4.2. Estimating the Market Heterogeneity of GAM-PEP Graphs
Using these three typical graphs, the procurement personnel can estimate the range for the market heterogeneity (
Em) using the intersects of the tangent with the
X-axis and the
Y-axis. With the estimated
Em, the procurement personnel are able to determine which procurement method should be adopted. The following shows how to estimate the range of
Em using Marshall’s price elasticity theory; the three typical Q-P Diagrams are shown in
Figure 2.
(1) Type G-I: a highly heterogeneous market
Type G-I is located in the upper-left area of the first quadrant. Thus, the tangents of points on G-I first intersect the
Y-axis; the values for the PEP for the points are greater than 1 (for detailed analyses of PEP, please refer to Yu and Wang [
12]).
(2) Type G-II: market with low heterogeneity
The Type G-II graph is located in the lower-right area of the first quadrant. The tangents first intersect the
X-axis and then the
Y-axis; thus, the PEP for Points are less than 1 (please refer to Yu and Wang [
12] for detailed analyses).
(3) Type G-III: moderately heterogeneous market
When the Q-P Diagram resembles a Type G-III graph (see
Figure 2), the analysis is more complicated than that of G-I and G-II. For a Type G-III graph, two tangents pass through (or very close to) the origin. These two tangents can be represented as ‘
y =
mx’. The tangents contact the HQ and LQ curves at Point-C and Point-D, respectively. The PEP for a Point-C on the HQ curve in
Figure 3 is calculated using Equation (5).
Similar results can be obtained for ED by calculating the PEP of Point-D. The location of Point-C and Point-D on the X-axis (i.e., xD < xC) can be used to differentiate the two sub-graphs, as described in the following:
• Estimating the heterogeneity of Type G-III (1)
When
xD <
xC (as shown in
Figure 3), it is called Type G-III (1). For convenience in later analyses, the Type G-III (1) graph is divided into three sub-regions along the
X-axis (Q): (1) α, the sub-region to the left of Point-D, (2) β, the sub-region between Point-D and Point-C, and (3) γ, the sub-region to the right of Point-C.
The diagonal line (the dashed line) for Type G-III (1) usually intersects the X-axis and the Y-axis near the origin, O, which represents a moderate PEP procurement market with Em ≒ 1. Further analysis is necessary to determine the appropriate project procurement method.
In
Figure 3, four sub-regions (i.e.,
AF and
FC of the HQ and
DG and
GB of the LQ) are almost a heterogeneous procurement market, with
E > 1 and the other two sub-regions (i.e.,
CB of the
HQ and
AD of the LQ) are almost a homogeneous procurement market, with
E < 1. The heterogeneity (
E values) analyses of the sub-regions in Type G-III (1) are shown in
Table 2.
• Estimating the heterogeneity of Type G-III (2)
When
xD >
xC (as shown in
Figure 4), it is called Type G-III (2). For convenience in later analyses, the Type G-III (2) graph is also divided into three sub-regions along the
X-axis (Q): (1) α, the sub-region to the left of Point-C, (2) β, the sub-region between Point-C and Point-D, and (3) γ, the sub-region to the right of Point-D. As shown in
Figure 4, the diagonal line (the segmented line) for the Type G-III (2) also intersects the
X-axis and the
Y-axis near the origin,
O, which represents a moderately heterogeneous procurement market with
Em ≒ 1, so further analysis is necessary to determine the appropriate contractor selection method.
In
Figure 4, four sub-regions (i.e.,
CF and
FB of the HQ and
AG and
GD of the LQ) are almost a homogeneous procurement market, with
E < 1; and, the other two sub-regions (i.e.,
AC of the HQ and
DB of the LQ) are almost a heterogeneous procurement market, with
E > 1. The graphical analysis is shown in
Table 3.
4.3. Determination of a More Appropriate Project Procurement Method
(1) Type G-I & Type G-II
Based on the analyses of the three typical graphs for a GAM-PEP mentioned above, the procurement market for Type G-I is highly heterogeneous and BV is the most suitable method. Type G-II is highly homogeneous, so LT is the most suitable method.
(2) Type G-III graphs
In order to analyze the heterogeneity of Type G-III, the overall value of
Em for these three types of graphs must be calculated. The overall value for
Em for Type G-III is the sum of the values for
Em for all the sub-regions, α, β, and γ, as calculated by Equation (6):
Using Equation (6) to calculate the value of
Em for Type G-III (1) (
) in
Figure 3, results in Equation (7):
Similarly, using Equation (6) to calculate the value for
Em for Type G-III (2) (
) in
Figure 4, results in Equation (8):
where
is the overall market PEP for the procurement project;
represents the PEP of arc ‘
ARC’ and ‘
ARC’ =
AG,
GD,
CF,
FB,
DB, and
AC.
Using Equations (7) and (8), ‘
’ is calculated to compare the value of
Em for Types G-III (1) & (2), as follows:
(3) Differentiation between Type G-III (1) and Type G-III (2)
The critical criteria for differentiating Type G-III graphs are the X coordinates of Point-C and Point-D. An alternative approach is to use the Cartesian X-Y coordinate system and let
E = 1 for Equation (1), yielding the tangent that passes through the origin, as in Equation (11):
Using the second order quadratic function,
f(
x), to model the HQ and LQ curves, as suggested by Yu and Wang [
12], results in Equation (12).
where
a,
b, and
c are constant coefficients.
Solving the following simultaneous equations gives x
C:
where
a1,
b1, and
c1 are the constant coefficients for the quadratic equation of
y1.
Similarly, solving the following simultaneous equations for the LQ curve,
y2, and the tangent at Point-D gives
xD:
With the coordinates xC and xD, it is possible to determine if the procurement market belongs to Type G-III (1) or (2): if xC > xD, it belongs to Type G-III (1); otherwise, it belongs to Type G-III (2). In the proposed GAM-PEP, a simplified method to differentiate Type G-III (1) and Type G-III (2) is: (1) draw the straight line ‘ ’ connecting the two intersection points A and B of curves HQ and LQ; (2) draw tangent lines of HQ and LQ passing the origin point and locate the tangent points C and D; (3) make judgements with the rule “if xC > xD, it belongs to Type G-III (1); otherwise, it belongs to Type G-III (2)”.