logic scoring of preference (lsp) application to transportation …docs.trb.org/prp/13-1830.pdf ·...

1 Casper, Paz de Araujo & Paz de Araujo

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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

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Maureen Paz de Araujo 19 HDR Engineering 20

2060 Briargate Parkway, Suite 120 21 Colorado Springs CO 80920 22 Telephone: 719-272-8833 23


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


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Submitted for consideration for presentation or poster session at the 37

Transportation Research Board 38

Annual Meeting 39

January 2013 40

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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

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Words = 5778 words + (6 figures @ 250 each = 1500) = 5531 + 1500 = 7278 of 7500 permitted 47

TRB 2013 Annual Meeting Paper revised from original submittal.


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



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



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



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



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



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



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



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



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



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



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



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



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



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



REFERENCES 467

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