Two-Stage Bi-Objective Stochastic Models for Supplier Selection and Order Allocation Under Uncertainty

Abstract

In supply chain management practices, supplier selection (SS) is a critical strategic planning activity that usually constitutes an ex ante decision made under uncertainty, whereas order allocation (OA) represents a subsequent operational decision determined ex post, contingent upon both the selected suppliers and actual operational conditions observed during the execution phase—specifically, the realized scenarios of uncertain circumstances. The practical performance of an SS decision inherently depends on its subsequent OA outcomes, while the OA decision itself is constrained by the preceding SS choices. Nevertheless, existing studies typically tackle the SS and OA problems separately or formulate them within a single-stage programming model, failing to adequately capture their sequential interdependence and the impact of OA on SS evaluation. To address this gap, this study develops novel two-stage bi-objective stochastic programming models in which the first-stage SS decisions are evaluated based on two key criteria—total cost and purchasing value—both of which depend on the second-stage OA decisions in response to realized operational scenarios. The stochastic performance of a given SS scheme, arising from adaptive OA decisions under uncertainty, is measured by expected value and conditional value-at-risk. An integrated approach combining weighted-satisfaction sum, linearization, Monte Carlo simulation, and genetic algorithm is developed to solve the models. Computational experiments demonstrate the effectiveness of the proposed methodology and reveal the influence of objective preferences and risk-aversion levels on the optimal supplier selection.

1. Introduction

Supplier selection (SS) and order allocation (OA) are two crucial decisions that contribute to the performance of a company, as well as the competitiveness of the entire supply chain [1]. Supplier selection involves evaluating and identifying suppliers capable of providing the necessary resources or services to an organization, and it significantly impacts the supply chain’s overall effectiveness. The SS decision is typically strategic and not subject to frequent changes, as maintaining long-term supplier relationships offers multiple advantages, such as enhanced stability in strategy and planning, consistent product quality and delivery schedules, deeper supply chain integration, potential cost savings, and improved inventory utilization efficiency [2]. In subsequent operational phases, the efficient allocation of purchase orders to selected suppliers further helps optimize costs and mitigate supply chain risks [3]. The SS decision lays the foundation for determining OA, while effective OA, in turn, enhances the practical outcomes of the SS decision. Thus, these two decisions are closely interconnected and mutually influential.

However, earlier literature often treats supplier selection and order allocation as separate decision-making problems. For example, Hosseininasab and Ahmadi [2], Shyur and Shih [4] focused solely on supplier selection, while Esfandiari and Seifbarghy [5], Meena and Sarmah [6] addressed only the order allocation problem. These studies overlooked the close interdependence between the two decisions. In recent years, researchers have increasingly recognized this intrinsic connection and have accordingly proposed integrated approaches that jointly address both supplier selection and order allocation (SSOA). A common method first evaluates suppliers utilizing multiple criteria decision-making (MCDM) techniques, then uses the evaluation results as objective function coefficients in a single programming model that incorporates decision variables for both SS and OA [7,8]. Solving this model yields simultaneous optimal decisions for both problems. Although such an approach acknowledges the interdependence between SS and OA, it fails to account for their sequential decision-making nature. Moreover, particularly under uncertain environments, it neglects the impacts of adaptive OA decisions—made in response to actual operational conditions—on the SS evaluation.

In practice, supplier selection and order allocation occur in two distinct and sequential decision stages, each with different information availability. During the supplier selection phase, key information, such as future supply quality, delivery punctuality, and the firm’s product demand, remains uncertain. In the subsequent operational phase, more accurate information is revealed, enabling order allocation decisions to be made based on these realized conditions. While order allocation is informed by this updated information, it also exerts a feedback effect on the initial supplier selection. Nevertheless, suppliers must be chosen before such information is fully known. Therefore, accounting for information uncertainty during the supplier selection process is essential.

To address these challenges, this paper develops a novel two-stage bi-objective stochastic programming method for supplier selection and order allocation under uncertain environments. Within this framework, the first-stage supplier selection is conducted under uncertainty, with its evaluation objectives relying on the second-stage order allocation. The latter is formulated based on the selected suppliers and newly revealed information. Due to the uncertainty of information, the outcome of the second-stage OA decision remains ambiguous, which consequently renders the performance of the SS decision undetermined. As a result, it is difficult to compare the candidate supplier portfolios. This issue underscores the need for a quantitative measure to estimate the uncertain performance of an SS decision. The Expected Value (EV), a widely adopted measure, has been utilized in prior studies of the uncertain SSOA problem, e.g., [9,10]. Hence, it is applied to estimate the uncertain performance of SS decisions. Moreover, since supplier selection is made under uncertain conditions and carries long-term implications, it is essential to account for the associated risks. Conditional Value-at-Risk (CVaR) is a risk measure that evaluates the expected losses exceeding the Value-at-Risk (VaR) threshold, where VaR is defined as the worst loss of an investment or investment portfolio at a given confidence level [11,12]. CVaR focuses on extreme risk events and is, thus, suitable for assessing potential risks in supplier selection decisions. Previous studies, such as [3,13], have successfully applied CVaR in this field, supporting our decision to incorporate it as a performance measure for supplier selection. Furthermore, given that supplier selection constitutes a classical MCDM problem [14,15], this paper thoroughly evaluates suppliers using both cost and purchasing value. Accordingly, two bi-objective models with different treatments of uncertainty are formulated: the EV-based model (EVM) and the CVaR-based model (CVM).

The core contributions of this research are threefold. First, we propose a novel two-stage framework for modeling the integrated supplier selection and order allocation problem under uncertainty. In this framework, the evaluation objectives of the first-stage supplier selection model are contingent upon the adaptive OA decisions made in the second stage in response to realized operational conditions, while the OA decision space itself in the second-stage model is constrained by the preceding SS choices. Second, to handle the uncertainty in the supplier selection evaluation objectives arising from adaptive OA decisions under stochastic conditions, we employ EV and CVaR to assess the stochastic total cost and purchasing value of the supplier selection decision, thereby formulating stochastic programming models with two objectives and two measures of uncertainty for this problem. Third, we design an integrated solution approach, combining the weighted-satisfaction sum method, Monte Carlo simulation, linearization techniques, and a genetic algorithm (GA), together with the Gurobi solver, to tackle the specific structure of the two-stage bi-objective stochastic models. Numerical experiments are conducted to demonstrate the effectiveness of our proposed methodology.

The remainder of this paper is organized as follows. Section 2 reviews related works and reveals the differences between this study and previous works. Section 3 describes the SSOA problem under uncertainty in detail and formulates this problem as two-stage bi-objective stochastic programming models based on EV and CVaR measures. The solution methodology for the models is presented in Section 4 and demonstrated through extensive numerical examples in Section 5. Conclusions and future research directions are provided in Section 6.

2. Literature Review

In this section, we divide the literature related to our study into three categories: (1) supplier selection, order allocation, and their joint decision; (2) handling uncertainty in the joint decision of supplier selection and order allocation; and (3) decision criteria under randomness. Finally, we summarize the main differences between this paper and previous works.

2.1. Supplier Selection, Order Allocation, and the Joint Decision

The supplier selection process involves the discovery of potential suppliers, suppliers’ evaluation, and the setting up of contracts with the suppliers, playing a vital role in business growth [16]. Earlier studies of the SS problem mainly focused on the construction of supplier evaluation criteria and measurement of supplier performance. For instance, Shyur and Shih [4] presented seven potential criteria (e.g., on-time delivery, product quality, and price/cost) for supplier evaluation and applied the Analytic Network Process (ANP) and Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) to support the selection process. Not determined by the supplier eligibility at the time of SS decision, Hosseininasab and Ahmadi [2] proposed a two-phase SS procedure based on the long-term trend of value, stability, and the relationship of potential suppliers. Alavi et al. [17] further developed a dynamic decision support system for sustainable supplier selection in circular supply chains.

A few studies only addressed the problem of how to allocate orders among the selected suppliers. The OA problem is usually solved by building a mathematical programming model. For example, Esfandiari and Seifbarghy [5] proposed a bi-objective nonlinear integer programming model assuming that the buyer obtains multiple products from a number of predetermined suppliers. Meena and Sarmah [6] developed a mixed-integer non-linear programming (MINLP) model for OA decisions among a set of predetermined suppliers and illustrated that different optimal OA decisions can be derived from different supplier portfolios.

Order allocation in the operational process is based on the selection of strategic suppliers, and the practical performance of supplier selection is influenced by the allocation of purchase orders. These two decisions are inseparable from each other. Therefore, the joint decision of SS and OA has gained a lot of attention. In this framework, typically, the MCDM techniques are first employed to obtain supplier rankings or scores; then, these results are brought into a mathematical programming model that integrates SS and OA decisions. For instance, Mohammed et al. [8] employed analytical hierarchy process (AHP) and TOPSIS methods to evaluate suppliers with green and resilience criteria first, then developed a multi-objective programming model for order allocation by integrating the obtained weights from AHP and TOPSIS. In contrast with commonly used MCDM approaches for SS (such as AHP, ANP, and TOPSIS), a novel amalgamation of data envelopment analysis and fuzzy mixed aggregation methods was proposed recently by Ali et al. [18] to identify inefficient suppliers and rank competent ones in the supplier evaluation phase. Then, the optimal order allocation among incumbent suppliers is derived by solving a multi-choice goal programming method. For a more detailed and comprehensive survey of SSOA, readers are referred to ref. [19] and the numerous references therein.

2.2. Handling Uncertainty in the Joint Decision of SS and OA

The decisions of SS and OA rely on some key information, such as the product price, quality, and delivery time, that significantly impacts the decision outcome. However, in practice, this information is usually uncertain, posing a major obstacle to decision making.

Fuzzy set-based techniques are widely used to handle uncertainty, particularly the cognitive uncertainty in decision makers’ subjective preference and judgment on supplier performance. For instance, Prasannavenkatesan and Goh [20] developed a hybrid method to evaluate and rank suppliers in which fuzzy AHP (FAHP) was applied to determine criteria weights, and the fuzzy preference ranking organization method for enrichment evaluation was used to derive supplier rankings. Hamdan and Cheaitou [21] and Mohammed et al. [14] further combined fuzzy TOPSIS and AHP to assess and rank suppliers. Ali and Zhang [22] also employed FAHP to determine the weights of various factors (e.g., quality, lead time, and cost), which are used in the SS model. However, these factors themselves would be uncertain as well. Therefore, fuzzy numbers have been frequently employed to represent and quantify these uncertain parameters involved in SSOA, and consequently, fuzzy optimization models have been constructed to solve the problem (e.g., [23]).

Besides fuzzy approaches, many studies in the literature have used stochastic methods to tackle the uncertainty in SSOA. For instance, Guo and Li [24] presented an MINLP with stochastic Poisson demand to select suppliers and allocate orders among selected suppliers for a multi-echelon system. Moheb-Alizadeh and Handfield [25] constructed an MINLP model that considers stochastic demand, which it handles with the chance constraint approach. Babbar and Amin [26] developed a novel model that comprises a two-stage quality function deployment and stochastic multi-objective mixed-integer linear programming to select the best suppliers and assign order quantity. In this work, the demand and unit costs of the products were assumed to be random. Mousavi et al. [9] developed an MINLP model in which the lead times are constant and the demands of buyers follow a normal distribution for a two-echelon vendor–buyer supply chain network. Hosseini et al. [27] proposed an efficient solution to the resilient SSOA problem with the consideration of random disruption risk. The method consists of two main steps: first, a probabilistic graphical model is used to calculate the likelihood of disruption scenarios; then, a stochastic bi-objective mixed-integer programming model is constructed to determine the optimal SS and OA decisions. Hosseini et al. [28] employed a hybrid best–worst/evidential reasoning method to evaluate and rank suppliers, then proposed a bi-objective mathematical model considering stochastic suppliers’ availability and demand. Jadidi et al. [29] proposed a multi-supplier news vendor model to determine the optimal order quantity under random buyer demand for a single product. In contrast with these works, Taghavi et al. [30,31] further extended the SSOA problem by considering joint decisions with vehicle routing or production scheduling in the supply chain context, with disruption risk depicted by stochastic scenarios.

The robust optimization approach has also drawn considerable interest with respect to addressing the SSOA problem in recent years. For example, Jia et al. [32] considered uncertain cost, emissions, demand, supply capacity, and minimum order quality and proposed a distributionally robust sustainable SSOA goal programming model that characterizes the distribution uncertainty by ambiguity sets. Mirzaee et al. [33] and Nazari-Shirkouhi et al. [34] proposed multi-objective robust optimization models to handle the uncertainty of parameters. Recently, Asadi et al. [35] considered the SSOA problem under mixed uncertainty and developed a novel multi-stage hybrid model combining stochastic fuzzy BWM, stochastic fuzzy TOPSIS, and Lexicographic Chebyshev revised multi-choice goal programming (LCRMCGP) methods to address this problem, with the uncertainty being handled following the robust fuzzy stochastic programming approach.

2.3. Decision Criteria Under Randomness and Solution Approaches

While the SSOA problem is considered in a stochastic environment, the outcomes of the alternative decisions become random variables. Consequently, in order to explicitly define the best one, appropriate decision criteria should be employed. Expected value is the most commonly used measure. Meena et al. [36] identified the number of suppliers under catastrophic event disruption with the goal of minimizing the expected total cost, which includes purchasing cost, management cost, and expected supplier failure cost. Meena and Sarmah [6] proposed an MINLP model considering different suppliers with failure probability. The model, solved by a GA, aims to find the order allocation scheme that minimizes the expected total cost. Prasannavenkatesan and Goh [20] constructed an MILP model with the objective of minimizing the expected total cost, including management cost, purchasing cost, and the cost of expected loss, and maximizing the total purchase value. A multi-objective particle swarm optimization (PSO) algorithm was used to yield Pareto solutions. Bohner and Minner [37] formulated an MILP model that aims to minimize total expected cost for integrated supplier selection and order allocation with quantity and business volume discounts.

In practice, decision makers are often risk-averse, so two common risk-averse measures—VaR and CVaR—have attracted increasing attention in recent years. For resilient supplier portfolio selection under disruption risks, Sawik [3] proposed a mixed-integer programming (MIP) model with CVaR applied to control the risk of worst-case cost. Sawik [38] further developed four single-objective MIP models with the objectives of minimizing the expected cost, minimizing the CVaR of cost, maximizing the expected service level, and maximizing the expected worst-case service level for the problem of integrated supplier selection and customer order scheduling under a single and dual sourcing strategy, respectively. All models were solved by Cplex. Then, Fang et al. [11] constructed a multi-objective model that aims to find the solutions of supplier portfolio and order allocation that minimize operational risk, disruption risk, and total cost. The operational risk is measured by VaR, and the disruption risk is evaluated by CVaR. The three objectives are handled through TOPSIS, gray relational analysis, and a weighted max–min fuzzy model. Esmaeili-Najafabadi et al. [39] proposed a risk-averse MINLP model based on CVaR for SSOA in centralized supply chains under disruption risks, solved by a PSO algorithm whose efficiency was verified in comparison with a genetic algorithm (GA) and the commercial GAMS solver. Considering random suppliers and regional disruptions, Lee and Moon [40] constructed risk-neutral and risk-averse models by using EV and CVaR to measure the risk associated with disruptions. They also developed a weighted CVaR model that integrates various CVaR values with different tolerance levels. These models were solved using Cplex. Feng et al. [13] constructed polyhedral and box ambiguity sets to describe uncertain probabilities and employed the worst-case mean-CVaR criterion for the second-stage cost, forming a two-stage distributionally robust mean-CVaR model for the SSOA problem. Furthermore, combining VaR and CVaR, Lotfi et al. [41] proposed a new coherent risk criterion—weighted VaR—to handle demand fluctuation, risk, and robustness in a viable SSOA problem.

2.4. Research Gaps

The literature analysis reveals a wealth of research findings reported over the past decade on the supplier selection and order allocation problem under uncertainty.

Table 1. Related studies on SSOA under uncertainty.

Article	Optimization Model Features			Treatment of Uncertainty		Solution Techniques for the Optimization Model
	Structure	Finite Discrete Scenarios	Multi-Objective	Type	Decision Criteria
Guo and Li [24]	One-PM	Unrestricted		Stochastic	EV	MD, CSolver
Mousavi et al. [9]	One-PM	Unrestricted		Stochastic	EV	GA, PSO
Bohner and Minner [37]	One-PM	√		Stochastic	EV	CSolver
Sawik [3]	One-PM	√		Stochastic	EV, CVaR	CSolver
Sawik [38]	One-PM	√		Stochastic	EV, CVaR	CSolver
Esmaeili-Najafabadi et al. [39]	One-PM	√		Stochastic	EV, CVaR	PSO, GA, CSolver
Lotfi et al. [41]	One-PM	√		Stochastic	weighted VaR	CSolver
Feng et al. [13]	Two-PM	√		Ambiguity set	EV, CVaR	RO-TRA, CSolver
Taghavi et al. [31]	Two-PM	√		Stochastic	EV, CVaR	CSolver
Taghavi et al. [30]	Two-PM	√		Stochastic	EV, CVaR	CSolver
Moheb-Alizadeh and Handfield [25]	One-PM	Unrestricted	√	Stochastic	Opt-CC	PCM, WSA, CSolver
Firouzi and Jadidi [23]	One-PM	Unrestricted	√	Fuzzy	EV	WSA, IRM
Jia et al. [32]	One-PM	Unrestricted	√	Ambiguity set	Opt-CC	RO-TRA, CSolver
Nazari-Shirkouhi et al. [34]	One-PM	Unrestricted	√	Ambiguity set	Opt-CC	RO-TRA, WSA, CSolver
Babbar and Amin [26]	One-PM	√	√	Stochastic	EV	WSA, ECM, CSolver
Hosseini et al. [27]	One-PM	√	√	Stochastic	EV	FCM, ECM
Hosseini et al. [28]	One-PM	√	√	Stochastic	EV	DP, ECM, CSolver
Prasannavenkatesan and Goh [20]	One-PM	√	√	Stochastic	EV	PSO
Fang et al. [11]	One-PM	√	√	Stochastic	VaR, CVaR	TOPSIS, GRA, WMF
Lee and Moon [40]	One-PM	√	√	Stochastic	EV, CVaR	WSA, CSolver
Asadi et al. [35]	One-PM	√	√	Mixed	EV, Opt-CC	LCRMCGP, RFSOM
This paper	Two-PM	Unrestricted	√	Stochastic	EV, CVaR	WSA, MCS, GA, CSolver

One-PM: modeling SS and OA decisions in a single one-stage programming model; Two-PM: modeling SS and OA decisions in separate stages and constructing an integrated two-stage programming model; Opt-CC: optimization under chance (and expected) constraints or conservatism levels referring to uncertainty; MD: model decomposition; CSolver: commercial solver (Cplex, Gurobi, Xpress-MP, GAMS, etc.); RO-TRA: robust optimization with tractable reformulation/approximation; PCM: piecewise McCormick envelopes; WSA: weighted-sum (satisfaction/fuzzy) aggregation function; IRM: interactive resolution method; ECM: $\epsilon$-constraint method; FCM: fuzzy C-mean clustering; DP: dynamic programming; GRA: gray relational analysis; WMF: weighted max–min fuzzy model; RFSOM: robust fuzzy stochastic optimization method; MSC: Monte Carlo simulation.

lists some of the aforementioned works in this field, identifying several research gaps.

First, most of these studies model SS and OA decisions in a single one-stage planning model, which does not adequately reflect the real-life characteristics of this problem, i.e., that SS and OA are sequential decisions and interact with each other. In other words, the SS decision is made prior to OA and does not change frequently, and its effectiveness is influenced by subsequent adaptive OA decisions. OA decisions depend on the determined SS plan while also taking into account updates of some critical information. To model this sequential and interactive feature, the present paper constructs a two-stage programming model for the joint optimization of SS and OA.

Although some works (e.g., [13,30,31]) have explicitly considered the sequential two-stage nature of SS and OA decisions, since the uncertain state space was characterized by finite discrete scenarios, the proposed two-stage models were reformulated into a single one-stage mixed-integer programming model. In this paper, we consider the uncertainty represented by stochastic variables distributed in continuous spaces.

Furthermore, considering the complex structure of our proposed models—with conflicting objectives in both SS and OA decisions, in continuous stochastic environments with infinite potential scenarios, and subject to nonlinear constraints—it is still challenging to effectively solve them.

3. Problem Description and Formulation

In a two-echelon supply chain involving a single buyer and multiple potential suppliers within a make-to-order context, the supplier selection and order allocation problem can be described as follows: The buyer needs to procure various products over a long planning horizon, which is divided into multiple short periods. The potential suppliers can supply one or more products that the buyer needs, and they are distinct from each other in product quality, late delivery rate, production capacity, discount policies, etc. Because long-term partnership can help reduce cost and improve productivity, the buyer prefers to choose one or more potential suppliers (a supplier portfolio) to sign supply contracts and build long-term partnerships before the start of the planning horizon. After that (during the planning horizon), the buyer determines the quantity to order from each selected supplier according to current demand and suppliers’ information for each period. If the demand cannot be met by the selected suppliers, the buyer can purchase from the spot market, which offers all products without volume restrictions but at a higher unit price.

This problem clearly involves two stages of sequential decision making: first-stage strategic supplier selection, and second-stage operational order allocation, with the latter being contingent upon the former. In the supplier selection process, it is essential to comprehensively evaluate the performance of each feasible supplier portfolio based on multiple criteria to identify the optimal selection. In this paper, we employ two conventional criteria—cost and purchasing value—to assess the performance of the SS decision. However, the evaluation of each supplier portfolio on the basis of these criteria depends on some key parameters (e.g., demand, price, defect rate, and late delivery rate) and the order quantities determined in the operational stage. In other words, the performance of the SS decision is influenced by the parameter values of the subsequent operational stage. Considering this, we propose a two-stage bi-objective interactive modeling framework for the SSOA problem. In the first stage, the total cost (TC) and total purchasing value (TPV), encompassing the cost and purchasing value of the OA decision, are used to evaluate the SS decision. In the second stage, the OA decision is made based on the predetermined SS decision.

When the data used to assess supplier portfolio performances are deterministic, the definite TC and TPV of an SS decision can be derived; then, the decision maker can easily select out the optimal decision from all possible supplier portfolios by comparing their performance. However, in real life, when making the SS decision, future specific OA decisions and the associated costs and purchasing values are still unknown because precise knowledge about some key parameters for OA decisions cannot be gained in advance due to the volatility of the environment. Consequently, the TC and TPV of a supplier portfolio are not fixed and become stochastic variables dependent on uncertain parameters and adaptive OA decisions. To determine the optimal supplier portfolio, we introduce two measures—expected value and conditional value-at-risk—and develop two corresponding models—EVM and CVM—for the SS decision. Figure 1 illustrates the whole decision-making process for the SSOA problem under uncertainty.

We construct the two-stage bi-objective stochastic programming model under the following assumptions:

(1)

Inventory is not considered, as is common practice for perishable or time-sensitive goods with limited shelf lives. Such products should typically be sold or consumed within the current period and are unsuitable for carrying over for future use.

(2)

The spot market offers all products, and the supply of each product is unlimited but at a higher price.

(3)

Several suppliers can be selected simultaneously, but the total number of selected suppliers is limited.

(4)

Each supplier can supply at least one type of product with different price levels depending on order quantity, following an incremental discount policy.

(5)

For each product type, the total defect rate and late delivery rate must be maintained below acceptable thresholds. A penalty applies to each defective or late-delivered item.

(6)

The realizations of uncertain parameters in period $t+1$ are known after the OA decision in period t has been made and before making the OA decision for the current period.

The notations used to mathematically formulate the problem are displayed in

Table 2. Notation  description.

Sets:
I:	Set of suppliers, indexed by i;
J:	Set of products, indexed by j;
$L=\left\{l\right|l=1,2,\dots ,m\}$:	Set of price discount levels, indexed by l;
$T=\left\{t\right|t=1,2,\dots ,H\}$:	Set of the periods, indexed by t;
Deterministic parameters:
$F_i$:	Fixed cost of cooperating with supplier i;
$O_{i,j,t}$:	Cost of ordering product j from supplier i in period t;
$v_{i,t}$:	Unit transportation cost from supplier i in period t;
$N_j^{max}$:	Maximum number of suppliers that can be selected for product j;
$p_{j,t}^m$:	Emergency purchase price from the spot market for product j in period t;
$q_{i,j,t}^{min}$:	Minimum order quantity for product j from supplier i in period t;
$q_{i,j,t}^{max}$:	Supply capacity of supplier i for product j in period t;
$b_{i,j,t,l}$:	The l-th break point of the price discount provided by supplier i for product j
	in period t ($b_{i,j,t,0}=q_{i,j,t}^{min}$ and $b_{i,j,t,m}=q_{i,j,t}^{max}$);
$p_{i,j,t,l}$:	Price of product j purchased from supplier i in period t at discount level l;
${\overline{R}}_j$:	Largest acceptable defect rate for product j;
$c_j^r$:	Penalty cost for unit defective product j;
${\overline{A}}_j$:	Largest acceptable late delivery rate for product j;
$c_j^a$:	Penalty cost for unit late-delivered product j;
${\omega }_{i,j,t}$:	Value of purchasing product j from supplier i in period t;
Stochastic parameters:
${\tilde{D}}_{j,t}$:	Stochastic demand of product j in period t;
${\tilde{R}}_{i,j,t}$:	Stochastic defect rate of product j ordered from supplier i in period t;
${\tilde{A}}_{i,j,t}$:	Stochastic late delivery rate of product j ordered from supplier i in period t;
${\mathit{\xi }}_t$:	Stochastic vector in period t, ${\mathit{\xi }}_t=({\tilde{D}}_{j,t},{\tilde{R}}_{i,j,t},{\tilde{A}}_{i,j,t}|i\in I,j\in J)$;
$\mathit{\xi }$:	Stochastic vector, $\mathit{\xi }=({\mathit{\xi }}_1,{\mathit{\xi }}_2,\dots ,{\mathit{\xi }}_H)$;
Decision variables:
$x_i$:	$x_i=1$ if supplier i is selected and $x_i=0$ otherwise;
$y_{i,j,t}$:	$y_{i,j,t}=1$ if supplier i is selected to supply product j in period t
	and $y_{i,j,t}=0$ otherwise;
$y_{i,j,t,l}$:	$y_{i,j,t,l}=1$ if the quantity of product j supplied by supplier i in period t falls
	within the l-th interval of the price discount policy and $y_{i,j,t,l}=0$ otherwise;
$q_{i,j,t}$:	Quantity of product j ordered from supplier i in period t;
$q_{i,j,t,l}$:	Quantity of product j ordered from supplier i at discount level l in period t;
$q_{j,t}^m$:	Quantity of product j purchased from the spot market in period t.

. For the sake of discussion, we assume that some of the key parameters, i.e., product demand ($\tilde{D}$), defect rate ($\tilde{R}$), and late delivery rate ($\tilde{A}$) are independent stochastic numbers. The set of all scenarios with possible realizations of ${\tilde{D}}_{j,t}, {\tilde{R}}_{i,j,t}, {\tilde{A}}_{i,j,t}$ is denoted as S. A random scenario ($\mathit{s}\in S$) is composed of H sub-scenarios associated with H periods, denoted as $\mathit{s}=(s_1,s_2,\dots ,s_H)$. Incidentally, when other parameters, such as the emergency purchase price and transportation cost, are also uncertain, the proposed model and solution methodology can be similarly applied.

At the first stage, when $t=0$, the buyer is required to choose a supplier portfolio represented by $\mathit{x}=\left(x_i\right|i\in I)$ for the subsequent planning horizon without knowing the realizations of stochastic vector $\mathit{\xi }$. To fully evaluate the performance of a supplier portfolio, two criteria—cost and purchasing value—are applied. The total cost of a supplier portfolio includes the fixed cost of cooperating with the selected suppliers and the procurement costs associated with the second-stage operation. The total purchasing value is equal to the purchasing value of the second-stage operation. For a potential scenario ($\mathit{s}\in S$), the realizations of random vectors ${\mathit{\xi }}_t$ and $\mathit{\xi }$ are denoted as ${\mathit{\xi }}_t\left(s_t\right)$ and $\mathit{\xi }\left(\mathit{s}\right)=({\mathit{\xi }}_1\left(s_1\right), {\mathit{\xi }}_2\left(s_2\right), \dots , {\mathit{\xi }}_H\left(s_H\right))$, respectively. Let $C_t(\mathit{x},{\mathit{\xi }}_t\left(s_t\right))$ and $P{V}_t(\mathit{x},{\mathit{\xi }}_t\left(s_t\right))$ represent the procurement cost and purchasing value of period t in the second stage under scenario $s_t$. The total cost and total purchasing value for a given supplier portfolio ($\mathit{x}$) with respect to a scenario ($\mathit{s}$) can be expressed as Equations (1) and (2), respectively.

$$\mathit{TC}(\mathit{x},\mathit{\xi }\left(\mathit{s}\right))=\sum _{i\in I}{F}_i{x}_i+\sum _{t\in T}{C}_t(\mathit{x},{\mathit{\xi }}_t\left(s_t\right))$$

(1)

$$TPV(\mathit{x},\mathit{\xi }\left(\mathit{s}\right))=\sum _{t\in T}P{V}_t(\mathit{x},{\mathit{\xi }}_t\left(s_t\right))$$

(2)

Obviously, the $TC(\mathit{x},\mathit{\xi }(\mathit{s}\left)\right)$ and $TPV(\mathit{x},\mathit{\xi }(\mathit{s}\left)\right)$ formulated in Equations (1) and (2) are real-valued functions with respect to a real-valued vector ($\mathit{\xi }\left(\mathit{s}\right)$). Considering all possible realizations ($\mathit{\xi }\left(\mathit{s}\right),\mathit{s}\in S$), the total cost ($TC(\mathit{x},\mathit{\xi })$) and total purchasing value ($TPV(\mathit{x},\mathit{\xi })$) of a supplier selection decision ($\mathit{x}$) become random variables. Since the total cost and purchasing value of each supplier portfolio are random variables, they cannot be directly compared, and appropriate decision criteria are required to determine their optimal values. In this study, we apply two widely used measures—EV and CVaR—to compare the $TC(\mathit{x},\mathit{\xi })$ and $TPV(\mathit{x},\mathit{\xi })$ of an SS decision.

The expected value reflects the average level of a random variable, taking into account all possible scenarios. Using the EV measure, the overall performance of a supplier portfolio can be jointly determined by the expected total cost and expected total purchasing value. Thus, the first-stage EV-based supplier selection model (EVM) can be formulated as follows:

$$\begin{array}{cc}min Z_1=\hfill & \mathbb{E}\left[\mathit{TC}(\mathit{x},\mathit{\xi })\right]\end{array}$$

(3)

$$\begin{array}{cc}max Z_2=\hfill & \mathbb{E}\left[\mathit{TPV}(\mathit{x},\mathit{\xi })\right]\\ \mathrm{subject} \mathrm{to}:\hfill \end{array}$$

(4)

$$\begin{array}{c}{x}_i\in \{0,1\},\forall i\in I,\hfill \end{array}$$

(5)

where the objective functions aim to minimize the expected total cost and maximize the expected total purchasing value. Constraint (5) defines the domain for the decision variables.

In the EVM, the performance of a supplier selection decision is assessed by the mean level of total cost and purchasing value. However, in real life, most decision makers are risk-averse and have varying degrees of risk aversion. They usually focus more on the worst-case scenario of a decision. Considering this, we also apply CVaR, a risk measure that focuses on extreme risk events and assesses the expected losses exceeding the VaR threshold. For a random variable (G) representing a stochastic loss distribution and a specified confidence level ($\alpha \in (0,1]$), CVaR is defined by [42]

$${\mathrm{CVaR}}_{\alpha }\left(G\right)=\mathbb{E}\left[G\right|G\ge {\mathrm{VaR}}_{\alpha }\left(G\right)],$$

(6)

where ${\mathrm{VaR}}_{\alpha }\left(G\right)=min\left\{\lambda \right|\mathrm{Prob}[G\le \lambda ]\ge \alpha \}$ is the $\alpha$ quantile of the loss distribution, indicating the maximum potential loss at the given confidence level. The greater the specified confidence level ($\alpha$), the more attention paid to potential extreme losses. Based on the CVaR measure, the first-stage supplier selection model (CVM) that seeks to maximize the expected purchasing value while controlling extreme loss is expressed as follows:

$$\begin{array}{c}min{Z}_3={\mathrm{CVaR}}_{\alpha }\left(TC(\mathit{x},\mathit{\xi })\right)\hfill \\ max{Z}_2\hfill \\ \mathrm{subject} \mathrm{to}:\hfill \\ x_i\in \{0,1\},\forall i\in I.\hfill \end{array}$$

(7)

The CVM has two objective functions: minimizing the CVaR of $\mathit{TC}(\mathit{x},\mathit{\xi })$ at confidence level $\alpha$ and maximizing the EV of $\mathit{TPV}(\mathit{x},\mathit{\xi })$, subject to the same constraints as the EVM. It should be noted that, considering the criteria of EV and CVaR, there may be two other models: one with the objectives of minimizing ${\mathrm{CVaR}}_{\alpha }\left(TC(\mathit{x},\mathit{\xi })\right)$ and maximizing ${\mathrm{CVaR}}_{\alpha }\left(\mathit{TPV}(\mathit{x},\mathit{\xi })\right)$ and the other with the objectives of minimizing $\mathbb{E}\left[\mathit{TC}\right(\mathit{x},\mathit{\xi }\left)\right]$ and maximizing ${\mathrm{CVaR}}_{\alpha }\left(\mathit{TPV}(\mathit{x},\mathit{\xi })\right)$. Since their structures are similar to the EVM and CVM, they are not discussed in this paper.

After signing contracts with the suppliers selected in the first stage, the buyer transitions to the operational phase, where the partnership is maintained throughout the planning horizon, which is divided into H periods. During each period, the buyer must make order allocation decisions to ensure ongoing production and operations. At the beginning of each period, the values of ${\tilde{D}}_{j,t},{\tilde{R}}_{i,j,t},{\tilde{A}}_{i,j,t}$ are observed, meaning that ${\mathit{\xi }}_t\left(s_t\right)$ is known. Consequently, for given $\mathit{x}$ and ${\mathit{\xi }}_t\left(s_t\right)$, the order allocation decision (${\mathit{q}}_t$) for period t can be derived by solving the single-period order allocation model (OAM-t) as follows:

$$\begin{array}{c}\begin{array}{cc}\hfill min & C_t(\mathit{x},{\mathit{\xi }}_t\left(s_t\right))=\sum _{i\in I}\sum _{j\in J}\sum _{l\in L}{p}_{i,j,t,l}{q}_{i,j,t,l}+\sum _{j\in J}{p}_{j,t}^m{q}_{j,t}^m+\sum _{i\in I}\sum _{j\in J}{O}_{i,j,t}{y}_{i,j,t}\hfill \\ \hfill & +\sum _{i\in I}\sum _{j\in J}{v}_{i,t}{q}_{i,j,t}+\sum _{i\in I}\sum _{j\in J}{c}_j^r{\tilde{R}}_{i,j,t}\left(s_t\right)q_{i,j,t}+\sum _{i\in I}\sum _{j\in J}{c}_j^a{\tilde{A}}_{i,j,t}\left(s_t\right)q_{i,j,t}\hfill \end{array}\hfill \end{array}$$

(8)

$$\begin{array}{c}maxP{V}_t(\mathit{x},{\mathit{\xi }}_t\left(s_t\right))=\sum _{i\in I}\sum _{j\in J}{\omega }_{i,j,t}{q}_{i,j,t},\hfill \\ \mathrm{subject} \mathrm{to}\hfill \end{array}$$

(9)

$$\begin{array}{c}\begin{array}{c}\hfill q_{i,j,t,l}=(q_{i,j,t}-b_{i,j,t,l-1})y_{i,j,t,l}+(b_{i,j,t,l}-b_{i,j,t,l-1})\sum _{k=l+1}^m{y}_{i,j,t,k},\\ \hfill \forall i\in I,j\in J,l\in \{1,2,\dots ,m-1\}.\end{array}\hfill \end{array}$$

(10)

$$\begin{array}{c}{q}_{i,j,t,m}=(q_{i,j,t}-b_{i,j,t,m-1})y_{i,j,t,m},\forall i\in I,j\in J.\hfill \end{array}$$

(11)

$$\begin{array}{c}{q}_{i,j,t}^{min}{y}_{i,j,t}\le q_{i,j,t}\le q_{i,j,t}^{max}{y}_{i,j,t},\forall i\in I,j\in J.\hfill \end{array}$$

(12)

$$\begin{array}{c}\sum _{i\in I}{q}_{i,j,t}+q_{j,t}^m={\tilde{D}}_{j,t}\left(s_t\right),\forall j\in J.\hfill \end{array}$$

(13)

$$\begin{array}{c}{q}_{i,j,t,l}\le b_{i,j,t,l}-b_{i,j,t,l-1},\forall i\in I,j\in J,l\in L.\hfill \end{array}$$

(14)

$$\begin{array}{c}\sum _{l\in L}{y}_{i,j,t,l}=y_{i,j,t},\forall i\in I,j\in J.\hfill \end{array}$$

(15)

$$\begin{array}{c}{y}_{i,j,t}\le x_i,\forall i\in I.\hfill \end{array}$$

(16)

$$\begin{array}{c}\sum _{i\in I}{y}_{i,j,t}\le N_j^{max},\forall j\in J.\hfill \end{array}$$

(17)

$$\begin{array}{c}\sum _{i\in I}{\tilde{R}}_{i,j,t}\left(s_t\right)q_{i,j,t}\le {\overline{R}}_j{\tilde{D}}_{j,t}\left(s_t\right),\forall j\in J.\hfill \end{array}$$

(18)

$$\begin{array}{c}\sum _{i\in I}{\tilde{A}}_{i,j,t}\left(s_t\right)q_{i,j,t}\le {\overline{A}}_j{\tilde{D}}_{j,t}\left(s_t\right),\forall j\in J.\hfill \end{array}$$

(19)

$$\begin{array}{c}{q}_{i,j,t},q_{i,j,t,l},q_{j,t}^m\ge 0,\forall i\in I,j\in J,l\in L.\hfill \end{array}$$

(20)

$$\begin{array}{c}{y}_{i,j,t},y_{i,j,t,l}\in \{0,1\},\forall i\in I,j\in J,l\in L.\hfill \end{array}$$

(21)

The first objective Function (8) aims to minimize the order allocation cost in period t, which includes the cost of purchasing from the selected suppliers and the spot market, as well as ordering, transportation, and penalty costs for delayed and defective products. The second objective function (9) seeks to maximize the purchasing value for order allocation in period t. Constraints (10) and (11) show the relationship between the total order quantity of product j purchased from supplier i in period t and the respective quantity at each discount level. Constraint (12) requires that the total quantity of each type of product purchased from each supplier exceed the minimum acceptable order quantity and be within the supplier’s capacity. Constraint (13) ensures a balance between supplies and demands. Constraint (14) stipulates that the ordered quantity at each discount level must lie within the given quantity interval. Constraint (15) ensures that the order quantity associated with each supplier and each product type falls into, at most, one specific discount level. Constraint (16) ensures all available products of a supplier can be supplied, as long as this supplier is chosen at the first stage. For each type of product, constraint (17) limits the number of chosen suppliers to the maximum allowed. Constraints (18) and (19) ensure defect and late delivery rates remain within acceptable limits. Finally, constraints (20) and (21) impose integrality and non-negativity restrictions on decision variables.

By combining Equations (1)–(5) with Equations (8)–(21) and Equations (1) and (2) with Equations (4)–(21), we formulate two-stage bi-objective stochastic programming models based on the EV and CVaR measures, respectively. Owing to their complex structures—two stages, two objectives, and stochastic parameters—we are motivated to develop an effective solution methodology to solve these models in the following section.

4. Solution Methodology

In this section, we devise a solution methodology to solve the two-stage bi-objective stochastic programming models presented in the previous section.

4.1. Overall Solution Framework

Considering the structure of our proposed models, there are several challenges in effectively solving them:

• Two conflicting objectives should be considered throughout the whole planning horizon—not only for the first-stage SS decision but also for the OA in each period of the second stage. The treatment of these two objectives must remain consistent and logically sound throughout the entire decision-making process.
• In a continuous stochastic environment, the problem involves an infinite number of potential scenarios, making the accurate evaluation of the decision objectives computationally intensive.
• The constraints of the subproblem for the OA decision in each period of the second stage are nonlinear.

To address these issues, a simulation-based heuristic approach with a nested structure is proposed, corresponding to the two-stage and multi-period feature of the problem. The overall solution framework for the two-stage bi-objective stochastic programming models is summarized in Figure 2. At the top level, which solves the first-stage SS problem, a standard genetic algorithm is employed to iteratively search for the optimal solution among candidate solutions. The pseudocode of this procedure is presented in Algorithm 1, and its main steps are explained in Appendix A.

The first-stage models contain two objectives, with $\mathit{TC}(\mathit{x},\mathit{\xi })$ and $\mathit{TPV}(\mathit{x},\mathit{\xi })$ being random variables dependent on $\mathit{x}$ and $\mathit{\xi }$, so we cannot directly use the existing exact methods to solve the model. The challenges in determining the optimal supplier portfolio in the first stage include deriving the distributions of $\mathit{TC}(\mathit{x},\mathit{\xi })$ and $\mathit{TPV}(\mathit{x},\mathit{\xi })$ and addressing bi-objective optimization. Although the realized values of the second-stage stochastic parameters in $\mathit{\xi }$ are unknown at the time of making the SS decision, their distribution information is available in advance. Given an SS decision ($\mathit{x}$) and a particular scenario of $\mathit{\xi }$, the ${\sum }_{t\in T}{C}_t(\mathit{x},{\mathit{\xi }}_t\left(s_t\right))$ and ${\sum }_{t\in T}P{V}_t(\mathit{x},{\mathit{\xi }}_t\left(s_t\right))$ can be obtained by solving the OAM-t H times and summing up. Subsequently, the values of $\mathit{TC}(\mathit{x},\mathit{\xi }(\mathit{s}\left)\right)$ and $\mathit{TPV}(\mathit{x},\mathit{\xi }(\mathit{s}\left)\right)$ for the given $\mathit{x}$ and $\mathit{\xi }$ can be calculated using Equations (1) and (2). By running a large number of sample simulations, an approximate distribution of $\mathit{TC}(\mathit{x},\mathit{\xi })$ and $\mathit{TPV}(\mathit{x},\mathit{\xi })$ of a feasible $\mathit{x}$ can be derived. Based on these distributions, $\mathbb{E}\left[\mathit{TC}(\mathit{x},\mathit{\xi })\right]$, $\mathbb{E}\left[\mathit{TPC}(\mathit{x},\mathit{\xi })\right]$, and ${\mathrm{CVaR}}_{\alpha }\left(\mathit{TC}(\mathit{x},\mathit{\xi })\right)$ can be estimated.

In this process, solving the OAM-t subproblem also presents challenges due to its nonlinear constraints. These constraints are translated into linear constraints using the method outlined in Section 4.2.

To handle the two objectives of both SS and OA in a consistent way, the objective function values are transformed into satisfaction levels referring to the possible outcomes of the problem. Then, the two objectives are combined into a single objective using the weighted-satisfaction sum method. Consequently, the optimal SS decision can be identified by comparing the combined satisfaction levels of each SS decision. The details of the treatment of the objectives are given in Section 4.3. The evaluation of candidate solutions (each particular SS decision ($\mathit{x}$)) is elaborated upon in Section 4.4.

Algorithm 1: Pseudocode of solution algorithm for SSOA

Input: N, $N^{\prime }$, all data involved in the models and GA parameters Output: Optimal portfolio ${\mathit{x}}^{*}$ Import all needed data; Generate N samples of $\mathit{\xi }$, and $N^{\prime }$ samples of ${\mathit{\xi }}^{-}$ and ${\mathit{\xi }}^{+}$ respectively; $C^{\mathit{PIS}/\mathit{NBS}}$, $P{V}^{\mathit{PIS}/\mathit{NBS}}$, $E_{\mathit{TC}}^{\mathit{PIS}/\mathit{NBS}}$, $E_{\mathit{TPV}}^{\mathit{PIS}/\mathit{NBS}}$, and $\mathit{CV}a{R}_{\mathit{TC}}^{\mathit{PIS}/\mathit{NBS}}⟵$ Algorithm 2; Generate the initial population ${\mathit{POP}}_0$; Initialize $t=0$, ${\mathit{POP}}_0^E={\mathit{POP}}_0^C={\mathit{POP}}_0$; Initialize $\mathit{Dict}=\left\{\right\}$, Note: $\mathit{Dict}=\{\mathit{key}:\mathit{value}\}$, “key” is chromosome, “value” is $({\mu }^{1E},{\mu }^{1C})$; ${\mathit{UPOP}}_0⟵$ Get the unique item in ${\mathit{POP}}_0$; ${\mathit{UFIT}}_0^E$, ${\mathit{UFIT}}_0^C⟵$ Algorithm 3 (${\mathit{UPOP}}_0$); $\mathit{Dict}⟵$ Append key-value item (${\mathit{UPOP}}_0$, ${\mathit{UFIT}}_0^E$, ${\mathit{UFIT}}_0^C$); ${\mathit{FIT}}_0^E$, ${\mathit{FIT}}_0^C⟵$ Match ${\mathit{POP}}_0$ with the key of $\mathit{Dict}$ and return value;

4.2. Linearization of Nonlinear Constraints in OAM-t

Constraints (12) and (13) in OAM-t are nonlinear. To linearize these constraints, we introduce an auxiliary variable ($z_{i,j,t,l}=q_{i,j,t}{y}_{i,j,t,l}$) and a large constant (M). The resulting linear constraints are presented in Equations (22)–(26).

$$\begin{array}{c}{q}_{i,j,t,l}=z_{i,j,t,l}-b_{i,j,t,l-1}{y}_{i,j,t,l}+(b_{i,j,t,l}-b_{i,j,t,l-1})\sum _{k=l+1}^m{y}_{i,j,t,k},\forall i\in I,j\in J,l\in L\setminus m.\hfill \end{array}$$

(22)

$$\begin{array}{c}{q}_{i,j,t,m}=z_{i,j,t,m}-b_{i,j,t,m-1}{y}_{i,j,t,m},\forall i\in I,j\in J.\hfill \end{array}$$

(23)

$$\begin{array}{c}{z}_{i,j,t,l}\le q_{i,j,t}+(1-y_{i,j,t,l})M,\forall i\in I,j\in J,l\in L.\hfill \end{array}$$

(24)

$$\begin{array}{c}{z}_{i,j,t,l}\ge q_{i,j,t}-(1-y_{i,j,t,l})M,\forall i\in I,j\in J,l\in L.\hfill \end{array}$$

(25)

$$\begin{array}{c}-y_{i,j,t,l}M\le z_{i,j,t,l}\le y_{i,j,t,l}M,\forall i\in I,j\in J,l\in L.\hfill \end{array}$$

(26)

4.3. Coping with Dual Objectives

In both the first and second stages, the decision maker faces two conflicting objectives that cannot be optimized simultaneously. This research employs the weighted-satisfaction sum method to combine the dual objectives into a single objective. In this method, the satisfaction value for each objective function is first calculated using Equation (27). Each satisfaction value is then multiplied by its corresponding weight, and the weighted satisfaction values are summed.

$$\mu \left(g\right)=\left\{\begin{array}{cc}{\displaystyle \frac{g^{nbs}-g}{g^{nbs}-g^{pis}}},\hfill & \mathrm{for} \mathrm{minimization} \mathrm{type} \mathrm{objective} \mathrm{function}\hfill \\ {\displaystyle \frac{g-g^{nbs}}{g^{pis}-g^{nbs}}},\hfill & \mathrm{for} \mathrm{maximization} \mathrm{type} \mathrm{objective} \mathrm{function}\hfill \end{array}\right.$$

(27)

where $\mu \left(g\right)$ is the satisfaction value of the objective function (g), $g^{pis}$ is the positive ideal solution (PIS) or best solution of the objective function (g), and $g^{nbs}$ is a negative bound (NBS) of g such that the solution is non-inferior while taking multiple objectives into account. For example, for the minimization (maximization) objective, a solution with an objective value of $g>g^{nbs}$ ($g<g^{nbs}$) is an inferior solution with negative satisfaction. Note that here, the NBS is not the typical negative ideal solution used in the TOPSIS approach. This bound offers a satisfaction function ($\mu \left(g\right)$) with improved discrimination among Pareto-optimal solutions and the capability of identifying inferior solutions. A similar approach can also be seen in ref. [43].

For a deterministic model, the PIS and NBS for each objective function can be obtained by optimizing each objective separately. However, since our model involves stochastic parameters, multiple periods, and a two-stage framework, PIS and NBS cannot be derived directly using this approach. Considering that the OAM-t subproblem is deterministic under a specific scenario, we employ a scenario-based simulation to determine the PIS and NBS, as outlined in Algorithm 2. Note that it is easy to verify that for the OAM-t subproblem under a fixed scenario, the optimal values of both objectives—$C_t$ and $P{V}_t$—increase with respect to the realized demand. Consequently, their extreme values correspond to the scenarios with the lowest and highest demands. Therefore, to reduce computational effort, two types of scenario samples are generated for the simulation, denoted as ${\mathit{\xi }}^{-}$ and ${\mathit{\xi }}^{+}$, in which the realized demands are fixed in correspondence with the lower and upper bounds of ${\tilde{D}}_{j,t}\left(s_t\right)$, whereas the other stochastic parameters are randomly generated. In other words, we have ${\mathit{\xi }}_t^{-}=(d_{j,t}^{-},{\tilde{R}}_{i,j,t},{\tilde{A}}_{i,j,t})$ and ${\mathit{\xi }}_t^{+}=(d_{j,t}^{+},{\tilde{R}}_{i,j,t},{\tilde{A}}_{i,j,t})$, where $d_{j,t}^{-}$ and $d_{j,t}^{+}$ are the lower and upper bounds of ${\tilde{D}}_{j,t}\left(s_t\right)$, respectively.

Algorithm 2 provides the PIS and NBS for $C_t(\mathit{x},{\mathit{\xi }}_t\left(s_t\right))$ and $P{V}_t(\mathit{x},{\mathit{\xi }}_t\left(s_t\right))$ of the OA decision in each period; thus, we can calculate their satisfaction values based on Equation (27). Let $\beta$ denote the weight assigned to the satisfaction related to cost, while the weight assigned to the purchasing value is $1-\beta$. Consequently, we can transform the OAM-t sub-problem into an equivalent model as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}max{\mu }_t^2=\beta {\displaystyle \frac{C^{\mathit{nbs}}-C_t(\mathit{x},{\mathit{\xi }}_t\left(s_t\right))}{C^{\mathit{nbs}}-C^{\mathit{pis}}}}+(1-\beta ){\displaystyle \frac{P{V}_t(\mathit{x},{\mathit{\xi }}_t\left(s_t\right))-P{V}^{\mathit{nbs}}}{P{V}^{\mathit{pis}}-P{V}^{\mathit{nbs}}}}\hfill \\ \mathrm{subject} \mathrm{to}:\hfill \\ \left(12\right)–\left(21\right),\left(22\right)–\left(26\right).\hfill \end{array}\right.\end{array}$$

(28)

The resulting model (28) is a typical MILP model that can be solved by applying some popular optimization solvers, e.g., Gurobi, Cplex, and Lingo. In the present paper, we utilize Gurobi in Python environment to solve it.

Algorithm 2: Determining PIS and NBS for first-stage and second-stage models

Input: $N^{\prime }$ samples of ${\mathit{\xi }}^{-}$ and ${\mathit{\xi }}^{+}$ Output: $C^{\mathit{pis}}$, $C^{\mathit{nbs}}$, ${\mathit{PV}}^{\mathit{pis}}$, ${\mathit{PV}}^{\mathit{nbs}}$, ${\mathit{TC}}^{\mathit{pis}}$, ${\mathit{TC}}^{\mathit{nbs}}$, ${\mathit{TPV}}^{\mathit{pis}}$, ${\mathit{TPV}}^{\mathit{nbs}}$

4.4. Evaluation of Candidate Solutions

To evaluate the performance of candidate solutions, let N be a large number and denote the size of samples randomly generated for the stochastic vector ($\mathit{\xi }$) in the simulation. A particular sample in the simulation is indicated by $s^n$, $n=1,\dots ,N$. For a given supplier portfolio, the expected values of $TC(\mathit{x},\mathit{\xi })$ and $TPV(\mathit{x},\mathit{\xi })$ can be approximated by Equations (29) and (30). Note that the larger the value of N is, the higher the accuracy of the approximation is but the more computational efforts are required. Trade-offs should be made for parameter settings.

$$E\left(\mathit{TC}(\mathit{x},\mathit{\xi })\right)={\displaystyle \frac{1}{N}}\sum _{n=1}^N\mathit{TC}(\mathit{x},\mathit{\xi }\left(s^n\right))$$

(29)

$$E\left(\mathit{TPV}(\mathit{x},\mathit{\xi })\right)={\displaystyle \frac{1}{N}}\sum _{n=1}^N\mathit{TPV}(\mathit{x},\mathit{\xi }\left(s^n\right))$$

(30)

To calculate CVaR, the $\mathit{TC}(\mathit{x},\mathit{\xi }\left(s^n\right))$ values of all samples are arranged in order from smallest to largest. The sorted set is denoted as $\mathit{STC}(\mathit{x},\mathit{\xi }\left(s^{n^{\prime }}\right))$ and indexed by $n^{\prime }$. Thus, according to the definition of CVaR, the CVaR of $\mathit{TC}(\mathit{x},\mathit{\xi })$ at confidence level $\alpha$ is estimated as follows:

$${\mathit{CvaR}}_{\alpha }\left(\mathit{TC}(\mathit{x},\mathit{\xi })\right)={\displaystyle \frac{1}{(1-\alpha )N}}\sum _{n^{\prime }=\lceil \alpha ·N\rceil }^N\mathit{STC}(\mathit{x},\mathit{\xi }\left(s^{n^{\prime }}\right))$$

(31)

Although the objective function values in the EVM and CVM are calculated by (29)–(31), it is still unclear which supplier portfolio is better due to the presence of two objective functions. Therefore, the approach introduced in Section 4.3 is applied to derive a single indicator for each first-stage model, i.e., ${\mu }^{1E}$ for the EVM and ${\mu }^{1C}$ for the CVM, to evaluate the performance of a supplier portfolio. ${\mu }^{1E}$ and ${\mu }^{1C}$ are calculated formally as follows:

$${\mu }^{1E}=\beta \ast {\displaystyle \frac{\mathbb{E}\left[\mathit{TC}(\mathit{x},\mathit{\xi })\right]-T{C}^{\mathit{nbs}}}{T{C}^{\mathit{pis}}-T{C}^{\mathit{nbs}}}}+(1-\beta )\ast {\displaystyle \frac{{\mathit{TPV}}^{\mathit{nbs}}-\mathbb{E}\left[TPV(\mathit{x},\mathit{\xi })\right]}{{\mathit{TPV}}^{\mathit{nbs}}-{\mathit{TPV}}^{\mathit{pis}}}}$$

(32)

$${\mu }^{1C}=\beta \ast {\displaystyle \frac{{\mathit{CVaR}}_{\alpha }\left(\mathit{TC}(\mathit{x},\mathit{\xi })\right)-T{C}^{\mathit{nbs}}}{{\mathit{TC}}^{\mathit{pis}}-{\mathit{TC}}^{\mathit{nbs}}}}+(1-\beta )\ast {\displaystyle \frac{{\mathit{TPV}}^{\mathit{nbs}}-\mathbb{E}\left[\mathit{TPV}(\mathit{x},\mathit{\xi })\right]}{{\mathit{TPV}}^{\mathit{nbs}}-{\mathit{TPV}}^{\mathit{pis}}}}$$

(33)

where the determinations of ${\mathit{TC}}^{\mathit{pis}}$, ${\mathit{TC}}^{\mathit{nbs}}$, ${\mathit{TPV}}^{\mathit{pis}}$, and ${\mathit{TPV}}^{\mathit{nbs}}$ are given in the previous subsection.

The process of evaluating candidate solutions is summarized by Algorithm 3.

Algorithm 3: Evaluating candidate solutions (fitness of chromosomes)

Input: N samples of $\mathit{\xi }$, population $\mathit{POP}$, and empty fitness set ${\mathit{FIT}}^E$ and ${\mathit{FIT}}^C$ Output: Set ${\mathit{FIT}}^E$ and ${\mathit{FIT}}^C$

5. Numerical Experiment

In this section, extensive numerical experiments are conducted to demonstrate the effectiveness of the proposed two-stage bi-objective stochastic programming model and solution methodology. The solution algorithms were implemented in Python 3.7.4, and all experiments were conducted on an ordinary personal computer with an Intel i5 2.4 GHz processor, 8 GB RAM, and a 64-bit operating system.

5.1. Parameter Settings

In a two-echelon supply chain, the buyer works on selecting a supplier portfolio and allocating orders to ensure the supply of products in the planning horizon. The aforementioned models and solution algorithms are proposed to support buyers in decision making. To verify the feasibility of the proposed models and solution method, we designed instances of different sizes for experimental testing.

Table 3. Parameter ranges for generating instances.

Parameter	Range	Parameter	Range	Parameter	Range
$q_{i,j,t}^{\mathit{min}}$	0	${\omega }_{i,j,t}$	$\mathrm{U}(0.5,1)$	a of ${\tilde{A}}_{i,j,t}$	$\mathrm{U}(0,0.1)$
${\overline{T}}_j$	0.1	$v_{i,t}$	$\mathrm{U}(0,4)$	b of ${\tilde{A}}_{i,j,t}$	$\mathrm{U}(0.03,0.15)$
${\overline{r}}_j$	0.1	$c_j^r$	$\mathrm{U}(20,40)$	a of ${\tilde{R}}_{i,j,t}$	$\mathrm{U}(0,0.1)$
$F_i$	$\mathrm{U}(500,4000)$	$c_j^r$	$\mathrm{U}(20,40)$	b of ${\tilde{R}}_{i,j,t}$	$\mathrm{U}(0.03,0.15)$
$N_j^{\mathit{max}}$	$\mathrm{U}(1,4)$	$b_{i,j,t,1}$	$\mathrm{U}(0,400)$	a of ${\tilde{D}}_{j,t}$	$\mathrm{U}(200,800)$
$p_{j,t}^m$	$\mathrm{U}(20,35)$	$O_{i,j,t}$	$\mathrm{U}(300,800)$	b of ${\tilde{D}}_{j,t}$	$\mathrm{U}(400,1000)$
$p_{i,j,t,1}$	$\mathrm{U}(1,15)$	$q_{i,j,t}^{\mathit{max}}$	$\mathrm{U}(400,1000)$

presents the parameter ranges used to generate these instances.

Specifically, the unit base price ($l=1$) of product j purchased from supplier i, denoted as $p_{i,j,t,1}$, is generated within the range of $[1,15]$. At each subsequent discount level ($l+1$), the unit price is reduced by a discount rate generated within the range of $[0,0.2]$ from the previous level’s price (i.e., $p_{i,j,t,l}$). The breakpoints for each discount interval (l) are defined as follows: $b_{i,j,t,0}=q_{i,j,t}^{\mathit{min}}=0$; $b_{i,j,t,1}$ is generated within $[0,400]$; for $1<l<m$, $b_{i,j,t,l}$ is increased by a value generated from $[100,600]$ on the basis of $b_{i,j,t,l-1}$; and for $l=m$, $b_{i,j,t,m}=q_{i,j,t}^{\mathit{max}}$. In addition, the stochastic parameters (i.e., ${\tilde{D}}_{j,t}$, ${\tilde{R}}_{i,j,t}$, and ${\tilde{A}}_{i,j,t}$) are assumed to follow a truncated normal distribution bounded between a and b ($a<b$), which are randomly generated from the intervals specified in Table 3.

5.2. Tuning Parameters of the Solution Algorithm

Since the solution quality of the GA might be influenced by its parameter settings [9], we first conducted preliminary experiments using the Taguchi method to determine the optimal parameter configuration for our proposed GA-based algorithm. The Taguchi method is a widely used approach for tuning parameters of meta-heuristic algorithms, as demonstrated in studies such as Hajipour et al. [44,45], and Rayat et al. [46]. By employing orthogonal arrays rather than a full factorial experimental design, this method requires only a small number of tests, which saves considerable time and enhances the practicality of the research. Additionally, the Taguchi method aims to minimize the impact of noise factors while optimizing the level of signal factors [44], with the signal-to-noise ratio (S/N) as a statistical measure. For minimization objectives, the S/N ratio is calculated as follows:

$$S/N=-10lg\left({\displaystyle \frac{{\sum }_{i=1}^n{Y_i}^2}{n}}\right),$$

(34)

where n represents the number of tests for each parameter combination and $Y_i$ denotes the response in the i-th experiment. We employed the relative percentages deviation (RPD) to evaluate the performance of the proposed algorithm. RPD is the Y value in Equation (34). RPD is defined as follows:

$$\mathit{RRD}={\displaystyle \frac{{\mathit{Max}}_{\mathit{sol}}-{\mathit{Alg}}_{\mathit{sol}}}{{\mathit{Max}}_{\mathit{sol}}}}\times 100,$$

(35)

where ${\mathit{Alg}}_{\mathit{sol}}$ and ${\mathit{Max}}_{\mathit{sol}}$ are the solutions derived from algorithm and the best solution among all solutions for each case. The smaller the RPD, the better the algorithm performs; hence the type of this indicator is “smaller is better”.

We considered four main parameters (factors) at three levels (low, medium, and high) to conduct preliminary experiments. The parameters and their corresponding values at each level are reported in

Table 4. Parameters and their levels for our proposed algorithm.

Parameter	Range	Low (1)	Medium (2)	High (3)
A: Population size (${\mathit{Pop}}_{\mathit{size}}$)	50–100	50	70	100
B: Crossover probability ($P_c$)	0.6–0.8	0.6	0.7	0.8
C: Mutation probability ($P_m$)	0.05–0.15	0.05	0.1	0.15
D: Maximum iteration ($\mathit{Iter}$)	100–300	100	200	300

. Using the Taguchi method, we chose the orthogonal array (L27) shown in

Table 5. The orthogonal array (L27) for our proposed algorithm.

Trial	A	B	C	D	Trial	A	B	C	D
1–3	50	0.6	0.05	100	16–18	70	0.8	0.05	200
4–6	50	0.7	0.1	200	19–21	100	0.6	0.15	200
7–9	50	0.8	0.15	300	22–24	100	0.7	0.05	300
10–12	70	0.6	0.1	300	25–27	100	0.8	0.1	100
13–15	70	0.7	0.15	100

as the appropriate design for these parameter levels.

To reduce the randomness of the results, we generated three test cases of different sizes: (1) three suppliers, one product, and periods; (2) four suppliers, two products, and two periods; (3) five suppliers, two products, and three periods. For each parameter combination (trial), we solved each case five times under $\beta =0.5$ and $\alpha =0.5$. Consequently, 15 response data were obtained for each parameter combination. Based on Equation (34), we calculated the S/N at each level for each parameter, as depicted in Figure 3. According to the criterion of “less is better”, the best level of each parameter is the one with highest peak. Based on Figure 3, the best values for population size, crossover probability, mutation probability, and maximum iteration are 100, 0.7, 0.05, and 300, respectively. These values were used in the following experiments.

5.3. Comparison with the Benchmark Algorithm

To evaluate the effectiveness and robustness of our proposed algorithm, a comparative study was conducted against a benchmark approach for solving the EVM model. The benchmark algorithm employs the sample average approximation (SAA) method, which incorporates all discretized simulation scenarios into a transformed deterministic model to approximate the objective function, then solves the resulting model using Gurobi. To ensure a fair comparison, the weighted-satisfaction sum functions for handling of the dual objectives in both approaches were predetermined identically by utilizing Algorithm 2.

First, using the same case elaborated upon in Section 5.4, we compare the objective function values derived by the two algorithms and their changes under different sample sizes. The results of solving the same instance five times for each sample size (ranging from 100 to 2500), with a time limit of two hours per run, are summarized in the box plots shown in Figure 4, which demonstrates that our approach achieves acceptable accuracy compared to the benchmark algorithm and that the variability of the optimal objective values decreases with increasing sample size, indicating that the solution accuracy improves with larger sample sizes. Interestingly, an exception is observed for the benchmark algorithm at a sample size of 2500, where it exhibits noticeably larger variation. This occurs because the benchmark method fails to converge to an optimal or near-optimal solution within the two-hour time limit for such a large sample size. In contrast, our algorithm completes the computation for 2500 samples in approximately 300 s.

Second, we further compare the performance of the two algorithms across different problem instances with increasing scale. As shown in Figure 4, acceptable accuracy is attainable with a sample size of 500, beyond which marginal gains diminish with further increases in the sample size. Therefore, the sample size is fixed at 500 to evaluate the performance of both algorithms across five distinct instances. The corresponding objective values and computation times are summarized in

Table 6. Objective values derived by two algorithms for different instances.

$\left|\mathit{I}\right|\times \left|\mathit{J}\right|\times \left|\mathit{T}\right|$	Benchmark Algorithm			Our Algorithm
	Max.	Min.	Ave.	Max.	Min.	Ave.
$3\times 1\times 2$	0.5808	0.5795	0.5802	0.5826	0.579	0.5807
$4\times 2\times 2$	0.5829	0.5816	0.5824	0.5826	0.5815	0.5819
$5\times 2\times 3$	0.5997	0.5951	0.5979	0.5926	0.5908	0.5919
$6\times 3\times 3$	0.5946	0.5885	0.5908	0.5939	0.5934	0.5937
$7\times 3\times 4$	0.6082	0.6014	0.6055	0.6108	0.6100	0.6104

and

Table 7. Runtimes (in seconds) of two algorithms for different instances.

$\left|\mathit{I}\right|\times \left|\mathit{J}\right|\times \left|\mathit{T}\right|$	Benchmark Algorithm			Our Algorithm
	Max.	Min.	Ave.	Max.	Min.	Ave.
$3\times 1\times 2$	1072	657	928	70	65	67
$4\times 2\times 2$	1305	1011	1165	219	200	208
$5\times 2\times 3$	6438	4347	5361	939	886	927
$6\times 3\times 3$	14,432	5208	9487	2283	2160	2236
$7\times 3\times 4$	14,442	10,779	13,316	7030	6142	6623

, respectively. The results illustrate that while both algorithms achieve comparable solution accuracy, our algorithm outperforms the benchmark on running time. Notably, for the instance with a size of $7\times 3\times 4$, the benchmark algorithm requires over three hours to obtain an optimal solution. To maintain practical experimental duration, comparisons on even larger instances are not pursued.

The above comparisons demonstrate the effectiveness of our proposed algorithm in solving the EVM. Given that a similar solution methodology is applied to the CVM in our study, comparable performance is anticipated in solving the CVM. On the other hand, there is a lack of standard approach for addressing the proposed CVM, and it is difficult to directly solve using conventional solvers. Therefore, further comparisons of the CVM are omitted in this study.

Note that the first-stage supplier selection is a strategic planning decision that is not time-sensitive and allows for sufficient time to evaluate and make the decision, whereas the second-stage model, converted into a linear programming formulation, can be rapidly solved using common optimization solvers during the operational decision-making phase. Therefore, the entire solution methodology proposed in this paper is practically feasible.

5.4. Illustrative Instance and Management Insights

In this subsection, we present an illustrative instance with three suppliers, one product, and two periods to provide a detailed illustration of our proposed models and solution algorithms. In this case, we assume that the stochastic parameters follow a truncated normal distribution, denoted by $\psi (\mu ,\sigma ,a,b)$, where $\mu$ and $\sigma$ are the mean and standard deviation of the parent general normal probability density function and a and b specify the truncation interval. The product demand in each period follows truncated normal distributions of $(800,50,650,950)$. The parameters of the truncated normal probability density function for the stochastic defect rate and late delivery rate are provided in

Table 8. Truncated normal distribution for stochastic defect rate and late delivery rate.

Supplier	${\tilde{\mathit{R}}}_{\mathit{i},\mathit{j},\mathit{t}}$	${\tilde{\mathit{A}}}_{\mathit{i},\mathit{j},\mathit{t}}$
1	$\psi (0.05,0.005,0.02,0.09)$	$\psi (0.04,0.005,0.01,0.08)$
2	$\psi (0.10,0.008,0.07,0.13)$	$\psi (0.10,0.009,0.07,0.13)$
3	$\psi (0.05,0.001,0.04,0.06)$	$\psi (0.06,0.001,0.05,0.07)$

. The fixed costs of cooperating with each supplier are 3000, 1500, and 1000. Additional parameters are listed in Appendix B. For this instance, the PIS and NBS for the two objectives used in both stages are calculated as given in

Table 9. PIS and NBS of two objectives for both stages.

Parameter	${\mathit{C}}^{\mathit{pis}}$	${\mathit{TC}}^{\mathit{pis}}$	${\mathit{C}}^{\mathit{nbs}}$	${\mathit{TC}}^{\mathit{nbs}}$	${\mathit{PV}}^{\mathit{pis}}$	${\mathit{TPV}}^{\mathit{pis}}$	${\mathit{PV}}^{\mathit{nbs}}$	${\mathit{TPV}}^{\mathit{nbs}}$
Value	4828.1	11199.4	7654.3	33785.7	854.5	1696.0	421.6	809.8

. Based on these values, we converted the two objectives into a single objective. To account for varying decision-maker preferences for objectives and risk aversion, experiments were conducted with five objective weights ($\beta =\{0.1,0.3,0.5,0.7,0.9\}$) and five confidence levels ($\alpha =\{0.5,0.6,0.7,0.8,0.9\}$).

Given a supplier portfolio and a scenario sample, we can determine the corresponding total cost and total purchasing value by solving the equivalent OAM-t model and using Equations (1) and (2). We generated 3000 samples to assess each feasible supplier portfolio. For each portfolio, we computed the total cost and total purchasing value for each sample, thereby approximating the distribution of the total cost and total purchasing value for each supplier portfolio under different objective weights. In this case, eight alternative supplier portfolios are available, represented as (0,0,0), (1,0,0), (0,1,0), (0,0,1), (1,1,0), (1,0,1), (0,1,1), and (1,1,1). Figure 5 and Figure 6 display the distributions of total cost and total purchasing value for these eight portfolios. Note that when the supplier portfolio is (0,0,0), the total purchasing value is 0; thus, we omit its distribution to more clearly illustrate the distributions of the other portfolios.

Figure 5 shows that the total cost of not cooperating with any suppliers, i.e., the supplier portfolio (0, 0, 0), is always much higher than other choices. In addition, cooperating with supplier 2, i.e., (0,1,0), is optimal in most scenarios when only considering the cost minimization objective, while cooperation with supplier 1, i.e., (1,0,0), is optimal in most scenarios when only considering the purchasing-value maximization objective (see Figure 6). Furthermore, most supplier portfolios exhibit a relatively consistent trend of the change in total purchasing value. However, the supplier portfolio (0,0,1) remains unchanged because the lower bounds of demand exceed supplier 3’s maximum supply capacity. Therefore, regardless of the scenario, supplier 3 always provide all items, resulting in the same purchasing value.

5.4.1. EVM Optimal Solution

For each supplier portfolio, we calculate the average of the total cost and total purchasing value across the 3000 samples, denoted as $\mathbb{E}\left[\mathit{TC}\right(\mathit{x},\mathit{\xi }\left)\right]$ and $\mathbb{E}\left[\mathit{TPV}\right(\mathit{x},\mathit{\xi }\left)\right]$ for that supplier portfolio decision under the EVM. For each value of $\beta$, the comprehensive performance (${\mu }^{1E}$) of each supplier portfolio is evaluated according to Equation (32). By comparing ${\mu }^{1E}$ values across all supplier portfolios, we identify the optimal decision under this weight.

Table 10. Optimal solutions of the EVM under different objective weights.

$\mathit{\beta }$	0.1	0.3	0.5	0.7	0.9
$\mathit{x}$	(1,0,1)	(1,0,1)	(1,0,0)	(0,1,1)	(0,1,0)
$\mathbb{E}\left[\mathit{TC}\right(\mathit{x},\mathit{\xi }\left)\right]$	24,123.56	24,015.2	23,107.25	17,867.47	17,181.67
$\mathbb{E}\left[\mathit{TPV}\right(\mathit{x},\mathit{\xi }\left)\right]$	1450.52	1446.03	1420.02	1165.1	1094.42
${\mu }^{1E}$	0.6935	0.6323	0.5807	0.6136	0.6938
Std. of ${\mu }^{1E}$	0.0014	0.0009	0.0002	0.0004	0.0013

presents the mean values of $\mathbb{E}\left[\mathit{TC}\right(\mathit{x},\mathit{\xi }\left)\right]$, $\mathbb{E}\left[\mathit{TPV}\right(\mathit{x},\mathit{\xi }\left)\right]$, ${\mu }^{1E}$, and the standard deviation (Std.) of ${\mu }^{1E}$ across five experimental runs. As shown in Table 10, the optimal supplier portfolio varies with the weight ($\beta$). In addition, as the weight increases, the optimal expected total cost and expected purchasing value decrease. This is because when $\beta$ is small, the decision maker pays more attention to the purchasing-value objective and tends to select a portfolio with higher purchasing value. As $\beta$ increases, the decision maker pays more attention to the cost objective and will choose the scheme with the lower cost.

When $\beta$ is low (e.g., 0.1 and 0.3), the optimal decision is to choose suppliers 1 and 3. This can be explained as follows: supplier 1 has the highest purchasing value per unit, but when the demand exceeds 800 units, supplier 1 alone cannot meet the demand. Procuring the excess from the spot market yields no purchasing value and incurs high costs, so in addition to supplier 3, another supplier should be partnered to increase the purchasing value. Since supplier 3 provides higher purchasing value than supplier 2, the supplier portfolio (1,0,1) is particularly favorable when purchasing value plays an important role in the evaluation. As $\beta$ increases to 0.5, the increase in purchasing value contributed by supplier 3 is insufficient to offset the associated cost increase, resulting in (1,0,0) becoming optimal. When $\beta$ further increases to 0.7, the decision maker places greater emphasis on cost while still retaining moderate consideration for purchasing value. Under this setting, the optimal portfolio shifts to (0,1,1). This portfolio reduces the expected cost relative to portfolios (1,0,1) and (1,0,0) while still retaining the relative high purchasing value contributed by supplier 3. Finally, when $\beta$ reaches 0.9, cost becomes the dominant objective, prompting the choice of supplier 2, which has the lowest cost.

5.4.2. CVM Optimal Solution

For the CVM, we first calculate the CVaR of total cost and the expected purchasing value over 3000 samples at different confidence levels, then figure out the comprehensive performance under varying objective weights. For any fixed confidence level and objective weight, we compare the ${\mu }^{1C}$ of each supplier portfolio to determine the optimal decision. The results are reported in

Table 11. Optimal solutions of the CVM under different combinations of confidence level and objective weight.

$\mathit{\alpha }$	$\mathit{\beta }$	0.1	0.3	0.5	0.7	0.9
0.5	$\mathit{x}$	(1,0,1)	(1,0,1)	(1,0,0)	(0,1,1)	(0,1,0)
	${\mathit{CVaR}}_{\alpha }\left(\mathit{TC}(\mathit{x},\mathit{\xi })\right)$	25,032.87	24,889.6	24,081.22	18,723.37	18,252.99
	$\mathbb{E}\left[\mathit{TPV}\right(\mathit{x},\mathit{\xi }\left)\right]$	1450.52	1446.03	1420.02	1165.1	1094.42
	${\mu }^{1C}$	0.6895	0.6207	0.5591	0.5871	0.6510
	Std. of ${\mu }^{1C}$	0.0014	0.0008	0.0004	0.0004	0.0014
0.6	$\mathit{x}$	(1,0,1)	(1,0,1)	(1,0,0)	(0,1,1)	(0,1,0)
	${\mathit{CVaR}}_{\alpha }\left(\mathit{TC}(\mathit{x},\mathit{\xi })\right)$	25,220.02	25,077.82	24,318.37	18,893.66	18,489.4
	$\mathbb{E}\left[\mathit{TPV}\right(\mathit{x},\mathit{\xi }\left)\right]$	1450.52	1446.03	1420.02	1165.1	1094.42
	${\mu }^{1C}$	0.6886	0.6182	0.5539	0.5818	0.6416
	Std. of ${\mu }^{1C}$	0.0014	0.0008	0.0005	0.0004	0.0014
0.7	$\mathit{x}$	(1,0,1)	(1,0,1)	(1,0,0)	(0,1,1)	(0,1,0)
	${\mathit{CVaR}}_{\alpha }\left(\mathit{TC}(\mathit{x},\mathit{\xi })\right)$	25,445.1	25,290.92	24,597.89	19,082.52	18,758.9
	$\mathbb{E}\left[\mathit{TPV}\right(\mathit{x},\mathit{\xi }\left)\right]$	1450.52	1446.03	1420.02	1165.1	1094.42
	${\mu }^{1C}$	0.6876	0.6154	0.5477	0.5760	0.6309
	Std. of ${\mu }^{1C}$	0.0014	0.0008	0.0006	0.0004	0.0013
0.8	$\mathit{x}$	(1,0,1)	(1,0,1)	(1,0,0)	(0,1,1)	(0,1,0)
	${\mathit{CVaR}}_{\alpha }\left(\mathit{TC}(\mathit{x},\mathit{\xi })\right)$	25,735.77	25,561.67	24,955.06	19,308.43	19,100.98
	$\mathbb{E}\left[\mathit{TPV}\right(\mathit{x},\mathit{\xi }\left)\right]$	1450.52	1446.03	1420.02	1165.1	1094.42
	${\mu }^{1C}$	0.6863	0.6118	0.5398	0.5690	0.6172
	Std. of ${\mu }^{1C}$	0.0014	0.0008	0.0007	0.0004	0.0012
0.9	$\mathit{x}$	(1,0,1)	(1,0,1)	(1,0,1)	(0,1,1)	(0,1,1)
	${\mathit{CVaR}}_{\alpha }\left(\mathit{TC}(\mathit{x},\mathit{\xi })\right)$	26,089.5	25,954.94	24,798.71	19,625.4	19,618.48
	$\mathbb{E}\left[\mathit{TPV}\right(\mathit{x},\mathit{\xi }\left)\right]$	1450.52	1446.03	1403.17	1165.1	1159.44
	${\mu }^{1C}$	0.6848	0.6066	0.5337	0.5591	0.6040
	Std. of ${\mu }^{1C}$	0.0015	0.0008	0.0002	0.0007	0.0009

. As expected, for a given objective weight, $CVa{R}_{\alpha }\left(TC(\mathit{x},\mathit{\xi })\right)$ increases as the confidence level rises. Meanwhile, consistent with the trends observed in the EVM, when the confidence level is fixed, increasing the weight on the cost objective generally leads to lower values of both ${\mathit{CVaR}}_{\alpha }\left(\mathit{TC}(\mathit{x},\mathit{\xi })\right)$ and $\mathbb{E}\left[\mathit{TPV}\right(\mathit{x},\mathit{\xi }\left)\right]$, as decisions gradually shift toward cost-minimizing portfolios.

For confidence levels ranging from 0.5 to 0.8, the optimal portfolio remains stable across all values of $\beta$. However, when the confidence level increases to 0.9, shifts in the optimal decisions emerge for $\beta =0.5$ and $\beta =0.9$. For $\beta =0.5$, the optimal portfolio changes from $(1,0,0)$ at $\alpha =0.8$ to $(1,0,1)$, which exhibits a lower extreme-cost tail (see Figure 5). Similarly, for $\beta =0.9$, the optimal decision shifts from $(0,1,0)$ to $(0,1,1)$, again reflecting the benefit of reduced tail risk. These observations demonstrate that as risk aversion strengthens, the decision maker becomes more sensitive to tail risks and may prefer diversified portfolios.

5.4.3. Management Insights

The above numerical experiments offer several meaningful implications for managers involved in supplier selection and procurement decisions under uncertainty.

First, it is important to balance the conflicting objectives of cost and purchasing value. Results from the illustrative instance indicate that optimal supplier portfolios are highly sensitive to the decision maker’s preference between cost minimization and purchasing-value maximization. When greater emphasis is placed on purchasing value (i.e., smaller $\beta$), portfolios containing suppliers with higher value contributions (e.g., supplier 1 or 3) become more attractive. Conversely, as the weighting shifts toward cost reduction (larger $\beta$), portfolios involving low-cost suppliers, such as supplier 2, dominate. This underscores the importance of firms clarifying their strategic preferences and explicitly defining their priorities. In the context of emphasizing long-term cooperative relationships or product quality, a higher weight assigned to purchasing value may be appropriate. Alternatively, in cost-sensitive or budget-constrained situations, managers should assign greater priority to cost objectives.

Second, risk aversion shapes supplier selection strategies. Under the CVM, higher confidence levels ($\alpha$) lead to more conservative supplier selection decisions. When risk aversion increases, firms tend to avoid portfolios with high-cost tail risks and shift toward diversified combinations. For example, the shift from $(1,0,0)$ to $(1,0,1)$ while $\alpha$ increases to 0.9 in the illustrative instance demonstrates that incorporating a second supplier helps mitigate extreme cost scenarios. This suggests that firms facing high uncertainty, such as unreliable suppliers, should adopt more risk-aware designs, even if doing so slightly increases expected cost. CVaR-based models provide a useful tool for identifying these robust portfolios.

Additionally, compared to the EV criterion, the CVaR criterion offers decision makers greater flexibility in managing risk in uncertain environments. Notably, when the confidence level specified for CVaR approaches zero, this criterion converges to the EV criterion. Correspondingly, the CVM provides more flexible risk-handling capabilities than the EVM for risk-aware decision making, but a careful selection of the confidence level constitutes a critical prerequisite for the model’s validity and practical relevance.

6. Conclusions

In this paper, we formulated the supplier selection and order allocation problem under uncertainty as two-stage bi-objective stochastic programming models. In the models, the second-stage order allocation decision is subject to the first-stage supplier selection, and the performance of the SS decision is influenced by the OA decision. In addition, the performance of the supplier selection decision was jointly evaluated through cost and purchasing value using both EV and CVaR criteria. To solve the models, we then developed a hybrid algorithm integrating the weighted-satisfaction sum method, linearization techniques, Monte Carlo simulation, and a genetic algorithm. Extensive numerical examples were reported to illustrate the effectiveness of the proposed model and solution approach. The illustrative example shows that both the decision maker’s preferences for various objectives and their risk-aversion level influence supplier selection decisions, revealing valuable managerial implications for the practical supplier selection and procurement decision under uncertainty.

It is important to acknowledge several limitations of this study, which also point to directions for future work. First, our proposed models rely on some assumptions, such as independent demand for each period and a lack of inventory used across periods, which may not hold in more generalized cases. Relaxing these assumptions presents an interesting and significant issue for further exploration. Second, more sophisticated solution approaches should be developed for two-stage bi(multi)-objective stochastic models, particularly when applied to large-scale instances. Since multiple objectives should be treated in each stage formulated by sub-models with different solution spaces, the complex structure of these models makes this issue challenging.

Furthermore, given the complex and volatile circumstances, it would also be a compelling and critical direction in future research to incorporate other uncertainty modeling and optimization approaches alongside stochastic methods so as to provide uncertainty-aware decision support for supply chain operations. For instance, to capture the stochastic fluctuation in raw material price and the epistemic uncertainty regarding procurement value, both random and fuzzy variables could be utilized to model mixed uncertainties. In this context, fuzzy stochastic optimization [47,48] offers a sound theoretical foundation for tackling such problems. Finally, our proposed two-stage bi(multi)-objective modeling and solution framework could be adapted and applied to related problems involving similar interdependent two-stage decisions under uncertainty, such as location-allocation [49] and location-routing [50] problems.