Special Feature
I/EC
Whether you are seeking a plant site or the proper point for product distribution in a consumer area, it may serve xyou well to consider these . . .
FREIGHT RATE EQUATIONS Freight equations can be made as precise and consistent as the data upon which they are based by H. Leroy Thompson, Consulting Chemical Engineer 329 Brown-Marx Building, Birmingham 3, Ala. "HEN
FREIGHT DATA are
plotted
to show rate per ton or per hundredweight (cwt.) along the vertical scale vs. distance on the horizontal, the resulting line in most cases is a curve shaped like an inverted saucer. Three curves of this type are shown in Figure 1 where rail freight rates for certain chemicals are shown as functions of distance in certain territories of the United States (lines CS, CO, and AL). Curves of this shape can be fitted by equations of the general form : F = uMx + c (1) where F = freight cost per unit hauled M = distance hauled in miles u, x, and c = arbitrary constants to be derived T h e exponent of M usually is less than one, but in a few instances it has been found equal to one. In these cases, the graph is a straight line and u becomes in effect a constant cost per cwt.-mile (or tonmile), while c may be regarded as a fixed cost that is applied to shipments of any distance. Line BF of Figure 1 is an example of this type. Deriving Equations
For the special case where the graph is a straight-line, calculation of the constants is simple. 44 A
I n the general case, a plot of rate vs. distance is curved, and more advanced techniques are necessary to derive the constants. The methods described herein are adapted from Davis (Davis, Dale S., "Empirical Equations and Nomography," 1st éd., McGraw-Hill Book Co., Inc., New York City, 1943). T h e first step is to plot logarithms of rate vs. the corresponding logarithms of distance on ordinary graph paper or to plot rate vs. distance directly on log-log graph paper. If the resulting line is straight, c then equals zero, and the equation becomes : F = uMx,
or in logarithmic form:
log F = χ log M + log u
(2) (3)
Substitution of corresponding values for log F and log M from the curve or from selected points en ables calculation of the constants χ and u. A few equations have been en countered in practice where c equals zero but the majority of the loglog plots show some curvature. This requires application of a con stant correction to F, such that its logarithm is influenced significantly at small values and negligibly at
INDUSTRIAL AND ENGINEERING CHEMISTRY
large ones. If in the general form of the freight equation (Equation 1), c is transposed to the left, a n d log arithms taken of both sides, it becomes : log (F — c) = χ log M + log u
(4)
T h e first step is to find c, which is a function of three properly selected values of F, designated as F\, Ft, and F3. /"Ί and F2 are chosen near the beginning and end of the curve, respectively, a n d the corresponding values of Mi and M2 are noted. M% is defined as the square root of the product of the other two mile ages and F3 is the corresponding rate read from the curve. After selecting Fi and F2 and calculating to find F3: FxF2 - F3* Fl + F2 - 2F3
(5)
T h e other constants then are determined by appropriate sub stitution. Other mathematical forms were tried for these purposes, but they were found to be unwieldy and dif ficult to manipulate for the ap plications described later. T h e three variations in form are summarized as follows:
Shape of Rate-vs.-Distance
Graph
Equation
Form
F = uM + c
Straight-line, linear graph paper
Derivation Determine e and υ
x
Curved, linear paper; straight on log-log
F = uM
Determine w and χ
Curved, linear paper and log-log
F = uMx + c
Determine υ, χ, and e
Scattered Data
Formal mathematical procedures are available for determining the best fit in these cases, but freight equations mostly are not precise enough to require such treatment, a n d it is satisfactory to average F and M separately for a number of neighboring points, thus obtaining a composite point to represent them for plotting purposes. When the n u m b e r of points to be plotted is reduced to five or six in this manner, the resulting curve generally is a smooth one that fits the original data well enough. These averaged
points also may be used in deter mining values of the constants in the equation. In working with lines that are curved on a linear plot, it is nec essary to average on the basis of log F and log M so that the aver ages will reflect the effect of the exponent x. Air Miles and Route Miles
Freight tariffs relate primarily to highway or rail distances between points—i.e., route mileage. Prob lems in site selection and operations research may be studied more con veniently with freight rate equations
Figure 1 shows four arithmetic plots of freight rate against distance. Line BF, which repre sents some rail rates obtained a few years ago on bagged fertilizers, is straight when plotted to linear co-ordinates so that it corresponds to the first type above. Line AL, showing some rail rates on alkalies, is curved to linear co-ordinates but straight on log-log paper so that an equa tion of the second type will fit it. Lines CO and CS typify the general case (Equation 1) where the plots are curved on both types of graph paper, and all three constants must be derived. These two lines are based on some 1953 class rail rates covering several types of chemicals, CS for the Southern territory and CO for the Official (or Eastern) territory. Line BF. From the linear plot, Fi = 8 cents where Mi = 50 miles and F2 = 37 cents when M*= 300, whence: F = 0.116M + 2.2
(6)
Line AL, From the log-log plot, Fx = 19 where Mi is 100, and F2 = 63 when M2 is 900. χ = log F2/Fil-i- log MilΜλ = 0.520/0.954 = 0.545
(7)
Fx Mi»
Where the points are widely scattered after plotting, no single expression can be derived which will yield each value correctly. However, if sufficient care is ex ercised in calculating the constants, the average deviation of the equation can be m a d e quite small. Since the equations are useful mainly as
19 (lOO)»·8*6
19 = 1.54 12.35
F = 1.54Aλ·**
(9)
FxF2-FJ Ft + Ft - 2F3
~ ~ - ν», 53 ΛX. 150 (83)' 53 + 150 - 166
(10)
(11)
In a similar manner, the equation for Line cs is found to be: F = 3.57Λ/0·88 + 16.4
(12)
After determination of these constants, it frequently is possible to readjust them for more convenience in computation. This is done largely by trial and error. For example, line cs is fit about as well by the expression:
| CSj
CO,.
A l ^ ^
„2S7 m
F = 0.85M0·73 + 28.7
Arithmetic Plots of Freight Rate Vs. Distance ca^^r^
m
Insertion of (Flf Mi) and (F2, M2) into Equation 4 then leads to the desired expression:
F = 3Λ/ο.β -L. 23
CS-i^"^
(8)
Line CO. From the log-log graph, Fa = 53 when Mi = 100, and F2 = 150 when M2 = 900. M3 is the square root of Μα Χ M2 or of 90,000, which equals 300. At this value of M, F3 = 83, from the graph. By Equation 5 above:
^ ^
AL
Br/'
Accuracy and Precision
The equation for line AL is then
Log-log Plots ol Freight Rate versus Distance
"J^
in terms of straight-line distances between the same points, and such equations can be developed. Air distances corresponding to each given route distance are scaled from a map, tabulated, and used as a basis for plotting the rate-distance curve and deriving an equation. This equa tion is likely to differ from one based on route miles. Scattering of the points on the curve is then more pronounced, especially for short distances, and correlations between actual values and those calculated by the equation are less exact.
BF M. Distance in Route Miles from Origin
(12a)
as by Equation 12 above. The foregoing curve-fitting methods and illustrations were based on smooth data, where the deviations of individual points from the curves are in significant. In practice, one finds considerable scattering of points so there may be un certainty as to the best path for the curve.
VOL. 52, NO. 11 ·
NOVEMBER 1960
45 A
A practical simplification of freight rate data which has been found useful in a number of such problems is described in this article. It entails deriving empirical equations which relate freight rates to distance hauled for a given commodity or class and a given type of carrier. From these equations, one can fre quently develop generalizations which satis factorily answer certain freight questions. Reduction of data concerning freight rates to empirical equations which relate rate to dis
tools with which masses of data can be treated systematically rather than as means of estimating individual rates, a small average deviation is satisfactory. I n cases where the data fit closely to a smooth curve, the equation will reproduce individual values within an error of ± 1 % , even when a slide rule is used for the calcula tions. Since high accuracy is not essential, all of the multiplying, dividing, and logarithmic work can be done quickly by slide rule. Proportional Adjustments
T h e form of the equations is such that proportional adjustments can be made by changing only the two constants, u and c. For example, to convert F in Equation 11 to dollars per ton instead of cents per hundred weight, both u and c are multiplied by 0.2 (20 cwt. per ton divided by 100 cents per dollar), and the new equation is: F(»/ton) = 0.17M«·73 + 5.74
(13)
Similarly, to adjust an equation to a new freight rate that is, say, 1 5 % above the old, u and c are multiplied by 1.15. Class freight rates, especially, are customarily expressed as percent ages of the 1 0 0 % class rate, and in these cases an equation derived for the base rate can be converted to any of the dependent rates by applying the appropriate percent age correction to c a n d / o r u.
i
APPLICATIONS As mentioned earlier, freight equa tions have been applied to a num ber of practical problems where they were found to be helpful and saving of time and money. A few instances of such uses are outlined in this section. These are not ac tual cases, but simplified problems to illustrate methods of approach. Comparison of Rate Structures
Equation 11 for curve CO and Equation 12 for curve CS permit a comparison of rate structures for the same commodities in dif ferent territories. T h e Southern total rates are higher for all dis tances, but the "fixed charge," constant c, is considerably higher in Official territory. Also the ex ponent χ is greater in Official ter ritory, meaning that the rate F increases more nearly in proportion to mileage M than is true in South ern territory. Consequently, the difference in these rates is accounted for almost entirely by the much higher unit charge, u, in the South ern territory. / / might be expected that these differences in rate structure will affect the policies of shippers and con signees as to average length of haul and as to freight equalization. Weighted Average Freight Rates
For those cases where products are shipped out from a central location more or less uniformly into
Transportation cost is an important variable in many management and engineering prob lems such as selection of sites for processing and distribution facilities, comparison of com petitive products, processes, and carriers, determination of equitable delivered prices 1ère freight is to be absorbed, and setting of 40 A
tance can serve a number of useful functions for market research, location analysis, planning competitive strategy, operations research, and the like. The techniques for deriving and using such equations are straightforward and of only moderate mathematical complexity. The time and money saved and the relative accuracy gained by treating transportation cost data systematically in this way can more than repay s tthe trouble involved in devising and checking the formulas.
INDUSTRIAL AND ENGINEERING CHEMISTRY
the surrounding territory (or raw materials are shipped in under the same conditions), it is convenient to have an approximation of the weighted average freight rate for the whole territory. Such products could include fertilizer, cement, fuels, and others. T h e average freight per ton obviously will not be an arithmetical mean because more product will be shipped to the edges of the territory if the distribution is uniform or nearly so. Freight equations readily permit calcula tion of weighted average freight rates, which may be used, for ex ample, in setting a fair delivered price where this policy of pricing is to be followed. When the distribution is assumed to be uniform and the territory is assumed to be circular with the common origin (or destination, for inbound raw materials) at the center, the formula for the weighted aver age rate is found to be : 2
(14) 2 + χ uM* + c where Fwa = weighted average rate Mmax = maximum distance, or radius of territory u, x, c as defined before Fwa =
T h e derivation of this expression is rather long and is not reproduced herein for that reason. Equation 14 demonstrates that the exponent χ is the important variable in defining Fwa. Its value for any given rate structure sets a constant coefficient which applies to the variable term of the maximum rate for the territory. For example,
policy on freight equalization. Freight data typically consist of extensive tables showing individual rates broken down by commodities or classes, carriers, zones, and territories. Finding optimum solutions for problems involving such masses of data can be laborious and expensive.
SPECIAL taking the equation for curve CO, where χ = 0.73, Fwa = ~
(0.85)Mmax°·" + 28.7 = 0.73(0.85)Afmax°·" + 27.7 (15)
Thus the new coefficient fixes the variable term in the average rate at 7 3 % of that in the max imum. Assuming a territory of radius equals 300 miles, the max imum rate is F = 83, while the weighted average is Fwa = 0.62(300)°·" + 28.7 = 68
(16)
Thus the composite rate is nearly 80% of the maximum, instead of the 50% that would be obtained by arithmetical averaging. Weighted averages also can be derived for product distributions that vary with distance from the center instead of remaining con stant. If the relationship between distribution and distance is known or can be assumed reliably, the alge braic expression defining distribu tion as a function of distance is in serted into the derivation of Fma.
Integration and averaging by total tonnage then leads to an expression comparable to Equation 15. Equations based on air distances instead of route mileage usually are better for use in determining aver ages. Use of these weighted aver ages is not confined to whole-cir cle territories. They apply to any fraction of a circle for which the radius is known. Freight Equalization
Equalization of freight is a trade practice applicable to a variety of commodities for which the con signee pays freight charges in addi tion to an f.o.b. price for the product at its origin. When freight is equalized, any given consignee's freight cost is limited to the rate from its nearest source of supply (basing point). Shippers who are further away from the consignee than its basing point and who still wish to compete as a source of sup ply do so by absorbing the dif ference between this minimum freight rate and the actual rate from
FEATURE
the more remote origin. A freight equalization pattern is relatively simple to construct for a shipping point that is already established. Rates from each shipping point to each des tination are available from pub lished tariffs, and any shipper can chart its freight equalization posi tion with respect to competitors around it by comparing these known rates for each prospective consignee. Any given shipper is surrounded by a zone of zero freight equalization which is bounded roughly by per pendiculars drawn through the mid points of lines connecting the given shipper with each competitive bas ing point. This zone takes the form of an irregular polygon with the number of sides equal to the num ber of competitive basing points. T h e central shipper is subject to freight equalization for shipments to destinations outside this zone, the amount depending upon the rela tive distances covered by its ship ments and shipments to the same destinations from their nearest re spective basing points. With known
••••ϋΗΗΗ
Freight Equalization Boundary Curves
East
1200 Miles
VOL. 52, NO. 11
·
NOVEMBER 1960
47 A
tariffs, it is possible to sketch in zones of constant freight equalization around the central point or to de termine the cost of equalizing freight throughout a specified territory. I n the absence of established rates, as is the case when new production or distribution facilities are being planned or now being considered, definitive answers to freight equal ization problems are more dif ficult to obtain. Decisions as to facility site selection m a y hinge heavily on freight costs, including equalization, yet reliable freight data are least available during the planning stages. Typically, car riers serving the various sites under consideration are asked to estimate the rates which would be established from the proposed sites to selected destinations and these estimates are used for site comparisons. Dif ferent carriers may use different procedures for these estimated rates, or the estimates may be based upon routings which arbitrarily favor one carrier or method of transportation. I n these situations, freight rate equations can be used to simplify the analyses, at least to the extent of eliminating all but one or two site candidates which m a y be then studied more intensively. T h e procedure begins with a rate vs. distance plot for the same product or products from basing points already serving in the gen eral area under consideration, or in the same freight territory, using air distances. A carefully drawn curve which properly weighs all these points is likely to be as ac curate (on the average) as any tentative rates which could be ob tained otherwise, and has the further advantage of being applicable to any sites under consideration in the territory. An equation then may be derived for this curve. Freight equalization is assigned the symbol Q and is measured in the same units as F in Equation 1. T h e value of Q for any shipment outside the zero zone is found by subtraction : Q = Fi - F2 « u(My* - M»') (17)
the subscript 1 identifying the central point under consideration and other subscripts the appropriate com petitive basing points. I n the rare instances where the exponent of M is unity, Equation 17 defines a hyperbola for any constant value of 48 A
Q. T h e ratio Q/u is measured in miles and this quantity represents the difference in focal radii, one focus being the central point, 1, and the other the basing point, 2. (This general hyperbolic form for lines of constant equalization may be observed on charts plotted from published tariffs.) T h e shape of the resulting hyperbola depends upon the ratio Q/u and the focal distance between points 1 and 2. Consider line BF (for bagged fertilizer) of Figure 1, where u has been evaluated at 0.116 cent per cwt.-mile (Equation 6). Sup pose that a new plant location is being studied 150 miles west of an existing basing point and that the maximum freight equalization which can be allowed from the new plant is $2.50 per net ton (12.5 cents per cwt.). T h e Q/u is 12.5/0.116 or about 108 miles, and point 1 (the proposed facility) can ship over distances u p to 108 miles further than the other basing point, within the zero zone surrounding the latter. This means that point 1 can ship to within 42 miles of the competitor along the line connecting the two shipping points, and that the in tersections of circular arcs from points 1 and 2 such that Mi = M2 + 108 will define the rest of the curve for the $2.50 per ton equal ization zone. By triangulation, the intersections due north and south of point 2 are found to be about 50 miles from it. These three inter sections (north, south, and west of point 2) enable visualization of the curve. T o serve any destination with in the zero zone for point 2, the new facility must allow equaliza tion u p to the maximum, where Mi — M2 — D, the intervening dis tance of 150 miles; in this case Q/u must equal 150 and Q must be about $3.50. T h e fractional exponent for M in the general freight equation pre vents use of Q/u as a simple dis tance handicap corresponding to a given freight equalization allowance. For freight equalization studies in these cases it is convenient to reduce all quantities to ratios rather than absolutes. This reduction also fa cilitates comparison of multiple com petitive basing points with selected possibilities for new sites and makes slide rule computations easier. O n e first defines a maximum (theoreti cal) equalization, where point 1
INDUSTRIAL AND ENGINEERING CHEMISTRY
ships into the switching zone of point 2 so that Mi = D and M2 — 0. T h e n Qmax = UD*
(18)
and any other value of Q is expressed as a ratio of this maximum : q = Q/Qmax, or Q = quD*
(19)
Similarly, Mi and M2 are expressed as fractions of D: Mi = miD and M 2 = m2D
(20)
Substituting in Equation 17: Q = quDx = u(Mix - M2X) = uDx{mi* - mf)
(21)
which simplifies into : q = τηχ* — m2x
(22)
T h e pertinent range for q is from 0 to 1, and the minimum value for mi is 0.5 since smaller values rep resent points within the zero equal ization zone for point 1. For a n y assigned value of q, enough {τηχ, m2) pairs to define the required boundary curve can be found rather simply. Thus, for the one destina tion which corresponds to a given q a n d lies along the connecting line between points 1 and 2, mi + m2 = 1, and the equation q = m-ΐ — (1 — mxi) can be solved for mi, which then yields m2. Similarly, at those destinations which a r e perpendicularly north and south from point 2 (referred to the eastwest orientation previously assumed for the competing shippers), mi2 = m22 + 1 and q — m-i — (mi2 — \)x/2. Another convenient ex pedient is to assume varying ratios of m-i/mi, calling this ratio r and beginning with the minimum found when mi + m2 = 1. T h e n q = mix(\ — rx), which is solvable for mi and hence m2. With these (mi, m2) pairs once found for given q's, they can be a p plied to any values of D for the var ious competitive basing points, even tually yielding a pattern of inter secting pseudohyperbolas which de fine a specified freight equalization zone completely surrounding the location under study. (Above cer tain limiting values of q these curves are closed, so that mi = m2 + 1 along the extension of the connecting line between the two points. True hyperbolas of course do not close.) Similar boundary curves may be derived for cases where the com peting shippers are subject to dif ferent equations, but the calcula tions are more complicated.
