An improved rough set approach to evaluate auto-mobility systems

Abstract

To evaluate auto-mobility systems and find out which one can provide better environmental, social, economic benefits and better system performance, we improve a rough set approach proposed by [1] with the combination of Rough Analytic Hierarchy Process (R-AHP) and Rough Technique for Order Preference by Similarity to Ideal Solution (R-TOPSIS). There are three steps in the improved approach. In the first step, proper indicators are identified by means of literature review and expert interviews, and they are divided into two categories, benefit and cost. In the second step, weight for each indicator is determined by R-AHP. In the third step, different alternatives are compared and ranked by R-TOPSIS. A case study is provided to show the detailed procedure of the improved approach by comparing car-sharing, ride-sharing and car-owning. And sensitivity analysis experiments are conducted to figure out the effect of indicators on decision making process. We arrive at the conclusion that ride-sharing is the best auto-mobility option in most cases and indicators play a significant role in the evaluation of auto-mobility options.

Keywords

Rough analytic hierarchy process rough technique for order preference by similarity to ideal solution auto-mobility indicator

1 Introduction

Most urban areas are suffering from terrible air pollution and congestion because of the ever-increasing traffic on road. Governments and administrative branches are trying their best to change this kind of condition by various kinds of means. For example, Beijing, capital of China, restricts new purchase of private vehicles and charges more for parking lots and invests more money in public transportation construction such as subway, which turns to be the most crowded yet efficient means of transportation especially during rush hours. Public transportation definitely has its advantage over private cars in terms of energy efficiency and transporting capacity. However, people still need private transportation due to its flexibility and convenience.

With the growing purchasing power of general public, owning a private car is no longer a dream to the majority of Chinese citizens. Registered motor vehicles in China have reached 290 million by the end of 2016, among which 194 million are private cars [2]. However, most of the private cars are not used efficiently. McKinsey Center for Business and Environment [3] reports that a typical European car is parked during 92% of its lifetime, 1% sitting in congestion, 1.6% looking for parking, 5% driving. Moreover, a typical private car has 5 seats but carries 1.5 people per trip on average. This results in serious congestion and great waste of energy.

Car-sharing systems (CSS) advocate the idea of sharing cars with others, which provide a flexible alternative that meets diverse transportation needs across the globe and reduces the negative impacts of private vehicle ownership [4]. A car-sharing system has a relatively small fleet of vehicles available for a relatively larger group of members [5]. Car-Sharing systems, are a complementary to private vehicle ownership. As application of product service system in the field of urban mobility [6] and an alternative to public transportation, car-sharing has a positive impact on urban mobility, mainly because each vehicle is used more efficiently. Car-sharing systems have the potential to replace many private vehicles and can have further significant benefits if electric vehicles are used [7]. Ride-sharing is a system which matches up drivers and riders on very short notice or even en-route” [8], combining flexibility and speed of private cars with reduced travel cost and loss of convenience to some extent [9]. Yet it is a challenging task to match drivers and riders in a short period of time. Besides, passengers may not want to share a ride with strangers for safety or privacy reasons. Car-owning is the most conventional way of private transportation and it is the first choice of many people due to its convenience and mobility. However, many low-income families cannot afford a private car [10].

The purpose of this study is to compare three auto-mobility options, i.e., car-sharing, ride-sharing, and car-owning and find out which can provide better performances. It is no easy job due to lack of prior information and the subjectivity and vagueness existing in the evaluation process. Therefore, we improve the rough set approach proposed by [1] and apply it to evaluate auto-mobility systems (AMS) under uncertain environments.

The remainder of this paper is organized in the following way. Section 2 introduces related works as to car-sharing, ride-sharing, and application of rough set theory in decision-making. Section 3 is about fundamental notions of rough set theory. Section 4 describes the application of R-AHP in weights determination and R-TOPSIS in ranking different alternatives. Then, a framework to evaluate auto-mobility systems is proposed and analyzed in Section 5. A case study is provided in Section 6 to show the detailed procedure of the proposed framework. We conclude the paper and discuss the significance of the improved approach in the last section.

2 Related works

2.1 Car-sharing system

Car-sharing involves a small fleet of vehicles available at several stations to be used by a relatively large group of members [11]. It can be classified into two categories: “one-way” and “two-way”, according to different vehicle returning rules. Two-way system requires that users should return the rented vehicle at the location where they picked it up and one-way system is more flexible without restrictions as to the returning station [12]. Baptista et al. [13] conducted an in-depth analysis of a car-sharing system in Lisbon to quantify its energy and environmental impacts and comprehend the influencing factors as to its economic feasibility, which confirms that car sharing contributes to a more efficient and sustainable mobility and reduction of energy and environmental impacts.

Jorge et al. [14] developed an integer programming model to design one-way trips to supplement a car-sharing company presently operating round trips. There is a great potential to integrate round-trip and one-way car-sharing services. However, they did not consider flexibility of demand to price. And they did not take into account the problems of expanding the capacity of existing stations to deal with one-way car-sharing service and increasing the vehicle fleet as well. Fassi et al. [15] proposed a decision support approach to assist decision makers to assess different growth strategies of CSS. The proposed approach is able to maximize customer satisfaction level and minimize the number of vehicles as well.

Nair and Miller-Hooks [16] developed an equilibrium network to configure a flexible vehicle sharing program (VSP) in which users are able to check out vehicles and return them to nearby stations. Future works may focus on congestion effects on transit networks, joint optimization of prices and design, price sensitivity of users in route choice, and multiclass extensions. Wagner et al. [17] developed an innovative Business Analytics approach to support car-sharing decision making process and estimate spatially varying demand for car-sharing services, on the basis of two hypotheses.

2.2 Ride-sharing system

Ride-sharing is defined as “a system where an automated process employed by a ride-share provider matches up drivers and riders on very short notice [18]. Furuhata et al. [9] focused on the key aspects of present ride-sharing systems to develop a framework to identify key challenges in the adoption of ride-sharing. Agatz et al. [18] proposed optimization-based approaches to minimize the total system-wide vehicle distances and travel costs. They applied rolling horizon strategy to deal with uncertainty and construct computer simulations.

Agatz et al. [8] identified the objectives of a ride-sharing system: to minimize system-wide vehicle-miles, minimize the system-wide travel time, and maximize the number of participants. They listed dynamic ride-sharing problems as arrival of riders and drivers, anticipation of future requests, and deviation from planned trips. Savelsbergh [19] studied the role of dedicated drivers in a dynamic ride-sharing system and benefits, complexities, and costs of ride-sharing. The authors identified the following three factors as most influential: the number of participants in ride-sharing process; the time flexibility of participants; and the similarity between origins and destinations of the participants. And they conducted a computational study to show the benefits of dedicated drivers and identify the circumstances under which dedicated drivers are most helpful.

2.3 Application of rough set theory

Rough set theory was proposed by Pawlak [20] to deal with imprecision, vagueness and uncertainty. Different from fuzzy sets, evidence theory, and probability theory, the axiom system in rough set is definitely objective, eliminating any prior information that is not related with the data to be handled [21]. Pawlak [22] outlined basic concepts of rough set theory including indiscernibility relation, approximations, rough membership, dependency of attributes, and reduction of attributes. He proposed decision tables and decision algorithms and discussed its possible application in decision support process. Li et al. [23] integrated rough set, Kano’s model and AHP to rate the final importance of customer requirements in product planning house of quality. Aydogan [24] developed a conceptual performance evaluation framework with consideration of quantitative and qualitative factors by applying rough-AHP and fuzzy-TOPSIS.

Wenyan Song and his team from Shanghai Jiaotong University made a great contribution as to the application of rough set theory in evaluation process. Song et al. [25] proposed a rough set approach to assess customer requirements of industrial product service system with uncertainties and vagueness. Yet it is time-consuming for decision makers to adjust their judgments in order to pass the consistency test. Song et al. [26] proposed an innovative decision approach to choose the best new product options at concept design stage involving subjectivity and uncertainty. They integrated rough AHP and rough TOPSIS to improve the quality and efficiency of concept evaluation for new products development. However, they did not consider the interdependencies among all criteria. Song et al. [26] proposed an innovative approach to improve the effectiveness of failure mode and effect analysis using rough TOPSIS, demonstrating the power of rough set theory and TOPSIS to deal with vagueness. However, the proposed method also asks for additional knowledge in rough set theory, which is somewhat difficult for experts.

To sum up, we identify a research gap as to the willingness of adopting car-sharing or ride-sharing and giving up car-owning by means of a comprehensive comparison of auto-mobility systems. Besides, we simplify the computation process of relevant equations and improve the efficiency of the method proposed by [1] with application of MATLAB and extend the application of rough set theory in auto-mobility systems.

3 Fundamental notions of rough set theory

3.1 Rough boundary interval and rough number

Rough set theory is proved to be efficient in dealing with inconsistency problems [27]. Rough boundary interval and rough number are two important notions proposed on the basis of rough set theory, which are used to handle the subjectivity and vagueness in the decision making process [28].

Definition 1. Suppose U is a universe whose objects belong to n classes, that is to say, R = { C₁, C₂, ⋯ , C_n } . If the classes are ranked in the way of C₁ < C₂ < ⋯ < C_n, for any particular class C_i ∈ R, 1 ≤ i ≤ n, the lower and upper approximations of C_i can be given by Equations (1) and (2) respectively: $\underline{Apr} (C_{i}) = \cup {Y \in U / R (Y) \leq C_{i}},$ (1) $\bar{Apr} (C_{i}) = \cup {Y \in U / R (Y) \geq C_{i}} .$ (2)

Here, $\underline{Apr} (C_{i}), \bar{Apr} (C_{i})$ represent the lower and upper approximations of C_i respectively.

The boundary region of C_i is defined as

$\begin{matrix} Bnd (C_{i}) & = & \cup {Y \in U / R (Y) \neq C_{i}} \\ = & {Y \in U / R (Y) > C_{i}} \\ \cup {Y \in U / R (Y) < C_{i}} \end{matrix}$ (3)

Bnd (C_i) stands for the boundary region of C_i. The above definition describes the ordered classes using upper and lower approximations. In other words, for any class C_i in a set of ordered classes R, the lower approximation of C_i refers to all the objects whose class values equal to or less than C_i; similarly, the upper approximation of C_i refers to all the objects whose class values equal to or bigger than C_i; moreover, the boundary region of C_i contains all the objects whose class values are different from C_i. On the basis of the above definition, rough number, its lower and upper limits and rough boundary interval can be defined as well.

Definition 2. C_i can be represented by its rough number (RN) consisting a lower limit $(\underline{Lim} (C_{i}))$ and an upper limit $(\bar{Lim} (C_{i}))$ . $\underline{Lim} (C_{i}) = \frac{1}{M_{L}} \sum R (Y) | Y \in \underline{Apr} (C_{i})$ (4)

Here, M_L stands for the number of objects contained in the lower approximation of C_i, and $\bar{Lim} (C_{i}) = \frac{1}{M_{U}} \sum R (Y) | Y \in \bar{Apr} (C_{i})$ (5)

Here, M_U stands for the number of objects contained in the upper approximation of C_i.

Therefore, the rough boundary interval refers to the interval between the lower limit $(\underline{Lim} (C_{i}))$ and the upper limit $(\bar{Lim} (C_{i}))$ , which is represented by RBnd (C_i) as: $RBnd (C_{i}) = \bar{Lim} (C_{i}) - \underline{Lim} (C_{i})$ (6)

Definition 2 makes it possible to express a rough number with its lower limit and upper limit as follows: $RN (C_{i}) = [\underline{Lim} (C_{i}), \bar{Lim} (C_{i})]$ (7)

The rough boundary interval of a class means the difference between its upper and lower limits, representing its vagueness. A class with a bigger rough boundary interval is vaguer. Compared with fuzzy numbers, rough numbers have flexible boundary intervals, representing the subjective and vague cognition of decision makers while maintaining the objectivity of original data at the same time [29].

3.2 Calculating algorithm of rough numbers

Suppose RN_x = [L_x, U_x], RN_y = [L_y, U_y] are two rough numbers, where L_x, L_y are their lower limits while U_x, U_y are upper limits. Then, $\begin{matrix} {RN}_{x} + {RN}_{y} & = & [L_{x}, U_{x}] + [L_{y}, U_{y}] \\ = & [L_{x} + L_{y}, U_{x} + U_{y}] \end{matrix}$ (8) $\begin{matrix} {RN}_{x} - {RN}_{y} & = & [L_{x}, U_{x}] - [L_{y}, U_{y}] \\ = & [L_{x} - L_{y}, U_{x} - U_{y}] \end{matrix}$ (9) $\begin{matrix} {RN}_{x} \times {RN}_{y} & = & [L_{x}, U_{x}] \times [L_{y}, U_{y}] \\ = & [L_{x} \times L_{y}, U_{x} \times U_{y}] \end{matrix}$ (10) $k \times {RN}_{x} = k \times [L_{x}, U_{x}] = [{kL}_{x}, {kU}_{x}]$ (11)

4 R-AHP and R-TOPSIS

4.1 R-AHP

Step 1. Constructing hierarchy system for AMS evaluation indicators.

Step 2. Establishing pair-wise comparison matrix A_i. Suppose there are m indicators and l decision makers, then pair-wise comparison matrix A_i can be constructed as follows:

$\begin{matrix} A_{i} & = & [\begin{matrix} 1 & x_{12}^{k} & \dots & x_{1 m}^{k} \\ x_{21}^{k} & 1 & \dots & x_{2 m}^{k} \\ ⋮ & ⋮ & x_{ij}^{k} & ⋮ \\ x_{m 1}^{k} & x_{m 2}^{k} & \dots & 1 \end{matrix}], i = 1, 2, \dots, m; \\ j = 1, 2, \dots, m; k \in [1, l] \end{matrix}$ (12)

Here $x_{ij}^{k}$ stands for the relative importance of indicator j against indicator i using nine-score evaluation method. It is obvious that $x_{ji}^{k} = 1 / x_{ij}^{k}$ . Elements in the above matrix are crisp numbers instead of rough ones. Rough group decision matrix is established in the next step to deal with different opinions from different evaluators. Consistency should be checked to ensure the consistency of the constructed matrix by consistency index (CI) and consistency ratio (CR). $CI = (λ_{max} - n) / - (n - 1)$ (13) $CR = CI / RI$ (14)

RI stands for random index and can be obtained from Table 1 [30].

Table 1

Random index RI (n)

n	1	2	3	4	5	6	7	8	9	10
RI	0	0	0.58	0.9	1.12	1.24	1.32	1.41	1.45	1.49

When CR ≤ 0.1, the matrix is ok with satisfying consistency. Otherwise, it should be revised.

Step 3. Constructing rough group decision making matrix and computing rough numbers for each indicator in the matrix. $A^{*} = [\begin{matrix} 1 & X_{12}^{*} & \dots & X_{1 m}^{*} \\ X_{21}^{*} & 1 & \dots & X_{2 m}^{*} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ X_{m 1}^{*} & X_{m 2}^{*} & \dots & 1 \end{matrix}]$ (15)

Where $X_{12}^{*} = {x_{12}^{1}, x_{12}^{2}, \dots, x_{12}^{l}}$ and other elements in the matrix can be obtained in the same way. For instance, the rough number of $X_{12}^{*}$ can be obtained by Equations (1–7). $RN (x_{12}^{k}) = [x_{12}^{k -}, x_{12}^{k +}]$ (16)

Where $x_{12}^{k -}, x_{12}^{k +}$ are the lower limit and upper limit of $RN (x_{12}^{k})$ ; k ∈ [1, l]. Therefore,

$\begin{matrix} RN (X_{12}^{*}) \\ = {[x_{12}^{1 -}, x_{12}^{1 +}], [x_{12}^{2 -}, x_{12}^{2 +}], \dots, [x_{12}^{l -}, x_{12}^{l +}]} \end{matrix}$ (17)

Using Equations (8–11), the average rough interval is $RN (X_{12}) = [x_{12}^{-}, x_{12}^{+}]$ (18) $\begin{matrix} x_{12}^{-} = (x_{12}^{1 -} + x_{12}^{2 -} + \dots + x_{12}^{l -}) / l, \\ x_{12}^{+} = (x_{12}^{1 +} + x_{12}^{2 +} + \dots + x_{12}^{l +}) / l . \end{matrix}$ (19)

Step 4. Constructing rough pair-wise comparison matrix. $X = [\begin{matrix} [1, 1] & [x_{12}^{-}, x_{12}^{+}] & \dots & [x_{1 n}^{-}, x_{1 n}^{+}] \\ [x_{21}^{-}, x_{21}^{+}] & [1, 1] & \dots & [x_{2 n}^{-}, x_{2 n}^{+}] \\ ⋮ & ⋮ & ⋱ & ⋮ \\ [x_{n 1}^{-}, x_{n 1}^{+}] & [x_{n 2}^{-}, x_{n 2}^{+}] & \dots & [1, 1] \end{matrix}]$ (20)

Here, $[x_{ij}^{-}, x_{ij}^{+}]$ is a rough number with $x_{ij}^{-}, x_{ij}^{+}$ representing its lower and upper limits respectively.

Step 5. Calculating weight for each indicator.

First, we divide X into rough lower limits matrix X^- and rough upper limits matrix X⁺. $X^{-} = [\begin{matrix} 1 & x_{12}^{-} & \dots & x_{1 n}^{-} \\ x_{21}^{-} & 1 & \dots & x_{2 n}^{-} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ x_{n 1}^{-} & x_{n 2}^{-} & \dots & 1 \end{matrix}],$ (21) $X^{+} = [\begin{matrix} 1 & x_{12}^{+} & \dots & x_{1 n}^{+} \\ x_{21}^{+} & 1 & \dots & x_{2 n}^{+} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ x_{n 1}^{+} & x_{n 2}^{+} & \dots & 1 \end{matrix}]$ (22)

Then we calculate their eigenvalues and eigenvectors with the aid of MATLAB software and get relative weight for each indicator.

$\begin{matrix} W^{-} & = & [w_{1}^{-}, w_{2}^{-}, \dots, w_{n}^{-}]^{T}, \\ W^{+} & = & [w_{1}^{+}, w_{2}^{+}, \dots, w_{n}^{+}]^{T} \end{matrix}$ (23)

Thirdly, we get the normalized weight by $f_{i}^{-} = w_{i}^{-} / \sum_{i = 1}^{n} w_{i}^{-}, f_{i}^{+} = w_{i}^{+} / \sum_{i = 1}^{n} w_{i}^{+}$ (24)

The weight of indicator C_i can be computed by $f (C_{i}) = \frac{1}{2} (| f_{i}^{-} | + | f_{i}^{+} |)$ (25)

Step 6. Suppose we divide all the indicators into different dimensions, then composition of synthetic weight of each dimension and indicator by: $f^{*} (ω_{i}) = f (D_{i}) \times f (C_{i})$ (26)

f^* (ω_i) , f (D_i) represent the synthetic weight of indicator C_i and weight of dimension D_i.

4.2 R-TOPSIS

Step 1. Establish crisp AMS evaluation matrix.

Suppose there are n AMS alternatives A_j (j = 1, 2, ⋯ , n) to be assessed against m indicators C_i (i = 1, 2, ⋯ , m). Decision makers apply conventional crisp scores (1, 2, 3, ⋯ 9) to give their ratings of alternatives as to indicators C_i (i = 1, 2, ⋯ , m). If there are l decision makers in an expert group, the AMS alternatives ranking can be constructed as a multi-criteria decision-making problem, being represented by the evaluation matrix D as: $D = \begin{matrix} \begin{matrix} A_{1} & A_{2} & \dots & A_{j} \end{matrix} \\ \begin{matrix} C_{1} \\ C_{2} \\ ⋮ \\ C_{i} \end{matrix} & [\begin{matrix} x_{11}^{k} & x_{12}^{k} & \dots & x_{1 n}^{k} \\ x_{21}^{k} & x_{22}^{k} & \dots & x_{2 n}^{k} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ x_{m 1}^{k} & x_{m 2}^{k} & \dots & x_{mn}^{k} \end{matrix}] \end{matrix},$ (27) $\begin{matrix} (i = 1, 2, \dots, m; j = 1, 2, \dots, n; \\ k = 1, 2, \dots l) \end{matrix}$

Where $x_{ij}^{k}$ stands for the rating of the kth decision maker for the jth alternative against indicator i.

Step 2. Convert the crisp factor $x_{ij}^{k}$ in group decision matrix D into rough number and construct rough group decision-making matrix D^* according to Equations (1–7). $D^{*} = \begin{matrix} \begin{matrix} A_{1} & A_{2} & \dots & A_{j} \end{matrix} \\ \begin{matrix} C_{1} \\ C_{2} \\ ⋮ \\ C_{i} \end{matrix} & [\begin{matrix} X_{11}^{*} & X_{12}^{*} & \dots & X_{1 n}^{*} \\ X_{21}^{*} & X_{22}^{*} & \dots & X_{2 n}^{*} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ X_{m 1}^{*} & X_{m 2}^{*} & \dots & X_{mn}^{*} \end{matrix}] \end{matrix},$ (28) $\begin{matrix} (i = 1, 2, \dots, m; j = 1, 2, \dots, n; \\ k = 1, 2, \dots l) \end{matrix}$

Where $X_{12}^{*} = {x_{12}^{1}, x_{12}^{2}, \dots, x_{12}^{l}}$ and other elements in the matrix can be obtained in the same way. For instance, the rough number of $X_{12}^{*}$ can be obtained by: $RN (x_{ij}^{k}) = [x_{ij}^{k -}, x_{ij}^{k +}]$ (29)

Here k ∈ [1, l] and $x_{ij}^{k -}, x_{ij}^{k +}$ represent the lower limit and upper one of rough number $RN (x_{ij}^{k})$ .

Therefore, rough number $RN (X_{ij}^{*})$ can be obtained by:

$\begin{matrix} RN (X_{ij}^{*}) \\ = {[x_{ij}^{1 -}, x_{ij}^{1 +}], [x_{ij}^{2 -}, x_{ij}^{2 +}], \dots, [x_{ij}^{l -}, x_{ij}^{l +}]} \end{matrix}$ (30)

Using Equations (8–11), the average rough interval is $RN (X_{ij}) = [x_{ij}^{-}, x_{ij}^{+}]$ (31) $\begin{matrix} x_{ij}^{-} = (x_{ij}^{1 -} + x_{ij}^{2 -} + \dots + x_{ij}^{l -}) / l, \\ x_{ij}^{+} = (x_{ij}^{1 +} + x_{ij}^{2 +} + \dots + x_{ij}^{l +}) / l . \end{matrix}$ (32)

$x_{ij}^{-}, x_{ij}^{+}$ are lower limit and upper limit of rough number RN (X_ij).

Then, the rough group decision matrix R is established as follows: $R = [\begin{matrix} [x_{11}^{-}, x_{11}^{+}] & [x_{12}^{-}, x_{12}^{+}] & \dots & [x_{1 n}^{-}, x_{1 n}^{+}] \\ [x_{21}^{-}, x_{21}^{+}] & [x_{22}^{-}, x_{22}^{+}] & \dots & [x_{2 n}^{-}, x_{2 n}^{+}] \\ ⋮ & ⋮ & ⋱ & ⋮ \\ [x_{m 1}^{-}, x_{m 1}^{+}] & [x_{m 2}^{-}, x_{m 2}^{+}] & \dots & [x_{mn}^{-}, x_{mn}^{+}] \end{matrix}]$ (33)

Step 3. Normalize rough group decision matrix by the following equations [1].

For benefit indicators,

$\begin{matrix} x_{ij}^{' -} & = & x_{ij}^{-} / - {max}_{j = 1}^{n} (x_{ij}^{-}, x_{ij}^{+}), \\ x_{ij}^{' +} & = & x_{ij}^{+} / - {max}_{j = 1}^{n} (x_{ij}^{-}, x_{ij}^{+}) \end{matrix}$ (34)

For cost indicators,

$\begin{matrix} x_{ij}^{' -} & = & {min}_{j = 1}^{n} (x_{ij}^{-}, x_{ij}^{+}) / - x_{ij}^{+}, \\ x_{ij}^{' +} & = & {min}_{j = 1}^{n} (x_{ij}^{-}, x_{ij}^{+}) / - x_{ij}^{-} \end{matrix}$ (35)

$x_{ij}^{' -}, x_{ij}^{' +}$ stand for the lower and upper limits of normalized interval $[x_{ij}^{' -}, x_{ij}^{' +}]$ .

Step 4. Applying the indicator weights obtained in Section 4.1, compute the weighed normalized decision matrix in rough numbers as follows:

$\begin{matrix} W_{ij}^{-} & = & f^{*} (ω_{i}) \times x_{ij}^{' -}, i = 1, 2, \dots, m, \\ j = 1, 2, \dots, n . \end{matrix}$ (36) $\begin{matrix} W_{ij}^{+} & = & f^{*} (ω_{i}) \times x_{ij}^{' +}, i = 1, 2, \dots, m, \\ j = 1, 2, \dots, n . \end{matrix}$ (37)

Step 5. Positive Ideal Solution (PIS) and Negative Ideal Solution (NIS) can be determined by:

$S^{+} (i) = {max}_{j = i}^{n} (W_{ij}^{+}); S^{-} (i) = {min}_{j = i}^{n} (W_{ij}^{-})$ (38)

S⁺ (i) , S^- (i) are PIS and NIS values as to indicator i.

Step 6. Calculating distances of each alternative from PIS $(d_{j}^{+})$ and NIS $(d_{j}^{-})$ by [26]: $\begin{matrix} d_{j}^{+} = \sqrt{\frac{1}{2} \sum_{i = 1}^{m} [(W_{ij}^{-} - S_{i}^{+})^{2} + (W_{ij}^{+} - S_{i}^{+})^{2}]}, \\ (i = 1, 2, \dots, m; j = 1, 2, \dots, n .) \end{matrix}$ (39) $\begin{matrix} d_{j}^{-} = \sqrt{\frac{1}{2} \sum_{i = 1}^{m} [(W_{ij}^{-} - S_{i}^{-})^{2} + (W_{ij}^{+} - S_{i}^{-})^{2}]}, \\ (i = 1, 2, \dots, m; j = 1, 2, \dots, n .) \end{matrix}$ (40)

Then, closeness coefficient (CC_j) can be computed by: ${CC}_{j} = d_{j}^{-} / - (d_{j}^{-} + d_{j}^{+}), j = 1, 2, \dots, n$ (41)

Therefore, we can rank the alternatives according to their CC_j in decreasing orders.

5 An improved rough set approach to evaluate AMS options

There are three steps in the improved approach to evaluate AMS options (as Fig. 1).

Fig.1

The improved rough set approach for AMS evaluation.

Step 1. Indicators selection.

Selection of appropriate indicators is a crucial step to evaluate auto-mobility systems, and we categorize the selected indicators into two categories, i.e., cost category and benefit category indicators. For cost category, the higher the value, the worse the option and for benefit category, the lower the value, the worse the alternative is.

Step 2. Weights determination.

We apply the method of R-AHP described in Section 4.1 to determine the importance of weights for each selected indicator.

Step 3. Comparison of different alternatives applying R-TOPSIS.

Firstly, decision makers including professional practitioners, experienced designers, and experts are invited to evaluate different alternatives with crisp ratings against each indicator. Secondly, rough evaluation matrix is established. Thirdly, R-TOPSIS is applied to calculate the closeness coefficient (CC_j) of each alternative. The one with the highest CC_j is chosen as the best AMS option.

6 Case study

6.1 Indicator selection

First of all, we identify a list of 48 indicators to evaluate auto-mobility systems [31, 32]. Then we delete those repeated items and choose the most representative 24 indicators by means of expert interviews. Finally, we identify a list with 4 dimensions and 24 indicators to evaluate the performance of AMS. The four dimensions are economic, environmental, auto-mobility system, and social dimension, each with 6 indicators, as shown in Fig. 2. It is obvious that C₁ to C₃ and C₁₄ to C₂₄ belong to the benefit category, that is to say, the higher the value, the better the alternative. C₄ to C₁₃ are cost ones, which means that the lower the value, the better the alternative.

Fig.2

Hierarchy structure for evaluation of auto-mobility systems.

6.2 Weight determination by means of R-AHP

Step 1. Constructing hierarchy system for AMS evaluation indicators as Fig. 2.

Step 2. Establishing pair-wise comparison matrix A_i.

Suppose there are 5 decision makers participating in the evaluation process. For indicators C_1 - 6 in economic dimension, 5 pair-wise comparison matrixes can be constructed as follows: $\begin{matrix} A_{11} = \begin{matrix} \begin{matrix} C_{1} & C_{2} & C_{3} & C_{4} & C_{5} & C_{6} \end{matrix} \\ \begin{matrix} C_{1} \\ C_{2} \\ C_{3} \\ C_{4} \\ C_{5} \\ C_{6} \end{matrix} & [\begin{matrix} 1 & 6 & 4 & 9 & 4 & 5 \\ 1 & 1 / - 7 & 2 & 1 / - 6 & 1 / - 3 \\ 1 & 8 & 3 & 5 \\ 1 & 1 / - 6 & 1 / - 3 \\ 1 & 4 \\ 1 \end{matrix}] \end{matrix} \\ A_{12} = \begin{matrix} \begin{matrix} C_{1} & C_{2} & C_{3} & C_{4} & C_{5} & C_{6} \end{matrix} \\ \begin{matrix} C_{1} \\ C_{2} \\ C_{3} \\ C_{4} \\ C_{5} \\ C_{6} \end{matrix} & [\begin{matrix} 1 & 6 & 2 & 8 & 3 & 4 \\ 1 & 1 / - 8 & 3 & 1 / - 5 & 1 / - 3 \\ 1 & 6 & 2 & 4 \\ 1 & 1 / - 5 & 1 / - 2 \\ 1 & 3 \\ 1 \end{matrix}] \end{matrix} \end{matrix}$

$\begin{matrix} A_{13} = \begin{matrix} \begin{matrix} C_{1} & C_{2} & C_{3} & C_{4} & C_{5} & C_{6} \end{matrix} \\ \begin{matrix} C_{1} \\ C_{2} \\ C_{3} \\ C_{4} \\ C_{5} \\ C_{6} \end{matrix} & [\begin{matrix} 1 & 5 & 2 & 7 & 2 & 3 \\ 1 & 1 / - 6 & 4 & 1 / - 4 & 1 / - 2 \\ 1 & 5 & 3 & 4 \\ 1 & 1 / - 7 & 1 / - 4 \\ 1 & 2 \\ 1 \end{matrix}] \end{matrix} \\ A_{14} = \begin{matrix} \begin{matrix} C_{1} & C_{2} & C_{3} & C_{4} & C_{5} & C_{6} \end{matrix} \\ \begin{matrix} C_{1} \\ C_{2} \\ C_{3} \\ C_{4} \\ C_{5} \\ C_{6} \end{matrix} & [\begin{matrix} 1 & 7 & 3 & 4 & 3 & 4 \\ 1 & 1 / - 5 & 2 & 1 / - 2 & 1 / - 2 \\ 1 & 7 & 5 & 2 \\ 1 & 1 / - 3 & 1 / - 2 \\ 1 & 3 \\ 1 \end{matrix}] \end{matrix} \\ A_{15} = \begin{matrix} \begin{matrix} C_{1} & C_{2} & C_{3} & C_{4} & C_{5} & C_{6} \end{matrix} \\ \begin{matrix} C_{1} \\ C_{2} \\ C_{3} \\ C_{4} \\ C_{5} \\ C_{6} \end{matrix} & [\begin{matrix} 1 & 5 & 2 & 7 & 2 & 6 \\ 1 & 1 / - 9 & 2 & 1 / - 4 & 1 / - 2 \\ 1 & 4 & 2 & 3 \\ 1 & 1 / - 6 & 1 / - 3 \\ 1 & 5 \\ 1 \end{matrix}] \end{matrix} \end{matrix}$

Consistency indexes of the five matrixes are calculated according to Equations (13 and 14). $\begin{matrix} {CR}_{1} & = & 0.084, {CR}_{2} = 0.052, {CR}_{3} = 0.061, \\ {CR}_{4} & = & 0.098, {CR}_{5} = 0.056 . \end{matrix}$

Step 3. Constructing rough group decision making matrix and computing rough numbers for each indicator in the matrix.

$A_{1}^{*} = [\begin{matrix} 1, 1, 1, 1, 1 & 6, 6, 5, 7, 5 & 4, 2, 2, 3, 2 & 9, 8, 7, 4, 7 & 4, 3, 2, 3, 2 & 5, 4, 3, 4, 6 \\ \frac{1}{6}, \frac{1}{6}, \frac{1}{5}, \frac{1}{7}, \frac{1}{5} & 1, 1, 1, 1, 1 & \frac{1}{7}, \frac{1}{8}, \frac{1}{6}, \frac{1}{5}, \frac{1}{9} & 2, 3, 4, 2, 2 & \frac{1}{6}, \frac{1}{5}, \frac{1}{4}, \frac{1}{2}, \frac{1}{4} & \frac{1}{3}, \frac{1}{3}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2} \\ \frac{1}{4}, \frac{1}{2}, \frac{1}{2}, \frac{1}{3}, \frac{1}{2} & 7, 8, 6, 5, 9 & 1, 1, 1, 1, 1 & 8, 6, 5, 7, 4 & 3, 2, 3, 5, 2 & 5, 4, 4, 2, 3 \\ \frac{1}{9}, \frac{1}{8}, \frac{1}{7}, \frac{1}{4}, \frac{1}{7} & \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{2}, \frac{1}{2} & \frac{1}{8}, \frac{1}{6}, \frac{1}{5}, \frac{1}{7}, \frac{1}{4} & 1, 1, 1, 1, 1 & \frac{1}{6}, \frac{1}{5}, \frac{1}{7}, \frac{1}{3}, \frac{1}{6} & \frac{1}{3}, \frac{1}{2}, \frac{1}{4}, \frac{1}{2}, \frac{1}{3} \\ \frac{1}{4}, \frac{1}{3}, \frac{1}{2}, \frac{1}{3}, \frac{1}{2} & 6, 5, 4, 2, 4 & \frac{1}{3}, \frac{1}{2}, \frac{1}{3}, \frac{1}{5}, \frac{1}{2} & 6, 5, 7, 3, 6 & 1, 1, 1, 1, 1 & 4, 3, 2, 3, 5 \\ \frac{1}{5}, \frac{1}{4}, \frac{1}{3}, \frac{1}{4}, \frac{1}{6} & 3, 3, 2, 2, 2 & \frac{1}{5}, \frac{1}{4}, \frac{1}{4}, \frac{1}{2}, \frac{1}{3} & 3, 2, 4, 2, 3 & \frac{1}{4}, \frac{1}{3}, \frac{1}{2}, \frac{1}{3}, \frac{1}{5} & 1, 1, 1, 1, 1 \end{matrix}]$

As to $X_{12}^{*}$ , we can compute the rough number of score ‘5’ as follows: $\begin{matrix} {\underline{X}}_{12}^{*} & = & {6, 6, 5, 7, 5} \\ Lim (5) & = & [R (x_{12}^{3}) + R (x_{12}^{5})] / 2 \\ = & (5 + 5) / 2 = 5.000 \\ \bar{Lim} (5) & = & [R (x_{12}^{1}) + R (x_{12}^{2}) + R (x_{12}^{3}) \\ + R (x_{12}^{4}) + R (x_{12}^{5})] / 5 \\ = & (6 + 6 + 5 + 7 + 5) / 5 = 5.800 \end{matrix}$

Therefore, the rough boundary interval is calculated as follows: $\begin{matrix} RBnd (5) & = & \bar{Lim} (5) - \underline{Lim} (5) \\ = & 5.800 - 5.000 = 0.800 \end{matrix}$

Rough number $RN (5) = [\underline{Lim} (5), \bar{Lim} (5)] = [5.000, 5.800]$

Similarly, we can obtain the rough number for grades ‘6’ and ‘7’: $\begin{matrix} RN (6) = [\underline{Lim} (6), \bar{Lim} (6)] = [5.500, 6.333] \\ RN (7) = [\underline{Lim} (7), \bar{Lim} (7)] = [5.800, 7.000] \end{matrix}$

Then,

$\begin{matrix} RN (X_{12}^{*}) \\ = {\begin{matrix} [\underline{Lim} (6), \bar{Lim} (6)], [\underline{Lim} (6), \bar{Lim} (6)], [\underline{Lim} (5), \bar{Lim} (5)], \\ [\underline{Lim} (7), \bar{Lim} (7)], [\underline{Lim} (5), \bar{Lim} (5)] \end{matrix}} \\ = {\begin{matrix} [5.500, 6.333], [5.500, 6.333], [5.000, 5.800], \\ [5.800, 7.000], [5.000, 5.800] \end{matrix}} \end{matrix}$

Using Equations (8–11) and (18, 19), the average rough interval of $X_{21}^{*}$ can be got as: RN (X₁₂) = [5.360, 6.253].

With the aid of MATLAB, the above procedures can be computed as:

$\begin{matrix} A & = & [6, 6, 5, 7, 5] \\ RN (X_{12}^{-}) & = & (sum (A (A < = 6)) / sum (A < = 6) \\ + sum (A (A < = 6)) / sum (A < = 6) \\ + sum (A (A < = 5)) / sum (A < = 5) \\ + sum (A (A < = 7)) / sum (A < = 7) \\ + sum (A (A < = 5)) / sum (A < = 5)) / 5 = 5.36 \\ RN (X_{12}^{+}) & = & (sum (A (A > = 6)) / sum (A > = 6) \\ + sum (A (A > = 6)) / sum (A > = 6) \\ + sum (A (A > = 5)) / sum (A > = 5) \\ + sum (A (A > = 7)) / sum (A > = 7) \\ + sum (A (A > = 5)) / sum (A > = 5)) / 5 \\ = & 6.253 \end{matrix}$

Step 4. Constructing rough pair-wise comparison matrix using Equation (20) as follows:

$X_{1} = [\begin{matrix} [1.000, 1.000] & [5.360, 6.253] & [2.170, 3.060] & [5.900, 8.000] & [2.360, 3.253] & [3.747, 5.080] \\ [0.162, 0.188] & [1.000, 1.000] & [0.128, 0.172] & [2.170, 3.060] & [0.211, 0.348] & [0.393, 0.473] \\ [0.358, 0.475] & [6.000, 8.000] & [1.000, 1.000] & [5.000, 7.000] & [2.400, 3.667] & [2.920, 4.253] \\ [0.129, 0.185] & [0.358, 0.475] & [0.148, 0.210] & [1.000, 1.000] & [0.166, 0.247] & [0.326, 0.427] \\ [0.326, 0.443] & [3.323, 5.040] & [0.305, 0.441] & [4.480, 6.213] & [1.000, 1.000] & [2.747, 4.080] \\ [0.205, 0.277] & [2.160, 2.640] & [0.246, 0.378] & [2.360, 3.253] & [0.261, 0.391] & [1.000, 1.000] \end{matrix}]$

Step 5. Calculating weight for each indicator.

First, we divide X into rough lower limits matrix X^- and rough upper limits matrix X⁺ using Equations (21 and 22).

$\begin{matrix} X_{1}^{-} & = & [\begin{matrix} 1.000 & 5.360 & 2.170 & 5.900 & 2.360 & 3.747 \\ 0.162 & 1.000 & 0.128 & 2.170 & 0.211 & 0.393 \\ 0.358 & 6.000 & 1.000 & 5.000 & 2.400 & 2.920 \\ 0.129 & 0.358 & 0.148 & 1.000 & 0.166 & 0.326 \\ 0.326 & 3.323 & 0.305 & 4.480 & 1.000 & 2.747 \\ 0.205 & 2.160 & 0.246 & 2.360 & 0.261 & 1.000 \end{matrix}] \\ X_{1}^{+} & = & [\begin{matrix} 1.000 & 6.253 & 3.060 & 8.000 & 3.253 & 5.080 \\ 0.188 & 1.000 & 0.172 & 3.060 & 0.348 & 0.473 \\ 0.475 & 8.000 & 1.000 & 7.000 & 3.667 & 4.253 \\ 0.185 & 0.475 & 0.210 & 1.000 & 0.247 & 0.427 \\ 0.443 & 5.040 & 0.441 & 6.213 & 1.000 & 4.080 \\ 0.277 & 2.640 & 0.378 & 3.253 & 0.391 & 1.000 \end{matrix}] \end{matrix}$

Then we calculate the eigenvalues and eigenvectors with the aid of MATLAB.

For matrix $X_{1}^{-}, λ_{max}^{-} = 5.6343$ , X₁ = (-0.7529, - 0.1020, - 0.5350, - 0.0690, - 0.3244, - 0.1626) ^T. For matrix $X_{1}^{+}, λ_{max}^{+} = 7.4457$ , X₂ = (-0.7388, - 0.0992, - 0.5494, - 0.0698, - 0.3346, - 0.1601) ^T. Normalized weights can be computed by Equation (24). $\begin{matrix} [f_{1}^{-}, f_{2}^{-}, f_{3}^{-}, f_{4}^{-}, f_{5}^{-}, f_{6}^{-}] \\ = [0.3869, 0.0524, 0.2749, 0.0355, \\ 0.1667, 0.0836] \\ [f_{1}^{+}, f_{2}^{+}, f_{3}^{+}, f_{4}^{+}, f_{5}^{+}, f_{6}^{+}] \\ = [0.3785, 0.0508, 0.2815, 0.0358, \\ 0.1714, 0.0820] \end{matrix}$

The weight of indicator C_i can be computed by Equation (25). For example, $f (C_{1}) = \frac{1}{2} (| f_{1}^{-} | + | f_{1}^{+} |) = \frac{1}{2} (0.3869 + 0.3785) = 0.3827$ Similarly, we can get the weights of the remaining indicators.

Step 6. Composition of synthetic weight of each dimension and indicator by Equation (26). For instance, f^* (ω₁) =0.25 × 0.3827 = 0.0957.

For the other three dimensions (D₂-D₄), the above steps are repeated and the final rough weights for all selected indicators are listed in Table 2. It is noteworthy that we assume the same weight (0.25) to the four dimensions due to their relative equal importance and the difficulty of deciding which one is more important than the other. Decision matrixes for D₂-D₄ are not listed here due to space limit.

Table 2
Weights of indicators C₁-C₂₄

Dimension (weight) Indicator f (C_i) f^* (ω_i)

Economic C₁ 0.3827 0.0957

   (0.25) C₂ 0.0516 0.0129

C₃ 0.2782 0.0696

C₄ 0.0356 0.0089

C₅ 0.1691 0.0423

C₆ 0.0828 0.0207

Environmental C₇ 0.4265 0.1066

   (0.25) C₈ 0.2666 0.0667

C₉ 0.0872 0.0218

C₁₀ 0.0311 0.0078

C₁₁ 0.1353 0.0338

C₁₂ 0.0532 0.0133

Auto-mobility system C₁₃ 0.4047 0.1012

   (0.25) C₁₄ 0.0968 0.0242

C₁₅ 0.0551 0.0138

C₁₆ 0.2554 0.0639

C₁₇ 0.0334 0.0083

C₁₈ 0.1546 0.0386

Social C₁₉ 0.1650 0.0413

   (0.25) C₂₀ 0.0313 0.0078

C₂₁ 0.0976 0.0244

C₂₂ 0.0562 0.0140

C₂₃ 0.2603 0.0651

C₂₄ 0.3897 0.0974

Dimension (weight)	Indicator	f (C_i)	f^* (ω_i)
Economic	C₁	0.3827	0.0957
(0.25)	C₂	0.0516	0.0129
	C₃	0.2782	0.0696
	C₄	0.0356	0.0089
	C₅	0.1691	0.0423
	C₆	0.0828	0.0207
Environmental	C₇	0.4265	0.1066
(0.25)	C₈	0.2666	0.0667
	C₉	0.0872	0.0218
	C₁₀	0.0311	0.0078
	C₁₁	0.1353	0.0338
	C₁₂	0.0532	0.0133
Auto-mobility system	C₁₃	0.4047	0.1012
(0.25)	C₁₄	0.0968	0.0242
	C₁₅	0.0551	0.0138
	C₁₆	0.2554	0.0639
	C₁₇	0.0334	0.0083
	C₁₈	0.1546	0.0386
Social	C₁₉	0.1650	0.0413
(0.25)	C₂₀	0.0313	0.0078
	C₂₁	0.0976	0.0244
	C₂₂	0.0562	0.0140
	C₂₃	0.2603	0.0651
	C₂₄	0.3897	0.0974

6.3 Comparison of different alternatives using R-TOPSIS

Step 1. Construct crisp AMS evaluation matrix.

Invite 5 decision makers to give their crisp ratings of the three alternatives as to each indicator.

Step 2. Convert the crisp factor $x_{ij}^{k}$ in group decision matrix D into rough number and establish rough group decision-making matrix D^* according to Equations (1–11) and (18, 19), as Table 3.

Table 3
Rough evaluation matrix for auto-mobility systems

Indicator Alternatives

A₁ A₂ A₃

C₁ (5.360,6.253) (5.747,7.080) (3.747,5.080)

C₂ (5.467,6.533) (3.457,4.980) (1.747,3.080)

C₃ (6.360,7.253) (4.747,6.080) (2.747,4.080)

C₄ (5.360,6.253) (4.747,5.640) (2.747,4.080)

C₅ (4.747,6.080) (6.020,7.543) (2.360,3.253)

C₆ (2.467,3.533) (1.360,2.253) (6.920,8.253)

C₇ (2.747,4.080) (2.747,4.080) (7.020,8.543)

C₈ (4.747,5.640) (2.747,4.080) (7.360,8.253)

C₉ (3.747,5.080) (2.467,3.533) (6.457,7.980)

C₁₀ (6.920,8.253) (2.747,3.640) (1.650,2.350)

C₁₁ (5.457,6.980) (2.360,3.253) (6.360,7.253)

C₁₂ (5.747,7.080) (2.457,3.980) (4.020,5.543)

C₁₃ (5.360,6.253) (4.000,6.000) (4.543,6.960)

C₁₄ (5.920,7.253) (1.747,3.080) (7.747,8.640)

C₁₅ (5.467,6.533) (2.747,4.080) (6.360,7.253)

C₁₆ (4.020,5.543) (5.360,6.253) (5.573,7.503)

C₁₇ (5.360,6.253) (3.920,5.253) (1.360,1.840)

C₁₈ (4.747,5.640) (1.747,2.640) (6.920,8.253)

C₁₉ (4.287,6.870) (6.920,8.253) (1.747,2.640)

C₂₀ (6.360,7.253) (7.747,8.640) (1.920,3.253)

C₂₁ (2.747,3.640) (5.360,6.253) (1.360,1.840)

C₂₂ (4.747,5.640) (5.920,7.253) (7.360,8.253)

C₂₃ (6.467,7.533) (2.747,4.080) (5.360,6.253)

C₂₄ (5.920,7.253) (4.457,5.980) (6.747,7.640)

Indicator	Alternatives
C₁	(5.360,6.253)	(5.747,7.080)	(3.747,5.080)
C₂	(5.467,6.533)	(3.457,4.980)	(1.747,3.080)
C₃	(6.360,7.253)	(4.747,6.080)	(2.747,4.080)
C₄	(5.360,6.253)	(4.747,5.640)	(2.747,4.080)
C₅	(4.747,6.080)	(6.020,7.543)	(2.360,3.253)
C₆	(2.467,3.533)	(1.360,2.253)	(6.920,8.253)
C₇	(2.747,4.080)	(2.747,4.080)	(7.020,8.543)
C₈	(4.747,5.640)	(2.747,4.080)	(7.360,8.253)
C₉	(3.747,5.080)	(2.467,3.533)	(6.457,7.980)
C₁₀	(6.920,8.253)	(2.747,3.640)	(1.650,2.350)
C₁₁	(5.457,6.980)	(2.360,3.253)	(6.360,7.253)
C₁₂	(5.747,7.080)	(2.457,3.980)	(4.020,5.543)
C₁₃	(5.360,6.253)	(4.000,6.000)	(4.543,6.960)
C₁₄	(5.920,7.253)	(1.747,3.080)	(7.747,8.640)
C₁₅	(5.467,6.533)	(2.747,4.080)	(6.360,7.253)
C₁₆	(4.020,5.543)	(5.360,6.253)	(5.573,7.503)
C₁₇	(5.360,6.253)	(3.920,5.253)	(1.360,1.840)
C₁₈	(4.747,5.640)	(1.747,2.640)	(6.920,8.253)
C₁₉	(4.287,6.870)	(6.920,8.253)	(1.747,2.640)
C₂₀	(6.360,7.253)	(7.747,8.640)	(1.920,3.253)
C₂₁	(2.747,3.640)	(5.360,6.253)	(1.360,1.840)
C₂₂	(4.747,5.640)	(5.920,7.253)	(7.360,8.253)
C₂₃	(6.467,7.533)	(2.747,4.080)	(5.360,6.253)
C₂₄	(5.920,7.253)	(4.457,5.980)	(6.747,7.640)

Step 3. Normalize rough group decision matrix by Equations (34, 35). For instance, the rough normalized rating for alternative A₁ as to indicator C₁ (productivity) which is a benefit indicator can be computed as: $\begin{matrix} x_{11}^{' -} & = & \frac{x_{11}^{-}}{{max}_{j = 1}^{3} (x_{1 j}^{-}, x_{1 j}^{+})} = \frac{5.360}{7.080} = 0.757; \\ 1 pt x_{11}^{' +} & = & \frac{x_{11}^{' +}}{{max}_{j = 1}^{3} (x_{1 j}^{-}, x_{1 j}^{+})} = \frac{6.253}{7.080} = 0.883 . \end{matrix}$

The rough normalized rating for alternative A₁ as to indicator C₄ (public expenditure) which is a cost indicator can be given by: $\begin{matrix} x_{41}^{' -} & = & \frac{{min}_{j = 1}^{3} (x_{4 j}^{-}, x_{4 j}^{+})}{x_{41}^{+}} = \frac{2.747}{6.253} = 0.439; \\ x_{41}^{' +} & = & \frac{{min}_{j = 1}^{3} (x_{4 j}^{-}, x_{4 j}^{+})}{x_{41}^{-}} = \frac{2.747}{5.360} = 0.513 . \end{matrix}$

Similarly, we can compute the rough normalized values of the three alternatives against all other indicators as Table 4.

Table 4

Rough normalized matrix for auto-mobility systems

Indicator	Alternatives
	A₁	A₂	A₃
C₁	(0.757,0.883)	(0.812,1.000)	(0.529,0.718)
C₂	(0.837,1.000)	(0.529,0.762)	(0.267,0.471)
C₃	(0.877,1.000)	(0.654,0.838)	(0.379,0.563)
C₄	(0.439,0.513)	(0.487,0.579)	(0.673,1.000)
C₅	(0.388,0.497)	(0.313,0.392)	(0.725,1.000)
C₆	(0.385,0.551)	(0.604,1.000)	(0.165,0.197)
C₇	(0.333,0.495)	(0.673,1.000)	(0.159,0.194)
C₈	(0.310,0.368)	(0.673,1.000)	(0.212,0.237)
C₉	(0.268,0.363)	(0.698,1.000)	(0.170,0.211)
C₁₀	(0.200,0.238)	(0.453,0.601)	(0.702,1.000)
C₁₁	(0.338,0.432)	(0.725,1.000)	(0.325,0.371)
C₁₂	(0.347,0.428)	(0.617,1.000)	(0.443,0.611)
C₁₃	(0.640,0.746)	(0.667,1.000)	(0.575,0.880)
C₁₄	(0.685,0.839)	(0.202,0.356)	(0.897,1.000)
C₁₅	(0.754,0.901)	(0.379,0.563)	(0.877,1.000)
C₁₆	(0.536,0.739)	(0.714,0.833)	(0.743,1.000)
C₁₇	(0.857,1.000)	(0.627,0.840)	(0.217,0.294)
C₁₈	(0.575,0.683)	(0.212,0.320)	(0.838,1.000)
C₁₉	(0.496,0.795)	(0.838,1.000)	(0.202,0.306)
C₂₀	(0.762,0.869)	(0.897,1.000)	(0.230,0.390)
C₂₁	(0.439,0.582)	(0.857,1.000)	(0.217,0.294)
C₂₂	(0.575,0.683)	(0.717,0.879)	(0.892,1.000)
C₂₃	(0.858,1.000)	(0.365,0.542)	(0.712,0.830)
C₂₄	(0.775,0.949)	(0.583,0.783)	(0.883,1.000)

Step 4. Compute the rough weighed normalized decision matrix by Equations (36, 37). We take advantage of the rough normalized values in Table 4 and synthetic weight f^* (ω_i) of each indicator in Table 2 to accomplish this step. For example, the rough weighted normalized value for indicators C₁ (productivity) is given by: $\begin{matrix} W_{11}^{-} = f^{*} (ω_{1}) \times x_{11}^{' -} = 0.0957 \times 0.757 = 0.072; \\ W_{11}^{+} = f^{*} (ω_{1}) \times x_{11}^{' +} = 0.0957 \times 0.883 = 0.085 . \end{matrix}$

Likewise, we can calculate the rough weighted normalized values of the alternatives as to the selected indicators as Table 5.

Table 5

Rough weighted normalized matrix for auto-mobility systems

Indi-	Alternatives			S^- (i)	S⁺ (i)
cator	A₁	A₂	A₃
C₁	(0.072,0.085)	(0.078,0.096)	(0.051,0.069)	0.051	0.096
C₂	(0.011,0.013)	(0.007,0.010)	(0.003,0.006)	0.003	0.013
C₃	(0.061,0.070)	(0.046,0.058)	(0.026,0.039)	0.026	0.070
C₄	(0.004,0.005)	(0.004,0.005)	(0.006,0.009)	0.004	0.009
C₅	(0.016,0.021)	(0.013,0.017)	(0.031,0.042)	0.013	0.042
C₆	(0.008,0.011)	(0.012,0.021)	(0.003,0.004)	0.003	0.021
C₇	(0.036,0.053)	(0.072,0.107)	(0.017,0.021)	0.034	0.107
C₈	(0.021,0.025)	(0.045,0.067)	(0.014,0.016)	0.022	0.067
C₉	(0.006,0.008)	(0.015,0.022)	(0.004,0.005)	0.007	0.022
C₁₀	(0.002,0.002)	(0.004,0.005)	(0.005,0.008)	0.002	0.008
C₁₁	(0.011,0.015)	(0.025,0.034)	(0.011,0.013)	0.011	0.034
C₁₂	(0.005,0.006)	(0.008,0.013)	(0.006,0.008)	0.005	0.013
C₁₃	(0.065,0.076)	(0.067,0.101)	(0.058,0.089)	0.058	0.101
C₁₄	(0.017,0.020)	(0.005,0.009)	(0.022,0.024)	0.005	0.024
C₁₅	(0.010,0.012)	(0.005,0.008)	(0.012,0.014)	0.005	0.014
C₁₆	(0.034,0.047)	(0.046,0.053)	(0.047,0.064)	0.034	0.064
C₁₇	(0.007,0.008)	(0.005,0.007)	(0.002,0.002)	0.002	0.008
C₁₈	(0.022,0.026)	(0.008,0.012)	(0.032,0.039)	0.008	0.039
C₁₉	(0.020,0.033)	(0.035,0.041)	(0.008,0.013)	0.009	0.041
C₂₀	(0.006,0.007)	(0.007,0.008)	(0.002,0.003)	0.002	0.008
C₂₁	(0.011,0.014)	(0.021,0.024)	(0.005,0.007)	0.005	0.024
C₂₂	(0.008,0.010)	(0.010,0.012)	(0.012,0.014)	0.008	0.014
C₂₃	(0.056,0.065)	(0.024,0.035)	(0.046,0.054)	0.024	0.065
C₂₄	(0.075,0.092)	(0.057,0.076)	(0.086,0.097)	0.057	0.097

Step 5. Positive Ideal Solution (S⁺ (i)) and Negative Ideal Solution (S^- (i)) can be determined by Equation (38) as shown in the last two columns of Table 5.

Step 6. Calculating distances of each alternative from PIS $(d_{j}^{+})$ and NIS $(d_{j}^{-})$ by Equations (39, 40). For example, for alternative A₁ and indicator C₁, the distances can be calculated as: $\begin{matrix} d_{1}^{+} & = & \sqrt{\frac{1}{2} [(W_{11}^{-} - S_{1}^{+})^{2} + (W_{11}^{+} - S_{1}^{+})^{2}]} \\ = & \sqrt{\frac{1}{2} [(0.072 - 0.096)^{2} + (0.085 - 0.096)^{2}]} \\ = & 0.018 \\ d_{1}^{-} & = & \sqrt{\frac{1}{2} [(W_{11}^{-} - S_{1}^{-})^{2} + (W_{11}^{+} - S_{1}^{-})^{2}]} \\ = & \sqrt{\frac{1}{2} [(0.072 - 0.051)^{2} + (0.085 - 0.051)^{2}]} \\ = & 0.028 \end{matrix}$

Similarly, we can get the distances of each alternative from PIS $(d_{j}^{+})$ and NIS $(d_{j}^{-})$ as to the remaining indicators (Table 6).

Table 6

Distances $(d_{j}^{+})$ and $(d_{j}^{-})$ of each alternative

Indicator	$(d_{j}^{-})$			$(d_{j}^{+})$
	A₁	A₂	A₃	A₁	A₂	A₃
C₁	0.028	0.037	0.013	0.018	0.013	0.037
C₂	0.008	0.005	0.002	0.001	0.005	0.008
C₃	0.039	0.026	0.009	0.006	0.019	0.037
C₄	0.000	0.001	0.004	0.005	0.004	0.002
C₅	0.006	0.002	0.024	0.024	0.027	0.008
C₆	0.007	0.014	0.000	0.011	0.006	0.017
C₇	0.058	0.058	0.005	0.025	0.025	0.069
C₈	0.014	0.035	0.002	0.031	0.015	0.043
C₉	0.006	0.012	0.001	0.010	0.005	0.014
C₁₀	0.000	0.003	0.005	0.006	0.004	0.002
C₁₁	0.003	0.019	0.001	0.021	0.007	0.022
C₁₂	0.001	0.007	0.003	0.008	0.004	0.006
C₁₃	0.013	0.031	0.022	0.032	0.024	0.032
C₁₄	0.014	0.003	0.018	0.006	0.018	0.002
C₁₅	0.006	0.002	0.008	0.003	0.007	0.001
C₁₆	0.009	0.016	0.023	0.024	0.015	0.012
C₁₇	0.006	0.004	0.000	0.001	0.002	0.006
C₁₈	0.016	0.003	0.027	0.014	0.028	0.004
C₁₉	0.020	0.029	0.003	0.015	0.005	0.030
C₂₀	0.004	0.006	0.001	0.002	0.001	0.005
C₂₁	0.007	0.017	0.001	0.012	0.002	0.018
C₂₂	0.001	0.003	0.005	0.005	0.003	0.001
C₂₃	0.037	0.008	0.027	0.007	0.036	0.015
C₂₄	0.028	0.014	0.035	0.016	0.032	0.008
∑C_i	0.333	0.355	0.240	0.302	0.306	0.402

For example, for alternative A₁ and indicators C₁ to C₂₄, the distances $(d_{1}^{+})$ and $(d_{1}^{-})$ can be got as follows: $\begin{matrix} d_{1}^{+} & = & \sqrt{\frac{1}{2} [(0.072 - 0.096)^{2} + (0.085 - 0.096)^{2}} \\ + \sqrt{\frac{1}{2} [(0.011 - 0.013)^{2} + (0.013 - 0.013)^{2}} \\ + \dots \\ + \sqrt{\frac{1}{2} (0.075 - 0.097)^{2} + (0.092 - 0.097)^{2}} \\ = & 0.302 \\ d_{1}^{-} & = & \sqrt{\frac{1}{2} [(0.072 - 0.051)^{2} + (0.085 - 0.051)^{2}} \\ + \sqrt{\frac{1}{2} [(0.011 - 0.003)^{2} + (0.013 - 0.003)^{2}} \\ + \dots \\ + \sqrt{\frac{1}{2} (0.075 - 0.057)^{2} + (0.092 - 0.057)^{2}} \\ = & 0.333 \end{matrix}$

Then, closeness coefficient CC_j can be computed by Equation (41).

For example, the closeness coefficient of alternative A₁ is given by: ${CC}_{1} = \frac{d_{1}^{-}}{d_{1}^{-} + d_{1}^{+}} = \frac{0.302}{0.302 + 0.333} = 0.525$

Similarly, CC_j for alternatives A₂ and A₃ are calculated as presented in Table 7.

Table 7

(CC_j) of the three alternatives

	A₁	A₂	A₃
$(d_{j}^{-})$	0.333	0.355	0.240
$(d_{j}^{+})$	0.302	0.306	0.402
(CC_j)	0.525	0.537	0.374

Then, we can rank the alternatives according to their CC_j (A₂ ≻ A₁ ≻ A₃). Therefore, A₂ (ride-sharing) is the best auto-mobility option in a comprehensive sense of meaning.

6.4 Sensitivity analysis

We carry out sensitivity analysis experiments to figure out how sensitive the decision process is to slight changes in individual weights. The values of the weights are changed slightly to work out their influence on the final decision. Under uncertain environments, sensitivity analysis is particularly useful in determining the indicator weights and selecting appropriate AMS options.

Thirty-one experiments are conducted to analyze sensitivity of each indicator to alternative ranking. In the first 5 experiments, we set weights of all indicators equal to one another, i.e. 1, 3, 5, 7, and 9 respectively. In experiments 6–29, the weight of one indicator is the highest (9) while the others the lowest (1), to find out the most influential indicator in decision making. Then, in experiment 30, we set the weights of benefit indicators as the highest (9) and those of cost indicators to be lowest (1). In experiments 31, we set the weights of benefit indicators as the lowest (1) and those of cost indicators to be highest (9). Figure 3 displays the results of sensitivity analysis experiments, from which we can easily find that out of 31 experiments, alternative A₂ (ride-sharing) has the highest score in 21 experiments at a ratio of 67.74%. Alternative A₁ (car-sharing) is better than A₃ (car-owning) at a ratio of 77.42%. In experiments 10, 19, 20, and 23, alternative A₃ ranks better than A₂. Therefore, we can arrive at the conclusion that alternative A₂ (ride-sharing) can be recommended as the best transportation mode and that indicators weights play an important role in the evaluation of auto-mobility systems.

Fig.3

Results of sensitivity analysis experiments.

7 Conclusions

An improved rough set approach to evaluate auto-mobility systems with application of R-AHP and R-TOPSIS under uncertain environments is developed in this paper. There are three steps in the approach. In step 1, appropriate indicators of auto-mobility systems are identified. They are divided into four dimensions, economic, environmental, auto-mobility system, and social dimension. Indicators C_1 - 3 and C_14 - 24 belong to benefit category while the others (C_4 - 13) are cost ones. In step 2, decision makers are invited to assess the weight for each indicator by means of R-AHP. In step 3, we invite decision makers to compare different alternatives and rank them with the aid of R-TOPSIS. Then we perform a case study to show the applicability of the improved approach and conduct sensitivity analysis experiments to figure out the effect of each indicator on decision making process.

The approach is capable of evaluating different auto-mobility alternatives with uncertainty and vagueness. It is flexible to reflect the subjectivity and vagueness of decision makers since the rough methodology does not need too much priori information. Moreover, the rough evaluation intervals can be modified easily according to slight changes in decision makers’ preferences. However, the proposed framework also has some limitations. Firstly, the proposed method does not take interdependencies of different indicators into account. Secondly, the 24 selected indicators may not represent the actual case comprehensively. Research can be conducted in the future to handle the above-mentioned issues.

Footnotes

Acknowledgments

This research was supported by National Key Technology R&D Program, China under Grant No. 2015BAH21F01; and the 111 Project under Grant No. B13044. We’d like to thank Conny Bakker from Delft University of Technology for her constructive advice. Many thanks also go to the anonymous reviewers for their help to improve this work.

References

Song

, Ming

and Wu

, An integrated rough number-based approach to design concept evaluation under subjective environments, Journal of Engineering Design 24(5) (2013), 320–341.

http://www.hbgajg.com/html/2016/0126/28440.html accessed on 15-03-2016, Traffic Management Bureau of The Ministry of Public Security of the People’s Republic of China, 2016.

GROWTH WITH: A circular economy vision for a cometitive Europe, in, http://www.ellenmacarthurfoundation.org/news/circular-economy-would-increase-european-competitiveness-and-deliver-better-societal-outcomes-new-study-reveals accessed on 14-03-2016. McKinsey Center for Business and Environment, 2015.

Kriston

, Szabó

and Inzelt

, The marriage of car sharing and hydrogen economy: A possible solution to the main problems of urban living, International Journal of Hydrogen Energy 35(23) (2010), 12697–12708.

Kek

A.G.H.

, Cheu

R.L.

, Meng

and Fung

C.H.

, A decision support system for vehicle relocation operations in carsharing systems, Transportation Research Part E: Logistics and Transportation Review 45(1) (2009), 149–158.

Cavalieri

and Pezzotta

, Product–Service Systems Engineering: State of the art and research challenges, Computers in Industry 63(4) (2012), 278–288.

Zhu

, Li

, Liu

and Yang

, Optimization Approach to Depot Location in Car Sharing Systems with Big Data, 2015, pp. 335–342.

Agatz

, Erera

, Savelsbergh

and Wang

, Optimization for dynamic ride-sharing: A review, European Journal of Operational Research 223(2) (2012), 295–303.

Furuhata

, Dessouky

, Ordóñez

, Brunet

M.-E.

, Wang

and Koenig

, Ridesharing: The state-of-the-art and future directions, Transportation Research Part B: Methodological 57 (2013), 28–46.

10.

Chen

, et al., Form gene clustering method about pan-ethnic-group products based on emotional semantic, Chinese Journal of Mechanical Engineering 29(6) (2016), 1134–1144.

11.

Correia

G.H.D.A.

, Jorge

D.R.

and Antunes

D.M.

, The added value of accounting for users’ flexibility and information on the potential of a station-based one-way car-sharing system: An application in lisbon, Portugal, Journal of Intelligent Transportation Systems 18(3) (2014), 299–308.

12.

Boyaci

, Zografos

K.G.

and Geroliminis

, An optimization framework for the development of efficient one-way car-sharing systems, European Journal of Operational Research 240(3) (2015), 718–733.

13.

Baptista

, Melo

and Rolim

, Energy, environmental and mobility impacts of car-sharing systems, Empirical Results from Lisbon, Portugal, Procedia - Social and Behavioral Sciences 111 (2014), 28–37.

14.

Jorge

, Barnhart

and de Almeida Correia

G.H.

, Assessing the viability of enabling a round-trip carsharing system to accept one-way trips: Application to Logan Airport in Boston, Transportation Research Part C: Emerging Technologies 56 (2015), 359–372.

15.

Fassi

A.E.

, Awasthi

and Viviani

, Evaluation of carsharing network’s growth strategies through discrete event simulation, Expert Systems with Applications 39(8) (2012), 6692–6705.

16.

Nair

and Miller-Hooks

, Equilibrium network design of shared-vehicle systems, European Journal of Operational Research 235(1) (2014), 47–61.

17.

Wagner

, Brandt

and Neumann

, In free float: Developing Business Analytics support for carsharing providers, Omega 59 (2016), 4–14.

18.

Agatz

N.A.H.

, Erera

A.L.

, Savelsbergh

M.W.P.

and Wang

, Dynamic ride-sharing: A simulation study in metro Atlanta, Transportation Research Part B: Methodological 45(9) (2011), 1450–1464.

19.

Lee

and Savelsbergh

, Dynamic ridesharing: Is there a role for dedicated drivers? Transportation Research Part B: Methodological 81 (2015), 483–497.

20.

Pawlak

, Rough sets, International Journal of Computer & Information Sciences 11(5) (1982), 341–356.

21.

Qingyuan

, Anhui

and Jinjin

, A rough set method for the vertex cover problem in graph theory, Journal of Intelligent & Fuzzy Systems 30(4) (2016), 2003–2013.

22.

Pawlak

, Rough set approach to knowledge-based decision support, European Journal of Operational Research 99 (1997), 48–57.

23.

, Tang

, Luo

and Xu

, An integrated method of rough set, Kano’s model and AHP for rating customer requirements’ final importance, Expert Systems with Applications 36(3) (2009), 7045–7053.

24.

Aydogan

E.K.

, Performance measurement model for Turkish aviation firms using the rough-AHP and TOPSIS methods under fuzzy environment, Expert Systems with Applications 38(4) (2011), 3992–3998.

25.

Song

, Ming

, Han

and Wu

, A rough set approach for evaluating vague customer requirement of industrial product-service system, International Journal of Production Research 51(22) (2013), 6681–6701.

26.

Song

, Ming

, Wu

and Zhu

, A rough TOPSIS approach for failure mode and effects analysis in uncertain environments, Quality and Reliability Engineering International 30(4) (2014), 473–486.

27.

Greco

, Matarazzo

and Slowinski

, Rough sets theory for multicriteria decision analysis, European Journal of Operational Research 129(1) (2001), 1–47.

28.

Zhai

L.-Y.

, Khoo

L.-P.

and Zhong

Z.-W.

, Design concept evaluation in product development using rough sets and grey relation analysis, Expert Systems with Applications 36(3) (2009), 7072–7079.

29.

Zhai

L.-Y.

, Khoo

L.-P.

and Zhong

Z.-W.

, A rough set enhanced fuzzy approach to quality function deployment, The International Journal of Advanced Manufacturing Technology 37(5-6) (2008), 613–624.

30.

Golden

B.L.

, Wasil

E.A.

and Harker

P.T.

, Springer Verlag, The analytic hierarchy process: Applications and studies 1989.

31.

Haghshenas

and Vaziri

, Urban sustainable transportation indicators for global comparison, Ecological Indicators 15(1) (2012), 115–121.

32.

Moldan

, Janoušková

and Hák

, How to understand and measure environmental sustainability: Indicators and targets, Ecological Indicators 17 (2012), 4–13.