Transportation Models and Theories (Part One)

By Alfonso Llanes

September 1, 2016

Abstract

In order to accommodate all concerning materials and still keep this paper length-readable it is made available in two parts.

Freight rates in nodal format based on origin-destination published by carriers; leave out minor ports in the network.  Using a minimal approach of 50 nodal points to cover the entire world, carriers can reduce the number of individual rates to (50)²= 2,500 for each service type that can easily be kept and updated in a data base thus, avoiding storage and updates of millions of rates. It fallows that with this recipe, carriers can use algorithms for covering the ports in the network not included in the nodes. The balancing act relies in developing an algorithm that works across all service types within the 50 nodes. For instance, if 4 service types are considered it would render 4*2,500= 10,000 rates with all its attributes.

Even though this method can work for carriers, how does a freight broker keep track of all the possible combinations of routes and rates? How does a manufacturer find out about a particular freight rate before committing to a production-delivery contract? The list goes on to include traders, bankers, governments and individuals who need to know freight rates as accurately as possible in any part of the world in the age of the Internet with instantaneous response for a management board in session?

Introduction

Over the years many methods, theories and models have been introduced by many researchers in various fields of study such as economic geography, transportation economics, economics, theoretical dynamic system analysis and fractals applied to transportation studies. This paper makes a collection of many of the methods and models developed by academics.  The most prevalent models will be cited here with the intent of making the reader familiar with the contribution made over the years by many scholars. This paper will introduce a model developed by the Barcelona Field Studies adapted to fit an ocean transportation model and conclude with a new model proposed by the author of this paper as a contribution to the field. The narrative of the adapted Barcelona model will be derived from Dijkstra’s algorithm.  By constructing Dijkstra’s “closest neighbor” algorithm and the Barcelona “alternative nearest neighbor” a new model can be developed.

Distance and density variables functions model

Elvira Kurmanalieva in her paper Transport Costs in International Trade, 2006 writes about a transport density that can be “broadly interpreted as an efficiency of transportation network and infrastructure”.  In her paper she states that inspiration came from the “shortest-path-problem” Adding that the number of shortest paths between two countries is an approximate estimation of transport density and therefore, the model is estimated as a function of distance and density variables.

FOB/CIF model ratio

This model is used by the International Monetary Fund----“If something is being transported from two international markets then the FOB and CIF prices must differ only by transport costs between one country and another wherein CIF-FOB equals the transport factor”. The problem with this approach is that it depends on shipper’s declaration of value and Custom entry form that more often than not, are not accurate. As a result a unit percent [(CIF/FOB)]-1*100% is only an academic exercise if declared and entry values are not dependable.

Iceberg Theory

The Iceberg Theory (Samuelson 1952) bases the cost of shipping on a relative price rather than relative quantity. In the words of David Hummels traditional “iceberg” formulation, transport is treated as an exogenous friction (r) that is fixed and proportional to the value shipped, with the value‐added of transportation services treated as pure waste, or “melt”.   Krugman-- (1991a, 1991b) formulation of the iceberg transport of costs is: T (d) = erd economic geography where d= distance and r = is the melt.

Static theory

This theory treats shipping markets as a static mechanism where a system of variables must link together supply and demand into balance. According to Hofstra University this model represent a well-functioning transport markets where supply meets transport demand.  Most theories, which were dedicated in market’s equilibrium, come from this static notion about shipping economy. A “stochastic process” is a random process changing with time. Directly, in probability theory, a stochastic process is a time-sequence representing the progression of some system characterized by a variable that varies as a subject of a random difference.

 Additive and multiplicative trade costs theory

In 2011 Alfonso Irarrazabal, Andreas Moxnes, and Luca David Opromolla published a paper that introduces the idea of additive costs as a constant monetary cost per unit which departs from Samuelson’s framework. This model incorporates variable trade costs as comprising both a multiplicative (iceberg) and an additive part. Multiplicative costs are defined as a constant percentage of the producer price per unit traded.  With data collected from Norwegian Customs these authors built a model of international trade with heterogeneous firms.

Time in transit theory

Kiyoyasu Tanaka 2010. Tanaka concentrates his argument around the issue of tradeoff between freight- cost-time and timely delivery to build this model. Using the Japanese Census of Logistics, his paper examines the cost influence of distance and time across shipping modes.  Tanaka states that he found the results “puzzling because business enterprises are likely to pay more for shout-distance shipments by truck, ship and railroad transportation”.  Tanaka’s statement is in itself puzzling because the effect of

short distances on rates as per-mile freight rates tend to decline with distance as the ratio decreases.

Effect of distance on rates.

System Dynamics Theory

In 2010 M. Jurčević, F. Mitrović, M. Nadrljanski introduced the concept of System dynamics and Theory of Chaos in Freight Rate Forming in Shipping with significant fractal characteristic  as self-resemblance. Fractal organization is already known in literature. The biggest impediment to the building blocks of a system is not an engineering problem but a managerial one. That is because management must deal with social issues which are harder to understand and administer.  Jay W. Forrester, 1992 and the history of system dynamics In the logistics and supply chain context between fractals and feedback models examines a simple reinforcing and balancing loop in a system.  

These collections of theories should give the reader a good idea of the different approaches that have been tried over the years.

 At this juncture we need to retake the narrative of the adapted Barcelona model and the derivation of the Dijkstra’s algorithm.  It follows that the graphic/photo representation of these ideas will provide a better understanding of “closest neighbor” algorithm the description and the Barcelona “alternative nearest neighbor” as new models.

Below is an example of a generic system of origin-destinations nodes and its geographic structure for world coverage constructed with 48 nodes or (48)²= 2,304 combinations.

The “alternative nearest neighbor” used by the Barcelona Field Studies applied to forest distribution studies can also be applied to transportation nodes for we already know the distribution of ports around the world.

This algorithm measures the distributions according to whether they are clustered, random or regular. The nearest neighbor formula will produce the following distribution patterns from a continuum:

   The formula used by the Barcelona Field Studies is as follows:

Methodology

1. Select an area using random points in a quadrant. This should be sufficient to obtain a minimum number of points with the corresponding values.

2. Measuring the coordinate’s distance of each point within the quadrant to its nearest neighbor and assigning a corresponding value based on the statistical average distribution of points and values.

The following alternate model of the “nearest neighbor” is measured from Adak to the destination port of the Vladivostok, Russia quadrant using Google Earth. This particular port is chosen because is not located in the node for Russia, for the country has three different water outlets: The Baltic Sea, the Black Sea and its Eastern Seaboard where these ports need to be calculated independently from the Russian node in the western part of the country. In this case, neither the bound Barcelona method nor the Dijkstra’s are favored over other closest neighbor techniques available. Either technique can be placed on a graph and be analyzed with discrete mathematics.

REFERENCES

Samuelson, P. A., 1952. "The Transfer Problem and Transport Costs: The Terms of Trade When Impediments are Absent," The Economic Journal, Vol. 62, No.246 (Jun., 1952), pp. 278-304.

Irarrazabal, Alfonso Moxnes, Andreas Opromolla, Luca David(2010). The Tip of the Iceberg: Modeling Trade Costs and Implications for Intra-Industry Reallocation.

Engelen, S., Meersman, H., Van der Voorde, E.: Using system dynamics in maritime economics Maritime Policy Management, 33 (2), 2006.

Abbas, K.A., and Bell, M.G.H. (1994). System dynamics applicability to transportation modeling.

Randers, J. and Göluke, U. (2007) Forecasting turning points in shipping freight rates: lessons from

30 years of practical effort. System Dynamics Review Vol. 23, No. 2/3, (Summer/Fall 2007): 253–284.

Aizenman, Joshua (2004), ‘Endogeneous pricing to market and financing cost’, Journal of Monetary Economics 51(4), 691–712.

Evans, Carolyn and Harrigan James (2005), “Distance, Time, and Specialization” American Economic

Review.

Barcelona Field Studies Center

https://www.geography-fieldwork.org/geographyfieldwork.com

Fractals, Graphs and Transportation Networks

In the summer of 1998, I was contemplating entering a PhD program in the school of Arts and Sciences as I attracted the attention of Thomas A. Breslin, Professor in Department of Politics & International  Studies arisen from my interest in fractals and a couple of research papers I wrote about them as they  relate to many aspects of evolution.  Following this line of thought--by introducing a small change in a natural or man-made dynamic process the result over time, would be in both cases, an entirely new organism.

My view then was that in a space of time where dual processes are taking place, one natural and random and the other planned and man-made would lead to the conclusion of two independent processes that could either lead to entropy or continue to change ad infinitum with just an infinitesimal amount of variation on each cycle.

Probably, the first person to observe this dynamic process was Edward Lorenz when working at MIT using a three variable model for forecasting the weather in the 1960’s. At this time, computers were pretty slow and cumbersome, so, when he decided to extend his weather forecast for a few more days he rounded off some numbers into his model’s equation expecting only minor change in the model. He ran the model again and to his surprise the minute differences had made a startling difference in the forecast. Lorenz’s discovery revealed that in the immense dynamic s of a weather system all parts are connected by feedback to all the other parts affecting the way a system ends up after interacting with its component parts.

Subsequently to Lorenz’s discovery many researches embarked in trying it out in other dynamic systems from printed electric boards to brain functions and in the process they found new laws.

In the 1970’s an IBM researcher, Benoit Mandelbrot conceived a new geometry which he named “fractal” to suggest perhaps fractured or fractional with its main focus on scaly, flaky and uneven surfaces. Fractal geometry was meant to describe the roughness of nature, its dynamic process and its use of energy. As fractals patterns are studied other secrets are revealed such as scaling and self-similarity when visuals of an image at a given scale are reproduced as the scale is change up or down.

Nevertheless as previously mentioned, if a fractional change is introduced in a pattern its interaction with the whole will eventually render a new pattern or a new scale of the same pattern. This suggests that a nonlinear co-existence is taking place in a circular continuum with no beginning or end in a multidimensional space.

Modern computers have made possible the visualization of many fractals the most famous being the Mandelbrot set and also the most famous object in modern mathematics and a leap departure from the tradition of Euclidian geometry.

“The set itself is a mathematical artifact…clustered in a complex number plane.”  As said by John Briggs, in his 1992 book “Fractals the Patterns of Chaos”. As we know from the study of the mathematics of complex numbers there are two parts to it: One real and the other imaginary.

During the 1980’s Mandelbrot and others were using simple iterative equations to observe and study the behavior of numbers in the complex plane. Many were using a three slot equation described by :

[Changing number]+ [Fixed number} = [New number} where this new number is plugged into the first slot to become again the changing number, i.e. 0+1=1, it follows that 1+1=2 and so on. This simple equation of number iteration can be accomplished with whole or imaginary numbers each rendering different visual representation when running in a computer.

In a previous paragraph it was mentioned that my personal take on fractals was twofold one natural and random and the other planned and man-made. Since randomness is all around us and self-explanatory only man-made planned change will be consider for now.

Having in mind a sample of a digital photograph or an idea that can be digitized for study and observation, and considering the components of this picture/idea made out of pixels that can be viewed over a interval of time: From (zero time) to (completion time) then the pixels can be spread out along a dimension of time-distance. 

As a planned iteration among the pixels takes place a complete photograph or idea emerges at the end of the time slot selected. This action plan once executed will render a completed objected in the tradition of Euclidian space of shapes that model nature.

Fractal geometry provides a closer view at nature’s subtlety change in the un-noticeable slow motion of time. Notwithstanding, we all know that if an event can be fast forward the slow change can become possible and visualize its entire process to its terminal entropy.

During this period in my pursuit of fractals, my wife died and I became too depressed to continue with academic work. As years went by my curiosity was aroused again only this time I thought applying my background on transportation networks to my previous fascination with fractals I started viewing the two as one neural system that could be graphed and be placed in a system of inputs and inputs with varying attributes throughout the network that could provide a utilitarian mission for international transportation.

So, from iterative non-linear equations I moved to the study of graph theory and its mathematics.  This time I became convinced that such neural system could be built over a geographic information system such as Google Earth and run in real time in layers of ground, water and air transportation networks in Euclidian spaces using vehicular speed displacement in the graph attributes to adjust for the topology of space transforms.

Since I am not a computer programmer I needed to enlist the expertise of one qualified to build the idea into a tangible object. After several months of frustrating research on the subject I came across Kevin Chugh a Ph.D. computer scientist and his team. At first, he was reluctant that such a system could be built let alone function  but after several sessions of discussion with me and his team he decided they could develop such a system using the parallels of graphic interaction embedded in social networks commonly used today.

In fact and in theory transportation networks should be easier to build as only vehicles move but not individuals. Also, all vehicles move in a predetermined time-space within the confines of our planet. Other components of the networks such as seaports, airports, cities and depots exist in a fixed location and can easily be graphed.

The question that remains is whether a utilitarian application is feasible to solve many of the problems burdening the transportation industry today? We’ll try to answer this question and others in a future paper once a model is available.