Discussion Paper No.297
Analysis of derivation of the transportation andproduction cost minimum place
Toshiharu Ishikawa
Faculty of Economics
Chuo University
January 2018

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, Japan

Analysis of derivation of the transportation and production cost minimum place

Toshiharu Ishikawa

1.Introduction
The spatial range of an economic activity has spread from the late 20th century, and the influence ofthe spread has expanded all over the world and has entered people life deeply. This economic
expansion is caused by not only a drastic decrease in transportation costs and but also by the decline
in office processing costs that is generated by various kinds of innovations. Indeed, technologicalinnovation in transportation reduced shipping costs, decreased its economic influence, and increased
the effects of other economic factors relatively. It should be noted, however, that the decline intransportation costs does not mean that the essential function of transportation costs is eliminated ineconomic world. While the manner of impact of transportation costs on economic activity is different
from before globalization economy. The expenses for the movement of goods and services play adecisive role in not only location of economic activities but also behavior style of economic agentseven as economic activity becomes globalized. Therefore, when individual companies strategicallyplan the location of a factory, naturally they first investigate the level of transportation cost, labour
cost and infrastructure in the location candidates.
When an individual firm decides the location of a factory, the firm selects a place that maximizes
its profit. If there is little relation between revenue and factory location, the place where production
cost is minimized is selected by the firm. Furthermore, when the production cost of goods has little
influence on the factory location, the location of the factory is determined at the place wheretransportation costs are minimized. First, this chapter takes up the transportation cost which has been
played an important role in determining the location of the factory. The two methods of derivation of
the place of minimum transportation cost of a factory are explained and the place is derived by themethods. Secondly, this paper offers the concept of the area of minimum production costs; the analysis
is enlarged by incorporating the factory’s production function to search the area of minimum
production costs including transportation cost, and the analysis explains the meaning the area has indetermination of factory’s location. Eventually, there are two purposes in the paper: One is to find out
the place of the minimum transportation cost of a factory. Another is to determine the area of minimum
production cost of a factoryThis paper is organized as follow. Numerous methods for deriving the place that minimizes factory’stransportation cost have been made so far. The scholars who systematically considered this method
for the first time are Launhardt (1882) and Weber (1909). they devised a geometric method to obtain
the place of minimum transportation cost for a factory. Thus, the section 2 explains their traditional
geometric method. The section 3 clarifies this geometric method by analytical approach: The validity1

of the geometric method developed by Launhardt and Weber is clearly verified by the analytical
approach. The section 4 explains the concept of the area of minimum production costs includingtransportation cost. The section 5 summarizes the results derived in the paper.

2. Derivation of minimum transportation cost place by geometric method
According to the framework of Launhardt (1882) and Weber (1909), this section explains the methodof deriving the place that minimizes factory’s transportation cost by using the Weight triangle.

2.1. Assumptions
The method of deriving the minimum transportation cost by Weight triangle is explained based on thefollowing assumptions.
1) A factory uses the two kinds of raw materials M 1 and M 2 to manufacture the final product M 3 .
2) The raw materials M 1 and M 2 are produced at the points of A1 and A2, respectively, and aretransported to the factory. The final product a 3 is transported from the factory to the market place
A3.
3) The weights of raw materials M1 and M2 are fixed and are given by m 1 and m 2 , respectively. The
weight of the product M 3 is fixed as m3.
4) The freight rates of the raw materials a 1 and a 2 and the product a3 are constant and they are all set
to 1. Therefore, the transportation cost borne by the factory is proportional to the weight of these
raw materials and products and their transport distances.
5) The factory should be located at the place that minimizes the transportation cost borne by thefactory.

2.2. Explanation of method of weight triangle
Now, suppose that the sites of the raw material sites M1 and M2 and the market place M3 are specified
by Figure 1. A figure which is formed by connecting the three sites of the market place and the raw
materials is named as location figure. The figure shown in Figure 1 is also called a location triangle.
In Figure 1 r 1 , r 2 , and r 3 note the apex angles at A1, A2 and A3. In such location triangle, the derivationof point minimizing factory shipping costs is one of the important tasks in traditional location theory.
The traditional theory uses the method of the weight triangle to find a point where factory’stransportation cost is minimized. This method is explained as follows. As explained in above
assumptions, the two raw materials M 1 and M 2 used by the factory are produced at points A1 andA2,
respectively, and their weights are m 1 and m 2 . And then, the product M 3 is sold at the point A3 and its
weight is m 3 .

According to the weights m 1 , m 2 , m 3 , let us give the force attracting the factory, thetraction force, to the points A1, A2, A3 shown in Figure1.

2yA3 (X0, Y0)

r3

r1

r2
x

A1(0,0)A2 (c,0)Figure 1. Location figure
The site that the factory’s transportation cost is minimized is determined as point P where thesethree traction forces are balanced. The situation of point P is explained by using Figure 2: Threevectors, V 1 , V 2 , and V 3 are assumed so as to be in proportion to weight m 1 , m 2 , m 3 , which imply thetraction forces of three points. Then, each vector is set from the point P toward each of points A1, A2,
A3 as shown in Figure 2.

y
A3

V3

a1

a2
P
a3

V2

V1
x
A1

A2

Figure 2 Balance of traction powers

3

The three vectors V 1 , V 2 , and V 3 are balanced at point P that the factory’s transportation cost is
minimized. The fact that these vectors are balanced at the point P means the following state: A
parallelogram having two sides of vectors V1 and V2 shown in Figure 2 is created with the point P asa starting point. The length from point P to its diagonal is given by the sum of the above two vectors,
V1 and V2. This new vector V 4 is the same in length as that of vector V3 and its direction is opposite.
That is, the sum of this new vector V 4 and vector V3 is zero. In Figure 2, the angle formed by thevectors V1 and V2 around the point P is noted as a3, the angle formed by the Vectors V2 and V3 is noted
as a1, and the angle made by the Vectors V1 and V3 is noted as a2, respectively.
Now, from the arrangement of the three vectors under the location triangle as shown in Figure 2, itis possible to form a so-called weight triangle by the three vectors. Figure 3 shows this weight triangle.

𝐚𝐚 *1
P

V2

a*3

V3

V1

a*2
Figure 3 Weight triangle
The length of the three Vectors of this weight triangle follows the ratio of weight m1, m2, m3. From
the weight triangle the diagonal of each side V1, V2, V3 is derived as a*1, a*2, a*3. Each angle issupplementary angle for each of angles, a1, a2, a3 formed around point P in Figure 2. Hence, if theangles at vertexes of the weight triangle formed by the three vectors V1, V2, V3 are known as shown in
Figure3, the angles a1, a2, a3 around point P in Figure 2 can be obtained. Eventually it is concluded thatpoint P minimizing the transport cost of the factory is a place at which the straight lines connecting
each of points A1, A2, A3 and point P forms three angles a1, a2, a3, which are derived from the
supplementary angles a*1, a*2, a*3 in the weight triangle.

2.3. Derivation of minimum transportation cost place by location circles
Let us geometrically derive the place of minimum transportation cost based on the location triangle
and the weight triangle revealed in the above subsection. When drawing a circle passing through points
A1 and A2 in Figure 2 and having a central angle twice the angle a3 which is the circumferential angle
4formed on the circle, this circle passes through point P. This circle is appropriate to be named as
location circle. In a similar way, when drawing a circle passing through points A1 and A3 and having
a central angle twice as large as the angle a2 which is the circumferential angle, this circle passes
through point P. This circle also is the location circle. Therefore, point where such two location circles
intersect indicates the place that minimizes the factory’s transportation cost.
Now, suppose that the coordinates of points A 1 and A 2 are specifically (0, 0), (5, 0), and let thecoordinates (X 0 , Y 0 ) of point A3 be (2.5, 5): And the weights m 1 and m 2 of the raw materials M 1 andM 2 handled by points A1 and A2 are assumed to be 1.5 tons and 1 ton, respectively: The weight m 3 ofthe product M 3 is 1 ton. In this situation, by using the location triangle and the weight triangle, twolocation circles, C 1 and C 2, are drawn in Figure 4. The two location circles intersect at point P whichis shown by the mark,. Point P indicates the place that factory’s transportation cost is minimized.
A3

C2

P

C１

A2

A1

Figure 4 Two location circles and the place of minimum transportation cost

3. Analytical explanation of the weight triangle
3.1. Derivation of minimum transportation cost place by analytical geometrics
This section analyzes the method which depicts two location circles to find out the place of minimumtransportation cost. It is assumed based on the assumptions in the previous section that the location
figure is given by the triangle like Figure 4, and the weight assigned to each point A1, A2, and A3 isgiven as m 1 , m 2 , m 3 . Factory’s transportation cost T is shown by equation (1). Thus, the task in thissection is the analytical geometric derivation of the point P that minimizes this transportation cost T.
T= m 1 (x2+y2)0.5+ m 2 ((x - c)2+y2)0.5+ m 3 ((x -X 0 )2+(y-Y 0 )2)0.5 .

(1)Since each weight m 1 , m 2 , m 3 is given, transportation cost T is a function of the factory’s coordination,5

x and y. The place P that minimizes transportation cost must the following two necessary conditions,∂T/∂x = 0,(2)∂T/∂y = 0.

(3)

And point P must satisfy the sufficient condition, Z, as indicated by the following equation (4),∂2T/∂x2∂2T/∂x ∂yZ=∂2T/∂x ∂y∂2T/∂y2

〉

0.

(4)

These equations (2), (3), and (4) are developed and shown as following equations (2a), (3a), and (4a),∂T/∂x = m 1 x / (x2+y2)0.5 + m 2 (x - c) / ((x - c)2+y2)0.5 + m 3 ( x -X 0 ) / ((x -X 0 )2+(y-Y 0 )2)0.5 =0 (2a)∂T/∂y = m 1 y / (x2+y2)0.5 + m 2 y / ((x - c)2+y2)0.5 + m 3 ( x -Y 0 ) / ((x -X 0 )2+(y-Y 0 )2)0.5=0 (3a)Z = m 1 2 m 2 2(xy - (x - c)y )2 / ((x2+y2)1.5((x - c)2+y2)1.5)+ m 1 2 m 3 2(x(y-Y 0 ) - (x – X 0 )y )2 / ((x2+y2)1.5((x – X 0 )2+(y-Y 0 )2)1.5)+ m 2 2 m 3 2((x-c)(y-Y 0 ) - (x – X 0 )y )2 / (((x-c)2+y2)1.5((x – X 0 )2+(y-Y 0 )2)1.5) 〉 0(4a)

Since equation (4a) is positive, solving the simultaneous equations of equations (2a) and (3a) for x andy gives the coordinates of point that minimizes transportation cost of the factory 1.

This section

obtains point that minimizes transportation cost by developing equation (2a) and (3a) as follows. First,
by transferring the third term of equations (2a) and (3a) to the right side, the following equations (5)and (6) are derived,.
m 1 x / (x2+y2)0.5 + m 2 (x - c) / ((x - c)2+y2)0.5 = - m 3 ( x -X 0 ) / ((x -X 0 )2+(y-Y 0 )2)0.5(5)m 1 y / (x2+y2)0.5 + m 2 y / ((x - c)2+y2)0.5 = - m 3 ( x -Y 0 ) / ((x -X 0 )2+(y-Y 0 )2)0.5.

(6)

By squaring both sides of equations (5) and (6), and adding these two equations, and further
1

Point of minimum transportation cost can be directly derived by numerical calculation method.
For example, the method is introduced by Kuhn and Kunne (1962).
6

transforming, equation (7) is derived,
2m 1 m 2 (x(x - c)+y2) / ((x2+y2)0.5((x - c)2+y2)0.5) = m 3 2 - m 1 2 - m 2 2 .

(7)

By squaring equation (7), and then transforming it, equation (8) is obtained,
4 (m 1 m 2) 2(x(x - c)2+y2)2 = (x2+y2)( (x - c)2+y2) (m 3 2 - m 1 2 - m 2 2 )2 .

(8)

Equation (8) can be transformed to equation (8a),4 (m 1 m 2) 2 / ((m 3 2 - m 1 2 - m 2 2 )2) (x(x - c)2+y2)2 = (x2+y2)( (x - c)2+y2) .

(8a)Considering the following relationship shown by equation (9), equation (8a) can be transformed as
equation (10),((x2+y2)(x - c)2+y2)- (x(x - c)2+y2)2 = (x-(x - c)2)2y2 = c2y2,(9)(4 (m 1 m 2) 2 / (m 3 2 - m 1 2 - m 2 2 )2 – 1) (x (x - c)2+y2)2 = c2y2.

(10)Equation (10) can be transformed to equation (11),x(x - c)2+y2 = ∓ c/(K2-1)0.5y,(11)where K= 2m 1 m 2 / (m 3 2 - m 1 2 - m 2 2 ). And from equations (2a) and (3a), it is known that x and y are
positive. And the sign of (x (x - c)+y2) of the left side of equation (11) depends on (m 3 2 - m 1 2 - m 2 2 )in the right side of equation (7). Thus, sign “∓” is added in the right hand side of equation (11). Then,equation (11) is transformed to equation (12),(x – c/2)2 + ( y ∓ c/(2(K2-1)0.5))2 = (cK/((2(K2-1)0.5))2 .

(12)Equation (12) indicates that the place of minimum transportation cost is located on the circle, whichis called the location circle and is described as follows2: This location circle has its center on theperpendicular bisector of the line connecting the points A 1 and A 2 and passes through points A 1 andA2.
2

As can be seen from the structure of equation (10), equation (12) is not defined in the following
two cases: (1) 4 (m1 m2)2 = (m32 - m12 - m22 )2, (2) (m32 - m12 - m22 )=0.
7

By using the same method, another location circle is derived which has its center on theperpendicular bisector of the line connecting points A 1 and A 3 and it passes through the points A 1 andA 3 . It can be said, therefore, that the place of minimum transportation cost is indicated by theintersection of these two location circles; the place is derived by solving simultaneous equations
representing the two location circles for x and y.
Let us find the place of minimum transportation cost by using the above analytical geometry withreference to Figure 4. It is assumed as follows; the coordinates of point A 1 are (0, 0), point A 2 are (5,0), and point A 3 are (2.5, 5). The weight of goods handled by each point is m 1 = 1.5 tons, m 2 = m 3 =1 ton. From equation (12), the place of minimum transportation cost in Figure 4 is on the location
circle, C 1 , which passes point A 1 and A 2 . This location circle C 1 is indicated by the next equation,(x-2.5)2 + (y+2.8347)2 = (3.7796)2(13)

The equation of the location circle C 2 in Figure 4 which passes point A 1 and A 3 is derived as follows:First, the distance between the points A 1 to A 3 in Figure 4 is 5.5902. Thus, as shown in Figure 5, in
another coordinate system X - Y, A* 1 is placed at (0, 0) and A* 3 is placed at (-5.5902, 0). By using theequation (12) to describe a location circle C* 2 in this coordinate system X - Y.
Y

A*1(0,0）X

A*3(-5.5902,0)

C*2

Figure 5.

Location circle in X-Y coordination

Second, since the angle at point A is 63.43∘, this coordinate is rotated by -116.57∘. As a result, therelationship between the coordinate axes X and Y and the coordinate axes x and y is expressed by thefollowing equation,
X=x∙cos(-116.57∘) +y∙sin(-116.57∘),(14)8

Y= x∙sin(-116.57∘) +y∙cos(-116.57∘).

(15)

Therefore, the equation of the location circle C 2 in Figure 4 is given as equation (16),(x+1.5843)2 + (y-3.9175)2 = (4.2258)2.

(16)

Point P that minimizes the transportation cost is obtained by solving the simultaneous equations ofequations (13) and (16) representing the location circle for x and y. The coordinates of point P are
derived as X = 1.1499, y = 0.695, and are displayed at point P at which the two location circles intersect
in Figure 4.

3.2 Derivation of minimum transportation cost place at a vertex
There are cases that point where transportation is minimized is a vertex of the location figure. If
location figure is triangle, under what conditions it is known for some time that such a case arises 3.
As shown in Figure 1, vertex angles at points A 1 , A 2 , A 3 are indicated by r 1, r 2 , and r 3 . And theweights of the raw materials and products in charge of each point are noted as m 1 , m 2 , and m 3 .
In this situation, the condition for point A 1 to minimize the transportation cost is expressed by theequation (17),cos(r 1 ) ≤ (m 1 2 – m 2 2 – m 3 2 ) / (2 m 2 m 3 ).

(17)

The conditions for point A 2 and point A 3 to be the places to minimize the transportation cost are givenby the equation (18) and (19),cos(r 2 ) ≤ (m 2 2 – m 1 2 – m 3 2 ) / (2 m 1 m 3 ),(18)

cos(r 3 ) ≤ (m 3 2 – m 1 2 – m 2 2 ) / (2 m 1 m 2 ).

(19)

Furthermore, when the weight of the goods in charge of a certain point, for instance point A 1 , is thesame as or larger than the total of the weights in charge of other points, that is, when equation (20)holds, that point A 1 minimizes the transportation cost 4,

3

Nishioka (1976) explains the condition under which minimum transportation cost place is settled
at a vertex.
4 This fact is explained by Weber (1909,S.235).
9

m 1 ≥ ∑3𝑖𝑖=2 𝑚𝑚𝑖𝑖

R

.

(20)

4. Derivation of area of minimum production costs
4.1. Incorporation of the production function into the analysis
In the derivation of the above-mentioned transportation cost minimization point, the weight of the raw
materials and products to be used is given. Such assumption is of course effective in theoreticalanalysis in a rather short period of time. Under a rather long period of time, however, the theoreticaleffectiveness of this assumption declines. The prices of raw materials fluctuate in a somewhat long
period, and the amounts of raw materials used change accordingly. As a result, the ratio of the weightsof the raw materials used change and the transportation cost minimizing point moves. The change ofprices of the raw materials vary the combination of the weights of the raw materials used at the factory.
The problem of how the combination of raw materials is changed with respect to price changes of raw
materials is explained by incorporating the factory’s production function into the analysis.
By introducing the production function into the analysis, the object of the analysis is led to thederivation of the point which minimizes the production cost including the transportation cost.
Therefore, the assumption of analysis is expanded and the place where the production cost of thefactory is minimized is derived.
The assumptions used in the previous section are expanded. (1) Prices of material M 1 and M 2 aregiven as p 1 , p 2 . (2) Distances between the factory and each point A 1 , A 2 , A 3 are denoted by d 1 , d 2 ,and d 3 . (3) The freight rates of each material M 1 , M 2 and the final goods M 3 are given as t m1 , t m2
t m3 , respectively. (4) Production function of the factory is shown by equation (21),
Q= Am 1 αm 2 β .

(21)

where Q is production amount of the final goods M 3 , A denotes parameter to indicate production
efficiency of the factory, parameter α and β are assumed 0.4 for simplicity of calculation.
Making use of the law of equi-marginal productivity, that is, the ratio between the productivities of
the two raw materials should be equal to the ratio between their delivered prices, quantities of them
are derived as equations (22) and (23),m 1 = Q1.25A-1.25(p 2 +t m2 d 2 )0.5/(p 1 +t m1 d 1 )0.5 ,(22)m 2 = Q1.25A-1.25(p 1 +t m1 d 1 )0.5/(p 2 +t m2 d 2 )0.5

(23)

The factory’s production cost C p is given by equation (24),10

C p = 2Q1.25A-1.25(p 1 +t m1 d 1 )0.5(p 2 +t m2 d 2 )0.5 +Qt m3 d 3 .

(24)When the cost function is given as in equation (24), the production cost becomes a function of thefactory’s location. The factory’s production cost is varied as a function of x and y as shown in Figure
6. It is assumed in Figure 6 that points A 1 , A 2 , A 3 in Figure 4 are specified as A 1 = (0.0), A 2 = (5.0),
A 3 = (2.5.5), the production amount of the final goods M 3 ,Q=50, production efficiency; A=1, the pricesof the raw materials are given as p 1 =0.25, p 2 =0.75. the freight rates are given as t m1 =0.11, t m2 =0.15,
t m3 =0.57. As shown in Figure 6, the production cost of the factory is minimized at point (2.65, 3.49)
and rises as it goes away from that point.
y
Cpx

Figure 6. The relationship between the factory’s location and production cost
Then, as the production cost becomes a function of the factory’s location, by solving thesimultaneous equations of (25) and (26) for x and y, the place at which production cost is minimized
can be derived. In deviation of equation (25) and (26), A 1 = (0.0), A 2 = (5.0), A 3 = (2.5.5) are assumed.
∂C p /∂x =Qt m3 (x-2.5)/(((x-2.5)2+(y-5)2)0.5)+Q1.25t m1 x(p 2 +t m2 ((x-5)2+y2)0.5)0.5/(A1.25((x2+y2)0.5)(p 1 +t m1 (x2+y2)0.5)0.5)+ Q1.25t m2 (x-5) (p 1 +t m1 (x2+y2)0.5)0.5/(A1.25((x-5)2+y2)0.5)(p 2 +t m2 ((x-5)2+y2)0.5)0.5)=0,(25)∂C p /∂y = Qt m3 (y-5)/(((x-2.5)2+(y-5)2)0.5)+Q1.25t m1 y(p 2 +t m2 ((x-5)2+y2)0.5)0.5/(A1.25((x2+y2)0.5)(p 1 +t m1 (x2+y2)0.5)0.5)+ Q1.25t m2 y (p 1 +t m1 (x2+y2)0.5)0.5/(A1.25((x-5)2+y2)0.5)(p 2 +t m2 ((x-5)2+y2)0.5)0.5)=0.

(26)

If the parameters are given the same value used in the derivation of Figure 6, the place of minimizing
production costs is dived from equations of (25) and (26). The solution can be derived as (2.653,3.490).

11

4.2. The concept of area of minimum production costs
The curved surface shown in Figure 6 shows the relationship between the location of the factory andits production cost. Let's cut this curved surface several times at different height around the minimum
value with a plane parallel to the bottom. Each cut is projected onto x-y plane and they are indicatedby each closed curve in Figure 7. These curves may be called equal production cost curves. Amongthese equal production cost curves, the inside of the innermost equal cost curve is indicated by dark
blue. If the factory is located in an area indicated by dark blue, the production costs of the factory is
not higher than 309.7. That is, it can be said that it is not more than 0.1 higher than the minimum
production cost. Point (2.65, 3.49) of minimum production cost is indicated by the white mark “ * ”
in Figure 7.
y*x
Figure 7. Derivation of area of minimum production costs
Based on the considerations of production costs using Figure4, Figure 6 and Figure7, the following
inference is possible. If the amount of each of the raw materials used by the factory has been
determined and it is stable over a relatively long period of time, deriving a point that minimizes thetransportation expenditure is quite important in determining the location of the factory. In a situation,
however, where the prices of raw materials change in relatively long term, even if one place thatminimizes shipping costs is found, that place will shift quickly. Hence, it can be said that theimportance of finding such a point is relatively low.
Instead, it is significant in the factory’s location strategy for the factory to set an amount that allows
deviation from the minimum production cost level, and the factory sets the spatial area in which the12deviation of the production costs is allowed. And if the factory is in the area, the factory accepts its
location from the viewpoint of production cost. This area can be called as area of minimum production
costs. It could be said, thus, that the location issue of the factory is to set the area of minimum
production cost in a geographical space.
As mentioned above, it is predicted that changes in raw material prices will often occur. In response
to such raw material price change, it is thought that the idea of area of minimum production costs is
effective for the firm's location strategy. For instance, if the factory sets 0.1 that can deviate from theminimum production cost level in the situation assumed in the above analysis, the spatial area in which
the factory’s location is allowed from the viewpoint of the production costs is indicated by the area ofdark blue in Figure 7. This area is the area of minimum production costs.
Let us examine the significance of the area of minimum production costs. Suppose that the pricep 2 of the material M 2 is changed by 12 percent from the 0.25: if the price is increased to 0.28, otherparameters are kept the same level, point of minimum production costs shift from point (2.65,3.49) topoint (2.63,3.56) which is shown by the white diamond mark. In this situation, the new area ofminimum production costs is set as shown by the inside dark blue area of the innermost equal cost
curve in Figure 8. The existing location of the factory, which is shown by the white mark “ * ”, isincluded in the new area of minimum production costs in Figure 8. Thus, the factory does not need toconsider the shifting its location according to the increase of price of the material.
In sum, when the factory allows the concept of area of minimum production costs, it can be said
that even if the price of the raw material changes somewhat, the factory does not shift its location pointaccording to the change.
ｙ*xFigure 8 A new area of minimum production costs

13In addition, since within area of minimum production costs the production costs of the factory arealmost same irrelatively its location, the factory can have a lot of location selections: Within this area
the factory incorporates many location factors in the determination of the factory’s location such as
security, health care, and dwellings of workers and so on. The concept of area of minimum production
costs may offer usefulness to the factory’s location determination: the factory can flexibly consider
the its location and select a better location considering various location factors.

5. Concluding remarks
Factory’s shipping costs tend to decline due to intermittent technological progress in transportation
systems. In addition, the transportation costs have been greatly reduced by technological innovation
in office processing. Due to the decline in transportation costs, many production processes become tobe released from places where transportation costs are minimized. The locations of the production
processes have become more influenced by labor costs and the agglomeration economies, and thelocations of the factories are becoming diversified. Transportation costs have various impacts on thelocation determination of various economic activities in each region of the world. The economic role
played by transportation expenses has been relatively declining, but the way of its work has become
diversified as economic activity is geographically broadened.
This paper analyzes the location effect of transportation cost on factory: At the initial stage of
economic development, the production activities are relatively simple, and it is assumed that the pricesof the raw materials used by the factory are constant for a relatively long time. Under this assumption,
first, the paper explains the traditional theory to derive the place that minimizes the transportation costof the factory, and clarifies its validity by analytical method. Secondly, the globalized economy isassumed, in this stage of economic development production process is fragmented into small blocs,
and some of them are spatially scattered from the existing place to various places which provide
production condition suitable to the individual production blocs. In this stage, there are many raw
materials and intermediate good are used to produce final goods, and it is assumed that the prices ofthese materials and intermediate goods are not constant, but they change in a relatively short period.
Thus, this paper proposes the concept of area of production cost minimization. This concept implies
as follows: In determining the location of a factory, deriving place that minimizes transportation costs
is not very important as the prices of intermediate goods easily changes. Rather it is more important
for the factory to allow some deviation from the minimum transportation cost and to determine an
acceptable geographical range. By setting area of production cost minimization the factory does notneed to consider the shifting its location in response to mild price changes of the materials andintermediate goods. And the factory can flexibly consider the its location within this area and select abetter location by incorporating various location factors into the consideration.

14

References
Kuhn, H.W. and Kunne, R.E., (1962). An efficient algorithm for the numerical solution of thegeneralized Weber problem in Spatial economics, Journal of Regional Science,4, pp 21-33.
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