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1 Casper, Paz de Araujo & Paz de Araujo
1
2
3
Logic Scoring of Preference (LSP) Application 4
to Transportation Investment Portfolio Optimization: 5
A Case Study in Colorado Springs 6
7 8
Craig Casper 9 Corresponding Author 10
Pikes Peak Area Council of Governments 11 15 South 7
th Street 12
Colorado Springs, CO 80905 13 Telephone: 719-471-7080 x 105 14
Fax Number: 719-471-1226 15 [email protected] 16
17
18
Maureen Paz de Araujo 19 HDR Engineering 20
2060 Briargate Parkway, Suite 120 21 Colorado Springs CO 80920 22 Telephone: 719-272-8833 23
Fax Number: 719-272-8801 24 [email protected] 25
26 27
Carlos Paz de Araujo, Ph.D. 28 University of Colorado at Colorado Springs 29
5055 Mark Dabling Boulevard 30 Colorado Springs CO 80919 31 Telephone: 719-594-6145 32
Fax Number: 719-598-3437 33 [email protected] 34
35
36
Submitted for consideration for presentation or poster session at the 37
Transportation Research Board 38
Annual Meeting 39
January 2013 40
41
42
Key Words: performance-based project prioritization, investment prioritization, multi-criteria analysis, 43
MPO, Colorado Springs, transportation, innovative, application, Logic Scoring of Preference, neural 44
networks, TIP, project selection, 45
46
Words = 5778 words + (6 figures @ 250 each = 1500) = 5531 + 1500 = 7278 of 7500 permitted 47
TRB 2013 Annual Meeting Paper revised from original submittal.
2 Casper, Paz de Araujo & Paz de Araujo
Abstract 48
The evaluation and prioritization of transportation investments at the portfolio level presents a 49
complex decision-making problem for state and regional planning organizations. Historically, 50
transportation investment decisions have been dealt with as a series of stand-alone problems to be 51
resolved using straight-forward engineering solutions. In this context, improvement needs and project 52
solutions were identified based on simple criteria, such as traffic congestion levels. Investment portfolio 53
optimization was accomplished by listing projects in order of most to least congestion reducing, and then 54
allocating funding to projects by rank until funding is exhausted. Recently, increasing awareness of the 55
complex interdependencies among transportation, land-use, social, economic and ecological systems has 56
fostered implementation of multi-criteria analysis (MCA) investment prioritization approaches that 57
incorporate increasingly more complex goals and metrics. The simplest MCA decision model is the 58
Weighted Sum Model (WSM). More rigorous methods, such as the Analytical Hierarchy Process (AHP) 59
and the Technique for Ordered Preference by Similarity to Ideal Solution (TOPSIS), provide increased 60
functionality, and support prioritization that is driven by asset performance and financial return in 61
addition to engineering criteria. This paper examines the suitability of three decision models for the 62
optimization of transportation investment priorities across full programs/portfolios. A Logic Scoring of 63
Preference (LSP) approach is contrasted to the WSM approach that is currently used by the Pikes Peak 64
Area Council of Governments (PPACG), as well as to an enhanced linear programming optimization 65
(OPT) algorithm approach. Functionality, advantages, and disadvantages of each method are discussed, 66
and potential enhancements are identified. 67
TRB 2013 Annual Meeting Paper revised from original submittal.
3 Casper, Paz de Araujo & Paz de Araujo
1 INTRODUCTION 68
Selection of projects for inclusion in fiscally constrained long range plans and short term 69
improvement programs has historically been completed using engineering methods, using straight-70
forward engineering criteria such as crash and/or congestion reduction. Projects were typically ranked and 71
prioritized for funding in order of most to least improvement, with performance for cost occasionally 72
considered in project selection. With project rank thus identified, projects were funded in priority order 73
until available funding was exhausted (rank and cut). Recently, as both awareness of, and requirements to 74
consider other criteria have increased, there has been an increased use of multi-criteria analysis (MCA) 75
methods. MCA methodology has proven attractive for addressing these emerging considerations because 76
it provides the necessary platform to evaluate individual projects using a variety of transportation and 77
non-transportation asset performance measures. The MCA methodologies most commonly applied to 78
transportation problems include: the Weighted Sum Model (WSM), the Analytic Hierarchy Process 79
(AHP) (1) and the Technique for Ordered Preference by Similarity to Ideal Solution (TOPSIS) (2). Each 80
of these MCA methods is designed to provide decision-makers with the ability to make the best possible 81
individual investment decisions based on past, present and future predicted information. Transportation 82
MCA applications include: project prioritization/selection (3, 4, 5 and 6), performance monitoring (7), 83
and evaluation of both economic development linkages (8) and transportation sustainability (9). 84
To date similar rigor has been applied only on a limited basis to optimizing the performance of 85
the full portfolio of potential transportation projects. Rigorously analyzing portfolio performance is 86
common in other fields, as examples from stock portfolio/mutual fund analysis (10) and internet/computer 87
facilities investment portfolio analysis (11 and 12). Application of this approach to the optimization of 88
transportation investment represents an emerging focus in which transportation investment is no longer 89
viewed as a series of stand-alone projects to address specific issues, but rather as investment in an 90
integrated system that is also characterized by complex interdependencies with related land-use, 91
economic, ecological, and social systems. 92
Transportation investment portfolio optimization can be viewed, in its simplest form, as selecting 93
the best fiscally constrained combination of projects from a finite set of available projects using adopted 94
goals and metrics for either the 20+ year Region Transportation Plan (RTP) or the 4+ year Transportation 95
Improvement Program (TIP). This requires identification of a “best set of projects” that, when 96
implemented, will simultaneously minimize negative and maximize positive total final outcomes, as 97
measured by adopted performance metrics. To find this “best set of projects” optimal solution, sets of 98
projects drawn from the larger set of projects must be systematically analyzed using a formal decision 99
process. 100
The first and second order decision models presented in this paper apply metaheuristic 101
methodologies (13) to transportation investment portfolio optimization. In computer science, 102
metaheuristics is a computational method of finding an optimal solution by iteratively trying to improve a 103
future condition using some predetermined measures of quality (performance metrics). Metaheuristic 104
computational methods include: combinatorial optimization, evolutionary algorithms, dynamic 105
programming, and stochastic optimization. While the most common metaheuristic method is stochastic 106
optimization, the searching use of random variables is not suited for transportation portfolio optimization. 107
Similarly, dynamic programming refers to simplifying a complicated problem by breaking it down into 108
TRB 2013 Annual Meeting Paper revised from original submittal.
4 Casper, Paz de Araujo & Paz de Araujo
simpler sub problems that can be solved once, in a recursive manner. The interrelationships between 109
goals, and how each individual project impacts those interrelated goals means that dynamic programming 110
is ill-suited to portfolio optimization. Instead, combinatorial optimization processes that find an optimal 111
set from within a finite set of projects should be used. Evolutionary Algorithms are also useful, but their 112
added complexity make them computationally challenging for operational implementation. Due to the 113
complexity of the systems involved in the portfolio analysis it is likely that there is more than one 114
“optimal” portfolio. Identifying the trade-offs between different optimal potential portfolios may be a 115
second goal when setting up and solving transportation portfolio optimization. 116
2 PROBLEM STATEMENT 117
The Pikes Peak Area Council of Governments’ (PPACG’s) 2035 Moving Forward Update long 118
range transportation plan (LRTP) was developed using a collaborative process with non-traditional 119
resource agency participation (e.g. U.S. Fish & Wildlife Service, National Park Service). During the 120
visioning phase of the LRTP update, 17 performance-based evaluation criteria were identified as the basis 121
for project selection for inclusion in the fiscally-constrained LRTP, and weights were established for each 122
of the evaluation criteria. Each project was scored on the evaluation criteria using consistent, 123
predetermined metrics. A simple, WSM-based decision model was then used to calculate weighted scores 124
for 139 projects and select a shortlist of projects for inclusion in the LRTP. Using a rank and cut process, 125
modified only as necessitated by funding eligibility requirements, 27 projects were selected. After the 126
fact, the process that was used was viewed by some stakeholders as insufficiently transparent and as 127
inflexible. Deficiencies cited included the inability to account for synergies among projects or to respond 128
to unique circumstances or needs. The small number of projects “funded” was also viewed unfavorably 129
by some stakeholders. This paper explores alternative transportation portfolio optimization methodologies 130
that might be implemented to improve upon the decision process used by PPACG for the 2035 LRTP. 131
3 ALTERNATIVE DECISION MODELS 132
3.1 Zero Order Decision Model: Weighted Sum Model (WSM) 133
The zero order decision model used by the PPACG for 2035 LRTP fiscally constrained project 134
selection is commonly known as the Weighted Sum Model (WSM). In the WSM, each project (i) receives 135
a total weighted score (Vi) as the sum of each criteria score (Sj) for project (i) weighted by an a priori 136
weight (Wj), such that the project with the highest resulting total score, or max (Vi) is afforded highest 137
priority directly. Priority for the rest of the project set is set directly as well, according to rank per the 138
value of Vi. This can be written in short hand form as equation [1] below. 139
140
i ∑ ij j [Equation 1] 141
142
Where: Jma x = the total number of scores and associated weights for the ith project. 143
The simplicity of this method is attractive, especially when communicating with the public, but it contains 144
significant pitfalls. First, the set of weights may have a high degree of arbitrariness, as the choice of a 145
weight is already in itself a decision, but not necessarily an outcome of a formal decision process. 146
147
TRB 2013 Annual Meeting Paper revised from original submittal.
5 Casper, Paz de Araujo & Paz de Araujo
In order to have weights mean the same across projects, they must also be based on a formal set 148
of criteria. This is a very difficult task to achieve when many decision makers are involved and unknown 149
biases can influence given weight choices. Thus, an inclusive decision process that can be applied to 150
criteria selection, weight choices and ultimate transportation improvement portfolio selection and 151
optimization is needed if the preferences are to be valid. The PPACG WSM application utilizes criteria 152
and weights developed through an inclusive process, but one that lacks formality and transparency needed 153
to make it flexible or defensible. 154
Using the WSM, it is difficult to cross prioritize or “bundle” best choices of projects because 155
performance scores (Sij) are tied to individual projects and not be able to go across projects as they should. 156
That is, a performance j for project i may be different and of un-measureable value for a project i+1 157
which may, or may not contain the j criterion. In this case it is in fact completely random and naïve what 158
Vi really is and means to the decision process. Thus, the weighted average may give an initial “feeling” of 159
choice and preference, but it is insufficient to give a global and adjustable score without some logical 160
filters. 161
3.2 First Order Decision Model Approach: Linear Programming Optimization Algorithm (OPT) 162
A significant improvement over the zero order, WSM decision model used for the PPACG 2035 163
LRTP can be achieved by a first order linear programming optimization algorithm (OPT) proposed by An 164
and Zheng (14). Application of OPT for the 139 PPACG 2035 LRTP Update projects produced increases 165
in total benefit scores (15,232 vs. 6,051) and number of projects selected (133 vs. 27) as compared to zero 166
order the WSM-based approach used by PPACG for the 2035 LRTP. These results confirm that the OPT 167
would be effective in maximizing both number of projects selected and overall benefit. However, neither 168
the need to account for synergies among projects (if you build Project A would it make Project B more or 169
less beneficial), nor the danger of precluding a project from selection based on cost alone (is the project 170
critical; could it be phased) would be addressed directly by OPT. 171
The proposed OPT decision model would introduce restricted formal logic functions (FLF) to 172
provide a “yes/no” bias to the selection criteria as described below in equations [2 and 3]. 173
xm = 1, if Cm B [Equation 2] 174
0, if Cm B 175
176
xn = 1, if Cn | B - xmCm | [Equation 3] 177
0, if Cn | B - xmCm | 178
179
Where: B is the total allowed budget for all projects and the definition of Cm and Cn are the costs 180
associated to the ith project under two constraints that in a single strike “filter out” that project 181
from the selection set, if either it exceeds the budget, or there is no budget left. 182
Applying the first OPT formal logic function, as represented by equation [2], if the cost of an 183
individual project is more than the funds available (B), the project is immediately deselected (Xm = 0). At 184
first this seems reasonable; however cost negotiations, phasing accommodations or extreme necessity 185
(e.g. to replace a fallen bridge) could be blind-sided on the very first pass by application of this zero-186
bandwidth binary filter. Expansion of the OPT method such as formulations 3 and 4 as described in the 187
TRB 2013 Annual Meeting Paper revised from original submittal.
6 Casper, Paz de Araujo & Paz de Araujo
NCHRP 590 report (15), have improved the OPT functionality by creating extra constraints in a Linear 188
Programming environment whilst maintaining a zero-bandwidth logic filter. 189
The second function, as represented by equation [3], acts as an “empty bucket” filter. This filter 190
excludes any project whose incremental cost, Cn, exceeds what is left after other, higher priority projects 191
have been funded. Again, at first this seems reasonable, but if a critical project is not selected or costs 192
more than funds available, there is no method to de-select an already selected project to include this 193
critical project. This scheme also lacks a decision process that weighs in cross-linkages among projects 194
with a common methodology that is discriminating of many aspects of the decision process. In most cases 195
scores across links are intrinsically non-quantitative and yet need enough of a quantitative core to be 196
objective and practical. 197
After the implementation of the two formal logic functions, a linear programming optimization 198
algorithm (OPT) couples the constraints to these two formal logic functions Xm and Xn. Finally, the 199
constraints imposed by the total budget, until it is exhausted, are evaluated by doing a binary search that 200
relaxes when the maximum number of projects is funded and the budget is met. Equations [4 and 5], 201
below, show the linear programming hyperspace. 202
{∑ ∑ i ij j
} {∑ ( ) ( )
} [Equation 4] 203
Where: CPij = chosen projects and Vj is like Vi, but only of selected projects, such that: 204
∑ Ci i [Equation 5] 205
The products of CPij and Vi, when at a maximum, represent all the chosen projects and the 206
weighted sum of such projects. However, as described above, Xn and Xm filter out projects with respect to 207
constraints within narrow logic criteria. The fundamental limitation intrinsic in the first order model is the 208
use of formal logic functions that provide zero bandwidth to the local decision and filter out projects 209
without allowing the degrees of freedom that are inherently needed in any decision process. Thus, the 210
zero order and first order models are mathematically descriptive of mappings that could be seen as 211
objective and formal, but that fall short of the expected breadth and depth that decision-makers require of 212
an analytical approach to decision making that the public can accept as fair and careful of special 213
circumstances. Such mathematical descriptions may create a sense of modeling but do not constitute 214
formal models. Thus, the optimization process (OPT) is a support system for decision-making. While the 215
analyses results are useful guides to the final decision, they are not substitutes for the final judgment. On 216
seeing the optimized results, decision-makers are likely to consider constraints and objectives that were 217
not included in the analyses (thus making the model blind to these variables). It is the decision-maker’s 218
final selection – grounded in their paradigm but informed by the analyses – that is funded, implemented 219
and managed. 220
3.3 A Second Order Decision Model Approach: Logic Scoring of Preference (LSP) 221
The central technique used as the framework for the second order model is the Logic Scoring of 222
Preference (LSP) method for decision processes. The LSP method described in the seminal work of J. J. 223
Dujmovic (11) was originally developed for evaluation and selection of complex networks. The key point 224
of this method is the expansion of the bandwidth of the logic functions. Thus, the “yes/no” measure 225
TRB 2013 Annual Meeting Paper revised from original submittal.
7 Casper, Paz de Araujo & Paz de Araujo
becomes percentages of preference from 0 – 100%. Furthermore, the preferences can be directly linked to 226
a quantitative measure of “cost/benefit” ratio. 227
Starting with equation [1] from the WSM in which the ith project total score was: 228
J max 229
Vi = Sij Wj 230 j 231
The LSP ith project total score is generalized as: 232
I max J max 233
Vi (G(Sij)) = G(Sij ) [Equation 6] 234 i j 235
In this formulation, Sij is still a performance criterion score, but the weights are generated by a 236
mapping function G(Sij) between each performance criterion of the ith project, generating first an 237
elementary preference, EPj which is represented by the preference aggregation structures shown in 238
Figure 1. 239
240
This elementary preference is subject to further optimization. For the ith project, the aggregate preference 241
Ei, can be written by the Master Decision Equation (MDE, 7). 242
Ei = (G (Si1) (EP1) r + G (Si2) (EP2)
r + … + G ( ik) (EPk)
r [Equation 7] 243
For the zero-order model (Weighted Sum Model) with constant weights, G(Sij) = Wj, where r is 244
equal to 1. Also, the elementary preference is normalized such that 0 < EPi < 1 (or 0-100%). Thus, the 245
preference aggregate for the ith project is: 246
J max 247
Ei = W1 EP1 + … + Jmax EPJmax = (EPj) Wj [Equation 8] 248 j 249
Equation [8] is identical to equation [1] for Vi, provided that Sij is now a normalized score EPj for 250
the ith project. However, there is a fundamental conceptual adjustment from Sij to EPj. Firstly, EPj is a 251
preliminary score, not yet subjected to a logical function. Also, the power index r will be described as a 252
function of the logical function bandwidth, d. This is the core of the LSP methodology: the logical 253
function bandwidth in LSP varies from 0 to 1, whilst the formal logic has only two values (0, 1). Thus, in 254
a sense the bandwidth of formal logic is zero and of LSP infinite. 255
G1
SUM Ek
Si1
Si2
Sik
G2
Gk
Ei (ETOTAL for the ith project)
FIGURE 1 First Order Preference Aggregation Structure
TRB 2013 Annual Meeting Paper revised from original submittal.
8 Casper, Paz de Araujo & Paz de Araujo
Figure 2 shows how the formal logic function (FLF) of the first order OPT model is expanded in the 256
bandwidth such that the aggregation preference for the ith project varies between “very good” or d = 1, 257
and “very bad” or d = 0. The value of r depends on d such that a value of r can be chosen from 258
objectively calculated results where the formal logic values of “yes/no” or “0/1” are expanded to degrees 259
of preference. It is clear that d is the indicator of the average position between Emax and Emin.. 260
261
In the second order LSP decision model, various values of r are used to create degrees of logic 262
such as “quasi-AND” and “quasi-OR.” In the decision literature, AND is a “conjunction,” meaning that 263
E1 and E2 and…Ek are all necessary and need to be “added” to the aggregate preference. These are “must-264
have” performance criteria. It is here that one can see that the expansion of the logic bandwidth spread the 265
values of the preferences, allowing a better decision space. The formal logic function of the first order 266
model does not allow for the value of Ej to be a number between 0 and 1. Thus, the OPT model is very 267
restrictive and may filter out projects too early by having only Xj = (0, 1). The range of degrees of logic 268
function created by the values of r includes: AND (r = -), HARMONIC MEAN (r = -1), 269
ARITHMETIC MEAN (r = 1), MEAN SQUARE SUM (r = 2), and OR (r = +). 270
How the value of “r” leads to the tabulated quasi logic functions is well explained by references 271
(11 and 15). Here we will use these tabulated functions to create “structured decision circuits” which are 272
set as the standards for the decision processes. In this manner the a priori transparency of the circuits will 273
make the results for every project readily comparable. There are five types of circuits as described in 274
reference (11). These are: 275
1) CPA (Conjunction with partial absorption) 276
2) Quasi-AND (Quasi-conjunction) 277
3) Neutrality (A or arithmetic mean) 278
4) Quasi-OR (Quasi disjunction) 279
5) DPA (Disjunction with partial absorption). 280
The reader should readily see that circuit type 3, (Neutrality - A or arithmetic mean) has already 281
been discussed. In circuit type 1 (CPA - Conjunction with partial absorption), a conjunction shows that 282
“E1 AND E2, AND … Ek means that the preferences are mandatory. Figure 3 below is an example for 283
circuit (2, “C - +” - Quasi-AND) with the simplified assumption of simple weights for Sij (= Wij). In this 284
example, the values specified for r and d are specified for a set of three performance criteria/scores. 285
Circuit types 4 and 5 will be explained later. 286
Emax = max ( E1, E2, … Ek ) = “Positive Ideal Solution (PIS)”
Emin = min ( E1, E2, … Ek ) = “Negative Ideal Solution (NIS)”
e0 = neutral “preference”
d = 1
d = .5
d = 0
FIGURE 2 Expanded Second Order Preference Aggregation Structure
TRB 2013 Annual Meeting Paper revised from original submittal.
9 Casper, Paz de Araujo & Paz de Araujo
287
For each project, thus, for simplicity, in Sij, the i is dropped. Thus, the MDE for the elementary preference 288
(E0) in a quasi-conjunction (where simultaneity is required for performance criteria) is: 289
0 ( ∑ j jr )
1/r = ( 0.5 S1
-0.028 + 0.3 S2
-0.028 + 0.2 S3
-0.028 )
-1/0.028 290
The combined or aggregate preference for this example (S1=.7, S2=.8 and S3=.8) is E0=73% or 0.7, closer 291
to d=1, or the conjunction of the ith project is a strong preference, close to 100% (or d=1). Where as in the 292
first order model this would be Xn or Xm where it would be simply 100% (or d=1). From this example, the 293
value of the LSP method is clear. Since E0 is 73% and not 100%, the bandwidth leaves 27% for other 294
projects to move ahead of this one. A similarly revealing example for Quasi-OR is presented in reference 295
(11). It is this sliding scale within LSP that yields an immediate advantage over the first order model 296
(where all exponents would be just “1” or r=1). 297
The Quasi-logic circuits (QLC) are the basic units for optimization. In this paper only the Quasi-298
AND function, as shown in Figure 4, will be used for Pikes Peak area data, as the benchmark for all 299
initially mandatory preferences. However, for completeness and to show the richness of the LSP method, 300
the circuit diagram for the Disjunction function is also shown. The circuit diagram for the Quasi-AND 301
function shows the ability to add “Optional” preferences, while the circuit diagram for the Disjunction 302
function shows that “ ufficient” and “Desired” preference specifications can be added for that QLC 303
function. Decision networks with the use of QLCs provide greater flexibility than either the WSM or 304
OPT methods for development of a Formal Decision Process. QLC first does an “A” model; then it adds 305
the large bandwidth logic to automatically drive the preference. 306
The set of {Sj} or {Sij} can be connected to allow qualitative measures such as mandatory, 307
optional, sufficient, and desired can be added to the decision process. Thus, every project starts with the 308
purely mathematical weighted sum (box A) and the logic filtering is contained in the “Quasi” boxes 309
yielding more degrees of freedom and bandwidth to choices. Optimization of the decision is then an 310
integral part of exciting these circuits until the constraints are met. The validity of this method can be 311
justified in terms of mathematical formalisms, such as “Threshold Logic” or “Preference Neural 312
Networks” (16). The second set of weights, as shown in Figure 4, allows another degree of freedom that 313
is used to spread these qualitative preferences. 314
r = -0.028
d= 0.3126
W3 = 0.2
W1 = 0.5
W2 = 0.3 Quasi AND
AND
E0 (or Ei0 for the ith project)
S1
S2
S3
FIGURE 3 Quasi-AND Preference Aggregation Structure
TRB 2013 Annual Meeting Paper revised from original submittal.
10 Casper, Paz de Araujo & Paz de Araujo
315
4 PROJECT COSTING AND RISK ANALYSIS 316
As discussed above, LSP allows a larger bandwidth of preferences by expanding the rigid 317
filtering of formal logic (0, 1) to a more practical infinite set of “percentage” scores for the of preferences 318
around the neutral value of a weighted sum scheme. This methodology is repeated for all projects within 319
the same Decision Network. As an extension of this approach that is not used in this paper, functional 320
descriptions for G (Sij) could also be developed in lieu of simple “weights.” ith final “scoring” 321
complete, budgetary constraints and the risk associated with determining the best cost/benefit ratios can 322
be evaluated as the second part of the decision process. This can be done within a single project circuit or 323
by using all projects in a Decision Network. 324
4.1 Cost and Budget Constraints 325
There are many ways to quantify the budget constraints such that decisions are focused only on 326
projects that fulfill the following criteria: (1) projects that are not too expensive, so that the largest 327
number of projects are selected for inclusion in the Regional Transportation Plan; and (2) that all chosen 328
projects have a maximum “aggregate preference” (max (E0)) and best cost/benefit ratio. This is another 329
great advantage of the LSP process. It actually allows one more degree of freedom to the decision process 330
– that is, after “all the votes are in,” or all aggregate preferences are chosen, the preferences can be 331
weighted further based on budgetary conditions, and clear benefit/cost ratios of each project or by the 332
aggregate. 333
334
S1
S2 E0 Quasi
AND
W1
(1 - W2)
W2
(1 - W1) A
Mandatory
Optional
Quasi-AND Circuit Diagram
Sufficient
Desired
Disjunction Circuit Diagram
S1
S2 E0 Quasi
OR
W1
(1 - W2)
W2
(1 - W1) A
FIGURE 4 LSP Quasi-logic Circuit Diagrams
TRB 2013 Annual Meeting Paper revised from original submittal.
11 Casper, Paz de Araujo & Paz de Araujo
In this paper we use the three cost models presented by J. J. Dujmovic (11), and generalize them with a 335
novel “best use of money” law for the Global Criterion. 336
The simplest rule for “Cost/Benefit” is given by equation [9], below (the linear model). 337
Q = E / C (for the ith project) [Equation 9] 338
339
Where: E is the preference score for the ith project, C is its associated cost, and 340
Q-1
“cost/benefit” ratio. 341
Thus, the function Q means that the score of the jth criterion, resulting from a combination of 342
these quasi-logic functions, versus its cost, is the preference score-to-cost ratio, or inverse cost/benefit 343
ratio. We use Q = (cost/benefit)-1
to stay consistent with the quantitative methods to follow. Figure 5 344
shows a project (i=1), with four performance criteria (Ej = 1,2,3,4) and their associated costs (Cj, 345
j=1,2,3,4). 346
347
e have purposely shown that the “Q” in this example is linear because, in most cases, E vs. C 348
plots are not linear and it is clear that the best rating (slope of the line) is the one with the lowest Q in the 349
E versus C plot. Therefore, the combined rating, or Global Criterion, is parameterized by just the slope of 350
the curve of performance criteria scores versus cost for each project. Thus, it is possible to quantitatively 351
determine if this is the best choice (high Q) or a bad choice (low Q). Conditioning E versus C to be linear 352
can be accomplished through regression techniques or other functional approximation. However, it is not 353
necessary in general that the curve be linear, although it facilitates calculations. 354
Finally it is obvious that the preliminary budget constraint is: 355
n 356
Ci B [Equation 10] 357 i=1 358
Where: B is the total budget and i = 1, …n are for all chosen projects. 359
slope = E =|Q|= 1/(cost/benefit ratio)
C
C1 C2 C3 C4 Ci ($ cost per project)
E4
E3
E1
E2
Ei
(% per project)
FIGURE 5 Preference Score to Cost Relationship
TRB 2013 Annual Meeting Paper revised from original submittal.
12 Casper, Paz de Araujo & Paz de Araujo
However, equation [10] can also be used in different ways within a single project. It is clear that 360
we can also use the aggregate “Q” for the ith project, and the aggregate preferences to show that the best 361
cost/benefit ratio for total budget allocation has been met when equation [11] is satisfied. 362
n 363
Ei /Qi B [Equation 11] 364 i=1 365
Equation [11] shows that the usually qualitative cost/benefit analysis is now quantitatively linked 366
to the budget constraints, and the optimization constraints of the first order model are now met with such 367
quantitative decision parameters embedded in the logic system used. 368
In today’s economy, to maximize the number of projects within budget is difficult to attain. 369
Thus, decision-makers need another degree of freedom to allocate resources marginally to each project 370
using a performance parameter that can “spread the money around” as much as possible and at the same 371
time, show the benefit of one project versus another. Possible cross project bundling and other partial or 372
time-dependent budget factors that are also allowed here are considerations that must be supported in the 373
evaluation of funding allocation alternatives. 374
The required second layer final tuning can be achieved by introducing a final preference score p, 375
to the Q score as shown in equation [12] below. 376
Qk = p (Ek/Emax) + (1-p) (Cmin/Ck), k 1,… n [Equation 12] 377
Where: 0 p 1. 378
In this case, k can mean the ith project or just the j
th criterion taking a larger role (or not) because 379
the cost Ck may exceed an a priori determined minimum cost that the agency or decision-makers are 380
willing to pay. This is a way to quantify the phrases “this is just too expensive at this time to have this 381
performance criterion,” and “not now.” Thus, if such a situation occurs, a new E versus C curve and a 382
new preference-based Q can go up or down depending on the purely resource allocation preferences at the 383
final level of decision. 384
The parameter p is also an “importance parameter” – that is, just how important this or that 385
project is in this budgeting year is the bias on Q by p. Thus, Q can be by project Qi or the Global 386
Criterion. Equally important to know is just how much project i (or criterion j) takes money from the 387
budget or adds to the cost of enforcing such a criterion. This is accomplished by: 388
Qk = pEk + (1-p) Cmax - Ck [Equation 13] 389
Cmax 390
Where: k = i for projects or k = j for criteria within a project 391
For k = i, the Q-index for the project goes up or down depending on the difference Cmax – Ck, or 392
how much the cost of a project takes from the budget. Thus, in this case Cmax = B. In the case k = j, the 393
maximum accepted cost of enforcing criterion j of project i, shows how much that criterion pushes Q up 394
TRB 2013 Annual Meeting Paper revised from original submittal.
13 Casper, Paz de Araujo & Paz de Araujo
or down. Thus, it allows the ability to modify this criterion (in this case Cmax B). In the Colorado 395
Springs data, this step is omitted as p is not yet used in the examples given. 396
Another quantitative correction to the decision process is that all costs can be associated with the 397
present value (PV) of the total cost over time. Thus, 398
Cpresent = Cfuture [Equation 14] 399
(1 + i%)n 400
401
Where: Cpresent = cost this budget year 402
Cfuture = cost after conclusion 403
i% = rate of inflation 404
n = the number of time cycles (months, years, etc.) 405
Thus, all Ci can be analyzed as cost-over-time. This is also a useful measure for Qk if one simply waits 406
and makes no decisions at the current budget versus make the decision and know the future cost. Risk is 407
introduced in the analysis in two ways: 408
(1) The risk of not making a decision or delaying a project as Ck increases over time, and 409
(2) The risks of pitching the wrong projects or choosing projects with diminished 410
synergy – “burning the budget” with insignificant projects 411
For case (1), calculation of two or more values of Q over different PVs suffices. For case (2), the 412
importance index can be used by sorting Q values – that is run a binary search of Qt+1 – Qt = Q (where: 413
the t-indices mean different times) and define the performance, index as: 414
p = Qmin/Q [Equation 15] 415
Where: Qmin means the smallest Q or best cost/benefit. It also means that now the final priority (or 416
performance) index, p, is a function of time. 417
4.2 Law of Best Effectiveness 418
The Law of Best Effectiveness in the Decision Process is described here as the “Decision Power.” 419
It is possible to define just how responsive (effective) the set of decisions were that made one 420
budgetary/project choice better than another. In this paper, “Decision Power” is shown as the curve of Q 421
versus C. Assuming that the combined Q, whether biased by p-indices or not, still follow a curve 422
described generally by: 423
Qi = Ei Ci-
[Equation 16] 424
Where is now the effectiveness parameter, which before was only equal to “1.” This 425
parameter can be used as the only parameter needed for optimization. The higher the value of the , the 426
better is the choice. Further, using a log-log plot of the equation, the slope is equal to . 427
TRB 2013 Annual Meeting Paper revised from original submittal.
14 Casper, Paz de Araujo & Paz de Araujo
Since in a log-log plot, the y-axis is y = ln Q, then when x = ln B, y = 0, or Q = 1. Thus, if all Qi 428
fall on the line or close to it, the budget is met with the best cost/benefit ratio. The budget is completely 429
met and the best set of projects is funded when: 430
= b [Equation 17] 431
ln B 432
433
Where: b = ln Q0, and Q0 = slope of the Ek versus Ck plot for the total number of projects 434
In non-linear plots, Q0 is the derivative of the function approximating E versus C. 435
5 APPLICATION of SIMPLIFIED LSP MODEL TO COLORADO SPRINGS DATA 436
Data for nine sample PPACG 2035 LRTP projects was used to run an Excel platform LSP 437
decision model using the Quasi-AND logic function. For this test adopted weighting and normalized 438
scores for nine projects, for which costs ranged from $12,000 to $80,000,000, were processed in a step-439
wise fashion, using separate, linked worksheets. Data input and processing was completed for five 440
selection criteria at a time, and then linked in a final LSP calculation step. The test project set included 441
roadway, transit and non-motorized mode improvements, demonstrating the necessary broad applicability 442
of the model across modes and project types. The results from test application, shown in Figure 6, 443
demonstrate the feasibility of applying LSP for transportation investment portfolio optimization as it has 444
been applied to other complex optimization problems. However, the simple Quasi-AND test does not 445
begin to tap the potential of an LSP-based decision model to address complex interactions among projects 446
with respect to benefit. The flexibility available within LSP to incorporate less rigid logic functions, such 447
as the Disjunction function, will support development of a logic network that provides the full 448
functionality needed for transportation investment portfolio optimization. 449
450
15.3694
1.3959
0.0573 2.1751
0.0733
0.0454 0.5496
0.2064
0.0074
Benefit/Cost Ratios for Selected Projects Woodland Park Electronic SpeedSignals
SH 105 Sidewalk - Phase I
Sand Creek Corridor TrailImprovements
Highway 85 Widening
Squirrel Creek Road Extension
Ruxton Avenue Pedestrian andDrainage Improvements
Colorado Avenue Reconstruction
Academy Blvd. Corridor Improvements(ABC Great Streets Study)
US 24 West: I-25 to Edlowe Rd.LSP Application Using Quasi-AND Logic Function
FIGURE 6 Results of LSP Quasi-AND Application Test
for PPACG 2035 LRTP Projects
TRB 2013 Annual Meeting Paper revised from original submittal.
15 Casper, Paz de Araujo & Paz de Araujo
6 CONCLUSIONS 451
The zero order WSM decision model commonly used for transportation investment project 452
selection is essentially an individual project evaluation tool that lacks the functionality required for 453
transportation investment portfolio optimization. The inability of the eighted um Model’s formulation 454
to adapt to address exceptional circumstances or to factor in project interactions were concerns during 455
PPACG 2035 LRTP development that marred an otherwise exceptional collaborative planning process. 456
Application of a proposed enhance decision model using a linear programming optimization algorithm to 457
the full 2035 LRTP project set demonstrated significant advantage in maximizing total benefit associated 458
with selected projects as well as the number of selected projects. However, the OPT decision model does 459
not have adequate “bandwidth” to provide flexibility needed to accommodate funding of high cost 460
projects, funding of “critical” projects, nor does it have functionality to address project interactions in a 461
way that is needed for full transportation investment portfolio optimization. A carefully crafted LSP 462
decision network can provide the framework for a broader bandwidth decision model that can include full 463
functionality needed for transportation investment portfolio optimization. Although this approach 464
represents an improved approach for transportation investment, it is an approach that has been 465
successfully implemented for computer system and investment portfolio applications. 466
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16 Casper, Paz de Araujo & Paz de Araujo
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TRB 2013 Annual Meeting Paper revised from original submittal.