1 Highlights
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Group decision making process is proposed for evaluations of infrastructure projects.
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The paper deals with the uncertainties present at the early stage of project developments.
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A novel evaluation approach utilizes combination weighting and compromise ranking.
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Subjective and objective weighting methods are integrated to specify criteria weights.
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VIKOR method extended under fuzzy environment is used to rank project alternatives.
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The approach is tested within the real-world project of railway line reconstruction.
2 Introduction
3 Literature review
3.1 Related work
3.2 Knowledge gap
4 Preliminary definitions
4.1 Fuzzy set theory and linguistic variables
4.2 Entropy concept
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Step 1: Normalize the decision matrix and calculate projected outcomes as follows:
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Step 2: Calculate the entropy of attribute j using the following equation:
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Step 3: Define the degree of divergence:
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Step 4: Obtain the entropy weight of attribute i as follows:
4.3 VIKOR method
5 The proposed method
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Step 1: Define the problem scope.The first step in decision making process is to gather relevant information in order to identify the problem scope. The general aim of this paper is to evaluate and select a favourable railway route from a set of draft route alternatives at the project early stage. In particular, railway lines, as complex infrastructural objects, demand not only huge investments for their construction or reconstruction but also a large amount of work on the project development. Therefore, it is very important to find a suitable draft railway route alternative at the early stage of the project as the conceptual base for further project development and detailed evaluations. The selected alternative has to make balance between infrastructure investments, transportation service quality, economic benefits and environmental protection on one side and land use restrictions on the other side.
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Step 2: Define group decision making attributes.Firstly, it is required to form a decision committee considering the problem scope. Generally various decision makers could be involved in an assessment of railway route alternatives in order to contribute their professional background or expertise in evaluating different aspects, such as structure design, rail traffic design, project economics, risks and impacts. Afterwards, it is required to identify attributes (criteria and alternatives). The criteria have to be determined considering the objectives defined in the problem scope in order to properly assess potential railway route alternatives. Based on the gathered information at the project early stage, which is mostly low quality data, it is not possible to create a mathematical model for the railway route generation and its optimization. Instead of a mathematical modelling, it is a rather common procedure to design a finite set of alternatives by varying basic technical and operational elements of the route. Afterwards, the railway route alternatives could be aligned and adjusted to existing spatial structures and conditions.
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Step 3: Select linguistic variables and corresponding fuzzy numbers.
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Step 4: Survey decision makers.Using predefined linguistic variables, decision makers answer the questions regarding the importance of criteria and ratings of alternatives relevant for the observed infrastructural project. The survey of decision makers is a resource for acquiring criteria weights and ratings of alternatives from each decision maker and for aggregating them into the decision matrix.
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Step 5: Determine criteria weights.In this step, we use both subjective weights aggregated from surveys and objective weights derived applying the entropy-based method. The combination weight of criterion Cj could be calculated using the following equation:
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Step 6: Determine the fuzzy best \( {f}_j^{\ast } \) and the fuzzy worst \( {f}_j^{-} \) values for each criterion.
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Step 7: Calculate the normalized fuzzy distance dij (i = 1, …m, j = 1, …, n) as follows:
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Step 8: Calculate Si and Ri using the following equations:
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Initially, we set the value of parameter v to 0.5. Similarly, as in the case of parameter φ, we later analyze the stability of ranking results changing the value of parameter v in range [0,1].
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Step 10: Rank the alternatives in three descending lists (by values S, R and Q).
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Step 11: Propose a compromise solution.The alternative A′ which is the best ranked by the value Q could be proposed as a compromise solution only if it satisfies the following conditions:
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Condition 1: The best ranked alternative A′ must have the acceptable advantage over the second ranked alternative A′′, i.e. Q(A′′) − Q(A′) ≥ DQ, DQ = min {0.25, 1/(m − 1)} where m stands for the number of alternatives.
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Condition 2: The alternative A′ must also be the best ranked by values S or R making the compromise solution stabile in decision making.
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if Condition 1 is not satisfied: Alternatives A′, A′′, …A(M), where A(M) is obtained using the relation Q(A(M)) − Q(A′) < DQ.
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if only Condition 2 is not satisfied: Alternatives A′ and A′′.
6 An illustrative example
6.1 Implementation of the proposed approach
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DM
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| VH | H | H | VH | H | M | M |
DM
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| H | VH | VH | M | VH | VH | L |
DM
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| M | M | H | VL | M | L | VH |
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DM
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A0 | P | VP | VP | G | P | P | P |
A1 | VG | P | P | VG | VP | G | P |
A2 | F | G | F | G | G | F | P |
A3 | VP | VG | VG | VP | F | VG | VP |
DM
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A0 | P | VP | P | G | P | VP | P |
A1 | VG | P | P | G | P | P | G |
A2 | G | G | G | F | VG | G | F |
A3 | F | VG | VG | P | G | VG | P |
DM
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A0 | F | P | F | G | P | P | VP |
A1 | G | F | G | G | P | F | G |
A2 | G | G | F | G | G | G | P |
A3 | P | G | P | P | G | G | VP |
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w
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| (0.25, 0.75, 1) | (0.25, 0.75, 1) | (0.50, 0.83, 1) | (0, 0.50, 1) | (0.25, 0.75, 1) | (0, 0.58, 1) | (0, 0.58, 1) |
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| (0, 0.33, 0.75) | (0, 0.08, 0.50) | (0, 0.25, 0.75) | (0.50, 0.75, 1) | (0, 0.25, 0.50) | (0, 0.17, 0.50) | (0, 0.17, 0.50) |
A
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| (0.50, 0.92, 1) | (0, 0.33, 0.75) | (0, 0.42, 1) | (0.5, 0.83, 1) | (0, 0.17, 0.50) | (0, 0.50, 1) | (0, 0.58, 1) |
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| (0.25, 0.67, 1) | (0.50, 0.75, 1) | (0.25, 0.58, 1) | (0.25, 0.67, 1) | (0.50, 0.83, 1) | (0.25, 0.67, 1) | (0, 0.33, 0.75) |
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| (0, 0.25, 0.75) | (0.50, 0.92, 1) | (0, 0.75, 1) | (0, 0.17, 0.5) | (0.25, 0.67, 1) | (0.5, 0.92, 1) | (0, 0.08, 0.5) |
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Subj. weights (wjs) | |||||||
Defuzzified | 0.67 | 0.67 | 0.78 | 0.50 | 0.67 | 0.53 | 0.13 |
Normalized | 0.15 | 0.15 | 0.18 | 0.12 | 0.15 | 0.12 | 0.12 |
Obj. weights (wjo) | |||||||
ej | 0.95 | 0.91 | 0.98 | 0.94 | 0.90 | 0.94 | 0.94 |
divj | 0.05 | 0.09 | 0.02 | 0.06 | 0.10 | 0.06 | 0.06 |
Normalized | 0.11 | 0.21 | 0.04 | 0.14 | 0.22 | 0.14 | 0.13 |
Comb. weights (wjc) | 0.13 | 0.18 | 0.11 | 0.13 | 0.19 | 0.13 | 0.13 |
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\( {f}_j^{\ast } \)
| (0.50, 0.92, 1) | (0.50, 0.92, 1) | (0.25, 0.58, 1) | (0.50, 0.83, 1) | (0.50, 0.83, 1) | (0.50, 0.92, 1) | (0, 0.58, 1) |
\( {f}_j^{-} \)
| (0, 0.25, 0.75) | (0, 0.08, 0.50) | (0, 0.25, 0.75) | (0, 0.17, 0.50) | (0, 0.17, 0.50) | (0, 0.17, 0.50) | (0, 0.08, 0.50) |
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d(fj*-fj
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| 0.93 | 1.00 | 1.00 | 0.09 | 0.94 | 1.00 | 0.92 |
d(fj*-A
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| 0.00 | 0.74 | 0.62 | 0.00 | 1.00 | 0.63 | 0.00 |
d(fj*-A
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| 0.41 | 0.15 | 0.00 | 0.31 | 0.00 | 0.34 | 0.50 |
d(fj*-A
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| 1.00 | 0.00 | 0.62 | 1.00 | 0.31 | 0.00 | 1.00 |
d(fj*-A
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| 0.93 | 1.00 | 1.00 | 0.09 | 0.94 | 1.00 | 0.92 |
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Condition 1: Q(A2) − Q(A1) = 0.6 which is higher than DQ (DQ = 0.25) and
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Condition 2: the alternative A2 is the best ranked by the measures S and R also.
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S
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| 0.85 | 0.48 | 0.23 | 0.51 |
R
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| 0.18 | 0.19 | 0.06 | 0.13 |
Q
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| 0.98 | 0.70 | 0 | 0.50 |
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S
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| 4 | 2 | 1 | 3 |
R
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| 3 | 4 | 1 | 2 |
Q
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| 4 | 3 | 1 | 2 |
6.2 Result discussions
Ranking | ||||
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by DM1 |
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