimate. The supply chain cost related to the distribution of pharmaceutical items can be reduced by about 25.26%. As a result, using this model can enable the company to reduce its expenses by an overwhelming amount. According to the results, the customer zones represented by districts in the country can be effectively served with only 10 local distribution centers and a few customer zones directly supplied by the main distribution center. This result complies with the results of the research carried out on the same company from another perspective by USAID [48], in recommending fewer local distribution centers.

6.2.2. Solution for the Model under Demand Uncertainty

Handling Uncertainty

In the preceding subsection, a solution was presented for a deterministic model. In a deterministic model, the customer demand $d_{ik}$ is assumed to be constant, and the goals involve minimizing costs and maximizing unmet demand while determining the placement of local distribution centers and distribution links. This subsection addresses the case where product demand is not precisely known and is subject to uncertainty. Consequently, we solve the problem described in Equations (5) and (6) (with the incorporation of the simplified version, Equation (12) and constraints (13) and (14)), along with its first corresponding constraints (Equations (7a)–(7m)). Before providing the solution, however, the handling of uncertainty is discussed.

The uncertainty in demand is captured by categorizing the observed demand fluctuations into three scenario realizations. These scenarios represent situations where demand is assumed to increase by 10%, 20%, and 30%.

Table 12. Scenario definition with the corresponding probability of occurrence.

Scenario (s)	Definition	Probability (${\mathit{\rho }}_{\mathit{s}}$)
Scenario 1	Demand is considered to be increased by 10%	0.20
Scenario 2	Demand is considered to be increased by 20%	0.35
Scenario 3	Demand is considered to be increased by 30%	0.45

provides the probabilities associated with each scenario.

There are two main reasons for considering demand uncertainty through scenarios, as outlined in Table 12. First, the demand pattern, as revealed by the analysis of existing data, does not follow a normal distribution for certain products. Second, the company’s expenditures on pharmaceutical items have been increasing over the past 11 years, indicating a growing need. It should also be noted that these numbers only reflect the capacity the company can provide and not the actual demand from customers, as per discussions with the top manager of the company. The population size is also increasing, further contributing to the existing demand. Additionally, there is an increasing public awareness of the advantages of utilizing modern health facilities. Currently, the overall utilization of modern health services in the case country is about 0.25 visits per person per year, which is significantly below the WHO’s recommendation of 3 visits as part of the Millennium Development Goals [59]. Consequently, it is expected that utilization will rise, leading to an increased demand for pharmaceutical items. Considering these factors, the uncertainty in demand is captured by employing three scenarios, all of which indicate an increase in demand (refer to Table 12).

To handle the uncertainty, a scenario-based optimization approach is utilized, as described in Section 5.3. The scenario-based optimization approach is divided into two parts: the first deals with a model where demand is uncertain but the distribution pattern is fixed, while the second part addresses a model where both demand and distribution pattern are uncertain. Each part is solved separately.

Demand Uncertainty and Fixed Distribution Pattern

In the first part, fixed variables included location variables $Y_j$, distribution link variables $X_j$, $Z_{jk}$, and $M_k$, and the storage capacity variable ${\mathsf{\Omega }}_j$. These variables were determined before the uncertainty occurred, while other variables $f_{ij}$, $L_{ijk}$, and $V_{ik}$ were determined after the uncertainty realization. A fixed distribution pattern can provide a stable delivery plan and facilitate communication among nodes, as customers know in advance who will handle their demand regardless of the scenario.

This model was also solved using the same solver as in the deterministic case. Both weighted and hierarchical or lexicographic approaches of the Gurobi solver were employed to solve the model, and the summary of the results is shown in

Table 13. Computational statistics for the weighted approach of the model under demand uncertainty.

Weight *	Obj. 1	Obj. 2	Gap	CPU Time
Highest/Lowest	4.941103393284 × 10${}^8$	0.00	0.0075%	2018.41 s
Equal weight	4.941103393284 × 10${}^8$	0.00	0.0075%	1561.31 s
Lowest/Highest	4.941103393284 × 10${}^8$	0.00	0.0075%	2027.04 s

* For the first objective/For the second objective.

and

Table 14. Computational statistics of the hierarchical approach of model under demand uncertainty.

	First Case		Second Case
Parameters	Obj. 1	Obj. 2	Obj. 1	Obj. 2
Priority	Highest	Lowest	Lowest	Highest
Best value	4.941115 × 10${}^8$	0.00	4.941115 × 10${}^8$	0.00
Integrality Gap	0.0079%	0.0000%	0.0.0079%	0.0000%
CPU time (sec)	541.27	543.03	684.70	279.95
Nodes explored	1145	0	1145	1
Variables	2,456,521 continuous and 17,440 binary

, respectively.

Table 13 shows that, regardless of the weighting scheme used, the optimal values for both objectives were similar, and the weighted approach provided a slightly better optimal value for the first objective compared to the hierarchical approaches (Table 14).

Furthermore, the allocation of customer zones within the distribution network for this scenario-based model is presented in

Table 15. Allocation of customer zones to local DC j for the scenario-based model.

Local Distribution Center (j)	Number of Customer Zones Allocated
LDC03 ≡ Semera	95
LDC04 ≡ Bahir Dar	47
LDC05 ≡ Gondar	58
LDC06 ≡ Jimma	95
LDC07 ≡ Nekemte	79
LDC08 ≡ Dessie	66
LDC09 ≡ Assosa	58
LDC11 ≡ Hawassa	60
LDC12 ≡ Arba Minch	78
LDC13 ≡ Negele Borena	88
LDC14 ≡ Dire Dawa	109
LDC17 ≡ Addis Ababa	17
LDC18 ≡ Adama	40
MDC * ≡ Addis Ababa	26
Total	916

* Direct delivery from the main distribution center to customer zones.

. It is worth noting that, in the deterministic model, only 10 local distribution centers were selected, compared to the existing 18, to meet customer demands. However, considering demand uncertainty, 13 local distribution centers are required to ensure the necessary demand fulfillment from customer zones.

Demand Uncertainty and Variable Distribution Pattern

In this section, the decision variables $Y_j$ (location binary variable) and ${\mathsf{\Omega }}_j$ (storage capacity of the local distribution center) were considered to be known in advance of uncertainty realization, while all other variables were determined in the second stage after uncertainty realization. The constraints from Equations (7a) to (7j), related to the binary variables of distribution links, were replaced by modified constraints, Equations (8a) to (8f).

For the second objective of minimizing the maximum unmet demand, the same simplified model as in the Section 6.2 was used. Similar to the previous section, the scenario-based model with uncertain demand and variable distribution patterns was solved using both the weighted and hierarchical approaches of the Gurobi solver. The summary of the results is presented in

Table 16. Computational statistics for weighted approach when distribution pattern is variable.

Weight *	Obj. 1	Obj. 2	Gap	CPU Time
Highest/Lowest	4.821993464381 × 10${}^8$	0.00	0.0098%	8944.20 s
Equal weight	4.821993464381 × 10${}^8$	0.00	0.0098%	6580.38 s
Lowest/Highest	4.821993464381 × 10${}^8$	0.00	0.0098%	7325.70 s

* For the first objective/For the second objective.

and

Table 17. Computational statistics for hierarchical approach when distribution pattern is variable.

	First Case		Second Case
Parameters	Obj. 1	Obj. 2	Obj. 1	Obj. 2
Priority	Highest	Lowest	Lowest	Highest
Best value	4.8219831 × 10${}^8$	0.00	4.8219786 × 10${}^8$	0.00
Integrality Gap	0.0092%	0.0000%	0.0090%	0.0000%
CPU time (s)	10,890.21	10,891.44	10,031.69	594.60
Nodes explored	30,180	0	35,519	1
Variables	2,456,521 continuous and 52,284 binary

.

The results of the scenario-based model when considering uncertain demand and variable distribution patterns indicate an improvement in the supply chain cost compared to the model only subjected to demand uncertainty, but at high computational cost (as illustrated in Figure 7). Furthermore, the hierarchical solution approach in both priority settings yields better results compared to the weighted solution approach. However, the computational time required for the weighted approach is significantly lower than that of the hierarchical approaches.

Similar to the case when the distribution pattern is fixed, in this scenario as well, the customer demand can be satisfied with 13 local distribution centers and a few direct deliveries from the main distribution center. However, there is a difference in the selection of local distribution centers. In the previous part, LDC${}_{09}$ was selected and LDC${}_{02}$ was not selected, while in this part, LDC${}_{02}$ was selected and LDC${}_{09}$ was not selected. The overall distribution links within the network for each realization of scenarios are summarized in

Table 18. Allocation of customer zones to local distribution center j for the scenario-based model when demand is uncertain and distribution pattern is variable in all scenarios.

	Number of CZs Allocated in Each Scenario
Local Distribution Center (j)	s = 1	s = 2	s = 3
LDC02 ≡ Shire	2	23	44
LDC03 ≡ Semera	39	61	96
LDC04 ≡ Bahir Dar	58	55	53
LDC05 ≡ Gondar	70	58	54
LDC06 ≡ Jimma	114	105	96
LDC07 ≡ Nekemte	106	103	88
LDC08 ≡ Dessie	76	69	59
LDC11 ≡ Hawassa	75	68	54
LDC12 ≡ Arba Minch	84	87	83
LDC13 ≡ Negele Borena	49	83	85
LDC14 ≡ Dire Dawa	136	113	109
LDC17 ≡ Addis Ababa	41	32	28
LDC18 ≡ Adama	39	30	26
MDC * ≡ Addis Ababa	27	29	41
Total	916	916	916

* Direct delivery from the main distribution center to customer zones.

, and the detailed results are provided in

Table A1. Full distribution link for the scenario-based model when demand is uncertain and distribution pattern is variable in all scenarios.

LDC (j)	Allocated Customer Zone (k)
Scenario, s = 1
LDC02	k647, k793
LDC03	k001, k002, k015, k026, k033, k035, k047, k094, k106, k109, k131, k141, k150, k185, k204, k219, k239, k314, k332, k341, k347, k352, k390, k418, k466, k489, k547, k585, k589, k623, k648, k691, k735, k765, k769, k835, k844, k895, k899
LDC04	k023, k051, k070, k077, k110, k119, k120, k124, k148, k181, k193, k226, k227, k228, k247, k251, k252, k253, k262, k271, k282, k289, k338, k342, k343, k350, k354, k355, k363, k364, k425, k429, k437, k438, k439, k507, k514, k522, k605, k636, k645, k675, k676, k701, k727, k731, k757, k762, k766, k789, k803, k815, k857, k872, k876, k893, k907, k913
LDC05	k008, k024, k025, k031, k032, k042, k095, k107, k147, k205, k212, k245, k246, k270, k277, k299, k324, k325, k327, k349, k370, k384, k389, k426, k427, k428, k446, k451, k491, k509, k518, k543, k566, k575, k594, k595, k604, k617, k655, k656, k671, k674, k677, k685, k709, k730, k740, k758, k759, k763, k775, k798, k825, k826, k827, k828, k829, k830, k831, k833, k837, k851, k852, k869, k879, k880, k881, k883, k884, k914
LDC06	k006, k010, k038, k041, k045, k050, k055, k056, k062, k068, k071, k132, k133, k134, k146, k149, k152, k156, k167, k171, k176, k188, k194, k197, k202, k209, k210, k243, k255, k256, k259, k260, k290, k298, k311, k340, k344, k353, k368, k369, k377, k383, k386, k387, k388, k391, k393, k401, k410, k412, k416, k423, k434, k447, k448, k455, k504, k510, k515, k525, k534, k536, k545, k556, k558, k573, k582, k613, k620, k621, k625, k629, k630, k632, k638, k642, k651, k652, k657, k664, k665, k672, k686, k687, k693, k694, k698, k705, k716, k717, k718, k726, k747, k761, k768, k774, k786, k787, k791, k801, k813, k823, k838, k839, k843, k846, k848, k870, k886, k887, k890, k902, k904, k905
LDC07	k004, k007, k012, k039, k061, k067, k072, k073, k098, k099, k101, k113, k118, k121, k123, k129, k137, k138, k158, k159, k166, k184, k186, k187, k196, k203, k213, k215, k216, k233, k250, k261, k276, k278, k279, k292, k329, k351, k374, k385, k394, k395, k396, k398, k399, k411, k436, k442, k443, k445, k459, k460, k479, k485, k486, k499, k500, k501, k511, k513, k516, k519, k521, k529, k531, k532, k533, k535, k551, k562, k567, k579, k591, k598, k599, k601, k611, k619, k628, k631, k635, k637, k662, k663, k683, k712, k713, k715, k719, k722, k744, k755, k777, k778, k780, k794, k800, k804, k849, k863, k865, k867, k889, k900, k906, k911
LDC08	k005, k046, k048, k049, k059, k076, k078, k079, k086, k087, k091, k100, k130, k173, k177, k199, k238, k240, k244, k249, k268, k274, k283, k286, k287, k288, k328, k339, k375, k376, k397, k407, k440, k462, k463, k493, k495, k517, k530, k544, k550, k554, k560, k561, k569, k578, k584, k592, k603, k606, k607, k609, k643, k650, k654, k666, k667, k668, k669, k678, k689, k699, k725, k734, k748, k760, k782, k832, k834, k842, k861, k868, k875, k877, k878, k885
LDC11	k003, k016, k020, k037, k053, k084, k090, k093, k097, k140, k165, k170, k174, k178, k191, k192, k195, k214, k218, k231, k293, k294, k295, k300, k306, k315, k372, k379, k403, k421, k431, k433, k469, k471, k474, k481, k487, k488, k505, k506, k570, k571, k574, k583, k597, k600, k622, k634, k639, k649, k680, k696, k707, k714, k736, k738, k776, k779, k784, k785, k788, k799, k805, k806, k841, k847, k871, k873, k874, k891, k892, k908, k909, k915, k916
LDC12	k057, k058, k074, k082, k083, k112, k128, k139, k163, k164, k168, k172, k179, k180, k189, k198, k220, k221, k222, k223, k237, k254, k269, k280, k281, k284, k285, k296, k297, k301, k303, k312, k313, k316, k317, k322, k323, k337, k378, k381, k392, k422, k424, k464, k470, k472, k502, k508, k538, k542, k548, k552, k555, k564, k565, k580, k581, k586, k612, k624, k633, k659, k660, k670, k673, k695, k697, k720, k724, k743, k745, k764, k767, k783, k802, k810, k811, k814, k824, k836, k840, k855, k897, k912
LDC13	k017, k027, k028, k029, k030, k036, k085, k089, k127, k143, k153, k161, k200, k201, k236, k241, k242, k272, k305, k308, k309, k310, k362, k406, k408, k409, k415, k430, k441, k449, k453, k454, k457, k476, k478, k503, k572, k615, k646, k702, k703, k704, k711, k723, k728, k739, k856, k859, k860
LDC14	k011, k034, k063, k064, k065, k066, k069, k081, k088, k102, k103, k104, k105, k108, k111, k115, k116, k117, k135, k136, k142, k145, k151, k154, k157, k160, k190, k206, k207, k211, k224, k230, k232, k234, k235, k257, k258, k263, k264, k265, k266, k275, k291, k302, k304, k318, k321, k326, k330, k333, k334, k335, k336, k346, k348, k356, k357, k358, k361, k365, k366, k367, k371, k373, k382, k405, k413, k414, k417, k419, k420, k432, k435, k452, k456, k458, k461, k465, k468, k473, k475, k477, k480, k482, k483, k484, k490, k494, k520, k527, k528, k537, k539, k540, k541, k546, k549, k557, k559, k577, k587, k590, k602, k640, k641, k644, k653, k658, k661, k681, k682, k688, k708, k721, k729, k732, k733, k741, k742, k773, k790, k792, k795, k796, k797, k812, k816, k822, k853, k854, k864, k866, k882, k896, k898, k910
LDC17	k009, k014, k021, k022, k052, k054, k075, k080, k144, k182, k183, k225, k248, k267, k345, k359, k360, k404, k444, k498, k526, k553, k563, k568, k576, k588, k593, k610, k618, k700, k746, k749, k753, k754, k770, k818, k819, k820, k821, k894, k901
LDC18	k013, k018, k019, k040, k043, k044, k092, k096, k122, k155, k162, k169, k175, k208, k273, k307, k319, k380, k400, k450, k467, k496, k497, k512, k523, k596, k616, k626, k627, k679, k692, k706, k710, k737, k752, k807, k809, k817, k903
MDC *	k060, k114, k125, k126, k217, k229, k320, k331, k402, k492, k524, k608, k614, k684, k690, k750, k751, k756, k771, k772, k781, k808, k845, k850, k858, k862, k888
Scenario, s = 2
LDC02	k008, k025, k031, k032, k042, k095, k107, k349, k370, k446, k491, k575, k594, k647, k677, k709, k740, k793, k798, k827, k828, k829, k831
LDC03	k001, k002, k005, k015, k026, k033, k035, k046, k047, k048, k063, k076, k088, k094, k100, k103, k106, k109, k131, k141, k150, k185, k199, k204, k217, k219, k239, k268, k314, k321, k332, k339, k341, k347, k352, k390, k418, k466, k489, k495, k547, k561, k569, k584, k585, k589, k623, k643, k648, k691, k725, k734, k735, k760, k765, k769, k822, k835, k844, k895, k899
LDC04	k070, k072, k077, k101, k110, k119, k120, k124, k129, k148, k181, k193, k226, k227, k228, k247, k250, k251, k253, k262, k271, k276, k289, k338, k343, k350, k354, k363, k425, k429, k436, k437, k438, k439, k507, k514, k516, k522, k593, k636, k645, k676, k701, k727, k731, k762, k766, k789, k804, k815, k857, k872, k876, k907, k913
LDC05	k024, k051, k147, k205, k212, k245, k246, k252, k270, k277, k282, k299, k324, k325, k327, k355, k364, k384, k389, k426, k427, k428, k451, k509, k518, k543, k566, k595, k604, k605, k617, k655, k656, k671, k674, k675, k685, k730, k758, k759, k763, k775, k803, k825, k826, k830, k833, k837, k851, k852, k869, k879, k880, k881, k883, k884, k893, k914
LDC06	k006, k010, k038, k041, k045, k050, k055, k062, k071, k132, k134, k146, k149, k152, k156, k167, k171, k176, k188, k194, k197, k202, k209, k210, k243, k255, k256, k259, k260, k290, k298, k311, k340, k344, k351, k353, k368, k369, k377, k383, k386, k387, k391, k393, k401, k410, k412, k416, k423, k434, k447, k455, k504, k510, k512, k515, k525, k529, k534, k536, k545, k558, k582, k620, k621, k629, k638, k642, k651, k652, k664, k665, k686, k687, k693, k698, k705, k716, k717, k718, k726, k747, k752, k753, k761, k768, k774, k786, k787, k791, k801, k813, k823, k838, k839, k843, k846, k848, k870, k886, k887, k890, k902, k904, k905
LDC07	k004, k007, k012, k014, k039, k061, k067, k073, k098, k099, k113, k118, k121, k123, k133, k137, k138, k158, k159, k166, k184, k186, k187, k196, k203, k213, k215, k216, k233, k261, k278, k279, k292, k329, k374, k385, k394, k395, k396, k398, k399, k411, k442, k443, k445, k459, k460, k479, k485, k486, k498, k499, k500, k501, k511, k513, k519, k521, k531, k532, k533, k535, k551, k562, k567, k579, k591, k598, k599, k601, k611, k613, k619, k628, k631, k632, k635, k637, k662, k663, k683, k690, k712, k713, k715, k719, k722, k744, k755, k777, k778, k780, k794, k800, k849, k863, k865, k867, k888, k889, k900, k906, k911
LDC08	k009, k049, k059, k075, k078, k079, k086, k087, k091, k092, k130, k173, k177, k238, k240, k244, k249, k274, k283, k286, k287, k288, k328, k342, k345, k375, k376, k397, k407, k440, k462, k463, k467, k493, k517, k530, k544, k550, k554, k560, k563, k578, k592, k603, k606, k607, k609, k650, k654, k666, k667, k668, k669, k678, k689, k699, k700, k746, k748, k782, k832, k834, k842, k861, k868, k875, k877, k878, k885
LDC11	k003, k013, k016, k020, k044, k053, k084, k090, k093, k097, k165, k169, k174, k178, k191, k192, k195, k214, k218, k231, k293, k294, k295, k300, k306, k315, k379, k431, k450, k469, k481, k487, k488, k496, k505, k506, k556, k570, k571, k573, k574, k583, k600, k616, k622, k634, k639, k649, k680, k692, k696, k707, k779, k784, k785, k788, k799, k806, k809, k841, k847, k871, k873, k874, k908, k909, k915, k916
LDC12	k056, k057, k058, k068, k074, k082, k083, k112, k128, k139, k163, k164, k168, k172, k179, k180, k189, k198, k220, k221, k222, k223, k237, k254, k269, k280, k281, k284, k285, k296, k297, k303, k312, k313, k316, k317, k322, k323, k337, k381, k388, k392, k422, k424, k448, k464, k472, k502, k508, k538, k542, k548, k552, k564, k565, k580, k581, k586, k612, k624, k625, k630, k633, k657, k670, k672, k673, k694, k695, k720, k724, k743, k745, k764, k767, k783, k802, k810, k811, k814, k824, k836, k840, k855, k892, k897, k912
LDC13	k017, k027, k028, k029, k030, k036, k037, k085, k089, k127, k140, k143, k153, k161, k170, k200, k201, k224, k236, k241, k242, k272, k301, k305, k308, k309, k310, k333, k335, k336, k358, k362, k372, k378, k403, k406, k408, k409, k413, k414, k415, k421, k430, k433, k441, k449, k453, k454, k457, k470, k471, k474, k476, k478, k483, k494, k503, k549, k555, k559, k572, k597, k615, k646, k659, k660, k697, k702, k703, k704, k708, k711, k714, k723, k728, k736, k738, k739, k776, k805, k856, k859, k860
LDC14	k011, k034, k065, k066, k069, k081, k102, k104, k105, k108, k111, k115, k116, k117, k135, k136, k142, k151, k154, k157, k160, k190, k206, k207, k211, k230, k232, k234, k235, k257, k258, k263, k264, k265, k266, k275, k291, k302, k304, k318, k326, k330, k334, k346, k356, k357, k361, k365, k366, k367, k371, k373, k382, k405, k417, k419, k420, k432, k435, k452, k458, k461, k465, k468, k473, k475, k477, k480, k482, k484, k490, k520, k527, k528, k537, k539, k540, k541, k546, k557, k577, k587, k590, k602, k640, k641, k644, k653, k658, k661, k681, k682, k688, k721, k729, k732, k733, k741, k742, k790, k792, k795, k797, k812, k816, k853, k854, k864, k866, k882, k896, k898, k910
LDC17	k021, k022, k052, k054, k080, k144, k182, k183, k225, k248, k267, k359, k360, k404, k444, k526, k553, k568, k576, k588, k610, k618, k706, k749, k754, k770, k818, k819, k820, k821, k894, k901
LDC18	k018, k019, k040, k043, k064, k096, k122, k145, k155, k162, k175, k208, k273, k307, k319, k380, k400, k497, k523, k596, k626, k627, k679, k710, k737, k773, k796, k807, k817, k903
MDC *	k023, k060, k114, k125, k126, k229, k320, k331, k348, k402, k456, k492, k524, k608, k614, k684, k750, k751, k756, k757, k771, k772, k781, k808, k845, k850, k858, k862, k891
Scenario, s = 3
LDC02	k005, k008, k024, k025, k031, k032, k042, k095, k100, k107, k109, k141, k219, k268, k299, k341, k349, k370, k446, k491, k543, k547, k561, k575, k589, k594, k595, k647, k677, k709, k740, k760, k765, k793, k798, k827, k828, k829, k831, k837, k851, k852, k869, k895
LDC03	k001, k002, k015, k033, k035, k046, k047, k048, k052, k063, k064, k069, k075, k076, k088, k091, k094, k096, k103, k106, k111, k131, k150, k157, k175, k177, k185, k199, k204, k208, k217, k234, k239, k240, k249, k314, k321, k328, k332, k339, k347, k352, k357, k367, k375, k376, k390, k397, k418, k420, k440, k461, k462, k466, k467, k477, k490, k495, k496, k497, k523, k526, k530, k544, k560, k569, k584, k585, k597, k603, k623, k643, k648, k655, k668, k679, k691, k692, k699, k721, k725, k734, k746, k763, k769, k770, k773, k807, k817, k822, k834, k835, k844, k878, k885, k899
LDC04	k077, k110, k119, k120, k124, k129, k148, k181, k184, k186, k187, k193, k226, k227, k228, k250, k251, k253, k262, k267, k271, k276, k289, k338, k354, k363, k404, k429, k436, k437, k438, k439, k500, k507, k514, k516, k522, k593, k628, k636, k645, k701, k727, k731, k762, k804, k815, k857, k872, k876, k889, k907, k913
LDC05	k026, k051, k070, k147, k205, k212, k245, k246, k247, k252, k270, k277, k282, k324, k325, k327, k350, k355, k364, k384, k389, k426, k427, k428, k451, k489, k509, k518, k566, k604, k605, k617, k656, k671, k674, k675, k685, k730, k758, k759, k766, k775, k803, k825, k826, k830, k833, k879, k880, k881, k883, k884, k893, k914
LDC06	k006, k010, k038, k041, k050, k055, k062, k071, k132, k133, k134, k146, k149, k152, k156, k167, k171, k176, k188, k194, k197, k202, k209, k210, k233, k243, k255, k256, k259, k260, k290, k298, k311, k344, k369, k377, k386, k387, k391, k393, k401, k410, k412, k416, k423, k434, k447, k455, k504, k510, k515, k525, k534, k536, k545, k558, k582, k620, k621, k629, k632, k638, k642, k649, k651, k652, k664, k665, k686, k687, k698, k716, k717, k718, k726, k747, k761, k768, k774, k786, k787, k791, k801, k813, k823, k839, k843, k846, k848, k865, k870, k886, k887, k902, k904, k905
LDC07	k004, k007, k012, k039, k061, k067, k073, k098, k099, k113, k118, k121, k123, k137, k138, k158, k159, k166, k196, k203, k213, k215, k216, k261, k278, k279, k292, k351, k368, k374, k385, k394, k395, k396, k398, k399, k411, k442, k443, k445, k459, k460, k479, k485, k486, k499, k501, k511, k513, k519, k521, k531, k532, k533, k535, k551, k567, k579, k591, k598, k599, k601, k611, k619, k631, k635, k637, k662, k663, k712, k713, k715, k719, k722, k744, k755, k777, k778, k780, k794, k800, k849, k863, k867, k890, k900, k906, k911
LDC08	k009, k040, k049, k059, k072, k078, k079, k086, k087, k092, k101, k130, k173, k238, k244, k274, k283, k286, k287, k288, k342, k343, k345, k400, k407, k425, k463, k493, k517, k550, k554, k563, k578, k588, k592, k596, k606, k607, k609, k610, k650, k654, k666, k667, k669, k678, k689, k700, k735, k748, k782, k789, k832, k842, k861, k868, k875, k877, k903
LDC11	k016, k020, k053, k054, k084, k090, k097, k174, k192, k195, k214, k218, k280, k293, k294, k295, k306, k315, k379, k431, k469, k487, k488, k506, k571, k573, k574, k583, k600, k622, k634, k639, k680, k696, k705, k707, k779, k783, k784, k785, k788, k799, k802, k806, k809, k841, k847, k873, k874, k891, k892, k909, k915, k916
LDC12	k056, k057, k058, k068, k074, k082, k083, k112, k128, k139, k163, k164, k168, k172, k189, k198, k220, k221, k222, k223, k237, k254, k269, k281, k284, k285, k296, k297, k303, k312, k316, k317, k322, k323, k337, k340, k353, k381, k383, k388, k392, k422, k448, k464, k472, k502, k508, k538, k542, k548, k552, k564, k565, k580, k581, k586, k612, k624, k625, k630, k633, k657, k670, k672, k673, k693, k694, k695, k720, k724, k743, k745, k764, k767, k810, k811, k814, k824, k836, k840, k855, k897, k912
LDC13	k003, k017, k027, k028, k029, k030, k036, k037, k085, k089, k127, k140, k143, k161, k165, k170, k178, k179, k180, k191, k200, k201, k231, k236, k241, k242, k272, k300, k301, k305, k308, k309, k310, k313, k333, k362, k372, k378, k403, k406, k408, k409, k415, k421, k430, k433, k449, k453, k454, k457, k470, k471, k474, k476, k478, k481, k503, k505, k555, k570, k572, k615, k646, k659, k660, k697, k702, k703, k704, k708, k711, k714, k723, k728, k736, k737, k738, k739, k776, k805, k856, k859, k860, k871, k908
LDC14	k011, k034, k065, k081, k102, k104, k105, k115, k116, k117, k135, k136, k142, k145, k151, k154, k160, k190, k206, k207, k211, k224, k230, k232, k235, k258, k263, k264, k265, k266, k275, k291, k302, k304, k318, k326, k334, k335, k336, k346, k348, k356, k358, k361, k365, k366, k371, k373, k382, k405, k413, k414, k417, k419, k432, k435, k452, k456, k458, k465, k468, k473, k475, k480, k482, k483, k484, k520, k527, k528, k537, k539, k540, k541, k546, k549, k557, k559, k577, k587, k590, k602, k640, k641, k644, k653, k658, k661, k681, k682, k688, k729, k732, k733, k741, k742, k790, k792, k795, k797, k812, k816, k853, k854, k866, k882, k896, k898, k910
LDC17	k014, k021, k022, k080, k114, k182, k183, k225, k248, k329, k359, k360, k444, k498, k576, k618, k683, k750, k753, k754, k796, k818, k819, k820, k821, k888, k894, k901
LDC18	k013, k018, k019, k043, k044, k093, k122, k144, k155, k162, k169, k273, k307, k319, k380, k450, k512, k556, k568, k616, k626, k627, k710, k749, k752, k808
MDC *	k023, k045, k060, k066, k108, k125, k126, k153, k229, k257, k320, k330, k331, k402, k424, k441, k492, k494, k524, k529, k553, k562, k608, k613, k614, k676, k684, k690, k706, k751, k756, k757, k771, k772, k781, k838, k845, k850, k858, k862, k864

* MDC is the main distribution center, and the assigned customer zones are those directly supplied from the main distribution center.

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It can be observed from the tables that the number of customer zones linked to each local distribution center varies slightly in all three scenarios, but the number of local distribution centers remains the same in all scenarios, since it is determined in the first decision stage before scenario realization. The geographical location of local distribution centers determined by the solution of the scenario-based model is also shown in Figure 8.

To further compare the values of the first objective function and visualize the differences, Figure 9 presents a graph. It illustrates the comparison between the values of the first objective in the model when only demand is considered uncertain and the model when both demand and distribution patterns are uncertain. Notably, the second objective function has a value of 0.00 in all cases at a 0.0001% integrality gap.

In terms of computation time, there is a notable difference between the two models. The model considering uncertain demand only has significantly faster computation times compared to the model with both uncertain demand and variable distribution patterns. This difference in computation time is depicted in Figure 10. These visualizations aid in understanding the comparison of objective values and computational efficiency between the two scenarios.

Figure 9 and Figure 10 provide valuable insights. They demonstrate that the scenario-based model performs better in terms of computational time when only demand uncertainty is considered, while it yields better optimal values for the objectives when both demand uncertainty and variable distribution patterns are taken into account. The variability of the distribution pattern, where distribution links differ in each scenario realization, may be perceived as a drawback by customers. As customers’ allocation can change over the scenarios, customers may not know exactly who is supplying them, introducing instability. Hence, decision-makers must consider the benefits and drawbacks of each case carefully before making decisions.

The delivery amounts for all models are not presented here due to large sizes but can be provided upon request. In

Table 19. Summary of cost components for the optimal solution of the scenario-based model when demand is uncertain and distribution pattern is variable.

	Cost Values in Each Scenario (in Relative Money Units)
Cost Components	s = 1	s = 2	s = 3
Supply cost (MDC to LDCs)	1.16 × 10${}^8$	1.35 × 10${}^8$	1.55 × 10${}^8$
Delivery cost (LDCs to CZs)	4.90 × 10${}^7$	5.35 × 10${}^7$	6.24 × 10${}^7$
Material handling cost (at LDCs)	3.41 × 10${}^7$	3.72 × 10${}^7$	4.03 × 10${}^7$
Fixed establishment cost (of LDCs)	1.75 × 10${}^8$
Direct delivery cost (MDC to CZs)	5.45 × 10${}^4$	5.00 × 10${}^4$	3.16 × 10${}^5$
Additional cost at MDC	2.13 × 10${}^5$	1.97 × 10${}^5$	3.62 × 10${}^5$

, the summary of the overall cost components throughout the distribution network in each scenario for the scenario-based model considering uncertain demand and variable distribution patterns is presented. This table offers a comprehensive overview of the costs involved in the system.

6.3. Discussion

This study aimed to address three objectives, which were achieved in different sections. Section 4 discussed and identified key parameters for supply chain performance optimization, including supply chain costs and customer service level. In Section 5, the models were developed in three stages, extending the basic deterministic model to scenario-based optimization models. These models were designed to minimize supply chain costs and maximize customer service levels. The models were implemented in a developing country’s pharmaceutical supply chain using real-world data from a representative company (Section 6). The results of the deterministic model showed that the proposed model effectively optimized the supply chain distribution network, achieving cost savings while maintaining customer service levels compared to the existing situation of the case company. The scenario-based models were developed and implemented in two stages: one stage considered only uncertain demand, while the other stage considered both uncertain demand and variable distribution patterns. The performance of these models was validated using real-world data. The study’s contributions include considering financial and non-financial performance measures, addressing demand uncertainty, and applying the models to a developing country context. The developed models can serve as a decision-making tool for supply chain management, and future research could incorporate additional factors such as environmental impact, safety, and disruptions risks. The research provides valuable insights and guidance for practitioners and researchers working on similar optimization problems in supply chain operations.

7. Conclusions

In conclusion, this study focuses on supply chain network design and optimization, aiming to achieve a balance between supply chain costs and customer service levels. The research develops a comprehensive model that incorporates both deterministic and scenario-based approaches to address uncertainties in demand. The model is applied to an example company in the pharmaceutical supply chain in Ethiopia, allowing for practical insights and recommendations.

The study begins by introducing the problems to be addressed and briefly discusses the company’s current situation. A comprehensive literature review is conducted, and the basic concepts of performance measures, including relevant measures and metrics from financial and non-financial categories, are presented. Building on this foundation, supply chain performance optimization models are developed. The optimization models consider various cost factors, such as establishment costs, handling costs, and transportation costs, while aiming to maximize customer demand satisfaction. The results demonstrate the effectiveness of the model in optimizing the supply chain’s performance. Comparisons between the deterministic model and the company’s current situation reveal potential cost reductions and improvements in customer service by optimizing the distribution network. The scenario-based model, which accounts for demand uncertainty, further emphasizes the model’s ability to handle real-world complexities and provide accurate recommendations for decision-making.

The optimization model offers several benefits to companies, including cost reduction, improved operational efficiency, and enhanced customer service. It enables companies to identify cost-saving opportunities, make informed decisions, and ensure the availability of products at the right time and place. However, there are some limitations, such as the lack of consideration for disruptions and other objectives, including environmental impact, which could be addressed in future research directions.