Freight Rates Optimization

Economists tell us that pricing strategy is contingent upon the highest level the market will bear but parallel to the constraints imposed by supply and demand. Conversely, “the market” is a plural as there are many markets and each market has to develop a unique pricing strategy, specially, in freight transportation--in order to endure its distinctive set of constrictions such as routes, traffic, competition, geography, ports, facilities, capacity, etc.

Specifically, freight markets are exclusive to each surface domain whether ground, air, water or a combination thereof. As a result, freight rate optimization must be implemented within an individual domain not only because the exercise takes place in a particular environment, but also, because the prerequisites required by the vehicles used for the implementation of conveyance with many of its variables.

It follows that a better question to ask when trying to optimize a freight rate would be: What is the optimal combination for a particular mode of transportation? How distance and geography affect the calculation after the cost of terminal facilities at each end are factored in? What about line haul, short haul or intermodal within or outside of a vehicle’s constraints? Perhaps the most difficult task to evaluate or measure is the cargo type which comes in many sizes, flavors and incumbent upon individual vehicle specifications. Cargo in its many forms can vary from bulk, breakbulk, containerized, liquid, dry, oversize, overweight, fresh, frozen or hazardous just to mention a few in the catalog. In addition, commodities have different packaging requirements and unique stowage factors that influence handling requirements and thus, cost. Moreover, the traditional economic principles of supply and demand, competition, efficiency, market specialization and so on are to be concerned with at the end of the day.  This universe of variable can be best expressed with a diagram where a definite integral depicts a generic market behaving within highs and lows of an average space for an interval of time.                                          

                                                              

This scenario suggests that a great deal of understanding, knowledge, training and capital is required for a new enterprise to enter a freight market. This paper suggests the economic challenge and capital exposure that each method of transportation represents; and, in order to evaluate revenue and cost based on optimal freight pricing in a competitive market those fundamental issues need to be address.

One very significant independent variable responsible for the many headaches in the industry is dimensional weight or stowage factor which is the area to volume ratio cm³/cm²*6 for a cube or 1/6 where 6000cm³=1 Kilo. In general, if we inscribe a cube Z in a sphere of radius R and optimize the cube’s volume in relation to the constriction of the sphere’s radius, then we have an optimization problem in three dimensions.

In this case the ratio is volume- to- volume within the constraint of the sphere’s dimensions and the equation isolating one variable becomes:     Where (r) is the radius of the sphere. Solving with partial derivatives for (x)     product rule & chain rule The maximum volume of a box inscribed in a sphere is therefore =   And

   

The point of calculating a cube inscribed inside a sphere is that the cube is the figure that has the most volume per surface area after the sphere. But, in the real world of transportation vehicles are not practical as spheres or cubes even though there may be the most efficient geometric surface. Builders must consider that these vehicles need to transit on roads or over rails. As a result, rectangular rail box cars and vans are the most common vehicles in use today. Whereas aircraft and ships are a special cases of SA/V and will be address later.

Historically, maximization or minimization became an issue during World War II as British military leaders asked scientists and engineers to analyze several military problems which resulted in what is now call Operations Research. Professor Morse at MIT was one of the pioneers of OR. 

In the setting of linear programming an objective equation model is arranged with a traditional number of constrictions (as was the case of the cube inscribed in a sphere of radius R mentioned above). Then, the objective equation is used to solve the problem. Furthermore, the Objective Equation is a component of the Simplex Method which is the common way for the numerical solution of linear programming problems. The Method was created by George B. Dantzig in 1947and John von Neumann, who independently developed the theory of duality and was the founder of the modern digital computer. This roundabout will get to freight optimization after all the recipe ingredients are accounted for.

Another component of the optimization paradigm is the addition of Freeform Transformations (or topology optimization) the mathematical approach for determining the optimum material layout for a given space which takes into account a number of design constraints such as the ratio of volume to area schemes previously mentioned. Historically, topology began with the investigation of certain questions in geometry. “Leonhard Euler's 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first academic treatises in modern topology.” Wikipedia.

According to Hofstra University, “transportation systems are commonly represented using networks as an analogy for their structure and flows.” Euler treatise has now been expanded to include other fields such as networks that belong to a wider category of spatial networks which led to the development of graph theory and its application to transportation network studies. These new engineering design challenges are there for improving the cost and fuel efficiency of vehicles using light weight designs and cost effective materials.

This paper will now focus on the vehicles currently in use for different modes of freight transportation and some of the constriction related to the operation of these vehicles.  The ideal vehicle is of course the one that cost the least and carries the most cargo. The “Greedy Algorithm” was developed for such purpose. Richard E. Bellman at the RAND Corporation applied Dynamic Programming to solve the “Knapsack problem” using the “Greedy Algorithm.”

Biographically, the “Knapsack problem” has been studied for over one hundred years. It is not known how the name  originated, though the problem was referred to as such in the early works of mathematician Tobias Dantzig (1884–1956, father of George Dantzig) suggesting that the name could have existed as a fable before a mathematical problem had been fully defined.

In reprise to the account of others, Eric Grimson, Professor of Computer Science and Engineering at the Massachusetts Institute of Technology states:

“In order to solve a given problem using a dynamic programming approach, we need to solve different parts of the problem then combine the solutions of the parts to reach an overall solution.”

However, the intent of this paper is to give only a brief historical background of how we got here and not to go any deeper into the fields of dynamic optimization, networks, topology, geometry, engineering or physical laws. Nonetheless, there is a need to describe settings, to compare and contrast some of the vehicles in use today by carries of freight by air, ground or water surfaces.

The scope of this paper is not to question why carriers select to enter into a specific mode of transportation because it would be as speculative as trying to read someone’s mind or pretending to know the background, knowledge, capital available and vision of the founder(s).

Besides, most carriers now days are corporations or capital managers who look at the fundamentals of a given carrier and purchase the assets for profit. Individual examples of this market dynamic are Warren Buffet who purchased BNSF rail transportation or founder of FEDEX, Frederick W. Smith who is now listed in the stock market as a public company.

At this juncture, it would be useful to review the basics of performance, specifications, and financials of the different vehicles that comprise a given transportation methodology.

Perhaps the best approach for comparing and contrasting methods and vehicles would be to use the statistical data available for each mode of transportation starting with energy efficiency.

Consumption of energy in Proportional Miles per Gallon (PMPG)

Transport Equipment	Gross Tonnage	Average PMPG	Ratio T/PMPG
Freight Ship (Handy size)	24,000	340.00	70.58
Freight Train (60’ Boxcar) (Unit train 50 cars)	6,500	190.50	34.12
Plugin Hybrid	N/A	N/A	N/A
Airplane (Boeing 747-8)	140	42.60	3.29
18-Wheeler (Truck)	36	32.20	1.12

Source: (PMPG) data comes from the U. S.  Bureau of Transportation Statistics.

Ratio data is provided by the author. All figures are provided in Metric Tons.

Specifications, Costs and Volume to Area Efficiencies.

RAIL CARS:

60' Standard Boxcar. (60’ 9”L x 9’ 4" W x 10’10” H). (Metric 18.51 L x 2.84 W x 3.3 H=173.48m³)

Area: (60’ 9”x9’ 4”  572’ 4” ft²). (Metric 18.51 x 2.84= 52.57m²)

6,085 ft. cubic capacity.

6,085/572  10.63 cubic ft. per unit of area. (Metric 173.48/52.57 3.3m³ per unit of area)

Cargo Weight: 100 Metric Tons

Source: CSX.

Cost per km: $9.1 million. (2015)

Cost per Vehicle, Diesel-electric locomotives: $7.5 million. (2015)

Cost per Vehicle Standard 60’ Boxcar: $135,000. (2015)

Source: Market Watch.

FINANCIALS

Revenue/Cost Ratios: 1.47 (2015) From Financial Statements

Source: Securities and Exchange Commission (SEC)

TRACTOR TRAILER DIMENSIONS 18 WHEELER:

Area: (52’x 8’25”  429’). (Metric 15.84 L x 3.07 W)=48.63m²

4050 ft. cubic capacity= (52’ L x 8’25” W x 9’17”H). (Metric 15.84 L x 3.07 W x 3.17 H=151.23m³)

(4050/429  9.44 cubic ft. per unit of area). (Metric 151.23/48.63 3.11m³ per unit of area).

Cargo Weight: 36 Metric Tons

Source: Wikipedia                                                                    

Cost per Vehicle Cab: $80,000 (2015)

Cost per Vehicle Van: $30,000 (2015)

Source: Freight liner trucks

FINANCIALS

Revenue/Cost Ratios: 1.55 (2015) From Financial Statements

Source: Securities and Exchange Commission (SEC)

AIRCRAFT FREIGHTER:

Make-Model               Payload MT          CBM                                   Cargo hold meters

Boeing 747-8               140                  upper 692.7 hold              (L 2.40 x W 3.20)34 x H 3.00        

----------------                --------              Lower 165.7 hold             (L 3.20 x W 2.40)12

                                                            Total    858.4 CBM

(2.4x3.20)34 + (3.2x2.4)12= 353.28m²

(692.7+165.7)/353.28= 2.43 CBM³ per unit of area.

Cargo Weight: 140 Metric Tons

Cost per Aircraft: $357.5 million (2015)

Source: Boeing 747-8 Freighter

FINANCIALS

Revenue/Cost Ratios: 1.92 (2015) From Financial Statements

Source: Securities and Exchange Commission (SEC)

International Convention on Tonnage Measurement of Ships

International Maritime Organization: Adopted 23 June 1969; Entry into force: 18 July 1982

Regulation 3 is based on two variables:

The gross tonnage (GT) of a ship shall be determined by the following formula: GT = K1V

Where: V = Total volume of all enclosed spaces of the ship in cubic meters,

K1 = 0.2 + 0.02log10V It is calculated by using the formula:

where V = total volume in m³ and K = a figure from 0.22 up to 0.32, depending on the ship’s size so that, for a ship of 10,000 m³ total volume, the gross tonnage would be 0.28 × 10,000 = 2,800. GT is consequently a measure of the overall size of the ship. For a ship of 80,000 m³ total volume the gross tonnage would be 0.2980617 × 80,000 = 23,844.94 GT or 23,844.94/.2980617=80,000 m³.

80,000/150x26= 20.51 CBM³ per unit of area.

CLASS: BV Bureau Veritas in Metric Tons LBP: Length between perpendiculars150.00 L. BREADTH: 26.00 W

Source: Wikipedia

Cost per Handy Size ship of 24,000 dwt Metric Tons: $25 million (2015)

Source: Lloyd’s Register

FINANCIALS

Revenue/Cost Ratios: 1.65 (2015) From Financial Statements

Source: Securities and Exchange Commission (SEC)

In closing, transportation network efficiency is a measure that can be used to assess the performance of the network in that it captures flows, costs, and mobility behavior information, along with its topology. Each mode of transportation network has its own particulars for efficiency analysis but in general:

The characteristic path length of a network is defined as the average of the shortest path lengths between any two nodes:  where D I, j is the shortest path length between i and j defined as the minimum number of links traversed to get from node i to node j.                                                                                                                                                                                                                                                                                                                                    

Source: Analysis of Networks

I. E. Antoniou and E. T. Tsompa

Finally, the conclusions remarks of this paper are about the most efficient rates for each method of transportation in terms of volume/area and energy efficiency are given in the table below:

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.

Logistics and Algorithms for Networks

Social networks and transportation networks share many of the same properties, although, there are some unique differences. Social networks are about people in diverse geographic locations linked to each other who link to other people on so on and on. A transportation network shares some of the same interconnections; some are mobile and others are fixed on geographic locations: i.e., seaports, airports, depots, and cities. However, ships, trucks, trains and airplanes are mobile in and out of the fixed networks and sometimes with nodes in between because of geography. In contrast, airplanes are free to move in a Euclidean space because they are not Earthbound.

Scale varies with mode of transportation i.e., commercial ports on the waterfront of countries are distinct from other ports such as fishing ports and marinas which are not included in this network. Nonetheless, a transportation network has a couple more unique properties: Nodes and edges are established "between" countries and "within" countries for all modes of transportation which generates another network known in the industry as inter-modal or multi-modal where one or more modes of transport are combined to complete a shipping transaction.

In this scenario nodes and edges must have attributes for each mode of transport such as distance, geography, freight rates, loops, links, all in a topological space.  Once the network is put together it must fulfill the functions and run with the properties of a network.

To be useful this kind of network must be able to locate the most efficient route within the network. Dijkstra's algorithm can be used for finding the shortest paths between nodes in a transportation network graph.  This method was originally published by Dutch computer scientist Edsger W. Dijkstra in 1956. The algorithm exists in many variants: The original variant was conceived for finding the shortest path between two nodes; however, a more common variant selects a single node as the fixed node and finds shortest paths from this node source to all other nodes in the graph, generating a shortest path tree.

For a given fixed node in the graph, the algorithm finds the shortest path between that node and every other node but it can also be used for finding the shortest paths from a single node to a single destination node by stopping the algorithm as soon as the shortest path to the destination node has been located. As an example, if the nodes of the graph represent ports and the edges path are weighted with cost values then cost is represent by the distances between pairs of ports and a direct routing protocol. Dijkstra's algorithm can also be used to find the shortest route between one city and all other cities.

For theoretical computer science and network routing, Suurballe's algorithm is used for finding two unconnected paths in a non- negatively-weighted directed graph, so that both paths connect the same pair of vertices with a total length. The algorithm was conceived by Danish J. W. Suurballe and published in 1974. The principal idea of Suurballe was to use Dijkstra's algorithm to find one path, then modify the weights of the graph edges, and run Dijkstra's algorithm a second time. A similar method, which allows negative edges in the network, was developed by American scientist Donald b. Johnson who published his algorithm in 1977. The objective is to suggest the minimum cost directional algorithms, where in this case there are two bidirectional edges connected and nodes that have units of weight… or values, time, distance, etc.

Shipping statistics indicate that ninety per cent of world trade by weight is carried by ocean going vessels and high shipping costs can significantly impede trade participation for some countries.

Transport costs also vary significantly among commodities and finished products. These transport costs are strong determinants of which countries can enter the market with the same kind of merchandise and pricing if these countries are not located on the trade routes; don’t have safe harbors with sufficient water depth or lack port infrastructure.

Shipping Economics.  The cost of transportation tends to be lower as distance increases and although it sounds counter-intuitive what happens is that the cost per-ton-mile decreases as the denominator gets larger. I,e., 1 ton/1 mile-- 1 ton/1000 miles which becomes the effect of distance on freight rates.

Another economics principle that applies is the 80-20 Pareto rule, where 80% of the cost is derived from 20% of the factors such as capital, energy and time which, can be depicted in a curve of central tendency if statistical analysis is used.

Port Infrastructure. The efficiency and capacity of transport modes and terminals has a direct effect on the landed cost of a commodity. These efficiencies include storage areas either open or enclosed, access roads and rail spurs to the docks, mechanical equipment like cranes and loaders, turn around basins for ships and sufficient dock space proportional to ship traffic and invariably, availability of dredging when more water depth is needed.

Geography of Transport. Its impacts mainly involve distance and accessibility. Distance is commonly the most basic condition affecting transport costs in a time-cost equation. However, the geography of the terminal is heavily influential on logistics of a transportation network.

Tanya Latty , Kai Ramsc, et als published in 2011 a study of transportation networks built by ants in a non-Euclidian topological spaces. “Structure and formation of ant transportation networks.”

The work of these authors establishes that “many biological systems use extensive networks for the transport of resources and information.” In essence, one can observe from the study that biological transportation networks are not unique to humans with advanced technology for a primitive natural system can create an efficient transportation and logistics network instinctually.

Commodities and Products. Bulk, semi-manufactured and manufacture goods require packaging, special handling, are bulky or perishable and thus, must treated as independent units of cost for any kind of analysis. The general rule to apply here is that the higher the value of the item the lower the percentage of transportation cost. i.e., a $1000 TV set will probably have cost of .01 of the unit value or $10 while $1000 worth of coffee beans will probably have a cost of .07 or $70. A TV set can fit in a box while coffee beans will require a much larger space which translates as the stowage factor of the merchandise.

A big issue for ocean container carriers is the asymmetry of trade because the trade imbalance implies the repositioning of empty containers that have to be taken into account in the total transport costs. Tramp carriers on the other hand, are on a better position to provide better rates since they don’t have to worry about positioning containers. However, most of these carriers operate without established routes or schedules and just go where the cargo is which makes it difficult for shippers to plan ahead on the inherently uncertainty of tramp carriers and bidding on spot movements and rates.