Bi-level programming model and KKT penalty function solution approach for reliable hub location problem

https://doi.org/10.1016/j.eswa.2021.115505Get rights and content

Highlights

  • The paper considers a reliable hub location problem.

  • The proposed model is formulated as a bi-level programming problem.

  • Minimizing the total cost for establishing the network and the total lost flow are objectives.

  • The KKT method and a two-step heuristic is proposed for solution procedure.

Abstract

Tactical and operational decisions must nowadays be made in production and distribution systems to allocate the best possible locations for the establishment of service centers. These systems seek to provide their services the fastest and the most reliably. In the meantime, hub location problems are classified as the most important categories of such decisions. These problems include locating hub facilities and establishing communication networks between facilities and demand centers. This paper aims to design a bi-level programming model to minimize the costs of establishing a hub network at the first decision-making level and reduce service loss due to disruption and failure in service processes at the second decision-making level. Therefore, the reliable bi-level hub location problem was analyzed, and an integer programming model was developed. The KKT method was then employed to solve the model, whereas a two-step heuristic method with a penalty function was proposed to first offer a feasible solution through an innovative algorithm. After that, a process was formulated to improve the feasible solution through the penalty function logic. Data of 37 major cities were collected from the Civil Aviation Organization of Islamic Republic of Iran to validate the proposed model. In brief, the developed hub location problem model managed to efficiently solve some real-world distribution problems through a bi-level programming approach. Moreover, the traffic reliability and total location routing cost of the network were incorporated into a mathematical model.

Introduction

With the ever-increasing international relations and large populations involved in these interactions nowadays, transportation and telecommunication must employ the updated scientific methods of scientifically managing their costs and security. The outcomes will be severe in the event of problems with these networks and the failure of any transactions. Due to the large number of interactions, the main outcome is the adverse effect on the levels of legitimacy and popularity of a brand in addition to financial costs. Therefore, the prioritized aspects must be taken into account in the decision-making process of establishing international and even national transportation and telecommunication networks (that will finally connect to international networks). A hub location problem (HLP) is a scientific area where these networks are developed. A hub is known as the center of aggregation and distribution (Farahani, Hekmatfar, Arabani, & Nikbakhsh, 2013). The HLP aims to direct the network flow in order to provide decision makers with the ideal state. These goals are usually classified as three general categories, i.e. financial priorities, full service delivery, justice orientation and general access (Farahani et al., 2013, Mohammadi et al., 2016). These goals affect the location of a hub facility and the ways of allocating demand spokes to hubs and even allocating hubs together. The HLP can be introduced as the strategy of establishing as many communication links as possible in a network. Hence, the demands of each node are definitely met through at least one hub.

There is fairly rich literature on these issues reviewed and classified by many studies such as the ones by O'Kelly and Miller, 1994, Klincewicz, 1998, Hale and Moberg, 2003, Alumur and Kara, 2008, Campbell and O'Kelly, 2012, and Farahani et al. (2013), who provided valuable research findings. They highlighted the importance and impact of HLP. Regardless of the leading cases, four sections of the literature are reviewed here.

The HLP was introduced as a mathematical model by O'kelly (1986), who later (1987) developed a new quadratic structure of the prior model. Establishing a network with a predetermined number of hubs and generally seeking to reduce costs, the P-hub median problem was proposed by Campbell (1994). Ernst and Krishnamoorthy (1996) added the discount factors to the HLP and gave the hubs a more prevalent role with respect to the capabilities of hub facilities. Ernst and Krishnamoorthy (1998) also developed an accurate and innovative approach. Along with the median models having an economic basis, centralized models were based on the fair distribution and delivery of services. Ernst, Hamacher, Jiang, Krishnamoorthy, and Woeginger (2009) designed one of such models in two single-allocation and multi- allocation structures. In addition to the capacity structures of facilities, service provision time was taken into account in cover-based models of subsequent development of HLPs. Kara and Tansel (2001) improved the hub issue by adding time constraints; however, Wagner (2004) upgraded the previous model by providing a corrective point emphasizing the longest network path.

Many efforts were made to improve the HLP solution and modeling. For instance, Chen (2007) achieved satisfactory results in solving non-capacity problems by developing a hybrid heuristic approach. Calık, Alumur, Kara, and Karasan (2009) designed an incomplete hub network model to eliminate unnecessary communications in the facility network. In addition to the development of HLPs, they proposed an innovative solution based on Tabu search. Contreras, Cordeau, and Laporte (2011) added contingency and uncertainty conditions to HLPs and took an effective step in bringing these problems closer to real conditions.

In recent decades, the reliability of communication paths and the location of hub facilities have been identified as one of the most influential factors in the decision-making process of creating hub networks. Kim and O'Kelly (2009) introduced two models to maximize the reliability of communication paths in hub networks. They are called the p-hub maximum reliability model and the p-hub mandatory dispersion model. They also determined the reliability of communication paths between the origin and destination nodes (ODs) with respect to the impacts of hubs. Finally, they evaluated the performance of models on telecommunication networks in the United States.

Davari, Zarandi, and Turksen (2010) designed a single-hub network to maximize the reliability of communication paths and showed that the reliability of each path followed a fuzzy variable. They employed the simulated annealing method to solve the problem. The model proposed by Davari et al. (2010) was redesigned by Zarandi, Davari, and Sisakht (2011) through the interactive fuzzy goal programming in order to establish an interactive relationship between decision-makers and the model to provide the appropriate design and solution.

Parvaresh, Hashemi Golpayegany, Moattar Husseini, and Karimi (2013) designed a bi-level game model for the multiple-allocation p-hub median problem. The primary problem intended to minimize the traffic flow caused by transportation, whereas the secondary problem aimed to maximize the successful flow of receiving services by selecting the best scenario. The effective method was the simulated annealing algorithm calibrated through the Taguchi method.

Parvaresh et al., 2013, Parvaresh et al., 2014 developed different bi-level models differing in the fact that there were two target functions at the upper level minimizing normal shipment and transportation costs in the worst-case scenario. They employed simulated annealing and Tabu search to solve the problems.

Eghbali, Abedzadeh, and Setak (2014) analyzed the multiobjective single-allocation hub-covering location problem by trying to balance the total cost and total number of intermediate links between all ODs to prevent the minimum reliability of all routes from exceeding a given minimum. They used the genetic algorithm to solve the problem.

In addition to checking the reliability of communication paths in HLPs, it is essential to consider to the possibility of interruptions in hubs. Hence, An, Zhang, and Zeng (2015) designed a reliable hub-and-spoke network to prevent any failure to respond to demand in the event of disruptions in hubs and alternative routes. The proposed model had a nonlinear mixed integer structure using the Lagrangian relaxation and branch-and-bound methods to solve the problem.

Emphasizing the necessity of considering reliability and speed, Mohammadi, Tavakkoli-Moghaddam, Siadat, and Dantan (2016) analyzed service provision in the single-allocation p-hub center-median problem. Their proposed model sought to determine uncertainty in flows, costs, times and hub operations and observe different transport modes in the allocation structure. They considered a fuzzy-queuing approach for the uncertainty problem. An algorithm was also designed to solve the model based on the game theory and the invasive weed optimization algorithm in comparison with the NSGA-II and PAES algorithms.

Mohammadi, Tavakkoli-Moghaddam, Siadat, and Rahimi (2016) also designed a reliable logistic network based on a hub location problem to minimize the total amount of nominal and expected failure costs. They proposed a novel hybrid metaheuristic algorithm based on genetic and imperialist competitive algorithms to solve the proposed model.

Based on the analysis of unreliability in the hub median problem, a mixed-integer nonlinear programming model was proposed by Tran, O’Hanley, and Scaparra (2016). This model was designed to minimize the expected weighted travel cost plus a penalty in case all hubs failed. The model was solved through the Tabu search algorithm.

Chen et al. (2017) assessed the transshipment hub selection from the perspectives of shippers and forwarders. Their study was based on a case of the direct transportation link policy in China. They concluded that costs, customs regulations, and connectivity were critical for all stakeholders and that every hub with short transport distance and near cargo sources might not override others.

de Sá et al. (2018) introduced an incomplete hub location problem with service time requirements by assuming that the travel time was uncertain. They also proposed a robust optimization problem. The robust model guaranteed that time requirements would not be violated.

The recent research approach focuses on bi-level programming (BLP) structures which actually address the major decision-making problem overshadowing any other issues. Therefore, all problems can be unified as one problem to achieve acceptable results by designing BLP models.

In recent years, may studies have been conducted on the BLP with regard to facility location and location-allocation routing.

Küçükaydin, Aras, and Kuban Altınel (2011) analyzed the competitive facility location problems (CFLPs) and proposed a bi-level mixed-integer nonlinear programming model in which the new facility would be deployed among others to maximize its profits by considering the competitive market conditions. MirHassani, Raeisi, and Rahmani (2015) studied CFLPs and developed a model in which two non-cooperative companies competed to maximize profits in a given market. They emphasized the development of a solution for CFLPs and proposed a simple and effective quantum binary particle swarm optimization method.

Given the importance of disorders caused by deliberate sabotage and terrorist attacks as a concern for governments, Aliakbarian, Dehghanian, and Salari (2015) designed a bi-level r-interdiction median model of critical hierarchical facilities to primarily minimize the impacts of the most disruptive attacks on unprotected facilities and secondarily moderate cost-related defender-attacker disruptions.

This study also focuses on the routing of public transport facilities, something which plays a key role in considering the hub facility in the public facility sector.

Raidl, Baumhauer, and Hu (2014) adopted the Benders’ decomposition method to solve the bi-level capacitated vehicle routing problem in large dimensions and showed that their algorithm yielded good results.

The timely delivery of humanitarian aid is a must in humanitarian logistics; hence, Camacho-Vallejo, González-Rodríguez, Almaguer, and González-Ramírez (2015) proposed a bi-level mathematical programming model that minimized shipping costs and maximized the service performance. They validated their model by using the data of the 2010 earthquake in Chile.

The school bus routing problem is effective in boosting public transportation systems. Parvasi, Mahmoodjanloo, and Setak (2017) introduced the BLP model to primarily locate appropriate bus stops and identify bus navigation routes. The secondary objective was to allocate students as a source of demand for transport systems or outsource them at a lower level.

Zhou et al. (2017) developed a bi-level programming model with type-2 triangular fuzzy variables. The general expectation reduction method was proposed to reduce the type-2 fuzzy variables, whereas the robust parametric optimization method was used as an effective decision-making technique.

Kolak et al. (2018) proposed a bi-level multiobjective traffic network optimization model with sustainability perspective. Its lower level was a stochastic user equilibrium model reflecting user decisions, whereas its upper level was a multiobjective model showing authority decisions.

Tong, Nie, Guo, Leng, and Xu (2019) proposed the bi-level programming model for the layout scheme of high-speed railway transfer hubs where the upper-layer problem was to optimize the passenger transfer hub layout scheme, whereas the lower-layer problem pertained to the passenger flow assignment problem on the railway physical network. Mahmoodjanloo, Tavakkoli-Moghaddam, Baboli, and Jamiri (2020) developed a multi-modal competitive hub location pricing model to design a transportation system for a company planning to enter a market with elastic demand. Khosravian, Shahandeh Nookabadi, and Moslehi (2019) dealt with the periodic variations of parameters and then proposed a bi-objective mathematical model for the single-allocation multi-period maximal hub. To solve the problem, they used the ε-constraint approach to offer non-dominated solutions. Since the single-objective problem found in the ε-constraint method is computationally intractable, they accelerated the solution process through the benders decomposition algorithm. Chalmardi and Camacho-Vallejo (2019) analyzed a sustainable supply chain network design by proposing a novel bi-level programming model to handle the problem. They also designed an algorithm based on the simulated annealing algorithm to find acceptable solutions. A dynamic maximal hub location covering problem was studied for a freight transportation system (Alizadeh & Nishi, 2019). Their model was a bi-level problem needing more efforts, for which they utilized two reformulations based on Karush–Kuhn–Tucker conditions and duality theory to reformulate the bi-level problem into a single-level one.

Basciftci and Van Hentenryck (2020) proposed a bi-level optimization model in which the leader problem was to design the network where each rider had a follower problem to decide on the best route through the ODMTS. Their bi-level model was solved through a decomposition algorithm that combined traditional benders cuts with combinatorial cuts to ensure the consistency of modal choices made by the leader and follower problems.

The BLP issues are NP-hard in the simplest possible case (Jeroslow, 1985); however, Jaumard et al. (1992) showed that BLPs were classified as strong NP-hard problems. Ben-Ayed (1993) proved that the non-convexity of BLPs would strongly result in the difficulty of solving them. Moreover, no subscription of answer between the two problems raised the possibility of not answering the question; thus, it was important to properly design these issues (Ben-Ayed, 1993).

Given the difficulty of BLPs, it would be distinct challenge to solve them. Hence, choosing an efficient way to solve these issues has always been a concern for researchers. An effective measure would be to facilitate the problem of using the Karush-Kuhn-Tucker (KKT) approach, which Bard (1983) employed to propose a one-dimensional grid search algorithm to solve BLP problems.

A branch-and-bound algorithm was also proposed by Bard and Moore (1990) to solve the linear quadratic problem.

One of the most valuable steps in solving BLPs is to employ the KKT optimality condition on the lower-level problem and convert the linear BLP into the corresponding single-level programming. This process was proposed by Lv, Hu, Wang, and Wan (2007).

Various methods have been discussed in Practical Bi-Level Optimization: Algorithms and Applications authored by Bard (2013) introducing how to use KKT in BLPs, which are very effective in solving such issues.

Complementarity constraints play a key role in the KKT method to simplify BLPs; therefore, they simplify the model by placing it in the target function and creating a penalty add-on. Allende and Still (2013) designed a simplified math structure for BLPs in the problems having the necessary conditions for minimizing their lower-level problems.

Farvaresh and Sepehri (2013) developed a new branch-and-bound algorithm to find the exact solution to the two-level discrete network design problem.

Kuo, Lee, Zulvia, and Tien (2015) proposed integrating the immune genetic algorithm with a vector-controlled particle swarm optimization to solve a bi-level linear programming problem in a supply chain.

Rahmani and MirHassani (2015) proposed an innovative method inspired by the Lagrangian relaxation method to solve BLPs. This method was applicable to problems with two specific features. First, the upper-level problem constraints should not include the lower-level problem variables, whereas the next one states that there is a feasible lower-level solution for each upper-level solution.

Saranwong and Likasiri (2017) developed a bi-level model for the distribution center problem in the supply chain network. Regarding the solution process, they proposed an innovative method for finding the initial grounding solution, and then introduced four innovative algorithms. Ultimately, it was embodied in the form of the steepest descent algorithm to find the optimal solution.

Mirzapour-Kamanaj, Majidi, Zare, and Kazemzadeh (2020) proposed a multi-follower bi-level optimization framework for the optimal interaction of energy hubs and distribution networks, the total operation cost of which was minimized due to network constraints in the upper-level problem, whereas the total cost of each energy hub being connected to the distribution network was minimized in the lower-level problem. Their bi-level model was solved by the Karush–Kuhn–Tucker optimality condition.

Yang, Luo, and Shi (2020) developed a bi-level programming model to analyze the scenario of coordinated subsidies in which a hypothetical network planner minimized the total subsidies for a given externality reduction target in the upper level. Moreover, the cargo owners minimized the total transportation cost in the lower level. They aimed to illustrate the problem with uncoordinated subsidies and propose an optimal subsidy scheme. Therefore, they applied the minimum cost the flow model to analyze three scenarios, i.e. no subsidies, the internalization of external costs, and uncoordinated subsidies.

Zeng et al. (2020) proposed a bi-level optimization model to guide the electrolytic aluminum load to provide demand-side responses. In their proposed optimization model, the upper-level problem was to minimize the system operational cost, whereas the lower-level problem was to maximize the aluminum plant profit. The Karush-Kuhn-Tucker optimality conditions were derived for the lower-level problem, whereas the bi-level problem was reformulated as the mix-integer quadratic programming problem. Karatas (2020) developed two competing multi-objective mixed integer non-linear program formulations, which were then solved by a commercial optimizer for a few cases through the branch-and-cut (B&C) procedure.

Offering fast and reliable delivery services has now become a vital issue associated with all shipment delivery systems (Mohammadi, Tavakkoli-Moghaddam, Siadat, & Dantan, 2016).

Therefore, it is essential to consider both the reliability and the structure of transportation networks in the high volume of interconnection. A literature review reveals a gap in the two categories of minimizing the costs of locating the hub facility and minimizing the volume of lost demand due to the disruption of the hub network in the form of a BLP. Therefore, this paper aims to minimize the costs of establishing hub networks in upper-level problem (ULP) by analyzing the HLP model for a bi-level reliable hub location problem (BL-RHLP) and in the lower-level problem (LLP). The demand flow volume was interrupted in the lowest possible mode, whereas the LLP strategy included a network design whose paths had the lowest chance of disruption and failure in the service flow. By designing this model, it is implicitly and indirectly proposed that the cost of lost services will also be reduced. This paper is mainly characterized by the strategic consideration of the establishment and creation of hub networks, the high costs of which made it difficult to change in the future. If required, every small change will cost greatly. Therefore, the effort to design a sustainable network with a minimum probability of failure is an integral part of an HLP.

When two objectives have the same spatial and strategic priority, they are presented in a multi-objective paradigm. However, by reviewing the literature, it can be realized that reducing the establishment cost of a hub is the main priority which is significantly different from such goals as reliability. It is because the potential costs of establishment and the actual costs of reliability may or may not occur. However, the establishment costs are inevitable, and the network must bear them. Therefore, the issue should be presented with the purpose of prioritizing the reduction of costs. In this regard, the bi-level method is one of the best modeling techniques.

As discussed earlier, the BLP issues are NP-hard in the simplest form (Jeroslow, 1985). Therefore, the use of an effective and efficient solution will play a significant role in achieving the appropriate results. The proposed KKT algorithm is proposed to solve the proposed model. In this method, the complementary and slackness conditions of the LLP are added as a penalty to the ULP target function. In a repeatable algorithm, a set of problems will then be solved to achieve an optimal solution.

In many distribution systems, hub facilities have been employed to concentrate and consolidate flows to gain economic profits. According to many studies of the hub networks, the main basis is the economic approach to the distribution network. At the same time, the other objectives such as reliability improvement are considered at lower levels and are not considered the first priority. In summary, the developed hub location problem model works effectively for some real-world distribution problems through a bi-level programming approach. At the same time, the traffic reliability and total location routing costs of networks are incorporated into the mathematical model. The research highlights of this paper can be classified as four categories: 1. In the theoretical field, it proposes a novel bi-level mathematical model of the reliable hub location problem; 2. It proposes a two-step innovative solution method inspired by the KKT algorithm for the NP-hard research problem; 3. In practice, it achieves a distribution network with an emphasis on prioritizing the established cost management and proposing the minimum probability of failure in the distribution network routes; 4. It implements a bi-level programming model for a reliable hub location problem with the real data obtained from 37 cities in Iran, adapts them to the status quo, and analyzes the expected results.

The rest of this paper consists of various sections. Section 2 describes the mathematical formulation of the BL-RHLP, whereas Section 3 presents the heuristic algorithm for improving the BL-RHLP solution. Section 4 solves a rational example based on the Civil Aviation Organization of Islamic Republic of Iran by using the proposed algorithm. Section 5 presents conclusion and discussion in detail.

Section snippets

Statement of the problem

This section presents the features of the BL-RHLP. This problem seeks to find the location of the hub facility and allocate demand nodes to them through two separate approaches. The main issue, or the ULP, is to try to minimize the cost of establishing a hub network. This includes the cost of establishing a hub facility, the cost of developing hub links, and the cost of setting links between non-fire nodes and hubs. In this economic network, there will be the state of the triangle inequality

Algorithm and solution procedure

In this paper, the problem of HLP has been studied in the form of a BLP, and as previously stated, Jeroslow (1985) showed that BLP problems is an NP-hard problem in the simplest form. Therefore, in order to achieve the desired results for this study, which is the study of Iran's Airport Aviation, the efficient methods should be used. Therefore, in this study, a two-step solving algorithm is proposed that in the first stage a preliminary solution can be obtained for both problems, and in the

Computational results

In this section, the Iranian Aviation Dataset (IAD) data set is used to validate and evaluate the capabilities of the model and the proposed solution method. The IAD includes the volume of passengers' traffic, distance, travel time and the cost of creating a hub and the cost of establishing a route between 37 cities according to the actual transportation information of Iran in a scientific framework. The IAD was introduced by Karimi and Bashiri (2011) for the first time. Some data, such as

Conclusions and discussion

In the modern world, transportation systems and telecommunication service companies are trying to improve the reliability of the service process due to the growing competition in the market. Any disturbances in the route of service provision will impose multiple costs on the system. Given the strategic level of this type of service in addition to the huge costs of lost sales and the re-launch of communication lines, the reflection of every mistake emerges quickly worldwide and greatly outweighs

CRediT authorship contribution statement

Ehsan Korani: Methodology, Investigation, Software, Writing - original draft. Alireza Eydi: Conceptualization, Supervision, Validation, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References (69)

  • A.T. Ernst et al.

    Efficient algorithms for the uncapacitated single allocation p-hub median problem

    Location science

    (1996)
  • A.T. Ernst et al.

    Exact and heuristic algorithms for the uncapacitated multiple allocation p-hub median problem

    European Journal of Operational Research

    (1998)
  • A.T. Ernst et al.

    Uncapacitated single and multiple allocation p-hub center problems

    Computers & Operations Research

    (2009)
  • R.Z. Farahani et al.

    Hub location problems: A review of models, classification, solution techniques, and applications

    Computers & Industrial Engineering

    (2013)
  • M. Karatas

    A multi-objective bi-level location problem for heterogeneous sensor networks with hub-spoke topology

    Computer Networks

    (2020)
  • H. Karimi et al.

    Proprietor and customer costs in the incomplete hub location-routing network topology

    Applied Mathematical Modelling

    (2014)
  • H. Karimi et al.

    Hub covering location problems with different coverage types

    Scientia Iranica

    (2011)
  • J.G. Klincewicz

    Hub location in backbone/tributary network design: A review

    Location Science

    (1998)
  • H. Küçükaydin et al.

    Competitive facility location problem with attractiveness adjustment of the follower: A bilevel programming model and its solution

    European Journal of Operational Research

    (2011)
  • R.J. Kuo et al.

    Solving bi-level linear programming problem through hybrid of immune genetic algorithm and particle swarm optimization algorithm

    Applied Mathematics and Computation

    (2015)
  • Y. Lv et al.

    A penalty function method based on Kuhn-Tucker condition for solving linear bilevel programming

    Applied Mathematics and Computation

    (2007)
  • M. Mahmoodjanloo et al.

    A multi-modal competitive hub location pricing problem with customer loyalty and elastic demand

    Computers & Operations Research

    (2020)
  • A. Mirzapour-Kamanaj et al.

    Optimal strategic coordination of distribution networks and interconnected energy hubs: A linear multi-follower bi-level optimization model

    International Journal of Electrical Power & Energy Systems

    (2020)
  • M. Mohammadi et al.

    Design of a reliable logistics network with hub disruption under uncertainty

    Applied Mathematical Modelling

    (2016)
  • M. Mohammadi et al.

    A game-based meta-heuristic for a fuzzy bi-objective reliable hub location problem

    Engineering Applications of Artificial Intelligence

    (2016)
  • A.H. Niknamfar et al.

    Opposition-based learning for competitive hub location: A bi-objective biogeography-based optimization algorithm

    Knowledge-Based Systems

    (2017)
  • M.E. O'kelly

    A quadratic integer program for the location of interacting hub facilities

    European Journal of Operational Research

    (1987)
  • M.E. O'Kelly et al.

    The hub network design problem: A review and synthesis

    Journal of Transport Geography

    (1994)
  • S.P. Parvasi et al.

    A bi-level school bus routing problem with bus stops selection and possibility of demand outsourcing

    Applied Soft Computing

    (2017)
  • S. Saranwong et al.

    Bi-level programming model for solving distribution center problem: A case study in Northern Thailand’s sugarcane management

    Computers & Industrial Engineering

    (2017)
  • H. Yaman

    The hierarchical hub median problem with single assignment

    Transportation Research Part B: Methodological

    (2009)
  • R. Alizadeh et al.

    Dynamic p+ q maximal hub location problem for freight transportation planning with rational markets

    Advances in Mechanical Engineering

    (2019)
  • G.B. Allende et al.

    Solving bilevel programs with the KKT-approach

    Mathematical Programming

    (2013)
  • G. Anandalingam et al.

    A solution method for the linear static Stackelberg problem using penalty functions

    IEEE Transactions on Automatic Control

    (1990)
  • Cited by (24)

    View all citing articles on Scopus
    View full text