This paper aims at being a guide to understand the different types of transportation problems by presenting a survey of mathematical models and algorithms used to solve different types of transportation modes (ship, plane, train, bus, truck, Motorcycle, Cars, and others) by air, water, space, cables, tubes, and road. Some problems are as follows: bus scheduling problem, delivery problem, combining truck trip problem, open vehicle routing problem, helicopter routing problem, truck loading problem, truck dispatching problem, truck routing problem, truck transportation problem, vehicle routing problem and variants, convoy routing problem, railroad blocking problem (RBP), inventory routing problem (IRP), air traffic flow management problem (TFMP), cash transportation vehicle routing problem, and so forth.
The Transportation Problems (TP) is the generic name given to a whole class of problems in which the transportation is necessary. The general parameters of TP are as follows.
This paper aims to be a guide to understand the Transportation Problems (TP) by presenting a survey of the characteristics, the algorithms used to solve the problems, and the differences of the variants of the Transportation problems. Section
The transportation problems are to minimize the cost of carrying resources, goods, or people from one location (often know as sources) to another location (often know as destinations) using diverse types of transportation modes (ship, aircraft, truck, train, pipeline, motorcycle and others) by air, water, road, aerospace, tube, and cable with some restrictions as capacity and time windows. The types of transportation problems are as follows.
The maritime transportation carries resources over long distances from locations to other locations using maritime routes composed of oceans, coasts, seas, lakes, rivers, and canals of ships, or similar routes.
The air transportation carries resources over long, medium, and short distances from locations to other locations using air routes by aircrafts, charter flights, planes or others.
The land transportation carries resources over long, medium, and short distances from locations to other locations using the road routes by vehicle or similar means of land transportation.
Rail transport carries resources over long, medium, and short distances by way of wheeled vehicles running on rail track or railway.
Space transportation carries resources from locations to other locations by suborbital and orbital flights in the upper atmosphere and the space by Hall Electric propulsion or similar.
Pipeline and cable transportation carries resources from locations to other locations by pipe and vehicles pulled by cables.
Intermodal transportation carries resources from locations to other locations using maritime, air, road, rail, cable, tube, and/or space routes by ships or similar, aircraft or similar, vehicle or similar, trains or similar, Hall Electric propulsion or similar, pipe and/or vehicles pulled by cables.
In the specialized literature there exist various variants of the Maritime Transportation Problems. The main variants of the Maritime Transportation Problems are RoRo ship stowage problem (RSSP) [
The RoRo ship stowage problem (RSSP) [
In (
In the specialized literature there exist various variants of the Air Transportation Problems. The main variants of the Air Transportation Problems are air traffic flow management problem (TFMP) [
The general problem of Air Transportation is represented in the mathematical model described by Li et al. [
The objective of (
In the specialized literature there exist various variants of the Land Transportation Problems. The main variants of the Land Transportation Problems are bus terminal location problem (BTLP) [
The school bus routing problem (SBRP) is a significant problem in the management of school bus fleet for the transportation of students; each student must be assigned to a particular bus which must be routed in an efficient manner to pick up (or return home) each of these students [
SBRP is formulated as mixed integer programming or nonlinear mixed integer programming models. The researchers are often not used to directly solve the problems; they use a relaxation of the problem to solve it. School bus schedules are important because they reduce costs to the universities or schools and bring added value to the students to have a quality transport.
In the specialized literature there exist various variants of the Rail Transportation Problems. The main variants of the Rail Transportation Problems are train formation problem (tfp) [
Train dispatching transportation problem, train meetandpass problem, or train timetabling problem is the process of handling a given set of desired train operating schedules and merging these requests as best as possible to a valid timetable [
In the specialized literature there exist various variants of the Space Transportation Problems. The main variants of the Space Transportation Problems are generalized location routing problem with space exploration or generalized location routing problem with profits (GLRPPs) [
The EarthMoon supply chain problem [
Equation (
In the specialized literature there exist various variants of the pipeline and cable Transportation Problems. The main variants of the Pipeline and Cable Transportation Problems are water distribution network (WDN) [
The coal, gas, water, and electricity production and transportation systems model [
The Intermodal Transportation Problems using more than one transportation mode are as follows. The main variants of the Intermodal Transportation Problems are intermodal multicommodity routing problem with scheduled services [
The railroad blocking problem (RBP) is a multicommodityflow, networkdesign, and routing problem [
The objective of the railroad blocking problem (RBP) is to minimize the sum of the costs of delivering each commodity using the blocking network formed by blocks for which
Various algorithms to solve the Transportation Problems (Table
Related Works.
Algorithm  Problem  Contribution 

Multiphase heuristic [ 
Multiperiod petrol station replenishment problem  A heuristic with a route construction and truck loading procedures, a route packing procedure, and two procedures enabling the anticipation or the postponement of deliveries for the MPSRP. 


Exact algorithm [ 
Petrol station replenishment problem  The algorithm decomposes the problem into truck loading problem and a routing problem. 


Genetic algorithm [ 
Air transportation scheduling problem  The Taguchi experimental design method is applied to set and estimate the proper values of gas parameters. 


Simulated annealing based heuristic algorithms [ 
Air transportation  The problem is formulated as a parallel machine scheduling problem with earliness penalties. 


Simulated annealing algorithm [ 
Arrival flight delays problem  Based on the characteristic of the flights and the thinking of system optimization, this paper builds up dynamic optimizing models of the flight delays scheduling with the objective function of delay cost. 


Greedy algorithm [ 
Singleairport groundholding problem (SAGHP)  A dynamic programming formulation with a corresponding backward solution algorithm. 


Coevolutionay genetic algorithm [ 
Multiairport groundholding problem  Survey model with dynamic capacity in details. 


Column generation based heuristic algorithm [ 
Helicopter routing problem  A MIP based heuristic with an add column generation procedures that improve the solution quality for the Brazilian State Oil Company (Petrobras). 


Heuristic algorithm [ 
Cash transportation vehicle routing problem  A solution algorithm based on a problem decomposition/collapsing technique, coupled with the use of a mathematical programming software. 


Tabu search [ 
Helicopter routing problem  Three routing policies are considered: a direct routing policy, a Hamiltonian routing policy, and a general routing policy. 


Hubandspoke configuration [ 
Helicopter routing problem  Mathematical model and theoretical results for route planning with a safetybased objective for helicopter routing in the Norwegian oil industry. 


Genetic algorithm [ 
Multiobjective helicopter routing problem  A variation of the clusterfirst routesecond method for routing helicopters. 


Transgenic algorithm [ 
Vehicle routing problem with time windows  Horizontal gene transfer based on the transformation mechanism and an intelligent mutation operator called Symbion operator. 


Vertical transfer algorithm [ 
School bus routing problem  A computer algorithm based on the mechanism of vertical gene transfer. 


Particle swarm optimisation [ 
Vehicle routing problem with time windows  An improved hybrid particle swarm optimisation (IHPSO) method with some postoptimisation procedures. 


Genetic algorithm [ 
Vehicle routing problem with time windows  Population preselection operators. 


Genetic algorithm [ 
Vehicle routing problem with time windows  A physical parallelisation of a distributed realcoded genetic algorithm and a set of eight subpopulations residing in a cube topology. 


Tabu search [ 
Vehicle routing problem with time windows and multidepot (VRPTW)  A unified Tabu search heuristic. 


Simulated annealing [ 
Vehicle routing problem with time windows  A twophase system (global neighbourhoods and local neighbourhood) of a parallel simulated annealing. 


Simulated Annealing [ 
Vehicle routing problem with time windows  2 interchanges with the bestacceptstrategy. 


Evolutionary Algorithm [ 
Vehicle routing problem with time windows  An individual representative called the strategy parameter used in the recombination and mutation operators. 


Tabu Search [ 
Vehicle routing problem with time windows  The Tabu search with a neighbourhood of the current solution created through an exchange procedure that swaps sequences of consecutive customers. 


Genetic Algorithm [ 
Vehicle routing problem with time windows  A genetic routing system or GENEROUS based on the natural evolution paradigm. 


GRASP [ 
Vehicle routing problem with time windows  A twophase greedy randomised adaptive search to solve VRPTW. 


BranchandBound method [ 
Vehicle routing problem with time windows  A BranchandBound method to solve VRPTW. 


A rural routing heuristic [ 
School bus routing problem  Constructing the initial route and then improving it by using a fixed tenure Tabu search algorithm. 


GRASP [ 
School bus routing problem  The solution method starts with a GRASPlike saving algorithm, after which a variable neighbourhood search algorithm is used to improve the initial solution. A modified version of the wellknown transportation problem helps the metaheuristic to quickly assign students to stops. 


Genetic algorithm [ 
School bus routing problem  Use the GENROUTER system to route school buses for two school districts. The routes obtained by GENROUTER system were superior to those obtained by the CHOOSE school bus routing system and the current routes in use by the two school districts. 


Simulated annealing [ 
Train formation problem  To explore the solution space, where the revised simplex method evaluates, selects, and implements the moves. The neighbourhood structure is based on the pivoting rules of the simplex method that provides an efficient method to reach the neighbours of the current solution. 


Genetic algorithm [ 
Train formation problem  The calibration and validation of the GA model are carried out for three different complexity levels of objective functions. 


Neural networks [ 
Train formation problem  A training process for neural network development is conducted, followed by a testing process that indicates that the neural network model will probably be both sufficiently fast and accurate, in producing train formation plans. 


Column generation based heuristic [ 
Generalized location routing problem with space exploration or generalized location routing problem with profits (GLRPPs)  The problem arises in exploration of planetary bodies where strategies correspond to different technologies. A description of the generalized location routing problem with profits and its mathematical formulation as an integer program are provided. Two solution methodologies to solve the problem—branchandprice and a threephase heuristic method combined with a generalized randomized adaptive search procedure—are proposed. 


Memetic algorithm [ 
Helicopter routing problem  The personnel transportation within a set of oil platforms by one helicopter that may have to undertake several routes in sequence. 


Genetic algorithm [ 
Locomotive routing problem  A clusterfirst, the routesecond approach is used to inform the multidepot locomotive assignment of a set of single depot problems and after that we solve each problem independently. Each single depot problem is solved heuristically by a hybrid genetic algorithm that in which push forward insertion heuristic (PFIH) is used to determine the initial solution and 


Genetic algorithm [ 
Locomotive routing problem  The proposed solution approach is tested with realworld data from the Korean railway. 


Branchandbound method [ 
Locomotive routing problem  Backtracking mechanism that can be added to this heuristic branchandprice approach. 


Heuristic algorithm [ 
Tour planning problem  A heuristic method based on local search ideas. 


Heuristic algorithm [ 
Team orienteering problem  Bilevel filterandfan method for solving the capacitated team orienteering problem. Given a set of potential customers, each associated with a known profit and a predefined demand, and the objective of the problem is to select the subset of customers as well as to determine the visiting sequence and assignment to vehicle routes such that the total collected profit is maximized and route duration and capacity restrictions are satisfied. 


Memetic algorithm [ 
Team orienteering problem  The memetic algorithm is a hybrid genetic algorithm using new algorithms. 


Branchand price algorithm [ 
Team orienteering problem  Includes branching rules specifically devoted to orienteering problems and adapts acceleration techniques in this context. 


Tabu search algorithm [ 
Team orienteering problem  A variable neighbourhood search algorithm turned out to be more efficient and effective for this problem than two Tabu search algorithms. 


Ant colony optimization [ 
Team orienteering problem  The sequential, deterministicconcurrent and randomconcurrent,and simultaneous methods are proposed to construct candidate solutions in the framework of ACO. 


Iterated local search heuristic [ 
Team orienteering problem  An algorithm that solves the team orienteering problem with time windows (TOPTW) 


Simulated annealing [ 
Team orienteering problem  Two versions of the proposed SA heuristic are developed and compared with existing approaches 


GRASP [ 
Team orienteering problem  A greedy randomised adaptive search Procedure for solving the Team orienteering problem. 


GLS, VNS, ILS [ 
Tourist trip design problems  Guided local search (GLS) and variable neighbourhood search (VNS) are applied to efficiently solve the TOP. Iterated local search (ILS) is implemented to solve the TOPTW. 


Tabu search [ 
Team orienteering problem  The Tabu search heuristic is embedded in an adaptive memory procedure that alternates between small and large neighbourhood stages during the solution improvement phase. Both random and greedy procedures for neighbourhood solution generation are employed and infeasible, as well as feasible, solutions are explored in the process. 


Tabu search [ 
Truck and trailer routing problem  A solution construction method and a Tabu search improvement heuristic coupled with the deviation concept found in deterministic annealing is developed. 


Simulated annealing [ 
Truck and trailer routing problem  The combination of a twolevel solution representation with the use of dummy depots/roots, and the random neighbourhood structure which utilizes three different types of moves. 


GRASP [ 
Truck and trailer routing problem  A hybrid metaheuristic based on GRASP, VNS and path relinking. 


Branchandcut [ 
Maritime inventory routing problem  A case study of a practical maritime inventory routing problem (MIRP) shows that the proposed neighbour and algorithmic framework are flexible and effective enough to be a choice of model and solution method for practical inventory routing problems 


Branchandcut [ 
Inventory routing problem  The algorithms could solve the instances with 45 and 50 customers, 3 periods and 3 vehicles. 


Branchandcut [ 
Inventory routing problem  The algorithm solves the IRP with several vehicles and with many products, each with a specific demand, but sharing inventory and vehicle capacities. 


Branchandcut [ 
Inventory routing problem  They implement a branchandcut algorithm to solve the model optimally. 


Branchandprice [ 
Inventory routing problem  A new branching strategy to accommodate the unique degeneracy characteristics of the master problem, and a new procedure for handling symmetry. A novel column generation heuristic and a rounding heuristic were also implemented to improve algorithmic efficiency. 


Local search [ 
Inventory routing problem  Our model takes into account pickups, time windows, drivers’ safety regulations, orders, and many other reallife constraints. This generalization of the vehiclerouting problem was often handled in two stages in the past: inventory first, routing second. 


Genetic algorithm [ 
Inventorydistribution problem  The delivery schedule represented in the form of a 2dimensional matrix and two random neighbourhood search mechanisms are designed. 


Genetic algorithm [ 
Bus Terminal Location Problem  A new crossover and mutation for the BTLP. 


Branchandprice method [ 
Maritime inventory routing problem  The method is tested on instances inspired from realworld problems faced by a major energy company. 


Variable neighbourhood search [ 
Inventory routing problem  A variable neighbourhood search (VNS) heuristic for solving a multiproduct multiperiod IRP in fuel delivery with multicompartment homogeneous vehicles, and deterministic consumption that varies with each petrol station and each fuel type. 


Branchandcut [ 
Airline crew scheduling problems  The branchandcut solver generates cutting planes based on the underlying structure of the polytope defined by the convex hull of the feasible integer points and incorporates these cuts into a treesearch algorithm that uses automatic reformulation procedures, heuristics and linear programming technology to assist in the solution. 


Simulated annealing [ 
Airline crew scheduling problems  Computational results are reported for some realworld shortto mediumhaul test problems with up to 4600 flights per month. 


Simulated annealing [ 
Airline crew scheduling problems  The first step uses the “pilotbypilot” heuristic algorithm to generate an initial feasible solution. The second step uses the Simulated Annealing technique for multiobjective optimization problems to improve the solution obtained in the first step. 


Genetic algorithms [ 
Airline crew scheduling problems  The development and application of a hybrid genetic algorithm to airline crew scheduling problems. The hybrid algorithm consists of a steadystate genetic algorithm and a local search heuristic. The hybrid algorithm was tested on a set of 40 realworld problems. 


Simulated annealing [ 
Train scheduling problem  They integrated the train routing the train routing problem and the train scheduling problem. They used simulated annealing to solve the problem. The objective is to minimize operational costs (fuel, crew, capital, and freight car rental costs) without missing cars. 


Genetic algorithm [ 
Train scheduling problem  They applied GA for solving the freight train scheduling problem in a single track railway system. 


Genetic algorithm [ 
Train scheduling problem  They solved the passenger train scheduling problem by attempting to minimize the waiting time for passengers changing trains. They proposed a GA with a greedy algorithm to obtain the suboptimal solutions. 


Genetic algorithm [ 
Train dispatching problem  A model for train dispatching on lines with double tracks. The model can optimize train dispatching by adjusting the order and times of train departures from stations, and then the efficiency of the method is demonstrated by simulation of the Guangzhou to Shenzhen highspeed railway. 


Genetic algorithm [ 
Train timetable problem  To obtain the optimal train timetables to minimize delay and changes of gates, they divided the railway network into multiple block, used the branchandbound method to determine the train sequence for each block, and calculate the train times. They applied GA to improve the solutions. 


ACO [ 
Railroad blocking problem  An ant colony optimization algorithm for solving RBP. The solution method is applied to build a car blocking plan in the Islamic Republic of Iran Railways. 


Very largescale neighbourhood [ 
Railroad blocking problem  An algorithm using a technique known as very largescale neighbourhood (VLSN) search that is able to solve the problem to near optimality using one to two hours of computer time on a standard workstation computer. 


ACO [ 
Railroad blocking problem  A new formulation for RBP in coal heavy haul rail network in north China. An improved ACO to solve a new formulation for RBP in coal heavy haul rail network in north China. They discussed the problem with direct train routing and frequencies and they did not consider the terminal capacity in handling classification process and maximum available blocks constraints. 


Multiobjective evolutionary algorithms [ 
Aeronautical and aerospace design problems  A taxonomy and a comprehensive review of applications of MOEAs in aeronautical and aerospace design problems. They provide a set of general guidelines for using and designing MOEAs for aeronautical and aerospace engineering problems. 


Genetic algorithms [ 
Aerospace problems  The paper uses GA to solve H2 and Hinfinity norm model reduction problems and helps obtain globally optimized nominal models. 


Genetic algorithms [ 
Military transport planning (MTP)  They study a logistics problem arising in military transport planning. A Niche genetic algorithm, together with a hybridized variant, is applied to the problem. 


GRASP [ 
School bus routing problem  A matheuristic that uses a GRASP construction phase followed by a variable neighbourhood descent (VND) improvement phase to solve 112 instances with 5 stops and 25 students to 80 stops and 800 students of the SBRP. 


ACO [ 
School bus routing problem  A hybrid evolutionary computation based on an artificial ant colony with a variable neighbourhood local search algorithm to solve the urban bus routing problem in the Tunisian case. 


Hybrid algorithm [ 
School bus routing problem  A mixed load improvement algorithm to solve 48 test instances for the SBRP with a number of schools 6, 12, 25, 50, and 100 and bus stops 250, 500, 1000, and 2000. 


Tabu search [ 
School bus routing problem  In addition to the minmax vehicle routing problem criterion imposed on the time it takes to complete the longest route, school districts are concerned with the minimization of the total distance travelled and they develop a solution procedure for this problem by applying Tabu search within the framework of Multiobjective Adaptive Memory Programming and compare it to an implementation of the Nondominated Sorting Genetic Algorithm— a wellknown approach to multiobjective optimization. 


Hybrid algorithm [ 
School Bus Routing Problem  For a school bus routing problem, called the MVTPPRC, which combines a bus stop selection and bus route generation with additional constraints on certain resources, we have developed a BCP algorithm as an implementation of a set partitioning formulation proposed for that problem. This formulation has been obtained from a DantzigWolfe decomposition of a threeindex variables model that describes the MVTPPRC. 
The paper survey mathematical models and algorithms used to solve different types of transportation modes by air, water, space, cables, tubes, and road. It presents the variants, classification, and the general parameters of the Transportation Problems.
As future work, we propose to investigate mathematical models of the space transportation problems, maritime transportation issues, and the creation of new algorithms that solve these problems.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by SEPPROMEP (Mexico) through Grant PROMEP/103.5/12/8060 and by UNACARDESDACI through grant POA 2014.