A sampling-based winner determination model and algorithm for logistics service procurement auctions under double uncertainty

Abstract

This paper studies the winner determination problem under disruption and demand uncertainty from the perspective of fourth-party logistics platform. To hedge risks brought by double uncertainty, a hybrid mitigation strategy that integrates temporary outsourcing strategy and fortification strategy is developed. With the objective of minimizing total cost, a new two-stage stochastic winner determination model under double uncertainty is constructed, which is further transformed into mixed-integer linear programming model by using an improved sample average approximation algorithm based on the chi-square test and the Latin hypercube sampling method. To address the challenges brought by numerous demand and disruption scenarios in model solving, combining dual decomposition Lagrangian relaxation algorithm and scenario reduction approach, a sampling-based heuristic algorithm is proposed. To validate the effectiveness of our model and algorithm, a real case and several numerical examples including cases generated by using the Combinatorial Auction Test Suite are provided. The results of numerical examples and real case indicate that the proposed algorithm has superior performance to CPLEX, validating the effectiveness of model and methods. Sensitivity analysis results show that demand fluctuations and the magnitude of disruption probability have significant impacts on the selection of risk response strategies.

Introduction

With the advancement of economic globalization, competition among enterprises has intensified significantly. Reducing costs and improving efficiency are the key to the survival and development of enterprises. To enhance core competitiveness, companies tend to outsource non-core business to specialized third-party logistics (3PL) providers. In recent years, with the emergence of numerous 3PL providers worldwide, one of the crucial challenges for enterprises is to integrate various logistics providers and offer comprehensive logistics solutions. The emergence of fourth party logistics (4PL) provides new opportunities for the intelligent transformation of logistics models. Comprehensive logistics solutions for enterprises can be provided through the integration and management of various technologies, resources, and capabilities by 4PL and 3PL providers. The 4PL model helps companies maintain a competitive edge and achieve sustainable development. 4PL providers such as Cainiao and UPS have provided comprehensive logistics supply chain solutions for e-commerce platforms such as Taobao and Tmall, as well as numerous manufacturing and retail enterprises. In the research field, many scholars conduct extensive research on 4PL, which can be generally classified into the following types: (a) advantages and conceptual analysis^1,2; (b) route planning^3,4,5; (c) network design^6,7,8; (d) logistics procurement auction^9,10. Nowadays, adopting reverse auctions for logistics service procurement has become an efficient model for resource acquisition.

It is noteworthy that affected by market fluctuations, seasonal changes, natural disasters and other factors, 4PL faces multiple uncertain risks in the process of logistics procurement. Logistics disruption risk cannot be ignored in the logistics procurement process of 4PL¹¹. The existing logistics system is susceptible to the threat of transportation service disruption caused by unexpected events, which may be caused by natural disasters, equipment failures, terrorist attacks, etc¹². For example, German chemical distribution company Brenntag stated that the Red Sea crisis caused the company’s logistics transportation time to be extended by two to three weeks, and profit margins decreased due to rising transportation costs. Severe weather conditions such as tsunamis and hurricanes, events in public health such as the COVID-19 pandemic, and political factors such as the Russia-Ukraine conflict are likely to lead to the disruption of logistics transportation services, posing significant harm to global trade and economic growth¹³. Therefore, considering the disruption risks of 3PLs in logistics service procurement auctions can help improve the efficiency and stability of logistics transportation, and build long-term strategic advantages.

In addition, due to factors such as seasonal changes and promotional activities, the logistics demand of companies often exhibits characteristics of uncertainty. For example, the Double Eleven promotional event on e-commerce platforms attracts a large number of consumers to shop intensively, leading to significant fluctuations in logistics demand. According to the international website Metro, ensuring the delivery of millions of parcels to more than 800 million shoppers worldwide is undoubtedly a significant challenge for Alibaba and other online retailers. The uncertainty of demand requires 4PL providers to have strong adaptability to accurately grasp changes in market demand while ensuring the stability and continuity of logistics services.

To address the multiple uncertainties faced in the logistics procurement process, this paper studies the winner determination problem (WDP) from the perspective of 4PL, considering both demand uncertainty and disruption risk simultaneously. To address the risk of disruption, a hybrid mitigation strategy that integrates temporary outsourcing strategy and fortification strategy is developed. Fortification strategy can ensure the stability and reliability of logistics services through measures such as facility protection and emergency backups¹⁴. With a limited fortification budget, designated 3PLs can be fortified in advance by 4PL to hedge disruption risks. In addition, 4PL can acquire temporary outsourcing service capabilities from other external 3PLs to expand transportation capacity and meet specific requirement of enterprises¹⁵. To solve this problem, the disruption risk is characterized in a scenario-based manner, and the uncertainty of logistics demand is characterized using random variables. A new two-stage stochastic winner determination model is constructed to minimize overall cost under double uncertainty. Considering the challenges posed by uncertainties on the solution of model, the framework of the sample average approximation (SAA) method is reconstructed based on chi-square test and Latin hypercube sampling (LHS) method. Further, we propose an improved sample average approximation (ISAA) algorithm that transforms the stochastic model into a deterministic mixed-integer linear programming (MILP) model. Since the constraints and decision variables contained in the model significantly increase due to numerous demand and disruption scenarios, a scenario reduction (SR) approach is adopted to handle disruption scenarios. Combined with dual decomposition Lagrangian relaxation (DDLR) algorithm, a sampling-based heuristic algorithm known as ISAA-SR-DDLR is proposed. The upper bound of optimal solution is derived from a hybrid ISAA and SR method, and the lower bound of optimal solution is derived from ISAA and DDLR algorithms. Numerical experiments are designed to validate the effectiveness of the proposed model and algorithm. The results of numerical experiments show that the gap between the lower and upper bounds derived from the proposed algorithm is acceptable, which verifies the applicability and effectiveness of our model and approach. The results of the sensitivity analysis of key parameters show that the magnitude of disruption probability and demand fluctuations have significant impacts on the outcome of the selection of risk response strategies. 4PL managers need to make decisions on the selection of 3PLs and the formulation of hybrid mitigation strategies based on the inherent interaction relationship between the two factors affecting strategy selection. The following outlines the key contributions of this paper compared to existing research:

• (1) The impact of uncertain demand and disruptions in logistics systems on WD decision-making has been widely demonstrated. However, existing research on WD mostly considers only a single uncertainty factor. This paper studies a new WDP for logistics service procurement auctions under double uncertainty, aimed at assisting 4PL in decision-making.

• (2) A two-stage stochastic winner determination model under double uncertainty is constructed, which simultaneously considers demand uncertainty and disruptions in logistics systems. Fortification and temporary outsourcing strategies are integrated to collaboratively address demand fluctuations and changes in logistics disruptions, providing management insights for 4PL.

• (3) To address the challenges brought by numerous demand and disruption scenarios in model solving, this study develops a general algorithmic framework suitable for solving a two-stage stochastic programming model with multiple scenarios. This study changes the traditional approach of randomly determining the sample size in the SAA method by utilizing the LHS method and chi-square test to determine the optimal sample size. Consequently, we reconstructs the traditional SAA algorithm framework and integrates SR and DDLR algorithms to propose the ISAA-SR-DDLR algorithm. The proposed algorithm can provide both lower and upper bounds for the original problem model, with the gap between the two reflecting the applicability of the algorithm to problem solving.

• (4) A real case and data from a 4PL company based in Shenzhen, China, are utilized to validate the effectiveness of the proposed model and algorithm in practice.

The following outlines the arrangement of this paper. An overview of the literature is given in Section 2. Section 3 presents the research problem and the constructed model. In Section 4, an ISAA-SR-DDLR algorithm is proposed. Section 5 sets up numerical experiments to validate effectiveness of our model and algorithm. Section 6 applies the proposed model and algorithm to practical cases and presents managerial insights. Finally, Section 7 summarizes the paper and explores possible future extensions.

Literature review

In recent years, WDP has important applications in auction, task allocation and scheduling, resource allocation and optimization, and other fields. For example, Lau and Li¹⁶ proposed WDP for an online B2B transportation matching platform, with the goal of maximizing platform profits while ensuring service levels; Triki et al.¹⁷ developed a student ride-sharing system based on combinatorial auctions to address traffic congestion on university campuses; Lee et al.¹⁸ considered demand uncertainty and developed a two-stage robust optimization model to optimize procurement decisions. Transportation service procurement involves multiple strategic links such as sourcing, negotiating, and contracting services, with the aim of optimizing costs in the supply chain, improving transportation efficiency, and ensuring logistics reliability¹⁹. Nowadays, the application of WDP in logistics transportation service procurement has become a common approach. This paper studies the WDP in logistics service procurement auctions from the perspective of 4PL, which considers both demand uncertainty and disruption risk simultaneously. Therefore, this section mainly reviews the relevant research on the application of WDP in logistics service procurement. Subsection 2.1 reviews WD problem and related models. In subsection 2.2, the mitigation strategies applied in supply chain risk management are examined. Subsection 2.3 focuses on the methods employed to address WD models. Finally, subsection 2.4 outlines the research gaps between our study and the current literature.

Winner determination problem

The application of WDP in logistics service procurement can be roughly divided into two categories. (i) WDP in deterministic environment. In this field, a relatively early study was conducted by Caplice and Sheffi²⁰, in which a mathematical model was developed aimed at solving the complex decision-making process of allocating transportation lanes to designated carriers. They also explored how to further extend this model to incorporate various constraints that may be encountered in actual business operations. Sheffi²¹ delved into the application of combinatorial auctions in reducing transportation costs and improving service levels. Through optimization methods, non-price attributes and system constraints can be effectively addressed to determine the winning bidders. To effectively balance service quality and transportation costs, Guo et al.²² explored the carrier allocation model in transportation procurement, considering the performance of carriers at transit points and penalty costs. Based on the need to achieve economies of scale, Chen et al.²³ developed an implicit WDP model that can effectively solve the exponential combination bidding problem faced in truckload procurement auctions. Rekik and Mellouli²⁴ proposed a WDP that emphasizes the importance of reputation in the selection of carriers. Experimental results showed that their approach achieved considerable savings in total transportation costs.

(ii) WDP in uncertain environment. This type of research can be further divided into: (a) WDP under parameter uncertainty. For instance, in response to the uncertainty of transportation volumes, many studies have established stochastic or robust WDP models. Ma et al.²⁵ first constructed a two-stage stochastic integer programming model to assist shippers in assigning transportation tasks to carriers under uncertain transportation volumes. The results showed that compared with deterministic models, stochastic solutions have higher application value. To address truck service procurement under uncertain shipment volumes, Zhang et al.²⁶ developed a two-stage stochastic programming model combined with Monte Carlo sampling. They verified that the stochastic model is more effective in limiting procurement costs relative to the deterministic model. Remli and Rekik²⁷ developed a two-stage robust optimization model to handle uncertainties in transport volume, and used an improved constraint generation algorithm as the solution method. Zhang et al.²⁸ constructed an uncertainty set based on the variance and mean of uncertain freight volumes, and introduced a new two-stage robust winner determination model. The test results showed that the solution of robust WDP can achieve lower procurement costs. Remli et al.²⁹ established a two-stage robust model to address full truckload transportation service procurement problems, considering the uncertainty of both transportation volume and carrier capacity, and developed an improved constraint generation algorithm with acceleration strategy as the solution method. Following the uncertainty of transportation volume, Lim et al.³⁰ studied how to minimize transportation costs while satisfying lane cost balancing constraint. Yang and Huang³¹ proposed a mixed-integer nonconvex programming (MINP) model that considers transportation cost discounts based on shipment distance and volume. To obtain the global optimum, they used advanced linearization techniques to transform the MINP model into a MILP model. Safari and Dolati³² proposed the WDP with uncertain bids’ prices in auctions. By using uncertainty theory, they transformed the complex uncertainty problem into a deterministic maximum weighted clique problem. Li et al.³³ proposed the WDP with procurement budget under uncertain transportation volume, and the results indicate that this model can help shippers effectively reduce the risk of cost overruns. Triki et al.³⁴ introduced green priority into the WDP with discount bidding, which helps enhance carriers’ market competitiveness and reduce carbon emissions. Qian et al.³⁵ introduced quantity discounts to the WDP in logistics service procurement auctions with uncertain demand, and constructed a two-stage stochastic nonlinear WDP model. Utilizing the SAA method and linearization techniques, the stochastic quantity-discount winner determination model was transformed into a deterministic MILP model, and a solution combining SAA and DDLR techniques was developed. Qian et al.³⁶ studied the WDP with sustainability-flexibility considerations in transportation service procurement auctions under uncertain demand from the perspective of 4PL. They considered both sustainability and flexibility attributes, and integrated outside option policy to deal with uncertainty in client demand. Furthermore, the bi-objective WDP was considered as a crucial extension direction. Besides the primary goal of minimizing total transportation costs, other important factors were also considered at the same time, such as minimizing the transit time^37,38,39, enhancing total transportation quality⁴⁰, maximizing the service-quality level⁴¹ and so on.

(b) WDP under disruption risk. Qian et al.⁴² introduced a two-stage stochastic winner determination model that considers the disruption risks of bidders and proposed a mixed mitigation strategy integrating reservation, fortification, and outside option strategies to hedge disruption risk. Yin et al.⁴³ constructed a stochastic nonlinear WDP model considering both unexpected disruptions and quantity discounts, and proposed a mixed strategy of integrated outside option policy and fortification policy to hedge disruption risk. Based on consideration of green WDP, Qian et al.⁴⁴ established a two-stage stochastic mixed-integer programming model which links sustainability and responsiveness with hybrid mitigation strategies to enhance the long-term competitive advantage of 4PL.

In Table 1, we list and categorize some of the literature on WDP. The above studies only consider parameter uncertainty or disruption risks from a single aspect, but research that simultaneously considers uncertain demand and disruption risks in the problem of logistics transportation service procurement auctions is not explicitly investigated in prior works.

Table 1 Summary of WDP in transportation service procurement.

Disruption mitigation strategy

In recent years, the emergence of significant events and changes in the international landscape have drawn increasing attention to disruption risk mitigation strategies. For example, Liu et al.⁴⁸ adopt a risk-sharing approach to select supply chain partners in order to address risks. Based on the studies by Jabbarzadeh et al.⁴⁹ and Govindan et al.⁵⁰, existing disruption mitigation strategies can be broadly classified into two categories: preventive mitigation strategies and responsive mitigation strategies. Preventive mitigation strategies typically include: fortification, backup capability, and strategic stock, among which fortification strategy is one of the most widely used. Fortification strategy focus on enhancing the reliability and stability of logistics systems through measures such as facility protection and the pre-deployment of emergency plans^14,51. Many studies regard reinforcement strategies as resilience measures to address the risks of supply chain network disruptions, with specific actions such as infrastructure reinforcement^52,53, expanding facility capacity⁵⁴, pre-positioning inventory⁵⁵ and so on.

In the study of logistics transportation service procurement, Qian et al.⁴² proposed that fortification policies can reduce the likelihood of disruptions by investing in key 3PL providers. Yin et al.⁴³ proved that the incorporation of mitigation strategies, such as fortification policies, can effectively tackle disruption risks in the WDP, which considers both disruptions and quantity discounts. Qian et al.⁴⁴ embedded fortification strategies into sustainable-responsive WD model, demonstrating that these strategies can save costs and enhance the resilience of the logistics system.

Responsive mitigation strategies mainly involve temporary outsourcing, temporary inventory, and substitute products. Outsourcing has been extensively implemented in different industries in recent years, as it can lower delivery risks and enhance the continuity of operations. Li et al.⁵⁶ proposed a dynamic model to evaluate maintenance service providers, helping companies make informed outsourcing decisions and enhance supply chain performance. Zhang et al.⁵⁷ and Deng and Xu⁵⁸ explored the selection of third-party remanufacturing models in competitive closed-loop supply chains. The results indicated that, in most cases, outsourcing strategies consistently contribute to the maximization of overall supply chain benefits. Hajipour et al.⁵⁹ proposed that outsourcing strategies can effectively mitigate the negative impacts of macroeconomic uncertainty in the Industry 4.0 environment, reducing the likelihood of project delays and cost overruns. Wang et al.⁶⁰ proposed an order planning method suitable for industrial production systems, which effectively reduced costs and improved order delivery rates through a series of operations, including outsourcing decisions. The research results indicated that the semi-finished product outsourcing strategy is effective in most cases. In addition, many studies incorporated outsourcing strategies into production scheduling problems, developing models and optimization methods to help manufacturers expand capacity and reduce costs^61,62,63,64. Sardar et al.⁶⁵ proposed an outsourcing strategy that comprehensively considers cost and capacity flexibility in the textile industry’s supply chain, providing a new perspective for the sustainable development of the textile industry. Jabbarzadeh et al.⁴⁹ incorporated outsourcing decisions into a stochastic bi-objective optimization model for supply chain network design to simultaneously enhance the resilience and sustainability of the supply chain in the event of disruptions. Shi and Ni⁶⁶ suggested that outsourcing strategies can effectively mitigate disruptions in supply chain facilities, and that they can significantly lower costs under high service level requirements. Heydari et al.⁶⁷ proposed that utilizing outsourcing strategies can effectively reduce the expected residual costs of the supply chain and enhance flexibility under demand uncertainty. Farghadani-Chaharsooghi et al.⁶⁸ incorporated factors such as outsourcing into the production-path problem under demand uncertainty. The experimental results indicate that the introduction of outsourcing provides new optimization strategies for supply chain management, significantly reducing costs and improving service levels. Yin et al.⁶⁹ integrated temporary outsourcing strategies into the study of 4PL network design under demand uncertainty. The results indicate that temporary outsourcing services can significantly reduce total costs and enhance customer satisfaction. The aforementioned study suggests that temporary outsourcing services serve as an effective strategy when dealing with disruptions or demand uncertainty.

Based on the above review, we can see that in the field of supply chain management, preventive and responsive strategies such as risk sharing, fortification, capacity backup, strategic inventory, and temporary outsourcing have been proposed to address supply chain risks. However, most existing studies primarily employ protective and preventive strategies to address uncertainties in issues such as supply chain partner selection, supply chain network design, planning and scheduling⁷⁰, and order production. Few studies explore the synergistic effects of fortification and temporary outsourcing strategies in simultaneously mitigating demand uncertainty and disruptions in logistics systems. Unlike existing models, this study primarily focuses on a new two-stage stochastic WD model that simultaneously considers uncertain demand and disruption risks. By analyzing the synergistic effects of fortification and temporary outsourcing strategies on mitigating demand fluctuations and logistics disruptions, we provide management insights for 4PL.

Solution method for WD

WDP in combinatorial auctions is computationally complex and involves numerous uncertainties, making it an NP-hard problem. Traditional exact algorithms cannot provide solutions in polynomial time when dealing with large-scale instances. Therefore, many studies have developed heuristic algorithms or approximation methods to solve the WDP. Although there is a slight loss in solution quality, heuristic algorithms and approximation methods are far superior to exact algorithms in terms of computation time. Thus, they can be considered practical methods for solving the WDP in practice^71,72. A brief overview of these two methods is presented as follows.

The first method is a heuristic algorithm for solving the WDP: For the single-objective WDP, Guo et al.²² developed a meta-heuristic method based on genetic algorithms (GA) and tabu search to solve the WDP, which provides effective solutions for large-scale problems. Boughaci et al.⁷³ proposed a memetic algorithm for solving the WDP by combining stochastic local search with specific crossover operators, and comparative experiments demonstrated that memetic algorithm outperforms traditional GA in terms of diversity and fitness. Lim et al.³⁰ developed a random move tabu search algorithm, and the results indicated that the proposed algorithm significantly improves solution quality and efficiency in solving large-scale WDP compared to CPLEX. Triki et al.⁷⁴ embedded clustering techniques into a simulated annealing (SA) algorithm to overcome the limitations of exact full-space methods for solving the WDP. Zhang et al.⁷⁵ introduced a local search algorithm called abcWDP, and its effectiveness and practicality were confirmed through comparisons with existing algorithms and CPLEX. Lee et al.⁷⁶ incorporated knowledge-based operators to enhance GA for solving the WD problem of multiple automated guided vehicles, demonstrating the effectiveness and flexibility of the proposed GA in addressing complex large-scale problems. Lin et al.⁷⁷ employed the cross-entropy method to solve the WDP, and the results indicated that the proposed method has advantages in both solution quality and computational time compared to GA and particle swarm optimization algorithms. Kazama et al.⁴⁵ proposed a new constraint-guided evolutionary algorithm and compared it with three other evolutionary algorithms, demonstrating the proposed algorithm’s superiority in handling complex constraints. In addition, existing research also focused on developing heuristic algorithms for solving the bi-objective WD problem, such as the two-stage multi-objective evolutionary algorithm³⁷, multi-objective genetic algorithm⁴¹, and $\epsilon$-constraint heuristic method³⁹.

The second method is an approximate algorithm for solving the WDP. Approximate algorithms can provide near-optimal solutions for NP-hard problems in polynomial time⁷⁸. Many researchers developed approximate algorithms to solve the WDP in logistics service procurement auctions. As an example, Remli and Rekik²⁷ designed a constraint generation algorithm, and their experimental results indicated that the method performs well in solving large-scale instances of the WDP. Zhang et al.²⁶ proposed a Monte Carlo approximation method to solve the stochastic WDP and validated the effectiveness of this method through numerical experiments under various uncertainty distributions. Garcia⁷⁹ proposed an approximate algorithm based on SA and random priority search to solve the WDP with sub-cardinality constraints, and they validated the robustness of proposed method through practical examples. Zhang et al.²⁸ developed a polyhedral uncertainty set to formulate a reconstruction method for WD, demonstrating the algorithm’s effectiveness through experimental results. Remli et al.²⁹ proposed a revised constraint generation algorithm by incorporating several acceleration strategies to address the dual uncertainty WD problem, considering transportation volume and carrier capacity. Yang et al.⁸⁰ provided an iterative approximate algorithm for WDP with quantity discounts, and the results indicated that the method effectively reduces service costs. Zhong et al.⁸¹ developed a fully polynomial-time approximation solution to solve the WDP for multi-resource allocation, aiming to maximize social welfare effectively. Yin et al.⁴³ proposed a scenario-based approximation method that combines SR and DDLR to solve the WDP with disruption and quantity discounts. The results showed that the proposed algorithm outperforms traditional heuristic algorithms in terms of computational time and solution quality. Qian et al.³⁵ combined SAA and DDLR to address the WDP considering uncertain demand and quantity discounts, and numerical experiments validated the effectiveness and practicality of the proposed algorithm.

Currently, approximate algorithms for solving the WDP have been widely developed. SR and DDLR have been used to address the WDP under demand uncertainty and the WDP under disruprion risks. Nevertheless, the simultaneous consideration of uncertain demand and disruption risks in the WDP results in an exponential increase in the number of model constraints and decision variables, posing significant challenges for problem solving. Therefore, the design of effective algorithms for solving the WDP under double uncertainty still requires further exploration.

Research gaps

From the literature review, we can see that although WD models and methods have been developed, research gaps can still be identified. Firstly, the impact of demand uncertainty and disruption risks in logistics systems on the WD model has been explored independently. These studies clearly indicate that uncertain demand and disruption risks are factors that cannot be overlooked in the WD decision-making process. However, in the existing research on the WDP in logistics service procurement auctions, the combined consideration of uncertain demand and disruption risks has not been explicitly studied. Secondly, the review of risk mitigation strategies reveals that there has been research on employing fortification and outsourcing strategies to handle disruption risks. However, most existing studies primarily adopt protective and preventive strategies to address uncertain risks in issues such as supply chain network design, planning and scheduling, and order production. Few studies explore the synergistic effects of fortification and temporary outsourcing strategies in simultaneously mitigating demand uncertainty and disruptions in logistics systems. Thirdly, although approximate algorithms for solving WDP have been developed, the simultaneous consideration of uncertain demand and disruption risks can lead to an exponential increase in model constraints and decision variables, posing significant challenges for problem solving. This study reconstructs the traditional SAA algorithm framework and integrates SR and DDLR algorithms to propose the ISAA-SR-DDLR algorithm. Finally, the proposed model and algorithm are validated in practice using a real case and data from a 4PL company in Shenzhen, China.

After identifying the research gap, this paper investigates a new WDP for logistics service procurement auctions under double uncertainty to assist 4PL in decision-making. Fortification strategies and temporary outsourcing strategies are integrated to address the risks posed by demand fluctuations and logistics disruptions. For this problem, a new two-stage stochastic WD model is constructed, which is further transformed into MILP model using a scenario approach. To address the challenges brought by numerous demand and disruption scenarios in model solving, we reconstructs the traditional SAA algorithm framework based on the chi-square test and the LHS method and integrates SR and DDLR algorithms to propose the ISAA-SR-DDLR algorithm. The ISAA-SR-DDLR algorithm offers the benefit of providing both lower and upper bounds for the original problem model, reflecting the applicability of the proposed algorithm for problem-solving based on the gap between the two bounds. Finally, several numerical examples and a real case are utilized to validate the effectiveness of the proposed model and algorithm, providing some management insights for 4PL.

Problem description and mathematical model

4PL (auctioneer) can use reverse auction to solicit bids from 3PLs (bidders) with the goal of enhancing service quality and reducing operating costs. Figure 1 illustrates the reverse auction process in the logistics platform. This paper conducts research on the WDP in reverse auctions of logistics service procurement, considering uncertain demand and disruption risk from the perspective of 4PL. In real life, the interference of external factors causes client demand to appear stochastic, and 4PL should formulate corresponding strategies to better adapt to changes in client demand. In addition, 4PL should consider potential disruption risks in logistics transportation during the auction decision-making process and develop response strategies to mitigate the negative impacts of disruptions. Therefore, we focus on studying how to integrate hybrid mitigation strategy with decision-making to minimize logistics transportation costs under stochastic demand and disruption risks.

Fig. 1

Reverse auction process in a logistics platform.

Specifically, the logistics task includes a set of lanes I, which can be viewed as edges from the starting point to the destination. The logistics transportation demand on each edge is denoted by a random variable ${\eta _i}$, where $i \in I$. A group of qualified 3PLs will be invited by 4PL to participate in the logistics service procurement auction. 4PL allows each 3PL $j \in J$ to submit a set of packages, but only one package will be selected at most. The maximum transportation capacity for package k submitted by 3PL j on lane i is denoted by ${US_{ijk}}$, and the transportation volume allocated by the 4PL to each 3PL cannot exceed the capacity limit. The bid price submitted by 3PL j for package k on lane i is denoted by $b_{ijk}$. The number of 3PLs that 4PL expects to win is limited, with the upper limit denoted by ${N_{\max }}$ and the lower limit denoted by ${N_{\min }}$. Let S represent the set of disruption scenarios. If the package is subject to disruption scenario $w \in S$, transportation service will be completely unavailable for the entire recovery period.

Let ${D_U}$ denote the set of packages that may experience disruption. All package in ${D_U}$ may be hit by a disruption or remain normal. For 3PL j, ${q_{jkw}}$ is a parameter that indicates whether package k experiences disruption under scenario w. If it is disrupted, ${q_{jkw}}$ equals 1; otherwise, ${q_{jkw}}$ equals 0. We can obtain the total number of disruption scenarios using $\left| S \right| = {2^{\left| {{D_U}} \right| }}$. Let ${\hat{r}_k}$ denore the probability that the kth package may experience disruption, $k \in {D_U}$. Referring to Snyder and Daskin⁸², the probability associated with disruption scenario w is obtained using ${r_w} = \prod {_{k \in {D_{{U_s}}}}} {\hat{r}_k}\prod {_{k \in \left\{ {{D_U}\backslash {D_{{U_s}}}} \right\} }} \left( {1 - {{\hat{r}}_k}} \right)$, where ${D_{{U_s}}}$ is the set of packages that may experience disruption under scenario w.

Due to the potential disruption of capabilities in 3PLs, which can prevent 4PL from providing efficient logistics services to clients as planned, 4PL must implement protective measures to ensure the stability of services. A hybrid mitigation strategy that integrates temporary outsourcing strategy and fortification strategy is developed to cope with risks brought by disruption. Specifically, to enhance the reliability of key packages from 3PLs, the 4PL will invest a maximum budget ${C_{\max }}$ to strengthen the capabilities of 3PLs. For the package k of 3PL j, the fortification cost ${c_{jk}}$ will be determined based on the size of the package and the capabilities of the 3PL, where $k \in {K_j}$, $j \in J$. Based on previous research¹², it can be assumed that even after encountering disruption events, the fortified packages of 3PL can continue to provide transportation services to clients. In addition, if the allocated transportation volume after disruptions cannot meet the specific demands of the clients, 4PL will purchase transportation services from other 3PLs at a unit outsourcing cost ${e_i}$ to compensate for the insufficient transportation capacity. Therefore, 4PL needs to weigh its budget and service level by selectively fortifying specific packages of 3PLs or purchasing transportation capacity from other external 3PLs as needed, in order to ensure flexibility in responding to clients’ stochastic demand and mitigate the adverse effects of disruptions. In Table 2, we provide a detailed introduction to the symbols required for model construction. Here, let ${v_j}$ denote transaction cost between 3PL j and 4PL, which includes costs incurred due to potential risks and service quality during transportation⁸³.

Table 2 Notations for mathematical model.

(M1) The first stage problem:

$$\begin{aligned} & \begin{array}{l} \mathrm{{min }}\sum \limits _{j \in J} {\sum \limits _{k \in {K_j}} {{c_{jk}}{x_{jk}}} } + \sum \limits _{j \in J} {\sum \limits _{k \in {K_j}} {{v_j}{p_{jk}} + } } {E_\eta }\left[ {f\left( {x,p,w,\eta } \right) } \right] \end{array} \end{aligned}$$

(1)

$$\begin{aligned} & \text {s.t.}\nonumber \\ & {\quad } {\sum \limits _{j \in J} {\sum \limits _{k \in {K_j}} {{c_{jk}}{x_{jk}}} } \le {C_{\max }}} \end{aligned}$$

(2)

$$\begin{aligned} & {\quad } {\sum _{k\in K_{j}}p_{jk}\le 1\quad \forall j\in J} \end{aligned}$$

(3)

$$\begin{aligned} & {\quad } {\sum \limits _{j \in J} {\sum \limits _{k \in {K_j}} {{p_{jk}}} } \ge {N_{\min }}} \end{aligned}$$

(4)

$$\begin{aligned} & {\quad } {\sum \limits _{j \in J} {\sum \limits _{k \in {K_j}} {{p_{jk}}} } \le {N_{\max }}} \end{aligned}$$

(5)

$$\begin{aligned} & {\quad } {x_{jk}} \in \left\{ {0,1} \right\} ,\mathrm{ }\forall j \in J,\forall k \in {K_j} \; \;\;\; \end{aligned}$$

(6)

$$\begin{aligned} & {\quad } {p_{jk}} \in \left\{ {0,1} \right\} \mathrm{{, }}\forall j \in J,\forall k \in {K_j} \; \;\;\; \end{aligned}$$

(7)

The second stage problem:

$$\begin{aligned} & \begin{array}{l} \mathrm{{ min }}f\left( {x,p,w,\eta } \right) = \sum \limits _{w \in S} {{r_w}} \left( {\sum \limits _{i \in I} {\sum \limits _{j \in J} {\sum \limits _{k \in {K_j}} {{b_{ijk}}{y_{ijkw}}\left( \upsilon \right) + \sum \limits _{i \in I} {{e_i}{\varphi _{iw}}\left( \upsilon \right) } } } } } \right) \;\;\; \end{array} \end{aligned}$$

(8)

$$\begin{aligned} & \text {s.t.}\nonumber \\ & {\quad } \sum \limits _{j \in J} {\sum \limits _{k \in {K_j}} {{y_{ijkw}}\left( \upsilon \right) } } + {\varphi _{iw}}\left( \upsilon \right) = {\eta _i}\left( \upsilon \right) ,\mathrm{ }i \in I,w \in S,\upsilon \in \bar{\upsilon }\end{aligned}$$

(9)

$$\begin{aligned} & {\quad } {q_{jkw}}{x_{jk}}U{S_{ijk}} + (1 - {q_{jkw}})U{S_{ijk}} \ge {y_{ijkw}}\left( \upsilon \right) ,i \in I,j \in J,k \in {K_j},w \in S,\upsilon \in \bar{\upsilon }\; \end{aligned}$$

(10)

$$\begin{aligned} & {\quad } M{p_{jk}} \ge {y_{ijkw}}\left( \upsilon \right) ,i \in I,j \in J,k \in {K_j},w \in S,\upsilon \in \bar{\upsilon }\end{aligned}$$

(11)

$$\begin{aligned} & {\quad } {y_{ijkw}}\left( \upsilon \right) \ge 0,\mathrm{ }\forall i \in I,\forall j \in J,\forall k \in {K_j},\forall w \in S,\upsilon \in \bar{\upsilon }\end{aligned}$$

(12)

$$\begin{aligned} & {\quad } {\varphi _{iw}}\left( \upsilon \right) \ge 0,\mathrm{ }\forall i \in I,\forall w \in S,\upsilon \in \bar{\upsilon }\end{aligned}$$

(13)

Eq. (1) is the objective function of the first stage, which focuses on minimizing the summation of cost associated with expected operations, transactions, and fortification under stochastic demand. Eq. (2) indicates that the total fortification cost is not allowed to exceed its upper limit. Eq. (3) indicates that only one package submitted by a 3PL can be selected at most. Eqs. (4) and (5) indicate that 4PL intends to limit the number of winning 3PLs to a specific range. Eqs. (6) and (7) ensure that the decision variables are binary integer variables, taking values of 0 or 1. Eq. (8) is the objective function of the second stage, which involves expected procurement cost and temporary outsourcing cost related to uncertain disruption in the case of realized stochastic demand. Eq. (9) indicates that the transportation demand of clients must be satisfied. Eq. (10) indicates that regardless of whether the packages of 3PLs are fortified or not, the allocated shipment volume is not allowed to exceed actual transport capacity of each 3PL. Eq. (11) indicates that if a certain amount of shipment volumes is allocated to the 3PL, it will definitely be the winner, with M defined as a sufficiently large constant. Eqs. (12) and (13) guarantee that decision variables are constrained to be non-negative.

Due to the difficulty in obtaining the probability distribution function of $f\left( {x,p,w,\eta } \right)$, it is very difficult to convert ${E_\eta }\left[ {f\left( {x,p,w,\eta } \right) } \right]$ in the objective function into an exact form. Therefore, we combine the LHS method and chi-square test to reconstruct the framework of SAA method, and propose an ISAA algorithm to enhance the computational efficiency of the lower and upper bounds of traditional SAA algorithms. The LHS method is adopted to extract samples, which can achieve the same or better results as Monte Carlo sampling (MCS) with only 1/5 of the sample size⁸⁴. Based on the LHS method, the above model is transformed into the following deterministic form:

(M2)

$$\begin{aligned} & \begin{array}{l} \mathrm{{min }}\sum \limits _{j \in J} {\sum \limits _{k \in {K_j}} {{c_{jk}}{x_{jk}}} } + \sum \limits _{j \in J} {\sum \limits _{k \in {K_j}} {{v_j}{p_{jk}}} } \mathrm{{ + }}\frac{1}{N}\sum \limits _{\upsilon = 1}^N {\left( {\sum \limits _{w \in S} {{r_w}\left( {\sum \limits _{i \in I} {\sum \limits _{j \in J} {\sum \limits _{k \in {K_j}} {{b_{ijk}}{y_{ijkw\upsilon }} + \sum \limits _{i \in I} {{e_i}{\varphi _{iw\upsilon }}} } } } } \right) } } \right) } \end{array} \end{aligned}$$

(14)

$$\begin{aligned} & \text {s.t.}\nonumber \\ & {\quad }\sum \limits _{j \in J} {\sum \limits _{k \in {K_j}} {{c_{jk}}{x_{jk}}} } \le {C_{\max }} \end{aligned}$$

(15)

$$\begin{aligned} & {\quad } \sum \limits _{j \in J} {\sum \limits _{k \in {K_j}} {{y_{ijkw\upsilon }}} } + {\varphi _{iw\upsilon }} = {\eta _{i\upsilon }},\mathrm{ }i \in I,w \in S,\upsilon \in \Lambda \end{aligned}$$

(16)

$$\begin{aligned} & {\quad } {q_{jkw}}{x_{jk}}U{S_{ijk}} + (1 - {q_{jkw}})U{S_{ijk}} \ge {y_{ijkw\upsilon }},i \in I,j \in J,k \in {K_j},w \in S,\upsilon \in \Lambda \end{aligned}$$

(17)

$$\begin{aligned} & {\quad } M{p_{jk}} \ge {y_{ijkw\upsilon }},i \in I,j \in J,k \in {K_j},w \in S,\upsilon \in \Lambda \end{aligned}$$

(18)

$$\begin{aligned} & {\quad } \sum \limits _{k \in {K_j}} {{p_{jk}}} \le 1,\mathrm{ }\forall j \in J \end{aligned}$$

(19)

$$\begin{aligned} & {\quad } {\sum \limits _{j \in J} {\sum \limits _{k \in {K_j}} {{p_{jk}}} } \ge {N_{\min }}} \end{aligned}$$

(20)

$$\begin{aligned} & {\quad } {\sum \limits _{j \in J} {\sum \limits _{k \in {K_j}} {{p_{jk}}} } \le {N_{\max }}} \end{aligned}$$

(21)

$$\begin{aligned} & {\quad } {y_{ijkw\upsilon }} \ge 0,\forall i \in I,\forall j \in J,\forall k \in {K_j},\forall w \in S,\upsilon \in \Lambda \end{aligned}$$

(22)

$$\begin{aligned} & {\quad } {\varphi _{iw\upsilon }} \ge 0,\forall i \in I,\forall w \in S,\upsilon \in \Lambda \end{aligned}$$

(23)

$$\begin{aligned} & {\quad } {p_{jk}} \in \left\{ {0,1} \right\} \mathrm{{, }}\forall j \in J,\forall k \in {K_j} \end{aligned}$$

(24)

$$\begin{aligned} & {\quad } {x_{jk}} \in \left\{ {0,1} \right\} ,\mathrm{ }\forall j \in J,\forall k \in {K_j} \end{aligned}$$

(25)

Since M2 is a MILP model, we can directly use commercial solvers to solve M2. However, it is important to emphasize that M2 incorporates both disruption and demand scenarios simultaneously, leading to a significant increase in decision variables and constraints, which poses significant challenges for solving the problem model. Therefore, an ISAA-SR-DDLR algorithm is proposed as a solution to the problem model in the next section.

A sampling-based heuristic algorithm

In this section, we propose an ISAA-SR-DDLR algorithm to solve the proposed MILP model. Traditional SAA methods usually use MCS to obtain samples and randomly set the sample size. The randomness of sample size requires extensive numerical testing to determine the optimal sample size for the SAA algorithm, which reduces its efficiency and increases the instability of solution quality. To improve the efficiency of model solving, we use the LHS method and chi-square test to determine the optimal sample size, thereby reconstructing the framework of the traditional SAA algorithm. We further integrate SR and DDLR algorithms to design the ISAA-SR-DDLR algorithm. The steps of our algorithm are illustrated in Fig. 2.

Fig. 2

Comparison details of different algorithms.

Specifically, SR method is used to obtain representative scenarios from a complete scenario set, simplifying the problem size. Based on the LHS method and chi-square test, an ISAA method is proposed to determine the number of demand scenarios. The upper bound of problem model is derived from a hybrid ISAA and SR method, and the lower bound of problem model is derived from ISAA and DDLR algorithms. Finally, the quality of the solution is assessed by calculating the gap between the upper and lower bounds. The advantage of the proposed algorithm lies in its ability to provide both the lower and upper bounds of the original problem model. The gap between these two bounds reflects the applicability of the proposed algorithm for solving the problem. The specific details are as follows.

Upper bound calculation

Scenario reduction approach

We use a scenario-based approach to characterize disruption risk in M2. It is worth noting that disruption scenarios grow exponentially. If n represents the number of packages that may experience disruption, then the disruption scenarios are ${2^n}$, which causes great difficulties in solving the model. Therefore, the SR approach proposed by Karuppiah et al.⁸⁵ is utilized to handle disruption scenarios in this section.

The SR approach seeks to employ a new subset as a replacement for the complete set of scenarios, while minimizing the deviation of the objective function values in the reduced subset and full scenarios as much as possible. Here, we provide a example to illustrate the process of SR techniques. We consider two uncertain parameters $X_1$ and $X_2$, each with two possible values. The value set for $X_1$ is {6, 9}, with corresponding probabilities of 0.4 and 0.6, respectively. The value set for $X_2$ is {30, 50}, with corresponding probabilities of 0.7 and 0.3, respectively. By combining these two uncertain parameters, we obtain the four scenarios shown in Table 3 and Fig. 3. Here, FS represents the original complete scenarios, and POS denotes the probabilities of the corresponding scenarios occurring.

Fig. 3

Full scenarios in the example.

Table 3 Full scenarios for the example.

To decrease the total scenarios, we can recombine the scenarios by reversing the values and probabilities of the uncertain parameters, while keeping the occurrence probabilities of each uncertain parameter unchanged. As shown in Fig. 4, by merging the scenarios (6, 30) and (6, 50) into a new scenario (6, 30), the occurrence probability of (6, 30) is equal to the probability of $X_1$ taking the value of 6. Table 4 presents the results after reducing the scenarios, where RS denotes the subset obtained from the reduction. For the original scenarios, the probability of $X_1$ taking the value of 9 is 0.6. In the reduced scenario set, $X_1$ takes the value of 9 in RS = 2′ and RS = 3′, with the total occurrence probability of RS = 2′ and RS = 3′ being 0.6 (0.3 + 0.3), ensuring that the occurrence probabilities of the uncertain parameter values remain unchanged. At the same time, it ensures that the total probability of the scenarios after reduction sums to 1.

Fig. 4

The scenario reduction process and results in the example.

Table 4 Reduced scenarios for the example.

For our problem, let $D = \left| {{D_U}} \right|$ denote the total number of packages likely to experience disruption. Let $\varepsilon = {\{ {\varepsilon _k}\} _{k = 1, \cdots ,D}}$ represent the uncertain parameter vector indicating whether package k experiences disruption, $k \in {D_U}$. We can assume that each ${\varepsilon _k}$ can take on values from the set ${\{ {\varepsilon _k}^{{d_k}}\} _{{d_k} = 1,2}}$.When ${\varepsilon _k}^{{d_k}} = 0$, it means that no disruption has occurred in the package k, while ${\varepsilon _k}^{{d_k}} = 1$ indicates that the package k has experienced a disruption. The probability corresponding to ${\varepsilon _k}^{{d_k}}$ assumed to be ${r_k}^{{l_k}}$, where $0< {r_k}^{{l_k}} < 1$, $k = 1,2, \cdots ,L$, ${l_k} = 1,2$. By using the following SR approach, we can obtain a new subset of scenarios after scenario reduction.

$$\begin{aligned} & \min f = \sum \limits _{{l_1} = 1}^2 {\sum \limits _{{l_2} = 1}^2 \cdots } \sum \limits _{{l_L} = 1}^2 {\left[ {(1 - r_1^{{l_1}}r_2^{{l_2}} \cdots r_L^{{l_L}}) \cdot {{\hat{r}}_{{l_1},{l_2},...,{l_L}}}} \right] } \end{aligned}$$

(26)

$$\begin{aligned} & \text {s.t.}\nonumber \\ & {\quad } \sum \limits _{{l_2} = 1}^2 {\sum \limits _{{l_3} = 1}^2 \cdots } \sum \limits _{{l_L} = 1}^2 {{{\hat{r}}_{{l_1},{l_2},...,{l_L}}}} = r_1^{{l_1}},\quad {l_1} = 1,2 \end{aligned}$$

(27)

$$\begin{aligned} & {\quad }\sum \limits _{{l_1} = 1}^2 {\sum \limits _{{l_3} = 1}^2 \cdots } \sum \limits _{{l_L} = 1}^2 {{{\hat{r}}_{{l_1},{l_2},...,{l_L}}}} = r_2^{{l_2}},\quad {l_2} = 1,2\mathrm{ } \end{aligned}$$

(28)

$$\begin{aligned} & {\quad } \vdots \nonumber \\ & {\quad } \sum \limits _{{l_1} = 1}^2 {\sum \limits _{{l_2} = 1}^2 \cdots } \sum \limits _{{l_{L - 1}} = 1}^2 {{{\hat{r}}_{{l_1},{l_2},...,{l_L}}}} = r_L^{{l_L}},\quad {l_L} = 1,2\mathrm{ } \end{aligned}$$

(29)

$$\begin{aligned} & {\quad } \sum \limits _{{l_1} = 1}^2 {\sum \limits _{{l_2} = 1}^2 \cdots } \sum \limits _{{l_L} = 1}^2 {{{\hat{r}}_{{l_1},{l_2},...,{l_L}}}} = 1\mathrm{ } \end{aligned}$$

(30)

$$\begin{aligned} & {\quad } 0 \le {{\hat{r}}_{{l_1},{l_2}, \ldots ,{l_L}}} \le 1,\quad \forall {l_1},{l_2}, \ldots ,{l_L} \end{aligned}$$

(31)

Here, ${\hat{r}_{{l_1},{l_2}, \ldots ,{l_D}}}$ denotes the probabilities allocated to each scenario in the subset of scenarios obtained after the reduction. Equation (26) represents incorporating scenarios with relatively high probabilities from the original full scenario set into the new scenario subset as much as possible. For each ${l_k} = 1,2$, Eqs. (27–29) that the total probability of scenarios related to the potentially disrupted package k equals the probability of disruption for package k in the subset of scenarios obtained after the reduction, $k \in {L_D}$. Equation (30) ensures that the total probability of the subset of scenarios in the reduced scenario result equals 1. Equation (31) represents that the new probabilities allocated to each scenario fall within the range of 0–1.

Based on the above SR approach, we can obtain a new set of disruption scenarios $\bar{S}$ and corresponding probability values ${\hat{r}_w}$. Therefore, M2 can be transformed into the following form:

(M3)

$$\begin{aligned} & \begin{array}{l} {\mathrm{{min }}\sum \limits _{j \in J} {\sum \limits _{k \in {K_j}} {{c_{jk}}{x_{jk}}} } + \sum \limits _{j \in J} {\sum \limits _{k \in {K_j}} {{v_j}{p_{jk}}} } \mathrm{{ + }}\frac{1}{N}\sum \limits _{\upsilon = 1}^N {\left( {\sum \limits _{w \in \bar{S}} {{{\hat{r}}_w}\left( {\sum \limits _{i \in I} {\sum \limits _{j \in J} {\sum \limits _{k \in {K_j}} {{b_{ijk}}{y_{ijkw\upsilon }} + \sum \limits _{i \in I} {{e_i}{\varphi _{iw\upsilon }}} } } } } \right) } } \right) } } \end{array}\nonumber \\ & \text {s.t.}\nonumber \\ & {\quad } (15)-(25) \end{aligned}$$

(32)

By using SR method to reduce the number of disruption scenarios, we further significantly decrease the number of constraints (Eqs. 16–18) and decision variables related to these scenarios, thereby greatly enhancing computational efficiency. To assess the impact of scenario loss on the quality of the model solution, we provide a method for calculating the gap between the upper and lower bounds of the original model in the next subsection. We use the size of the gap to reflect the impact of scenario loss on the quality of the solution in the SR method.

ISAA algorithm

Although we have reduced the number of disruption scenarios in M3, determining the sample size N after stochastic demand sampling remains a crucial issue. Traditional SAA methods usually use MCS to obtain samples and randomly set the sample size. The randomness of sample size requires extensive numerical testing to determine the optimal sample size for the SAA algorithm, which reduces its efficiency and increases the instability of solution quality. However, according to the literature⁸⁶, using the LHS method instead of simple MCS in importance sampling can save half of the computational workload. MCS usually requires a large sample size to approximate the demand distribution, while LHS is designed to accurately approximate the demand distribution with fewer sampling iterations⁸⁷. Fattahi et al.⁸⁸ pointed out that, with the same sample size, the LHS method covers a larger domain of random variables compared to MCS. Shi et al.⁸⁹ pointed out that LHS is recommended as a practical method for risk and uncertainty analysis in simulation experiments. According to Lo et al.⁹⁰, we find that the LHS method can achieve comparable or even superior results to the Monte Carlo sampling method using only 1/5 of the sample size. To verify the superiority of LHS, we present the chi-square test results for LHS and MCS under different sample sizes. By calculating the average deviation between the true mean and the sample mean, as well as between the true variance and the sample variance, we use chi-square test to evaluate the degree of deviation between the eigenvalues of the true distribution and those of the sample distribution. The variance and mean of real random variables are denoted by ${\sigma ^2},\mu$; The variance and mean of the sample are denoted by ${\bar{\sigma }^2},\bar{\mu }$. Assuming that all demands follow a uniform distribution, we set the coefficient of variation $CV = \sigma /\mu = 0.216$, $\mu = 400$. Table 5 displays the chi-square test results for different samples derived from LHS and MCS. Here, EM denotes $E\left( {\left| {\bar{\mu }- \mu } \right| } \right)$, EV denotes $E\left( {\left| {{{\bar{\sigma }}^2} - {\sigma ^2}} \right| } \right)$, CS denotes ${\chi ^2}$, PD denotes P-value.

Table 5 Chi-square test analysis.

From Table 5, we can see that LHS is significantly superior to the MCS method. For LHS, when N=100, the P-value and chi-square value are 1 and 0, respectively. This indicates that the deviation between actual observed values and theoretical values is very small. Therefore, the sample size is set to 100. Set $\tilde{N}$ as the sample size determined by the chi-square test, then M3 is expressed below: (M4)

$$\begin{aligned} & \begin{array}{l} {\mathrm{{min }}\sum \limits _{j \in J} {\sum \limits _{k \in {K_j}} {{c_{jk}}{x_{jk}}} } + \sum \limits _{j \in J} {\sum \limits _{k \in {K_j}} {{v_j}{p_{jk}}} } \mathrm{{ + }}\frac{1}{{\tilde{N}}}\sum \limits _{\upsilon = 1}^{\tilde{N}} {\left( {\sum \limits _{w \in \bar{S}} {{{\hat{r}}_w}\left( {\sum \limits _{i \in I} {\sum \limits _{j \in J} {\sum \limits _{k \in {K_j}} {{b_{ijk}}{y_{ijkw\upsilon }} + \sum \limits _{i \in I} {{e_i}{\varphi _{iw\upsilon }}} } } } } \right) } } \right) } } \end{array}\nonumber \\ & \text {s.t.}\nonumber \\ & {\quad } (15)-(25) \end{aligned}$$

(33)

Based on the SR approach, LHS, and chi-square test, M4 becomes an easily solvable MILP problem. The CPLEX solver allows us to obtain the optimal solution, denoted by $\left( {{{\bar{\textbf{X}},\bar{P}}}} \right)$.

Based on the optimal solution $\left( {{{\bar{\textbf{X}},\bar{P}}}} \right)$ of M4, we can provide an effective upper bound for M1. Given a large sample size of $\hat{N}$, ensuring that $\hat{N} > N$, solve M2 based on $\left( {{{\bar{\textbf{X}},\bar{P}}}} \right)$ and $\hat{N}$. Then record the corresponding objective function value ${v_{\hat{N}}}\left( {{{\bar{\textbf{X}},\bar{P}}}} \right)$, and calculate the variance of ${v_{\hat{N}}}\left( {{{\bar{\textbf{X}},\bar{P}}}} \right)$ using the following formula:

$$\begin{aligned} \sigma _{\hat{N}}^2\mathrm{{ = }}\frac{1}{{\hat{N}\left( {\hat{N} - 1} \right) }}\sum \limits _{n = 1}^{\hat{N}} {\left[ {{\mathbf{{C}}^T} \mathbf{{X + }}{\mathbf{{V}}^T} \mathbf{{P + }}f\left( {\mathbf{{X,P,w}},\mathbf{{\eta }}} \right) } \right] } \mathrm{ } \end{aligned}$$

(34)

By substituting $\left( {{{\bar{\textbf{X}},\bar{P}}}} \right)$ into M2, $\sum \limits _{j \in J} {\sum \limits _{k \in {K_j}} {{c_{jk}}{x_{jk}}} } + \sum \limits _{j \in J} {\sum \limits _{k \in {K_j}} {{v_j}{p_{jk}}} }$ becomes a constant, and M2 degrades into the following linear programming (LP) problem:

(M5)

$$\begin{aligned} & \begin{array}{l} {\mathrm{{min }}\frac{1}{{\hat{N}}}\sum \limits _{\upsilon = 1}^{\hat{N}} {\left( {\sum \limits _{w \in S} {{r_w}\left( {\sum \limits _{i \in I} {\sum \limits _{j \in J} {\sum \limits _{k \in {K_j}} {{b_{ijk}}{y_{ijkw\upsilon }} + \sum \limits _{i \in I} {{e_i}{\varphi _{iw\upsilon }}} } } } } \right) } } \right) } } \end{array}\nonumber \\ & \text {s.t.}\nonumber \\ & {\quad } (15)-(25) \end{aligned}$$

(35)

Although M5 is a LP problem, solving it directly with commercial software remains a significant challenge due to numerous scenarios. We can decompose M5 into multiple scenario-independent subproblems. For each given demand scenario v, the sub-problems of M5 can be expressed as:

(M5s)

$$\begin{aligned} & \begin{array}{l} \mathrm{{min }}\sum \limits _{w \in S} {{r_w}\left( {\sum \limits _{i \in I} {\sum \limits _{j \in J} {\sum \limits _{k \in {K_j}} {{b_{ijk}}{y_{ijkw\upsilon }} + \sum \limits _{i \in I} {{e_i}{\varphi _{iw\upsilon }}} } } } } \right) } \mathrm{ } \end{array}\nonumber \\ & \text {s.t.}\nonumber \\ & {\quad } (15)-(25) \end{aligned}$$

(36)

By solving these N independent sub-problems, we can obtain a point estimate of upper bound for M1 and the corresponding variance, denoted as ${v_{\hat{N}}}\left( {{{\bar{\textbf{X}},\bar{P}}}} \right)$ and $\sigma _{\hat{N}}^2$, respectively.

Lower bound calculation

The quality of solutions from the SAA model is evaluated by Kleywegt et al.⁹¹ through statistical analysis methods. Similarly, we employ this approach to estimate the upper and lower bounds of M1, thus allowing for an evaluation of the quality of the obtained solution.

To obtain the lower bound of M1, we first provide a simplified form of M2, which relaxes the constraint on transport capacity (Eq. 17). Setting ${q_{jkw}}\mathrm{{ = }}0$, we can obtain a simplified version of M2, represented by M2v, and the simplified problem is as follows:

(M2v)

$$\begin{aligned} & \begin{array}{l} {\mathrm{{min }}\sum \limits _{j \in J} {\sum \limits _{k \in {K_j}} {{c_{jk}}{x_{jk}}} } + \sum \limits _{j \in J} {\sum \limits _{k \in {K_j}} {{v_j}{p_{jk}}} } \mathrm{{ + }}\frac{1}{N}\sum \limits _{\upsilon = 1}^N {\left( {\sum \limits _{w \in S} {{r_w}\left( {\sum \limits _{i \in I} {\sum \limits _{j \in J} {\sum \limits _{k \in {K_j}} {{b_{ijk}}{y_{ijkw\upsilon }} + \sum \limits _{i \in I} {{e_i}{\varphi _{iw\upsilon }}} } } } } \right) } } \right) } } \end{array} \end{aligned}$$

(37)

$$\begin{aligned} & \text {s.t.}\nonumber \\ & {\quad } U{S_{ijk}} \ge {y_{ijkw\upsilon }},i \in I,j \in J,k \in {K_j},w \in S,\upsilon \in \Lambda \nonumber \\ & {\quad } (15)-(16), (18)-(25) \end{aligned}$$

(38)

We can obtain a point estimate of lower bound for M1 through multiple solutions of M2v. The specific details are as follows:

Generate M independent sample sets with a size of N. Repeat solving the M2v model for M times and record the optimal objective function value $\hat{v}_N^m$ for each iteration, $m = 1,2, \cdots ,M$.

Calculate the mean of objective function values and corresponding variance over M times. The formula details are given below:

$$\begin{aligned} \sigma _{{{\bar{v}}_{N,M}}}^2= & \frac{1}{{\left( {M - 1} \right) M}} \sum \limits _{m = 1}^M {{{\left( {\hat{v}_N^m - {{\bar{v}}_{N,M}}} \right) }^2}} \end{aligned}$$

(39)

$$\begin{aligned} {\bar{v}_{N,M}}= & \frac{1}{M}\sum \limits _{m = 1}^M {\hat{v}_N^m} \end{aligned}$$

(40)

The average optimal value of the objective function ${\bar{v}_{N,M}}$ serves as a point estimate of lower bound for M1^92,93. Since M2v needs to be computed multiple times, we utilize the DDLR method to decompose M2v to improve computational efficiency. In M2v, only the decision variables $\left( {\mathbf{{X,P}}} \right)$ are scenario-independent. We introduce decision variables $\left( {{\mathbf{{X}}_\mathbf{{1}}} \mathbf{{,}}{\mathbf{{P}}_\mathbf{{1}}}} \right) \mathbf{{,}}\left( {{\mathbf{{X}}_\mathbf{{2}}} \mathbf{{,}}{\mathbf{{P}}_\mathbf{{2}}}} \right) \cdots \left( {{\mathbf{{X}}_\mathbf{{N}}} \mathbf{{,}}{\mathbf{{P}}_\mathbf{{N}}}} \right)$ to ensure that M2v can be decomposed into N independent subproblems. To ensure equivalence with the simplified problem M2v, the non-anticipativity constraint $\left( {{\mathbf{{X}}_\mathbf{{1}}} \mathbf{{,}}{\mathbf{{P}}_1}} \right) \mathrm{{ = }}\left( {{\mathbf{{X}}_\mathbf{{2}}} \mathbf{{,}}{\mathbf{{P}}_2}} \right) \mathrm{{ = }} \cdots \mathrm{{ = }}\left( {{\mathbf{{X}}_\mathbf{{N}}} \mathbf{{,}}{\mathbf{{P}}_\mathbf{{N}}}} \right)$ must be added to the simplified problem M2v, ensuring that the decision variables remain equivalent across all subproblems. The decomposition formula is as follows:

$$\begin{aligned} & \begin{array}{l} \mathrm{{min }}\sum \limits _{\upsilon = 1}^N {\left( \begin{array}{l} \frac{1}{N}\sum \limits _{j \in J} {\sum \limits _{k \in {K_j}} {{c_{jk}}{x_{jk\upsilon }}} } + \frac{1}{N}\sum \limits _{j \in J} {\sum \limits _{k \in {K_j}} {{v_j}{p_{jk\upsilon }}} } \mathrm{{ + }}\\ \frac{1}{N}\sum \limits _{w \in S} {{r_w}\left( {\sum \limits _{i \in I} {\sum \limits _{j \in J} {\sum \limits _{k \in {K_j}} {{b_{ijk}}{y_{ijkw\upsilon }} + \sum \limits _{i \in I} {{e_i}{\varphi _{iw\upsilon }}} } } } } \right) } \end{array} \right) } \mathrm{ } \end{array} \end{aligned}$$

(41)

$$\begin{aligned} & \text {s.t.}\nonumber \\ & {\quad } (2)-(7), (9)-(13)\nonumber \\ & {\quad } {(\mathbf{{x}}_w^T,\mathbf{{p}}_w^T)^T} = {(\mathbf{{x}}_{w + 1}^T,\mathbf{{p}}_{w + 1}^T)^T},w = 1,2, \ldots ,|N| - 1\mathrm{ } \end{aligned}$$

(42)

$$\begin{aligned} & {\quad } {(\mathbf{{x}}_{|N|}^T,\mathbf{{p}}_{|N|}^T)^T} = {(\mathbf{{x}}_1^T,\mathbf{{p}}_1^T)^T} \end{aligned}$$

(43)

Where ${x_{jk\upsilon }}$ and ${p_{jk\upsilon }}$ are decision variables related to ${x_{jk}}$ and ${p_{jk}}$. Equations (42) and (43) ensure that ${x_{jk1}} = {x_{jk2}} = \cdots = {x_{jk\left| N \right| }}$ and ${p_{jk1}} = {p_{jk2}} = \cdots = {p_{jk\left| N \right| }}$.

Equations (42) and (43) can be reformulated as $\sum \limits _{\upsilon = 1}^N {{\mathbf{{H}}_\upsilon }\left( {{\mathbf{{X}}_\upsilon } \mathbf{{,}}{\mathbf{{P}}_\upsilon }} \right) } = \mathbf{{0}}\mathrm{ }$ based on Carøe and Schultz⁹⁴, where ${\mathbf{{H}}_\upsilon }$ is the corresponding transformation matrix and the specific detailed expression can be found in Meng et al.⁹⁵.

Given $\lambda$ as the vector of Lagrange multipliers, we use (LR) to denote the Lagrangian relaxation model as follows:

$$\begin{aligned}&\mathrm{{LR}}\left( \lambda \right) \mathrm{{ = min }}\sum \limits _{\upsilon = 1}^N {\left( \begin{array}{l} \frac{1}{N}\sum \limits _{j \in J} {\sum \limits _{k \in {K_j}} {{c_{jk}}{x_{jk\upsilon }}} } + \frac{1}{N}\sum \limits _{j \in J} {\sum \limits _{k \in {K_j}} {{v_j}{p_{jk\upsilon }}} } \mathrm{{ + }}{\mathbf{{\lambda }}^T}{\mathbf{{H}}_\upsilon }\left( {{\mathbf{{X}}_\upsilon } \mathbf{{,}}{\mathbf{{P}}_\upsilon }} \right) \mathrm{{ + }}\\ \frac{1}{N}\sum \limits _{w \in S} {{r_w}\left( {\sum \limits _{i \in I} {\sum \limits _{j \in J} {\sum \limits _{k \in {K_j}} {{b_{ijk}}{y_{ijkw\upsilon }} + \sum \limits _{i \in I} {{e_i}{\varphi _{iw\upsilon }}} } } } } \right) } \end{array} \right) } \nonumber \\&\text {s.t.}{\quad }(2)-(7),(9)-(13) \end{aligned}$$

(44)

The Lagrangian relaxation problem can be divided into various independent subproblems. By solving the Lagrange dual model (LD), we can find the lower bound of the Lagrange relaxation problem:

$$\begin{aligned} LD\left( \mathbf{{\lambda }} \right) = \mathop {\max }\limits _\mathbf{{\lambda }} LR\left( \mathbf{{\lambda }} \right) \end{aligned}$$

(45)

Note that (LD) is a concave maximization problem in which the objective function is non-differentiable. The steps for using the subgradient method to solve (LD) are shown below:

Step 1::

Define k to be 1, and initialize the Lagrangian multiplier vector as ${\mathbf{{\lambda }}^\mathbf{{1}}}$.

Step 2::

Given ${\mathbf{{\lambda }}^k}$, solve the subproblem of Lagrangian relaxation problem and calculate the corresponding subgradient $\sum \limits _{\upsilon = 1}^N {{\mathbf{{H}}_\upsilon }\left( {\mathbf{{x}}_\upsilon ^*\mathbf{{,p}}_\upsilon ^*} \right) }$. Then update the optimal objective function value that is currently available and denote it by ${\Phi ^*}$.

Step 3::

Design the following formula to update Lagrange multipliers:

$$\begin{aligned} {\mathbf{{\lambda }}^{\mathbf{{k + 1}}}}\mathrm{{ = }}{\mathbf{{\lambda }}^\mathbf{{k}}}\mathrm{{ + }}{\mathrm{{s}}^k}{\sum \limits _{\upsilon = 1}^N {{\mathbf{{H}}_\upsilon }\left( {\mathbf{{x}}_\upsilon ^*\mathbf{{,p}}_\upsilon ^*} \right) } _k}\mathrm{ } \end{aligned}$$

(46)

      The setting of step size refers to Bazaraa et al.⁹⁶, ${\mathrm{{s}}^k}\mathrm{{ = }}{\alpha ^k}\frac{{{\beta ^k} - LR\left( {{\mathbf{{\lambda }}^\mathbf{{k}}}} \right) }}{{{{\left\| {{{\sum \limits _{\upsilon = 1}^N {{\mathbf{{H}}_\upsilon }\left( {\mathbf{{x}}_\upsilon ^*\mathbf{{,p}}_\upsilon ^*} \right) } }_k}} \right\| }^2}}}$, where ${\alpha ^k}\mathrm{{ = }}\frac{{1\mathrm{{ + }}m}}{{k + m}}$, ${\beta ^k}\mathrm{{ = }}\left( {1\mathrm{{ + }}\varepsilon \left( k \right) } \right) {\Phi ^*}$. $\varepsilon \left( k \right)$ is a linear function that decreases with increasing values of k and is always positive. m is a fixed constant.

Step 4::

If the conditions listed below are satisfied, the algorithm will stop running:

$$\begin{aligned} \left| {\frac{{LR\left( {{\mathbf{{\lambda }}^{k\mathrm{{ + }}1}}} \right) - LR\left( {{\mathbf{{\lambda }}^k}} \right) }}{{LR\left( {{\mathbf{{\lambda }}^k}} \right) }}} \right| \le \varepsilon \mathrm{ } \end{aligned}$$

(47)

Where $\varepsilon$ is a sufficiently small number. Otherwise, increase k by 1 and continue to Step 2.

By solving the (LD) M times using the subgradient method, we can obtain a point estimate ${\bar{v}_{N,M}}$ for the lower bound of M1 and its corresponding variance $\sigma _{{{\bar{v}}_{N,M}}}^2$.

Based on the point estimates of the upper and lower bounds of M1, we can estimate the gap between. The formula are given below:

$$\begin{aligned} & \mathrm{{gap}}\left( {{{\bar{\textbf{X}},\bar{P}}}} \right) \mathrm{{ = }}{v_{\hat{N}}}\left( {{{\bar{\textbf{X}},\bar{P}}}} \right) - {\bar{v}_{N,M}} \end{aligned}$$

(48)

$$\begin{aligned} & \sigma _{gap}^2 = \sigma _{\hat{N}}^2 + \sigma _{{{\bar{v}}_{N,M}}}^2 \end{aligned}$$

(49)

The formula for calculating the 1-$\alpha$ confidence interval of the optimality gap is

$$\begin{aligned} {v_{\hat{N}}}\left( {{{\bar{\textbf{X}},\bar{P}}}} \right) - {\bar{v}_{N,M}}\mathrm{{ + }}{z_\alpha }\sqrt{\left( {\sigma _{\hat{N}}^2 + \sigma _{{{\bar{v}}_{N,M}}}^2} \right) } \end{aligned}$$

(50)

Here, ${z_\alpha }\mathrm{{ = }}{\varphi ^{\mathrm{{ - 1}}}}\left( {\mathrm{{1 - }}\alpha } \right)$, and more details can be found in Qian et al.³⁵.

Numerical experiments

The following numerical experiments are designed to evaluate the performance of the algorithm in this study. Firstly, we introduce the generation rules for problems of different scales in Section 5.1. In Section 5.2, the effectiveness of ISAA-SR-DDLR algorithm is validated by comparing it to CPLEX. In Section 5.3, the importance of the proposed mixed mitigation strategy is validated through sensitivity analysis, providing management insights for 4PL in logistics service procurement.

Problem instance generation

This study considers three numerical examples of different scales. For small-scale problems, 4PL serves 5 lanes and considers 10 3PLs, with each 3PL submitting bids for 2 packages. The maximum fortification budget is 15000. We assume that the upper and lower limits of winning 3PLs are 10 and 0, respectively. According to the literature⁹⁷, we randomly generate model parameter values using uniform distribution, which can be denoted by $U\left[ {a,b} \right]$. The average of demand is fixed at 400, CV=0.216. ${b_{ijk}}$ $\sim$ $U\left[ {70, 120} \right]$, ${c_{jk}}$ $\sim$ $U\left[ {750, 2000} \right]$, ${v_j}$ $\sim$ $U\left[ {2000, 5000} \right]$, $U{S_{ijk}}$ $\sim$ $U\left[ {10, 200} \right]$.

The bidding bundles submitted by 3PLs can be expressed as [label of 3PL,{package1},{package2 }], that is, [3,{3},{3,4}], [6,{1,3},{1,2}], [9,{5},{1}], [7,{2,3},{2,5}], [5,{5},{1,5}], [2,{2},{2,3}], [1, {1}, {1,2}], [4,{4},{4,5}], [10,{2},{4}], [8,{3,4},{3,5}]. The disrupted packages can be expressed as [label of possible disrupted packages, disruption probability vector, {(label of 3PL, label of package)}], i.e., [5, (0.7, 0.75, 0.8, 0.85, 0.9)$^T$, {(5, 2), (7, 2), (9, 1), (8, 2), (4, 1)}], [10, (0.7, 0.7, 0.75, 0.75, 0.8, 0.85, 0.85, 0.9, 0.9, 0.9)$^T$, {(3, 2), (6, 2), (4, 2), (9, 1), (10, 2), (5, 2), (8, 2), (1, 2),(2, 2), (7, 2)}].

Medium scale examples are presented in Appendix 1. Appendix 2 presents the data for large-scale examples from Combinatorial Auction Test Suite (CATS).

Performance of ISAA-SR-DDLR

Numerical experiments are carried out using CPLEX 12.6.3 on desktop PC that has 16 GB of RAM and 3.60 GHz CPU. With the given maximum fortification budget ${C_{\max }}$ = 15000 and unit outsourcing cost H = 100, the total costs under different problem scales and disruption scenarios are calculated and compared with the results provided by CPLEX. Table 6 presents the comparison of computational results. Meanwhile, Table 7 presents the lower and upper bounds obtained through ISAA-SR-DDLR algorithm for medium and large-scale solutions, as well as gap between the two. Here, the number of disruption scenarios is denoted by NOS. NA indicates that CPLEX is unable to provide the calculation result due to insufficient memory. LB denote the lower bound, UB denote the upper bound.

Use ISAA-SR-DDLR algorithm and CPLEX to solve small-scale problems, respectively. From Table 6, it is found that when handling a small set of scenarios, the proposed algorithm can obtain optimum solution consistent with CPLEX. However, as the number of scenarios increases, CPLEX fails to run due to insufficient memory, while the proposed algorithm is capable of providing numerical results that CPLEX is unable to solve. For the other two scale problems, the excessive number of scenarios makes CPLEX unable to operate, while the proposed algorithm can still provide satisfactory results. It is worth noting that M3 is a MILP model, and CPLEX can be used for solving it before scenario reduction. However, we can see that the number of scenarios is enormous before scenario reduction, which prevents CPLEX from running due to insufficient memory, making it impossible to provide an effective solution. After scenario reduction, we can use the proposed algorithm to obtain high-quality near-optimal solutions in a shorter time, demonstrating that the incorporation of SR techniques significantly enhances computational efficiency. The proposed algorithm can effectively solve the presented model.

To further validate the superiority of the algorithm, it is used for solving medium-scale and large-scale problems. From Table 7, we obtain the lower and upper bounds of total cost under disruption scenarios of 32 and 1024, as well as the gap between them. The extremely small gap between the upper and lower bounds further demonstrates the effectiveness and superiority of our proposed algorithm.

Table 6 Comparison of computational results between ISAA-SR-DDLR algorithm and CPLEX.

Table 7 Computational results for problem of different scales.

Sensitivity analysis

In this section, to examine the influence of changes in key parameters on the results, we conduct sensitivity analysis experiments to provide management insights for 4PL. Firstly, the effects of CV changes are examined on operating costs of 4PL and fortification strategy. Secondly, the impact of changes in disruption probability is examined on the selection of fortification strategy, validating that considering fortification strategy is essential.

Analysis of CV changes

To analyze the effects of CV changes on fortification strategy and operating costs of 4PL, for medium scale problems, the set of values for CV is given as {0.072, 0.144, 0.216}. The set of values for the unit outsourcing cost H is {100,150,200,300,500}. The parameter information for the medium-scale problem is presented in Appendix 1, and the calculated cost results as well as the selection of 3PLs by 4PL are shown in Table 8. The changes in the proportion of external costs and auction costs to the total cost are illustrated in Fig. 5. Here, OC denotes outsourcing costs, QS denotes the quantity of 3PLs selected by 4PL. AC denotes auction costs, which include expected procurement cost, transaction cost and fortification cost. QD denotes the quantity of 3PLs that may experience disruptions, TC denotes total costs, and QF denotes the quantity of fortified 3PLs.

From Table 8 and Fig. 5, we can observe that when the unit outsourcing cost is fixed, as CV increases, 4PL will invest more money to procure additional temporary outsourcing services from external 3PLs, resulting in higher overall costs. In this situation, the number of 3PLs that need to be fortified slightly increases, and temporary outsourcing strategy becomes more important. In the case of fixed CV, when the unit outsourcing cost increases, outsourcing services become expensive, and 4PL will choose more potential 3PLs to allocate transportation tasks to reduce costs. Intuitively, the number of 3PLs that need to be fortified will increase, indicating that fortification strategy is very important in situations where demand fluctuations are small.

When the cost of outsourcing per unit is lower, as CV increases, the costs allocated for auctions gradually decrease, indicating that the 4PL may rely more on temporary outsourcing services to meet the temporarily increased demand from clients. This means that the temporary outsourcing strategy becomes more important. Once disruption occurs, 4PL can outsource the unsatisfied demand to other 3PLs that did not participate in the bidding, thereby keeping the total cost relatively low. When the temporary outsourcing cost per unit is high, as CV increases, the costs allocated for auctions gradually increase, indicating that more logistics tasks will be completed through auctions. If a large number of other 3PLs that did not participate in the bidding are used during the disruption, the total cost will be relatively high, so a more economical fortification strategy is needed.

Based on the fortification results under different parameters, when the cost of outsourcing per unit is low, with the decrease of CV, the number of 3PLs that need to be fortified gradually decreases, indicating that when H is small, the smaller the demand fluctuations, the impact of disruptions is also reduced. However, when CV increases, the impact on the 4PL will be greater once disruption occurs. When the cost of outsourcing per unit is high, the number of fortified 3PLs equals the number of 3PLs selected for possible disruption, regardless of the value of CV. This indicates that when H is large, the occurrence of a disruption will have significant impacts on the results regardless of the value of CV. Therefore, it also shows that when H is large, considering fortification strategy is very necessary.

Table 8 The selection of 3PLs under different CV and H.

Fig. 5

Changes in outsourcing and auction costs under different CV and H.

Analysis of disruption probability changes

In this subsection, the impact of changes in the probability of disruption is analyzed by setting up different instances for a comparative analysis of the results. 4PL can select certain 3PLs that are likely to experience disruptions for fortification. Figure 6 illustrates changes in the number of fortified 3PLs under different CV and H values in low-probability scenario. Figure 7 shows changes in the number of fortified 3PLs under different CV and H values in medium-probability scenario. Figure 8 shows changes in the number of fortified 3PLs under different CV and H values in large-probability scenario.

Through comparison, we can find that: (1) The number of selected 3PLs that may experience disruptions is likely to decrease as disruption probability increases. For example, when H=100, the number of selected disrupted 3PLs decreases as the probability changes from low to medium. (2) The priority of fortification strategy increases as the disruption probability grows. In situations with low probability, the priority of the fortification strategy may decrease. As shown in Fig. 3, the number of fortified 3PLs may not be the same as the number of selected disrupted 3PLs. However, when the disruption probability increases, fortification strategy cannot be overlooked. As shown in Figs. 4 and 5, regardless of the values of H and CV, the number of selected disrupted 3PLs is exactly the same as the number of fortified 3PLs, which means that 4PL fortifies all selected 3PLs. With an increasing disruption probability, fortification strategy becomes increasingly important.

Fig. 6

Changes in the number of fortified 3PLs under low probability.

Fig. 7

Changes in the number of fortified 3PLs under medium probability.

Fig. 8

Changes in the number of fortified 3PLs under large probability.

A real case

The practical effectiveness of the proposed model and algorithm is examined using a real case from a 4PL company based in Shenzhen, China. The company primarily provides global transportation services for electronic products by integrating 42 3PL companies (such as FedEx, MAERSK, etc.) to accomplish the task allocation for 29 transportation lanes. To address the risks posed by uncertain factors, other freight companies located in Shenzhen, such as Full-Trans Logistics and Forever Air & Sea, can be temporarily selected to increase capacity. For this company, a simplified transportation task scenario is described as shown in Table 9. Based on the company’s historical data, the transportation demand on the 29 lanes is described using a uniform distribution, as shown in Appendix 3. The bidding prices, capacity, relationship management costs, and other parameters for 3PLs are generated based on historical data, company documents, and subjective estimates. The disruption scenario is set to 1024. We utilize the proposed model and algorithm to implement task allocation for 42 3PLs across 29 transportation lanes, thereby verifying the effectiveness of the proposed model and algorithm. Section 6.1 presents the comparison results of the proposed ISAA-SR-DDLR algorithm with CPLEX, while Section 6.2 analyzes the impact of changes in CV, H, and disruption probability on the results.

Table 9 Transportation lanes in the real case.

Performance of ISAA-SR-DDLR

The comparison between the ISAA-SR-DDLR algorithm and CPLEX under different H is provided in Table 10. NA indicates that CPLEX is unable to provide a valid solution due to insufficient computer memory. From Table 10, we can see that the ISAA-SR-DDLR algorithm outperforms CPLEX in both solution quality and computation time. The gap between the upper and lower bounds of the optimal solution obtained by the ISAA-SR-DDLR algorithm is less than 1.2%, indicating that the algorithm can achieve a high-quality near-optimal solution, making it suitable for solving this practical case.

Table 10 Comparison of ISAA-SR-DDLR algorithm with CPLEX in real case.

Sensitivity analysis

To analyze the impact of changes in CV, H, and disruption probability on the results, we present the selection of 3PLs under different parameters, the selection of potentially disrupted 3PLs, and the changes in fortified 3PLs, which are shown in Tables 11, 12, and 13. QS represents the quantity of 3PLs selected by 4PL, QD represents the quantity of 3PLs that may experience disruptions, and QF represents the quantity of fortified 3PLs. The generation of medium-scale CV and large-scale CV is obtained using $\mu + 3\sigma$ standard deviations and $\mu + 5\sigma$ standard deviations, respectively.

From Tables 11, 12, and 13, we can observe that the values of CV, H, and disruption probability have a complex interdependent effect on the task allocation results of 3PLs. Similar to the analysis results in Section 5.3, the magnitude of disruption probability plays a decisive role in the decision-making outcomes of WD problem.

When disruption probability is low, for any given CV, as the cost of outsourcing per unit H increases, QS, QD, and QF all gradually increase. It is noteworthy that when H is low, the number of fortified 3PLs is less than the number of potentially disrupted 3PLs selected. This indicates that 4PL can appropriately reduce the upper limit of the protective investment budget and utilize potentially disrupted 3PLs to complete logistics transportation tasks. Once disruption occurs, unmet demand can be outsourced to other 3PLs that did not participate in the bidding. As H increases, the number of fortified 3PLs equals the number of potentially disrupted 3PLs selected, indicating that 4PL should appropriately increase the protective investment budget and focus on the selection of reinforcement strategies.

For any given H, as CV increases, there is a trend for QD and QF to increase. When H is small, as CV increases, the difference between QD and QF tends to increase, indicating that the greater the CV, the smaller the impact of disruptions. At this point, temporary outsourcing strategies can be used to mitigate disruption risks. As H increases, a larger CV leads to a higher R value, indicating that the importance of fortification strategies also gradually increases with the rise in CV.

As the probability of disruption increases, regardless of the values of H and CV, the number of potentially disrupted 3PLs and the number of fortified 3PLs are exactly the same, meaning that the 4PL fotrifies all selected 3PLs. Therefore, the impact of disruption probability on fortification strategies is decisive; once the disruption probability is high, fortification strategies become particularly important.

For practical applications, management insights are proposed for 4PL in the context of demand uncertainty and disruption risks in logistics service procurement auctions, to ensure the continuity and stability of logistics services. Firstly, the impact of disruption probability on fortification strategies is decisive; with a higher disruption probability, fortification policies become crucial as they can mitigate the adverse effects of disruption while saving costs for the 4PL. In situations with a low disruption probability, the 4PL weighs appropriate risk mitigation strategies based on the variations in CV and H. By effectively combining fortification strategies with temporary outsourcing strategies and allocating them reasonably based on market conditions, the 4PL can minimize costs while ensuring service levels. In practice, the 4PL needs to assess the value of CV and the probability distribution of disruption based on historical data and expert forecasts, which will be one of the challenges faced in decision-making for mitigation strategies.

Table 11 Analysis of 3PL selection and fortification changes under low disruption probability.

Table 12 Analysis of 3PL selection and fortification changes under medium disruption probability.

Table 13 Analysis of 3PL selection and fortification changes under large disruption probability.

Overall, we can draw the following management insights: When the disruption probability is high, regardless of the values of CV and H, 4PL must implement fortification strategy for the selected 3PL to hedge risks brought by disruptions. When the disruption probability is low, if the temporary outsourcing cost per unit is relatively low, 4PL should focus more on meeting client demand through temporary outsourcing services, especially when CV is high. When the cost of outsourcing per unit is high, due to the expensive outsourcing services, 4PL should choose more economical fortification strategy and allocate logistics tasks to the winning 3PLs through auctions. When the cost of outsourcing per unit is low and the CV is smaller, 4PL should consider gradually reducing the number of 3PLs that need to be fortified. On the contrary, as CV increases, the potential impact of disruptions will become more severe, requiring 4PL to weigh the options between fortification strategy and temporary outsourcing strategy. When the cost of outsourcing per unit is high, regardless of the size of CV, the occurrence of disruptions poses a significant threat to 4PL, making fortification strategy a crucial measure to mitigate the potential impact of disruptions.

Conclusions

Due to the disruption risks and uncertain demand that 4PL providers face in logistics service procurement auctions, it is essential for these providers to possess strong flexibility and adaptability. This paper studies a new WDP in logistics service procurement auctions, considering uncertain demand and disruption risk from the perspective of 4PL. A hybrid mitigation strategy integrating temporary outsourcing and fortification strategies is proposed to hedge uncertainty risks. A two-stage stochastic winner determination model is constructed, and an ISAA algorithm based on the chi-square test and LHS methods is proposed for transformation from stochastic model to a deterministic MILP model. To deal with challenge posed by the increase in the number of scenarios for model solving, combining SR approach and DDLR algorithm, an ISAA-SR-DDLR algorithm is proposed.

Using the designed numerical examples for experiments, it is found that when faced with an excessive number of scenarios, CPLEX is unable to provide a computational result. In contrast, the proposed algorithm provides satisfactory solutions with very small gap between the upper and lower bounds. This validates effectiveness and applicability of our algorithm. The results of sensitivity analysis show that 4PL needs to implement the optimal risk mitigation strategy based on the changes in unit outsourcing costs and demand fluctuations. In the case of higher disruption probability, regardless of changes in unit outsourcing costs and demand fluctuations, 4PL must implement fortification strategy for the selected 3PL to cope with disruption risks. When faced with a small probability of disruption, the lower outsourcing cost per unit means that 4PL should focus more on using temporary outsourcing strategy to meet client demand, especially when the CV is large. But as the cost of outsourcing per unit becomes increasingly high, 4PL should choose the fortification strategy that is more favorable for cost savings. When the cost of outsourcing per unit is relatively small, a smaller CV indicates that the impact of disruptions is minimized, leading to a gradual reduction in the number of 3PLs that need to be fortified. However, as CV increases, the uncertainty in the supply chain also intensifies, requiring 4PL to carefully weigh the options between fortification strategy and temporary outsourcing strategy. When the cost of outsourcing per unit is relatively high, regardless of the size of the CV, it is necessary to consider fortification strategy to hedge the adverse impacts of disruptions.

There are related issues worthy of further research in the future. Firstly, It is assumed in this study that both 4PL and 3PL act with complete rationality. But in practical operation, decision-makers may exhibit characteristics of bounded rationality. In the future, 4PL can be examined from the perspective of bounded rationality, incorporating potential psychological preferences and constraints, such as regret sentiment, loss aversion, and broader principles of fairness. Secondly, this paper assumes that disruptions are independent. However, in real-world scenarios, the occurrence of a disruption event may propagate or affect other parts of the supply chain, thereby triggering related disruptions⁹⁸. Future research should consider the correlations of disruption events by improving optimization models and adopting new technological methods to more accurately assess and respond to disruption risks. For example, considering the ambiguity and uncertainty of probability distributions, a distributionally robust optimization framework⁹⁹ can be used to address the uncertainty of disruption risks. In addition, uncertainty theory can be used to address the uncertainties related to disruptions, taking into account various possibilities and combinations of disruption events during modeling.