1 Introduction
(1)
\[ \text{Weight space}=\Bigg\{({\omega _{1}},\dots ,{\omega _{n}})\mid {\omega _{1}}\geqslant {\omega _{2}}\geqslant \cdots \geqslant {\omega _{n}},\hspace{2.5pt}{\sum \limits_{j=1}^{n}}{\omega _{j}}=1,\hspace{2.5pt}{\omega _{j}}\geqslant 0\Bigg\}.\]2 Improved ROC (IROC)
2.1 The Proposed Idea
(2)
\[\begin{aligned}{}\text{ROC weights:}\left(\begin{array}{c}{\omega _{1}}\\ {} {\omega _{2}}\\ {} \begin{array}{c}\vdots \\ {} {\omega _{n}}\end{array}\end{array}\right)& =\frac{1}{n}\left(\begin{array}{c}1\\ {} 0\\ {} \vdots \\ {} 0\end{array}\right)+\frac{1}{n}\left(\begin{array}{c}1/2\\ {} 1/2\\ {} 0\\ {} \vdots \\ {} 0\end{array}\right)+\cdots +\frac{1}{n}\left(\begin{array}{c}1/n\\ {} 1/n\\ {} \vdots \\ {} 1/n\end{array}\right)\Rightarrow {\omega _{j}}\\ {} & =\frac{{\textstyle\textstyle\sum _{r=j}^{n}}(1/r)}{n}.\end{aligned}\](3)
\[\begin{aligned}{}\text{IROC weights:}\left(\begin{array}{c}{\omega _{1}}\\ {} {\omega _{2}}\\ {} \vdots \\ {} {\omega _{n}}\end{array}\right)& ={\varphi _{1n}}\left(\begin{array}{c}1\\ {} 0\\ {} \vdots \\ {} 0\end{array}\right)+{\varphi _{2n}}\left(\begin{array}{c}1/2\\ {} 1/2\\ {} 0\\ {} \vdots \\ {} 0\end{array}\right)+\cdots +{\varphi _{nn}}\left(\begin{array}{c}1/n\\ {} 1/n\\ {} \vdots \\ {} 1/n\end{array}\right)\Rightarrow {\omega _{j}}\\ {} & ={\sum \limits_{r=j}^{n}}({\varphi _{rn}}/r).\end{aligned}\]2.2 Determining the IROC Coefficients
Table 1
n | The coefficients ${\varphi _{jn}}$ ($j=1,2,\dots ,n$) of the vertices respectively |
2 | 0.51150, 0.48850, |
3 | 0.32415, 0.35529, 0.32056, |
4 | 0.23236, 0.26888, 0.26293, 0.23583, |
5 | 0.17918, 0.21316, 0.21710, 0.20416, 0.18639, |
6 | 0.14482, 0.17384, 0.18191, 0.17813, 0.16733, 0.15397, |
7 | 0.12136, 0.14634, 0.15571, 0.15561, 0.14946, 0.14063, 0.13089, |
8 | 0.10278, 0.12579, 0.13581, 0.13710, 0.13400, 0.12837, 0.12160, 0.11456, |
9 | 0.09000, 0.11021, 0.11944, 0.12201, 0.12101, 0.11760, 0.11215, 0.10655, 0.10102, |
10 | 0.07919, 0.09745, 0.10607, 0.10893, 0.10985, 0.10786, 0.10405, 0.10011, 0.09575, 0.09075, |
11 | 0.07104, 0.08682, 0.09547, 0.09880, 0.10022, 0.09916, 0.09680, 0.09375, 0.08982, 0.08595, 0.08218, |
12 | 0.06445, 0.07851, 0.08648, 0.09051, 0.09113, 0.09129, 0.08978, 0.08800, 0.08502, 0.08182, 0.07812, 0.07490, |
13 | 0.05837, 0.07143, 0.07857, 0.08266, 0.08450, 0.08437, 0.08405, 0.08215, 0.08000, 0.07760, 0.07482, 0.07192, 0.06955, |
14 | 0.05322, 0.06512, 0.07230, 0.07611, 0.07829, 0.07896, 0.07866, 0.07708, 0.07549, 0.07313, 0.07147, 0.06899, 0.06680, 0.06440, |
15 | 0.04916, 0.06070, 0.06690, 0.07044, 0.07264, 0.07378, 0.07352, 0.07305, 0.07161, 0.06962, 0.06781, 0.06597, 0.06366, 0.06155, 0.05959, |
16 | 0.04576, 0.05618, 0.06207, 0.06556, 0.06761, 0.06870, 0.06922, 0.06853, 0.06742, 0.06609, 0.06475, 0.06314, 0.06144, 0.05971, 0.05789, 0.05595, |
17 | 0.04233, 0.05180, 0.05767, 0.06100, 0.06332, 0.06430, 0.06497, 0.06507, 0.06417, 0.06322, 0.06213, 0.06082, 0.05920, 0.05739, 0.05582, 0.05432, 0.05247, |
18 | 0.03955, 0.04832, 0.05372, 0.05683, 0.05941, 0.06061, 0.06123, 0.06128, 0.06081, 0.06030, 0.05948, 0.05817, 0.05703, 0.05570, 0.05409, 0.05276, 0.05106, 0.04965, |
19 | 0.03706, 0.04535, 0.05026, 0.05364, 0.05600, 0.05729, 0.05785, 0.05823, 0.05778, 0.05759, 0.05684, 0.05593, 0.05480, 0.05350, 0.05224, 0.05097, 0.04961, 0.04807, 0.04698, |
20 | 0.03514, 0.04276, 0.04770, 0.05077, 0.05281, 0.05424, 0.05516, 0.05540, 0.05536, 0.05496, 0.05426, 0.05360, 0.05248, 0.05134, 0.05035, 0.04925, 0.04784, 0.04670, 0.04553, 0.04435, |
21 | 0.03302, 0.04005, 0.04460, 0.04766, 0.04987, 0.05113, 0.05218, 0.05247, 0.05278, 0.05260, 0.05203, 0.05146, 0.05056, 0.04956, 0.04861, 0.04769, 0.04678, 0.04584, 0.04485, 0.04377, 0.04249, |
22 | 0.03132, 0.03816, 0.04228, 0.04508, 0.04729, 0.04878, 0.04965, 0.05013, 0.05031, 0.05011, 0.04979, 0.04924, 0.04869, 0.04786, 0.04699, 0.04617, 0.04540, 0.04430, 0.04341, 0.04258, 0.04172, 0.04075, |
23 | 0.02974, 0.03638, 0.04029, 0.04292, 0.04479, 0.04650, 0.04725, 0.04770, 0.04800, 0.04810, 0.04793, 0.04743, 0.04693, 0.04634, 0.04555, 0.04478, 0.04390, 0.04301, 0.04221, 0.04144, 0.04047, 0.03964, 0.03871, |
24 | 0.02825, 0.03455, 0.03837, 0.04084, 0.04282, 0.04410, 0.04483, 0.04575, 0.04585, 0.04610, 0.04592, 0.04545, 0.04521, 0.04457, 0.04413, 0.04336, 0.04265, 0.04198, 0.04123, 0.04035, 0.03963, 0.03890, 0.03803, 0.03714 |
25 | 0.02702, 0.03286, 0.03681, 0.03932, 0.04108, 0.04234, 0.04323, 0.04371, 0.04399, 0.04423, 0.04410, 0.04386, 0.04348, 0.04318, 0.04267, 0.04193, 0.04134, 0.04059, 0.03984, 0.03933, 0.03855, 0.03778, 0.03696, 0.03624, 0.03556 |
2.3 Comparison
Table 2
HR | RC | ||||||||
Alt. | Cri. | ROC | IROC | Difference | Improvement (%) | ROC | IROC | Difference | Improvement (%) |
3 | 3 | 0.90312 | 0.90372 | −0.00060 | 0.066% | 0.75060 | 0.75068 | −0.00005 | 0.011% |
5 | 0.88866 | 0.89000 | −0.00134 | 0.151% | 0.71887 | 0.72027 | −0.00140 | 0.195% | |
7 | 0.89232 | 0.89388 | −0.00156 | 0.175% | 0.72545 | 0.72837 | −0.00292 | 0.403% | |
10 | 0.89836 | 0.89970 | −0.00134 | 0.149% | 0.73592 | 0.73932 | −0.00340 | 0.462% | |
15 | 0.90692 | 0.90636 | 0.00056 | −0.062% | 0.76072 | 0.76227 | −0.00155 | 0.203% | |
5 | 3 | 0.85868 | 0.85942 | −0.00074 | 0.086% | 0.53050 | 0.53127 | −0.00077 | 0.146% |
5 | 0.85920 | 0.85868 | 0.00052 | −0.061% | 0.52080 | 0.52058 | 0.00022 | −0.041% | |
7 | 0.85882 | 0.85998 | −0.00116 | 0.135% | 0.52694 | 0.53037 | −0.00343 | 0.650% | |
10 | 0.86860 | 0.86884 | −0.00024 | 0.028% | 0.54814 | 0.55313 | −0.00499 | 0.911% | |
15 | 0.87904 | 0.88114 | −0.00210 | 0.239% | 0.57459 | 0.57854 | −0.00395 | 0.686% | |
7 | 3 | 0.83384 | 0.83444 | −0.00060 | 0.072% | 0.36175 | 0.36164 | 0.00011 | −0.032% |
5 | 0.83800 | 0.83916 | −0.00116 | 0.138% | 0.35628 | 0.35628 | 0.00000 | 0.000% | |
7 | 0.84466 | 0.84542 | −0.00076 | 0.090% | 0.36715 | 0.37102 | −0.00387 | 1.054% | |
10 | 0.85362 | 0.85372 | −0.00010 | 0.012% | 0.39131 | 0.39515 | −0.00384 | 0.981% | |
15 | 0.86294 | 0.86418 | −0.00124 | 0.144% | 0.42541 | 0.43017 | −0.00476 | 1.119% | |
10 | 3 | 0.81382 | 0.81384 | −0.00002 | 0.002% | 0.17066 | 0.16968 | 0.00098 | −0.574% |
5 | 0.82344 | 0.82256 | 0.00088 | −0.107% | 0.16099 | 0.16252 | −0.00153 | 0.954% | |
7 | 0.82180 | 0.82338 | −0.00158 | 0.192% | 0.17266 | 0.17648 | −0.00382 | 2.210% | |
10 | 0.83904 | 0.84018 | −0.00114 | 0.136% | 0.20291 | 0.20872 | −0.00581 | 2.866% | |
15 | 0.84578 | 0.84652 | −0.00074 | 0.087% | 0.24436 | 0.25106 | −0.00670 | 2.743% | |
Mean | 0.85924 | 0.86026 | −0.00102 | 0.084% | 0.46230 | 0.46488 | −0.00258 | 0.557% |
(8)
\[\begin{aligned}{}& \left\{\begin{array}{l}{\mathrm{H}_{0}}:\hspace{2.5pt}{\mu _{\text{ROC}\hspace{0.1667em}\text{HR}}}-{\mu _{\text{IROC}\hspace{0.1667em}\text{HR}}}=0,\\ {} {\mathrm{H}_{1}}:\hspace{2.5pt}{\mu _{\text{ROC}\hspace{0.1667em}\text{HR}}}-{\mu _{\text{IROC}\hspace{0.1667em}\text{HR}}}<0,\end{array}\right.\end{aligned}\](9)
\[\begin{aligned}{}& \left\{\begin{array}{l}{\mathrm{H}_{0}}:\hspace{2.5pt}{\mu _{\text{ROC}\hspace{0.1667em}\text{RC}}}-{\mu _{\text{IROC}\hspace{0.1667em}\text{RC}}}=0,\\ {} {\mathrm{H}_{1}}:\hspace{2.5pt}{\mu _{\text{ROC}\hspace{0.1667em}\text{RC}}}-{\mu _{\text{IROC}\hspace{0.1667em}\text{RC}}}<0.\end{array}\right.\end{aligned}\](10)
\[\begin{aligned}{}& \left\{\begin{array}{l}{\mathrm{H}_{0}}:\hspace{2.5pt}\text{Differences}({\mu _{\mathrm{ROC}\hspace{2.5pt}\mathrm{HR}}}-{\mu _{\mathrm{IROC}\hspace{2.5pt}\mathrm{HR}}})\hspace{2.5pt}\text{are normally distributed},\\ {} {\mathrm{H}_{1}}:\hspace{2.5pt}\text{Differences}({\mu _{\mathrm{ROC}\hspace{2.5pt}\mathrm{HR}}}-{\mu _{\mathrm{IROC}\hspace{2.5pt}\mathrm{HR}}})\hspace{2.5pt}\text{are not normally distributed},\end{array}\right.\end{aligned}\](11)
\[\begin{aligned}{}& \left\{\begin{array}{l}{\mathrm{H}_{0}}:\hspace{2.5pt}\text{Differences}({\mu _{\mathrm{ROC}\hspace{2.5pt}\mathrm{RC}}}-{\mu _{\mathrm{IROC}\hspace{2.5pt}\mathrm{RC}}})\hspace{2.5pt}\text{are normally distributed},\\ {} {\mathrm{H}_{1}}:\hspace{2.5pt}\text{Differences}({\mu _{\mathrm{ROC}\hspace{2.5pt}\mathrm{RC}}}-{\mu _{\mathrm{IROC}\hspace{2.5pt}\mathrm{RC}}})\hspace{2.5pt}\text{are not normally distributed}.\end{array}\right.\end{aligned}\]Table 3
$n:$ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
ROC | 0.7500 | 0.6111 | 0.5208 | 0.4567 | 0.4083 | 0.3704 | 0.3397 | 0.3143 | 0.2929 | 0.2745 | 0.2586 | 0.2446 | 0.2323 | 0.2212 |
0.2500 | 0.2778 | 0.2708 | 0.2567 | 0.2417 | 0.2276 | 0.2147 | 0.2032 | 0.1929 | 0.1836 | 0.1753 | 0.1677 | 0.1608 | 0.1545 | |
0.1111 | 0.1458 | 0.1567 | 0.1583 | 0.1561 | 0.1522 | 0.1477 | 0.1429 | 0.1382 | 0.1336 | 0.1292 | 0.1251 | 0.1212 | ||
0.0625 | 0.0900 | 0.1028 | 0.1085 | 0.1106 | 0.1106 | 0.1096 | 0.1079 | 0.1058 | 0.1036 | 0.1013 | 0.0990 | |||
0.0400 | 0.0611 | 0.0728 | 0.0793 | 0.0828 | 0.0846 | 0.0851 | 0.0850 | 0.0844 | 0.0834 | 0.0823 | ||||
0.0278 | 0.0442 | 0.0543 | 0.0606 | 0.0646 | 0.0670 | 0.0683 | 0.0690 | 0.0692 | 0.0690 | |||||
0.0204 | 0.0335 | 0.0421 | 0.0479 | 0.0518 | 0.0544 | 0.0562 | 0.0573 | 0.0579 | ||||||
0.0156 | 0.0262 | 0.0336 | 0.0388 | 0.0425 | 0.0452 | 0.0471 | 0.0484 | |||||||
0.0123 | 0.0211 | 0.0275 | 0.0321 | 0.0356 | 0.0381 | 0.0400 | ||||||||
0.0100 | 0.0174 | 0.0229 | 0.0270 | 0.0302 | 0.0326 | |||||||||
0.0083 | 0.0145 | 0.0193 | 0.0230 | 0.0260 | ||||||||||
0.0069 | 0.0123 | 0.0165 | 0.0199 | |||||||||||
0.0059 | 0.0106 | 0.0143 | ||||||||||||
0.0051 | 0.0092 | |||||||||||||
0.0044 | ||||||||||||||
IROC | 0.7557 | 0.6086 | 0.5134 | 0.4464 | 0.3960 | 0.3574 | 0.3251 | 0.2998 | 0.2775 | 0.2591 | 0.2434 | 0.2290 | 0.2163 | 0.2057 |
0.2443 | 0.2845 | 0.2810 | 0.2673 | 0.2512 | 0.2360 | 0.2223 | 0.2098 | 0.1984 | 0.1881 | 0.1789 | 0.1706 | 0.1631 | 0.1566 | |
0.1069 | 0.1466 | 0.1607 | 0.1643 | 0.1628 | 0.1594 | 0.1547 | 0.1496 | 0.1447 | 0.1397 | 0.1349 | 0.1305 | 0.1262 | ||
0.0590 | 0.0883 | 0.1037 | 0.1109 | 0.1142 | 0.1149 | 0.1143 | 0.1129 | 0.1109 | 0.1087 | 0.1064 | 0.1039 | |||
0.0373 | 0.0591 | 0.0720 | 0.0799 | 0.0844 | 0.0870 | 0.0882 | 0.0882 | 0.0880 | 0.0874 | 0.0863 | ||||
0.0257 | 0.0421 | 0.0531 | 0.0602 | 0.0651 | 0.0681 | 0.0700 | 0.0711 | 0.0717 | 0.0718 | |||||
0.0187 | 0.0317 | 0.0406 | 0.0471 | 0.0516 | 0.0548 | 0.0571 | 0.0586 | 0.0595 | ||||||
0.0143 | 0.0245 | 0.0322 | 0.0378 | 0.0420 | 0.0451 | 0.0473 | 0.0490 | |||||||
0.0112 | 0.0197 | 0.0260 | 0.0310 | 0.0348 | 0.0377 | 0.0398 | ||||||||
0.0091 | 0.0161 | 0.0215 | 0.0259 | 0.0293 | 0.0319 | |||||||||
0.0075 | 0.0133 | 0.0181 | 0.0220 | 0.0249 | ||||||||||
0.0062 | 0.0113 | 0.0155 | 0.0188 | |||||||||||
0.0054 | 0.0097 | 0.0133 | ||||||||||||
0.0046 | 0.0084 | |||||||||||||
0.0040 |
3 The Proposed Methodology
Table 4
Coefficient $(T)$ | Consensus in ranks |
[0.0 0.1) | Very weak |
[0.1 0.3) | Weak |
[0.3 0.5) | Fair |
[0.5 0.7) | Strong |
[0.7 0.9) | Very strong |
[0.9 1.0) | Complete |
4 Study Case
4.1 A Brief Introduction
-
• Diesel engines/vehicles such as conventional diesel, ultra-low-sulfur diesel, bio-diesel (e.g. vegetable oil biodiesel, and animal fat biodiesel).
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• Gas engines/vehicles such as Compressed Natural Gas (CNG), Liquefied Propane Gas (LPG), Liquefied Natural Gas (LNG), Dimethyl Ether (DME), Gas-To-Liquid (GTL), and hydrogen fuel cell.
-
• Blend engines/vehicles such as methanol & gasoline blend, hydrogen & CNG blend or hythane, and bio-CNG blend.
-
• Electric engines/vehicles such as opportunity charging, direct electric charging, and exchangeable-battery electric.
-
• Hybrid engines/vehicles such as electric & gasoline hybrid, electric & diesel hybrid, electric & CNG hybrid, and electric & LPG hybrid.
4.2 The Criteria List
Table 5
1st level criteria | 2nd level criteria | Description |
Performance-based features | Energy supply | Yearly amount of energy which can be supplied. |
Energy efficiency | The efficiency of fuel/energy used in engine. | |
Traffic flow speed | The average speed of vehicle for definite traffic. | |
Vehicle capabilities | The capability of vehicle, such as speed and slope climbing. | |
Environmental features | Air pollution | The amount of release of pollutants into the air. |
Soil pollution | The amount of release of pollutants into the soil. | |
Water pollution | The amount of release of pollutants to the water, such as organic pollutants, inorganic pollutants, pathogens, suspended solids, nutrients, and agriculture pollutants. | |
Noise pollution | The noise made by operation of engine/vehicle. | |
Economical features | Distance to market | Average distance between the production factories and the consumption region of the related fuel. |
Transportation easiness | The degree of hardness of fuel/energy transportation. | |
Energy storage | The rate of hardness of fuel/energy to be stored. | |
Internal consumption trend | The consumption trend of fuel in the region under study. | |
World trend | The consumption trend of fuel in the world. Specifically, the focus point of the big oil companies. | |
Fixed price | The fixed price of fuel/energy. | |
Financial features | Purchase cost | The purchase cost of vehicle. |
Maintenance cost | The maintenance cost of engine/vehicle. | |
Infrastructural features | Road infrastructures | The road infrastructures required for the operation of vehicle. |
Industrial infrastructures | The existent industrial infrastructures to produce engine/vehicle. | |
Technological features | Maturity of technology | The maturity level of the relevant technologies. |
Safety aspects | The safety features of engine/vehicle. | |
Industrial relationships | The relationship between engine/vehicle industrial system and other industrial sectors. | |
Social features | Community acceptability | The extent to which the community’s people accept vehicle. |
Accessories | The accessories and other options of vehicle, in order to provide sense of comfort. | |
Risk-based features | Political risks | For example, regulatory, diplomacy, and entente risks. |
Economical risks | For example, inflation, rent, and sanctions risks. | |
Social risks | For example, risks concerned with culture, carriers, and psychology. | |
Technical risks | Risks and uncertainties related to technical and operational aspects, for example, maintainability. |
4.3 The Criteria Weights
Table 6
Ranks by the experts | ${R_{j}}$ | Overall rank | ||||||||||
Criteria | #1 | #2 | #3 | #4 | #5 | #6 | #7 | #8 | #7 | #9 | ||
Performance-based features | 1 | 2 | 1 | 1 | 3 | 1 | 2 | 1 | 2 | 1 | 14 | 1 |
Environmental features | 4 | 1 | 2 | 2 | 2 | 4 | 5 | 2 | 1 | 4 | 23 | 2 |
Economical features | 2 | 4 | 3 | 3 | 1 | 2 | 6 | 3 | 7 | 2 | 31 | 3 |
Financial features | 3 | 3 | 4 | 8 | 6 | 3 | 3 | 7 | 8 | 3 | 45 | 4 |
Infrastructural features | 6 | 5 | 7 | 6 | 7 | 5 | 1 | 6 | 3 | 6 | 46 | 5 |
Technological features | 5 | 6 | 5 | 7 | 8 | 8 | 8 | 8 | 4 | 5 | 59 | 8 |
Social features | 8 | 8 | 6 | 5 | 4 | 7 | 7 | 4 | 5 | 8 | 54 | 7 |
Risk-based features | 7 | 7 | 8 | 4 | 5 | 6 | 4 | 5 | 6 | 7 | 52 | 6 |
$\overline{R}=40.5$, $T=0.5309$ |
Table 7
1st level criteria | 2nd level criteria | Final weight | ||||
Title | Rank | Weight | Title | Rank | weight | |
Performance-based features | 1 | 0.3251 | Energy supply | 2 | 0.2810 | 0.0914 |
Energy efficiency | 1 | 0.5134 | 0.1669 | |||
Traffic flow speed | 4 | 0.0590 | 0.0192 | |||
Vehicle capabilities | 3 | 0.1466 | 0.0477 | |||
Environmental features | 2 | 0.2223 | Air pollution | 1 | 0.5134 | 0.1141 |
Soil pollution | 3 | 0.1466 | 0.0326 | |||
Water pollution | 2 | 0.2810 | 0.0625 | |||
Noise pollution | 4 | 0.0590 | 0.0131 | |||
Economical features | 3 | 0.1594 | Distance to market | 1 | 0.3960 | 0.0631 |
Transportation easiness | 2 | 0.2512 | 0.0400 | |||
Energy storage | 4 | 0.1037 | 0.0165 | |||
Internal consumption trend | 3 | 0.1643 | 0.0262 | |||
World trend | 6 | 0.0257 | 0.0041 | |||
Fixed price | 5 | 0.0591 | 0.0094 | |||
Financial features | 4 | 0.1142 | Purchase cost | 1 | 0.7557 | 0.0863 |
Maintenance cost | 2 | 0.2443 | 0.0279 | |||
Infrastructural features | 5 | 0.0799 | Road infrastructures | 2 | 0.2443 | 0.0195 |
Industrial infrastructures | 1 | 0.7557 | 0.0604 | |||
Technological features | 8 | 0.0143 | Maturity of technology | 1 | 0.6086 | 0.0087 |
Safety aspects | 2 | 0.2845 | 0.0041 | |||
Industrial relationships | 3 | 0.1069 | 0.0015 | |||
Social features | 7 | 0.0317 | Community acceptability | 1 | 0.7557 | 0.0240 |
Accessories | 2 | 0.2443 | 0.0077 | |||
Risk-based features | 6 | 0.0531 | Political risks | 1 | 0.5134 | 0.0273 |
Economical risks | 2 | 0.2810 | 0.0149 | |||
Social risks | 3 | 0.1466 | 0.0078 | |||
Technical risks | 4 | 0.0590 | 0.0031 |