Use of reasoning maps in evaluation of transport alternatives: inclusion of uncertainty and “I Don’t Know”: demonstration of a method

Abstract

Selection of a transport alternative is usually a messy process. The traditional approaches consider the relationships as either deterministic or probabilistic, neither of which incorporates the degree of ignorance (i.e., “I don’t know” opinion). Further, different stakeholders seek to justify their preferences with reasoning that suits their agenda. This paper proposes and demonstrates a method that evaluates the validity of the reasoning process and derives the degrees of belief that stated goals are achieved. The paper demonstrates a ‘reasoning map’ method for evaluating transport alternatives, where the analysts accept and employ the notion of “I don’t know” about an issue. The reasoning map depicts the relational chains from the attributes of an action to the stated goals, and recognizes the notion of “I don’t know”. This paper uses the theory of evidence to account for ignorance; it calculates the propagation of uncertainties along the reasoning chains. The context chosen for this demonstration is the selection of a public transit mode, personal rapid transit, over Bus, in a commercial complex in Washington DC. The paper has a limited objective and is not a comprehensive evaluation of alternatives. It merely explains how to compute a numerical value for the strength of reasoning, how to deal with analyst’s notion of “I don’t know,” how to interpret the overall reliability of the reasoning process, how to measure the goal achievement of an alternative, and how to find the critical paths linking the planning options to goals. For use in planning practice, consultation of experts and affected citizens and aggregation of their views is needed to develop the reasoning maps.

Appendices

Appendix A: the Dempster–Shafer theory of evidence

Probability theory is the established mathematical framework for measuring risk and uncertainty of proposition in nearly all scientific fields. It measures the degree of belief, “x is A,” based on the weight of the available evidence about x. The strength of opinion or evidence is the probability distribution about x, p(x). In this framework, each outcome x is a clearly defined set, such as head or tail of a coin. Probability theory, however, is not capable of measuring the outcome “I don’t know” or mixed outcomes. The weight of evidence must be defined for each well-defined outcome clearly, and the measures are additive, e.g. p(A) + p(Not A) = 1. As a result, if the weight of evidence for A is given as α, then the weight of evidence that for “Not A” is 1-α, although there may be no such evidence for “Not A.” In other words, the weight of evidence for “A or Not A,” i.e., “I don’t know” cannot be measured. This Appendix briefly introduces how the degree of belief is presented and the aspect of “I don’t know” is handled in the Dempster–Shafer theory of evidence.

Consider the following. There are three possible outcomes (A, B, and C) of an experiment. Figure 7 shows the weight of evidence toward possible outcomes: 0.2 for A; 0.3 for A or B (we cannot tell either A or B); 0.05 evidence for C, and 0.45 for unknown (we have no idea which outcome will occur.) Each piece of evidence is considered as the basic probability assignment, m(Z); it is the weight of evidence attached to statement Z, where Σm(Z) = 1 for all Z. Z is the power sets: A, B, C; A or B; A or C; B or C; and A or B or C.

Fig. 7

Measures of degrees of belief in Dempster–Shafer Theory

As shown in Fig. 7, the weight of evidence for A can be said in two ways, optimistically 0.95 (=0.2 + 0.3 + 0.45) and conservatively 0.2. The optimistic measure is based on weighting all positive or at least not negative evidence pointing to A; the conservative measure is based on only the positive evidence strictly pointing to A. These measures are called Plausibility (Pl) and Belief (Bel), respectively. In this formulation, m(A or B), m(A or C), m(B or C), and m(A or B or C) represent ignorance with respect to individual sets, since the evidence cannot be exclusively located in individual outcome A, B, or C. m(A or B) indicates the weight of evidence that the analyst cannot discern between A and B; in other words, I don’t know if the outcome is A or B.

This set of measures is introduced to appraise the strength of inference associated with causality or input–output relation, X → Y, where X is not completely clear, and also the causality (X → Y) is not clear and includes “I don’t know.” The operation is shown by examples in the following section.

When different experts or planners have different opinions about the degrees of belief of the proposition, Dempster’s rule of combination (DRC) aggregates different opinions. Unlike aggregation of probabilistic distribution, DRC brings out the degree of belief for those propositions that are agreeable and subdues those that are not agreeable. The traditional Bayesian based aggregation takes the average value of the evidence of each proposition, not accounting for the opinions of “I don’t know.” Application of these characteristics of Dempster–Shafer theory is explained using examples.

A useful feature of the outcome of Dempster–Shafer theory is that because the distribution m(Z) includes “I don’t know”, there are two measures of uncertainty about proposition. The plausibility measure (Pl) and belief measure (Bel).

Appendix B: calculation process

This Appendix presents the calculation process of the proposed approach. Two basic analysis steps are involved. One is the inference method which propagates the knowledge from premises to conclusions. The other is the Dempster’s Rule of Combination which aggregates the knowledge from different pieces of evidence (opinions.) These two steps are described using the application example.

Consider two reasoning chains as shown in Fig. 8. One proponent of a PRT system may argue that PRT will improve passengers’ travel time because it provides a short and direct trip with a door-to-door service for each passenger. Another may argue that PRT vehicles improve travel time because they will proceed to the destination without stopping due to its off-line stations, and as a result, PRT minimizes delay and improves travel time. The first chain connects between “Short and Direct Trip” (X) and “Short Travel Time,” (Z) and the second connects between “Non-Stop Operation” (Y) and “Short Travel Time” (Z). Each attribute has three states: “Agree,” “Disagree,” and “I don’t know.”

Fig. 8

Example of calculation process in the proposed reasoning

The beliefs about the inputs “Short and Direct Trip,” m(X), and “Non-Stop Operation,” m(Y), and the beliefs about causal relations connecting each of these two inputs to the outcome “Short Travel Time,” m(Z|X) and m(Z|Y) are given as shown in Fig. 8. The beliefs about the “Short Travel Time,” m(Z), are calculated as follows.

Using inference method, the beliefs about input, m(X), are combined with the beliefs about the relation X → Y, m(Y|X), by using matrix multiplication. Let X ₁, X ₂, and X ₁ or X ₂ are “Agree,” “Disagree,” and “I don’t know” about “Short and Direct Travel,” respectively. Z ₁, Z ₂, and Z ₁ or Z ₂ are “Agree,” “Disagree,” and “I don’t know” about “Short Travel Time,” respectively. The beliefs for the outcome “Short Travel Time” from the first reasoning chain m ₁ (Z) are calculated as

$$m_{1} \left( {Z_{1} } \right) = m\left( {Z_{1} |X_{1} } \right) \cdot m\left( {X_{1} } \right) + m\left( {Z_{1} |X_{2} } \right) \cdot m\left( {X_{2} } \right) + m\left( {Z_{1} |X_{1} {\text{ or }}X_{2} } \right) \cdot m\left( {X_{1} {\text{ or }}X_{2} } \right) = 0.52$$

$$m_{1} \left( {Z_{2} } \right) = m\left( {Z_{2} |X_{1} } \right) \cdot m\left( {X_{1} } \right) + m\left( {Z_{2} |X_{2} } \right) \cdot m\left( {X_{2} } \right) + m\left( {Z_{2} |X_{1} {\text{ or }}X_{2} } \right) \cdot m\left( {X_{1} {\text{ or }}X_{2} } \right) = 0.12$$

$$m_{1} \left( {Z_{1}\, {\text{or}}\,Z_{2} } \right) = m\left( {Z_{1} \, {\text{or}} \, Z_{2} |X_{1} } \right) \cdot m\left( {X_{1} } \right) + m\left( {Z_{1} \, {\text{or}} \, Z_{2} |X_{1} } \right) \cdot m\left( {X_{1} } \right) + m\left( {Z_{1} \, {\text{or}} \, Z_{2} |X_{1} {\text{or}}X_{2} } \right) \cdot m\left( {X_{1} \, {\text{or}} \, X_{2} } \right) = 0.36$$

Similarly, the degrees of belief of the outcome “Short Travel Time” from the second chain m ₂ (Z) are

$$m_{2} \left( {Z_{1} } \right) = m\left( {Z_{1} |Y_{1} } \right) \cdot m\left( {Y_{1} } \right) + m\left( {Z_{1} |Y_{2} } \right) \cdot m\left( {Y_{2} } \right) + m\left( {Z_{1} |Y_{1} {\text{ or }}Y_{2} } \right) \cdot m\left( {Y_{1} {\text{ or }}Y_{2} } \right) = 0.68$$

$$m_{2} \left( {Z_{2} } \right) = m\left( {Z_{2} |Y_{1} } \right) \cdot m\left( {Y_{1} } \right) + m\left( {Z_{2} |Y_{1} } \right) \cdot m\left( {Y_{1} } \right) + m\left( {Z_{2} |Y_{1} {\text{ or }}Y_{2} } \right) \cdot m\left( {Y_{1} {\text{ or }}Y_{2} } \right) = 0.13$$

$$m_{2} \left( {Z_{1} \, {\text{or}} \, Z_{2} } \right) = m\left( {Z_{1} \, {\text{or}} \, Z_{2} |Y_{1} } \right) \cdot m\left( {Y_{1} } \right) + m\left( {Z_{1} \, {\text{or}} \, Z_{2} |Y_{2} } \right) \cdot m\left( {Y_{2} } \right) + m\left( {Z_{1} \, {\text{or}} \, Z_{2} |Y_{1} \, {\text{or}} \, Y_{2} } \right) \cdot m\left( {Y_{1} \, {\text{or}} \, Y_{2} } \right) = 0.19$$

Using Dempster’s Rule of Combination, the degree of belief for the outcome from two reasoning chains, m ₁(Z) and m ₂(Z), are aggregated (see the table in Fig. 8). Figure 9 shows the distribution of belief values of Evidence 1, 2, and their combination. For the first reasoning chain, the degrees of belief are m ₁(Z ₁) = 0.52, m ₁(Z ₂) = 0.12, and m ₁(Z ₁ or Z ₂) = 0.36, and for the second m ₂(Z ₁) = 0.68, m ₂(Z ₂) = 0.13, and m ₂(Z ₁ or Z ₂) = 0.19. The combined degree of belief m(Z) is calculated as:

Fig. 9

Illustration of distributions of belief values in Dempster–Shafer theory

$$m\left( {Z_{1} } \right) = \frac{{m_{1} \left( {Z_{1} } \right) \cdot m_{2} \left( {Z_{1} } \right) + m_{1} \left( {Z_{1} } \right) \cdot m_{2} \left( {Z_{1} \,{\text{or}}\,{\text{Z}}_{2} } \right) + m_{1} \left( {Z_{1} \,{\text{or}}\,{\text{Z}}_{2} } \right) \cdot m_{2} (Z_{1} )}}{{1 - m_{1} \left( {Z_{1} } \right) \cdot m_{2} \left( {Z_{2} } \right) - m_{1} \left( {Z_{2} } \right) \cdot m_{2} \left( {Z_{1} } \right)}} = \frac{0.35 + 0.10 + 0.25}{1 - 0.06 - 0.09} = 0.82$$

$$m\left( {Z_{2} } \right) = \frac{{m_{1} \left( {Z_{2} } \right) \cdot m_{2} \left( {Z_{2} } \right) + m_{1} \left( {Z_{2} } \right) \cdot m_{2} \left( {Z_{1} \,{\text{or}}\,Z_{2} } \right) + m_{1} \left( {Z_{1} \,{\text{or}}\,Z_{2} } \right) \cdot m_{2} \left( {Z_{2} } \right)}}{{1 - m_{1} \left( {Z_{1} } \right) \cdot m_{2} \left( {Z_{2} } \right) - m_{1} \left( {Z_{2} } \right) \cdot m_{2} \left( {Z_{1} } \right)}} = \frac{0.02 + 0.02 + 0.05}{1 - 0.06 - 0.09} = 0.10$$

$$m\left( {Z_{1} \,{\text{or}}\,Z_{2} } \right) = \frac{{m_{1} \left( {Z_{1} \,{\text{or}}\,{\text{Z}}_{2} } \right) \cdot m_{2} \left( {Z_{1} \,{\text{or}}\,{\text{Z}}_{2} } \right)}}{{1 - m_{1} \left( {Z_{1} } \right) \cdot m_{2} \left( {Z_{2} } \right) - m_{1} \left( {Z_{2} } \right) \cdot m_{2} \left( {Z_{1} } \right)}} = \frac{0.07}{1 - 0.06 - 0.19} = 0.08$$

The numerator of DRC is the sum of the product of the belief values associated with outcomes of evidence that supports set X. The denominator of DRC is a normalizing factor which is the sum of the product of the belief values associated with all the possible combinations of evidence that are not in conflict.

In this example, both pieces of evidence support the outcome “Short Travel Time”; as a result, the truth value associated with “Short Travel Time” increases from 0.52 and 0.68 to 0.82, and the degree of uncertainty (I don’t know) decreases from 0.36 and 0.19 to 0.08. This characteristic is intuitive and consistent with human’s mind building up their belief. The degree of truth about “Short Travel Time” can be presented by two measures. In this case, the Belief (Bel) measure of “Short Travel Time” is 0.82, and the Plausibility (Pl) measure is 0.90 (=0.82 + 0.08).

Using the calculation process discussed, the degrees of belief in achievement of individual goals are derived. Figure 10 shows the beliefs for all inputs assigned by experts (in black) and the beliefs for all the outcomes determined by the proposed approach (in red) in the reasoning map considered in this study.

Fig. 10

Strength of reasoning