Optimizing Freight Rate Functions

Using discrete mathematics to describe the behavior of transportation networks has to include Riemannian geometry in consideration of geographical curved space and gradients. This issue alone can represent an immense challenge for developers who may be very competitive in computer science but not in the business side of transportation networks. Specially, when defining which method and what set of metrics are to be used for taking measurements across the entire spectrum of the system being developed.

Freight managers recognize that they need to allocate in separate sets direct and indirect costs for the vehicles, the routes, as well as the size, and capacity, for determining the unique transportation platform’s weight and cubic space available for the market at a price that defines the break-even point for a specific case in question.

The short cut for calculating a freight rate is of course to add all the fixed and variable costs and divide them by the vehicle’s capacity on either weight or space in order to establish a freight rate per unit of cost. Intuitively, this is a reasonable argument but other hidden costs can upset the balance in unexpected ways.

A more labored approach is to use a tool such as a modified Cobb-Douglas  production function used in economics to determine the marginal product of capital (K) and the marginal product of labor (L) and replace them with applicable variables for freight transportation. The method prosed here is to use a theoretical freight rate divided with fractional exponents that can result in a model that can measure  returns of scale with a given set of variables and within a specific route or market.

For example, if a freight vehicle makes a round trip but only carries cargo one way then the utilization factor is ½ its capacity. In this case, the backhaul is negative but the round trip cost is positive. So, if no backhaul cargo is available a round trip charge would need to be applied, however, this may not be possible because route competition or some other market condition don’t allow it. Freight component can be represented with a fractional exponent representing the utilization factor as it is applied to the theoretical model.

 The next problem to be assign a solution is stowage factor and freight managers will try to maximize the amount of cargo that can be carried for either weight or volume.  Under these circumstances, vehicle capacity becomes an issue of dimensional weight factoring. That is the factor at which the unit platform arrives at its weight or volume capacity for a given freight rate. So, another fractional exponent can be inserted into the cost function.

Determining maximum carrying value on a limited transportation platform by either weight or volume is an issue that can, in many cases, determine success or failure of a transportation company. The theory behind the problem has been endlessly studied with “the knapsack problem” which, relates to combinatorial optimization between weight, space and value.  Given a set of cargoes, each with a weight, a volume and value, freight managers must determine the number of each unit that can be included in a collection of items equal to a given limit where the total commodity-freight-value is as large as possible. Thus, the problem becomes an issue of optimizing a vehicle utilization factor for any given trip. As a result, another fractional exponent representing commodity value can be added to the function.

Distance from origin to destination can be a constant for a route but not for off route trips and must be considered accordingly as yet another exponential fraction that can be added to the overall equation. Once all the exponents have been determined the function should provide a theoretical optimized approximation to an original set previously defined by capital, labor and energy at a break-even of the function.

It follows that if a freight manager who targets optimization based using the modified Cobb-Douglas can assign random values to the function to perform an initial evaluation between the break-even and marginal cost incremental  for Utilization, Stowage, Commodity Value and  Distance. This ca be abbreviated as: Freight where (F)=   a fractional exponent like 1/n taken to the nth root and expressed on its decimal exponent equivalent of F= (  U^.25 S^.25 C^.25 D^.25   ).

If we make the assumption that this equation will hold for 90% of the rate optimization cases and the remaining 10% is treated as extraneous freight components that only apply to very few cases. For instance, where berthing facilities are poor, port infrastructure is inadequate and/or terminal expenses are not competitive with the rest of the world because of inefficiencies or corruption.

Under these circumstances, freight managers can assess the marginal effect of cost on freight rates when one or more of the factors of the proposed freight equation domain is increased or decreased from an initially evenly divided cost-freight-factors set at  1/4+1/4+1/4+1/4.

The following exercise using a generic freight rate of $12 for weight or measure per metric ton constrained by market rates in a given route is found to have a $4 utilization rate cost component. If the marginal cost needs to be measured in order to define if any rate flexibility exist on the intended route using the proposed equation, then, a freight manager must proceed as follows: 12/4=3 as the freight initial constant of cost in its decimal equivalent of F= (U^.25 S^.25 C^.25 D^.25).

= .25 ( ) ·²⁵ =    .27 so that the original cost component of $4 has a marginal increase/decrease of 4/1.27 3.15. In other words, the rate component could be dropped $0.85 cents per ton with little effect on the profit of the rate because the marginal utilization cost factor can be set at a smaller cost.

This equation can be applied not just to obtain marginal return on each freight cost component but also, to measure profit margin and overall productivity by replacing the corresponding exponent(s) with different testing values for the model.

 Comparing values obtained between the model and those calculated based on break-even point, rate per ton mile, rate per trip, rate per day and so on can vary with each method of calculation but once the factors are obtained and entered in a computer or interactive worksheet it can become a very useful tool for improving productivity.

Arguably, optimization can also be accomplish using LaGrange multipliers, although, it requires a lot more algebraic manipulation which makes its handling weighty,  moreover, with the proposed method individual variables can be isolate and evaluated and this feature cannot be replicated using the LaGrange multipliers.

Alfonso Llanes

May 20, 2016

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