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Journal of Applied Mathematics
Volume 2014 (2014), Article ID 848129, 17 pages
http://dx.doi.org/10.1155/2014/848129
Review Article

A Survey of Transportation Problems

1Universidad Autónoma del Carmen, Avenida 56 No. 4, Esq. Av. Concordia, Col. Benito Juarez, 24180 Ciudad del Carmen, CAM, Mexico
2Benemérita Universidad Autónoma de Puebla, Facultad de Ciencias de la Computación, Ciudad Universitaria, 14 sur y Avenida san Claudio, Edificio 104C, Colonia San Manuel, 72570 Puebla, PUE, Mexico
3Universidad Autónoma del Estado de Hidalgo, Av Universidad S/N, Centro, 42780 Tlahuelilpan, HGO, Mexico

Received 30 December 2013; Revised 5 February 2014; Accepted 6 February 2014; Published 27 March 2014

Academic Editor: Francisco Chiclana

Copyright © 2014 Ocotlán Díaz-Parra et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.

1. Introduction

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.(A)Resources. The resources are those elements that can be transported from sources to destinations. Examples of discrete resources are goods, machines, tools, people, cargo; continuous resources include energy, liquids, and money.(B)Locations. The locations are points of supply, recollection, depot, nodes, railway stations, bus stations, loading port, seaports, airports, refuelling depots, or school.(C)Transportation modes. The transportation modes are the form of transporting some resources to locations. The transportation modes use water, space, air, road, rail, and cable. The form of transport has different infrastructure, capacity, times, activities, and regulations. Example of transportation modes are ship, aircraft, truck, train, pipeline, motorcycle, and others.

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 2 presents the classification and the general parameters of the Transportation Problems, Section 3 the variants of the Transportation Problems, and Section 4 the algorithms used to solve the Transportation Problems, and the last section presents the conclusions.

2. Transportation Problems

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.

2.1. Maritime Transportation

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.

Resources. Bulk cargo (oil, coal, iron ore, grains, bauxite/alumina, phosphate, dry or liquid not packet); break-bulk cargo (bags, boxes, drums, all general cargoes that have been packaged); passenger vessels (passenger ferries, cruise ships); bulk carriers (liquid bulk vessels, dry bulk vessels, largest tankers of liquefied natural gas technology enabled); general cargo (vessels designed to carry nonbulk cargoes); Roll-on/Roll-off (RORO vessels, cars, trucks, and trains to be loaded directly on board); shrimp and seafood, hazardous materials; military-owned transportation resources; and goods (nonperishable goods, final manufactured goods, processed food, produce, livestock, intermediate goods, processed and raw materials).

Locations. Fishing port, warm water port, sea port, cruise port, cargo port, cruise home port, port of call, cargo ports, and oil platform.

Transportation Mode. Bulk carriers, container ships, tankers, reefer ships, Roll-on/Roll-off ships, coastal trading vessels, ferries, cruise ships, ocean liner, cable layer, tugboat, dredger, barge, general cargo ship, submarines, sailboat, jet boat, fishing vessels, service/supply vessels, barges, research ships, dredgers, and naval vessels.

2.2. Air Transportation

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.

Resources. Military-owned transportation resources, passengers, air taxi service, goods, supplies and equipment, mail, troops, and others.

Locations. Airport terminals, heliport, helipads, helistop, Helideck and Helispot, and area of military operations.

Transportation Mode. Fixed-wining aircraft, airplane, gyroplane, recreational aircraft, military cargo aircraft, helicopters, zeppelins, personal air transportation with jet packs and blimps, military transport helicopters, tactical and strategic airlift, air ambulance, and aerial refuelling.

2.3. Land Transportation

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.

Resources. Military-owned transportation resources, goods, people, hazardous materials, waste, or money.

Locations. Pizza restaurant, post office, university, schools, gas stations, warehouse, stores, markets, fish market, dump, bottling plants, malls, depots, houses, Landfill, incineration plant, waste container, banks, and others.

Transportation Mode. Buses; trucks; motorcycles; bicycles; cars and pickups; box trucks and dock highs; cargo and sprinter vans; less than truck Load (LTL); full truck load (FTL); flatbeds; reefers (refrigerated units); longer combination vehicles (LCV) with double semitrailer, recovery vehicle, scooters, and pedestrians; main battle tank; infantry fighting vehicles; armored personnel carriers; light armored vehicles; self-propelled artillery and anti-Air mine protected vehicles; combat engineering vehicles; prime movers and trucks; unmanned combat vehicles; military robot; joint light tactical vehicle (JLTV), utility vehicle, refrigerator truck, landfill compaction vehicle, garbage truck, waste collection vehicle; armored cash transport car; and security van.

2.4. Rail Transportation

Rail transport carries resources over long, medium, and short distances by way of wheeled vehicles running on rail track or railway.

Resources. Military-owned transportation resources, goods, passenger, containers, and bulks.

Locations. Stations, transit centre locations, park and ride locations, railway station, railroad station, goods stations, large passenger stations, smaller stations, early stations, central stations, railway platform (bay platform, side platform, island platform), metro station, train station, tram stop, station facilities, terminal, interchange station, tunnel stations, metro depot, maintenance depot, and light rail depot.

Transportation Mode. Trains, metro, subway, vactrain, magnetic levitation train, ground effect train, U-Bahn and S-Bahn, intercity trains, intercity rail, high-speed rail, high speed train, locomotive, pacer (train), freight car, goods train, railway passenger car, coach passenger car, intermodal freight transport, refrigerated railroad cars, light rail vehicles, suburban railway, urban railway, rapid transit, underground railway, elevated railway, metropolitan railway, carbody, ballast tamping machine, long welded rail cars, cleaning trains, concreting trains, rail grinders, ballast tamping machines, track recording cars, and rail grinders.

2.5. Space Transportation

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.

Resources. Military-owned transportation resources, cargo or passengers, personnel, fuel (LH2), oxydizer (LOX), and propellants (LOX and LH2 at given mixture ratio).

Locations. Earth spaceport (ES), Low Earth Orbit (LEO), Geostationary Earth Orbit (GEO), Lagrange Point L1 (L1), Low Lunar Orbit (LUO), Lunar SpacePort (LUS), Lagrange Point L2 (L2), Planetary escape mission (PLA), Mars Spaceport (MAS), Space Operations Center, Lunar Service Center, Lunar Propellant Production Facility, and others.

Transportation Mode. Rocket-powered aircraft, nuclear powered aircraft, spacecraft, space shuttle, space, space planes, rockets, missiles, and advanced Hall electric propulsion, crew exploration vehicle (CEV), automated transfer vehicle (ATV), evolved expendable launch (EELV), National Aerospace Plane (NASP), transatmospheric vehicle (TAV), orbital space plane, next generation launch technology, winged shuttle (WSLEO), expendable interorbital ferry vehicle (EIOFV), reusable interorbital ferry vehicle (RIOFV), and spaceship.

2.6. Pipeline and Cable Transportation

Pipeline and cable transportation carries resources from locations to other locations by pipe and vehicles pulled by cables.

Resources. Water, energy, electricity, petroleum products, telecommunications, chemicals, slurry coal, natural gas, sewage, beer, biofuels (ethanol and biobutanol), hydrogen, skiers, and passenger lift.

Locations. Residential and commercial areas, treatment plant, processing facility, gas stations, pump stations, terminals, tanks, storage facilities, partial delivery stations, inlet station, injection station, block valve station, regulator station, final delivery station, floors (levels, decks) of a building, vessel, or other structures.

Transportation Mode. Pipelines (gathering pipelines, transportation pipelines, distribution pipelines), ducts, oil pipelines, gas pipeline, transmission lines, electrical substations, pole-mounted transformed, generation stations, distribution systems, electricity transmission system, tubes, aerial lifts (aerial tramway, chairlift, funitel, gondola lift, ski lift), surface cable transportation (cable car, cable ferry, funicular, surface lift), and vertical transportation (Elevator).

2.7. Intermodal Transportation

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.

3. Variants of the Transportation Problems

3.1. Maritime Transportation Problems

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) [1]; ship routing problem (SRP) [2, 3]; ship routing problem of tramp shipping (SRPTP) [2]; inventory constrained maritime routing and scheduling problem for multicommodity liquid bulk [4]; vessel fleet scheduling/allocation [5, 6]; cargo routing problem [4]; maritime inventory ship routing problem [7]; oil-tanker routing and scheduling problem [8, 9]; maritime oil transportation problem [10]; industrial ship scheduling problem [7]; industrial ship scheduling problem with fixed cargo sizes [7]; tramp ship scheduling problem [7]; single product inventory ship routing problem [7]; multiple product inventory ship routing problem [7]; tramp ship routing and scheduling problem with speed optimization [11]; and maritime platform transport problem of solid, special and dangerous waste [12].

The RoRo ship stowage problem (RSSP) [1] to decide (which optional cargoes to carry, how stow all cargoes on board the ship, all long-term contracts are fulfilled) upon a deck configuration with respect of the height. The objective is to maximize the sum of revenue from optional cargoes minus penalty costs incurred when having to move cargoes. The mathematical model of RSSP [1] is formed by (1)–(18): where is the set of all cargoes, is the set of all mandatory cargoes, is the set of all optional cargoes, is the set of all decks, is the set of all potential lanes on each decks, is the set of all ports (except the last port on the route), is the set of ports from loading port of cargo to the port before the unloading port of cargo , is the set of cargoes , is the width of deck , is the length of deck , is the length of one vehicle in cargo , is the width of one vehicle in cargo , is the height of one vehicle in cargo , is the loading port of cargo , is the unloading port of cargo , is the lower bound for where deck can be placed, is the upper bound for where deck can be placed, is the revenue for transporting optional cargo , is the number of vehicles in cargo , is the cost incurred if cargo needs to be moved, is the maximum allowable torque on the ship from the cargo, is the highest allowable center of gravity of the laden ship, is the weight of one vehicle from cargo , is the lightweight of the ship, is the vertical distance from the ship’s bottom deck to its center of gravity when empty, : approximated horizontal distance of lane on deck from the ship’s center of gravity, is the approximated vertical distance of deck from the ship’s bottom deck, is the cargo, is the deck, is the lane, and is the port.

In (1) the objective is to maximize the sum of revenue from optional cargoes minus penalty costs incurred when having to move cargoes. Equation (2) links the binary indicator variables for if lane on deck is used from port to by cargo , to the integer variables for how many vehicles from cargo that are stowed in lane on deck from port to . Equation (3) ensures that there is enough vertical space on the deck where the cargoes are placed. Equation (4) shows the sufficient width of the lanes. Equation (5) makes sure that once a cargo has been placed, it remains unmoved until it is unloaded. In (6), the partitions of decks into lanes are restricted. Equations (7) and (8) link the integer variables for how many vehicles from cargo , that is, stowed in lane on deck from port to , to the number of vehicles from cargo , that is, carried, for respectively mandatory and optional cargoes. Equation (9) ensures that the length of a lane is not violated by the vehicles stowed in that lane. Equations (10) and (11) are restrictions on ship stability calculations and involve nonlinear equations. Equation (10) imposes that the torque from the cargo on the ship should be within the allowable limit to avoid rolling. The constants view the are approximations of the horizontal distance of a lane to the center of the ship, with negative values indicating a possible tilt to the port side and positive values indicating a tilt to the starboard side. Equation (11) ensures that the maximum allowable vertical distance from the ship’s bottom deck to the ship’s centre of gravity when loaded is not exceeded. When vehicles from cargo are loaded in front of vehicles from cargo and cargo is unloaded before cargo , there is an inconvenience as vehicles from cargo must be moved out of the way. Equation (12) makes sure that a corresponding penalty is added to the objective function. Equation (13) provided upper and lower bounds on the deck heights. Equation (14) ensured the nonnegativity of lane width. Equations (15), (16), and (18) make sure that the variables , , and take binary values. And (18) imposes nonnegativity and integrality on the number of vehicles carried in each lane.

3.2. Air Transportation Problems

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) [13], multiairport ground holding problem (MAGHP) [14, 15], air traffic flow management rerouting problem (tfmrp) [16], helicopter routing problem (HRP) [17], airline crew scheduling problem [18], and oil platform transport problem [19].

The general problem of Air Transportation is represented in the mathematical model described by Li et al. [20, 21], which presents an objective that is to minimize the overall total cost which consists of the total transportation cost of the orders allocated to normal flight capacity, the total transportation cost for the orders allocated to special flight capacity, and the total delivery earliness tardiness penalties cost. The mathematical programming formulation of the model is shown as follows: where , , and are the order or job index, ; or is the flight index, ; is the destination index, ; is the departure time of flight at the local airport; is the arrival time of flight at the destination; is the transportation cost for per unit product when allocated to normal capacity area of flight ; is the transportation cost for per unit product when allocated to special capacity area of flight ; is the available normal capacity of flight ; is the available special capacity of flight ; is the quantity of order ; is the delivery earliness penalty cost (/unit/h) of order ; is the delivery tardiness penalty cost (/unit/h) of order ; is the due date of order ; is the quantity of the portion of order allocated to flight normal capacity area; is the quantity of the portion of order allocated to flight special capacity area; is the order destination; is the flight destination; is a large positive number; is the processing time of order . The decision variables (, , and ) are nonnegative integer.

The objective of (19) is to minimize total cost which consists of transportation cost of orders allocated into normal flight capacity, transportation cost of orders allocated into special flight capacity, the delivery earliness penalty costs of orders, and the delivery tardiness penalty costs of orders. Equations (20) and (21) ensure that if order and flight have different destination, order cannot be allocated to flight . Equation (22) ensures that the quantity of the portion of order allocated into flight consists of quantities of the portion of order allocated into normal capacity area of flight and the portion of order allocated to special capacity area of flight . Equation (23) ensures that the normal capacity of flight is not exceeded. Equation (24) ensures that the special capacity of flight is not exceeded. Equation (25) ensures that order is completely allocated. Equation (26) ensures that allocated orders do not exceed production capacity. It ensures that allocated quantity can be supplied by sufficient assembly capacity.

3.3. Land Transportation Problems

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) [22], convoy routing problem (CRP) [23], inventory routing problem (IRP) [24], inventory routing problem with time windows (IRPTW) [25], school bus routing problem (SBRP) [26], tour planning problem (TPP) [27], truck and trailer routing problem (TTRP) [28], vehicle departure time optimization (VDO) problem [29], vehicle routing problem with production and demand calendars (VRPPDC) [30], bus terminal location problem (BTLP) [22], bus scheduling problem [31], delivery problem [32], combining truck trip problem [33], open vehicle routing problem [34], transport problem [35], truck loading problem [36], truck dispatching problem [37], convoy routing problem [23], multiperiod petrol station replenishment problem [38], petrol station replenishment problem [39], vehicle routing problem [40], capacitated vehicle routing problem (CPRV) [40], multiple depot vehicle routing problem (MDVRP) [40], periodic vehicle routing problem (PVRP) [40], split delivery vehicle routing problem (SDVRP) [40], stochastic vehicle routing problem (SVRP) [40], vehicle routing problem with backhauls (VRPB) [40], vehicle routing problem with pick-up and delivering (VRPPD) [40], vehicle routing problem with satellite facilities [40], vehicle routing problem with time windows (VRPTW) [40], waste transport problem (WTP) [41], cash transportation vehicle routing problem [42], team orienteering problem [43], military transport planning (MTP) [44], petrol station replenishment problem with time windows [45].

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 [26]. The characteristics of SBRP [46] are number of schools (single or multiple), surrounding services (urban or rural), problem scope (morning, afternoon, both), mixed Load (allowed or no allowed), special-education students (considered or not considered), fleet mix (homogeneous fleet or heterogeneous fleet), objectives (number of buses used, total bus travel distance or time, total students riding distance or time, student walking distance, load balancing, maximum route length, Child’s time loss), constraints (vehicle capacity, maximum riding time, school time windows, maximum walking time or distance, earliest pick-up time, minimum student number to create a route). The mathematical model of SBRP [47] is formed by (27)–(34): School buses are centrally located and have to collect waiting students at pick-up points and to drive them to school. The number of students that wait in pickup point is , (, ). The capacity of each bus is limited to students (). The objective function of the School Bus Problem is composed of two costs: (a) cost incurred by the number of buses used, (b) driving cost (fuel, maintenance, drivers salary, and others), subject to operational constraints, Costs (a) or (b) have to be minimized. For a given of buses, let ,  , , be variables that attain the value 1 if pickup points and are visited by the th bus and pickup point is visited directly after . Otherwise, is 0. Let , , be variables that may attain any value. The objective of the SBRP is to find variables and that minimize . Where = cost of driving from point to point , is a function of the distance between and and the driving time, , = driving time from point to point , = a quantity to be loaded (or unloaded) at , = set of constraints characterized by the nature of the problem, where . The three-dimensional assignment problem given in ((28), (29), (30), (31), and (32)) could be transformed into a regular assignment problem by duplicating times the row and column corresponding to city 0 and obtaining an assignment problem with dimensions () by (). Constraints ensure the formation of exactly tours, where each one passes through the school. The restriction of capacity is in (33) and the constraint of time is in (34).

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.

3.4. Rail Transportation Problems

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) [48], locomotive routing problem [49], tour planning problem by rail (tpp) [50], rolling stock problem (rsp) [51], yard location problem [52], and train dispatching problem [53].

Train dispatching transportation problem, train meet-and-pass 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 [53]: Equations (36) and (37) are the track capacity constraint, these equations ensure ensures that no two trains are scheduled that occupy the block at the same moment . Each binary variable takes the value of one if and only if the train occupies the block . The set contains all vectors that result in technically and logistically feasible schedules for the train .

3.5. Space Transportation Problems

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) [54], Earth-Moon supply chain problem [55], interplanetary transfer between halo orbits [56], and Hill’s restricted three-body problem (Hill’s R3BP) [57].

The Earth-Moon supply chain problem [55] considers the problem of delivering cargo units of water from low Earth orbit to lunar orbit and the lunar surface. The formulation requires that the architectural characteristics of the vehicle used to transport the packages to the destinations and the paths the vehicles travel are to be determined concurrently. The problem is solved using both traditional design optimization methods and a concurrent design optimization method: where () is the transfer starting at node traveling to node and terminating at node , is the number of vehicles on route (), is the capacity of vehicle on route (), is the vehicle initial mass, is the vehicle wet mass, is the number of packets that leave node equal to the supply at node (), and is the velocity change.

Equation (38) defines the main objective. The main objective of the system is to minimize the initial mass of the transportation system architecture. The is the number of vehicles that start at node , travel to node , and then terminate at node and is the initial mass of a vehicle on route (, , ). The initial vehicle mass () is determined by the vehicle capacity for each route and the actual initial mass is the wet mass () plus the amount of payload carried on that vehicle. Each route carries packages that each weighs . For each route, the initial mass is defined as and this is summed over all routes. The summation of over all routes is simply the amount of supply, which is a constant. Equation (39) contains a restriction of the network subsystem that determines the actual package flows from Low Earth Orbit (LEO) to the destination nodes. To ensure a feasible package flow, we must define the supply, demand, and capacity constraints for the space network. The supply constraints ensure that the number of packages () that leave node is equal to the supply at node (). Equation (40) ensures that the number of packages that arrive at node is equal to the demand of node (). Equation (41) ensures that the vehicle has enough capacity to accommodate the packages. Equations (42) and (43) contain the upper bound on the number of packages on each route. Equation (44) defines a binary decision variable, , which is equal to one if we stage after burn and zero otherwise. It can stage at most times, where is the total number of burns required for that route. It assumes that the vehicle stages after the last burn (). Equation (45) defines the variable to represent the type of fuel used during stage . The variable can take on integer values up to the number of different types of fuel available (the model do not allow hybrid stages, to ensure that the same type of fuel is used for consecutive burns in a single stage). First in (46) the total number of stages is computed (). Next using the staging locations, the amount of required for each stage () can be defined. The amount of in a given stage is the sum of the for each burn up to and including the first burn for which the vehicle stages (). Finally, the initial mass () of the vehicle is calculated using the rocket equation. Equation (47) computes the vehicle wet mass (the mass of the structure and fuel without the payload mass).

3.6. Pipeline and Cable Transportation Problems

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) [58, 59], bulk energy transportation networks [60], generalized network flow model or multiperiod generalized minimum cost flow problem [61, 62], water flow and chemical transport [63], CO2 pipeline transport [64, 65].

The coal, gas, water, and electricity production and transportation systems model [66] uses the fact that each of these subsystems depends on the integrated operation of a network together with a market, and it captures the strong coupling within and between the different energy subsystems. The mathematical framework using a network flow optimization model with data characterizing the actual national electric energy system as it exists today in the United States [66] is formed by (48)–(52): The objective function (48) is equal to the total production cost + total generation cost + total storage cost + total transportation cost for the gas and coal subsystems, subject to energy balance at the production nodes (49), energy balance at the storage nodes (50), energy balance at the generation nodes (51), and energy balance at the electric transmission nodes (52), where is the total cost (production, storage and transportation) of the energy over 1 year at weekly intervals; is the production node; is the generation node; is the electric transmission mode; is the storage node; is the transportation mode; is the transmission line, , , and are the per unit cost of extraction, generation (without including the fuel cost to avoid duplication), and storage; , , and are the per unit cost of gas or coal transportation from a production or storage node to a storage or generation node, using the transportation mode at time ; is the per unit cost of the electric energy transported by the transmission line at time ; is the total energy produced in the production node during time ; is the energy at the storage facility at the end of time ; is the total energy arriving to the generation facility at time ; , , and are the amount of energy going from a production or storage node to a storage or generation node, shipped using the transportation mode during the time ; is the amount of electric energy flow in the transmission line during the time ; is the forecasted energy demand in the electric node during the time ; and are the energy in the storage facility at the beginning and end of the scheduling horizon; is the efficiency of the energy transmission line ; is the energy from node to node ; is the energy from node to node ; is the energy from outside the system to node ; and is the energy from node to outside the system (electric demand).

3.7. Intermodal Transportation Problems

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 [67], tour planning problem (TPP) [50], tourist trip design problems [68], railroad blocking problem (RBP) [69], and intertemporal demand for international tourist air travel [70].

The railroad blocking problem (RBP) is a multicommodity-flow, network-design, and routing problem [71], and RBP is the railroad blocking problem which is one of the most important decisions in freight railroads. The mathematical model of RBP [69] is formed by (53)–(59): where is the graph with terminal set and potential blocks set , is the set of all commodities designated by an origin-destination pair of nodes, is the volume of commodity , is the origin terminal for commodity , is the destination terminal for commodity , is the origin of potential block , is the destination of potential block , is the capacity of potential block , is the per unit cost of flow on arc (assumed equal for all commodities), is the number of blocks which may originate at terminal , and is the volume which may be classified at terminal , , if commodity is flowing on block , 0 otherwise. if block is included in the blocking network, 0 otherwise.

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 (53). In (54), for each terminal there are balance equations for the flow of each commodity. For each potential block, equations in (55) prevent flow on blocks which are not built and enforce the upper bound on flow for blocks which are built. The constraints (56) enforce the terminal limit for the sum of the blocks which leave the terminal. The constraints (57), (58), and (59) model the volume of cars, which may be classified at each terminal.

4. Algorithms to Solve Transportation Problems

Various algorithms to solve the Transportation Problems (Table 1) may be found in the literature. We mention only some of the most popular algorithms to solve the Transportation Problems.

tab1
Table 1: Related Works.

5. Conclusions

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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by SEP-PROMEP (Mexico) through Grant PROMEP/103.5/12/8060 and by UNACAR-DES-DACI through grant POA 2014.

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