. In supply chains of perishable goods, paying attention to product freshness during delivery to customers is very important because perishable goods deteriorate during the transportation process and their quality decreases. Therefore, how to design the distribution system of perishable goods to transfer products from origins to destinations impacts the responsiveness of the transportation system. This paper presents a bi-objective model for designing a capacitated hierarchical hub network with multiple allocation for perishable goods. In the proposed model, transportation costs and maximum travel time (as a measure of network responsiveness) are optimized simultaneously. To get Pareto-optimal solutions, three exact methods, namely the augmented epsilon constraint method, Torabi-Hassini method, and augmented weighted Tchebycheff procedure, are applied. Validation of the proposed model is performed using some test problems on the Turkish network dataset. Also, sensitivity analyses are performed on the critical parameters of the model. The results show that multiple allocation can be effective in the variety of Pareto solutions. Also, designing the network with economic and responsiveness considerations costs more than designing the network with only economic considerations.

In hub-and-spoke systems, hub location and allocation decisions depend on some parameters, such as transportation demand. When demand is time-dependent, a function can be estimated that reflects demand changes over time. In real-world conditions, due to the lack of knowledge and the nature of transportation demand, the coefficients of this function cannot be determined precisely. In this case, these coefficients are considered as uncertain parameters. In this paper, we propose a multi-objective mixed-integer nonlinear programming model for a sustainable multi-period hub location problem when demand changes are based on a linear function of time, and the coefficients of this function are fuzzy parameters. In the proposed model, it is possible to increase the capacity of hubs and hub links and transfer the capacity between hubs through modules. Also, the model determines the best timing for implementing the decisions. The objectives include minimizing system costs, minimizing emissions, and maximizing job opportunities related to sustainability aspects. We linearize the model and present a crisp counterpart formulation. Then, the computational results and sensitivity analysis are presented. Results show that the cost of designing the hub network with economic and social considerations is higher than the cost of designing the network with economic and environmental considerations. Ignoring the uncertainty in the problem can lead to non-optimal solutions. Also, the impact of increasing the number of hubs on reducing emissions after a threshold is marginal. We also introduce some valid inequalities to strengthen the formulation. The results indicate the acceptable performance of valid inequalities.

This paper presents a bi-objective model for the design and optimization of a sustainable hierarchical multi-modal hub network. The proposed model focuses on sustainability by considering economic, environmental, and social aspects of the decisions in a hierarchical network. A case of Turkish network for freight transportation is used to validate the proposed model. To solve the small-sized problems, the augmented epsilon constraint method version 2 (AUGMECON2) is applied. It can be inferred from the Pareto-optimal set obtained by AUGMECON2 that the effect of increasing the number of hubs after a threshold is marginal. The current contribution proposes two multi-objective genetic algorithms (NSGA-II and NRGA), which incorporate LP solving and Dijkstra algorithm. The results show the superiority of NRGA compared to NSGA-II in terms of solution time. Also, we present an alternative, more efficient formulation to the problem. Based on the alternative formulation, in addition to AUGMECON2, we use two exact methods, including Torabi and Hassini (TH) method and augmented weighted Tchebycheff procedure (AWTP), to find Pareto-optimal solutions for small, medium, and large-sized problems (including the case study). The performance of the proposed solution methods is measured using some multi-objective indicators. The results show the superiority of AUGMECON2.

This paper presents a bi-objective, nonlinear mathematical model for designing a multi-period hub network, considering a polynomial function for time-dependent transportation demand. The proposed demand function can be used for various applications of the hub network design problem. Hubs' capacity in the proposed model is considered to be modular. Each module corresponds to a number of servers at hubs. Objectives of the problem include minimizing network costs and maximizing responsiveness by minimizing the sum of the maximum travel times in all time periods. The proposed model provides the best timing for implementing decisions during the planning horizon. To solve the model, NSGA-II, NRGA, PESA-II, and SPEA-II meta-heuristics are used. To compare the performance of the solution algorithms, some multi-objective metrics and statistical tests on the CAB, AP, and TR datasets are applied. We also perform sensitivity analysis on some critical parameters. The sensitivity analysis results show that the network design with economic and responsiveness considerations simultaneously has a higher cost than the network design with only economic considerations.

In hub-and-spoke systems, due to the changes in the input parameters affecting the system over time, transportation system providers should make the right and timely decisions about hub facilities. For this reason, multi-period hub location problems are especially important. This paper presents a comprehensive review of multi-period hub location problems from 1990 up to the most recent published studies. First, we study the developed models based on some characteristics: type of planning horizon (continuous-time and discrete-time), capacity constraints (uncapacitated, capacitated, and modular), type of problem (median and covering), type of services (single-level and hierarchical), number of commodities and modes, type of hub facility (mobile and virtual), type of assignment (single and multiple), and type of parameters (deterministic and uncertain). Also, practical applications of the models are investigated. Then, we survey the proposed solution methods and real-life case studies. Finally, future suggestions are presented for researchers.

Today, many transportation systems use hub-and-spoke structures to transfer flow from origin to destination. In such systems, making the right and timely decisions about facilities during the planning horizon is crucial for Decision-Maker (DM); because the parameters influencing decision-making change over the planning horizon, and the initial design of the transportation network may not be desirable for the future. In this paper, we present a novel mathematical programming model for designing a multi-period hub network in a continuous-time planning horizon, considering linear time-dependent demand. The proposed model is a non-linear programming model in which sustainability aspects are addressed through the following objectives: economic (minimizing network costs), environmental (minimizing emissions), and social (maximizing fixed and variable job opportunities). After linearizing the model, using the aggregation function of the Torabi–Hassini (TH) method, the model becomes a parametric single-objective model, and some valid inequalities are introduced. To solve the single-objective model, an accelerated Benders decomposition algorithm and a rolling horizon heuristic algorithm are proposed. The proposed heuristic method can solve problems with twenty-five nodes and six time periods on the CAB dataset. Pareto solutions provide optimal decisions about facilities location, adjusting the operational capacity of facilities (through modules), and routing flows for DM. Also, the best time to implement decisions and the satisfaction degree of sustainability objectives are determined for each solution. We also perform sensitivity analysis on important parameters. The results show that the cost of designing a network with economic and social objectives is more than the cost of designing a network with economic and environmental objectives. Also, by increasing the module capacity after a threshold value, the network costs remain constant.

In this paper, we study the hierarchical hub network design problem with a ring-star-star structure. In this problem, the hubs are located in two layers. In the first layer, central hubs (main hubs) are located in a ring structure, and in the second layer, secondary hubs are located. Each secondary hub is allocated to a central hub. Other demand points can also be allocated to each of these hubs (central hubs or secondary hubs). The objective is to minimize transportation costs on the network. This problem applies to communication networks when establishing direct links between demand nodes is not cost-effective, and there are two levels of service for customers. Also, the ring structure is used to reduce costs associated with full communication between central hubs. We present a mixed-integer programming model for the problem and report the results of the problem solving for a numerical example on the CAB dataset. Also, we present three solution methods for the problem. First, by exploiting the decomposable structure of the proposed model, we introduce an accelerated Benders decomposition algorithm. Then, we present a hybrid genetic algorithm that uses the Dijkstra algorithm to evaluate solutions. Next, we present a hybrid variable neighborhood search algorithm that uses the Dijkstra algorithm to calculate the cost of each solution. Also, a relax-and-fix algorithm is proposed to find near-optimal feasible solutions for the problem. Computational results are presented on the USA423 dataset. The results show that the proposed relax-and-fix algorithm can solve instances with up to 100 nodes. Also, the proposed hybrid algorithms can solve instances with up to 423 nodes. The performance of the proposed hybrid variable neigh-borhood search is better than the proposed hybrid genetic algorithm in solving large-sized instances.

This paper presents a mathematical programming model for designing a sustainable continuous-time multi-period hub network considering time-dependent demand. The present model can be used in situations where the distribution of parameters related to the demand function is unknown, and we only can determine the range of changes of these parameters. To model these conditions, we consider interval uncertainty for the demand function parameters. The proposed model is a nonlinear multi-objective model. The objectives of the model cover economic, environmental, and social aspects of sustainability. These objectives include minimizing total costs, minimizing emissions, and maximizing fixed and variable job opportunities. We linearize the model by using some linearization techniques, and then, with the help of Bertsimas and Sim’s method, we construct a robust counterpart of the model. We also present some valid inequalities to strengthen the formulation. To solve the proposed model, we use Torabi and Hassini method. From solving the proposed model, network design decisions and the best time to implement decisions during the planning horizon are determined. To validate the model, we solve a sample problem based on the Turkish dataset and compare the designed network in two cases: in the first case, the demand function parameters take nominal values, and in the second case, the value of these parameters can change up to 20% of their nominal values. The results show that in the second case, the total capacity selected for hubs and hub links is greater than the first case. To investigate changes in objective functions to parameters level of conservatism and probability of constraints violation, we perform sensitivity analysis on these parameters in both single-objective and multi-objective optimization cases and report the results.