A hybrid ranking approach based on fuzzy analytical hierarchy process and data envelopment analysis: Road maintenance and transport organization of Iran

Abstract

The continuous improvement of performance in organizations requires the continuous and purposeful evaluation of organizations. One of the most vital sectors that significantly affects the economic growth of countries is transportation; in this regard, performance assessment plays an important role in this sector. Therefore, a novel hybrid algorithm based on Fuzzy Analytical Hierarchy Process (FAHP) and Data Envelopment Analysis (DEA) is proposed here to measure the efficiency of product transportation in road fleets of Iranian provinces. FAHP method is used to define the weights of input and output criteria, and the priority of criteria is implemented as a constraint in a DEA model. The efficiency of each province in this domain will be determined by means of the proposed DEA model. The results obtained from the application of the proposed approach to Iran Road Maintenance & Transportation Organization vindicated the effectiveness and efficiency of the proposed approach.

Keywords

Transportation analytical hierarchy process data envelopment analysis fuzzy theory

1 Introduction

Transportation is one of the basic needs of humans. The fast rate of global transformations and changes as well as the continuous progress of communities, the diversity of products and services in the world, the competitiveness of economic activities, division of labor, product production and service delivery based on regions’ competitive advantages, etc. have created the increasing significance of transportation for the fast transfer of goods and people. On the other hand, transportation is known as one of the main prerequisites of development, and the economic growth of developed countries is indebted to the expansion of their transportation and communication systems. The enjoyment of a tactical and strategic assessment system is an important factor in the better development of this industry. Therefore, this study proposes a strategic approach for the performance assessment of the product transportation fleet in road sector in different provinces of Iran.

There are a large number of studies that have used Multi-Criteria Decision Making (MCDM) tools in various fields of evaluation and priority, such as Mina et al. [1], Rao et al. [2], Bafrooei et al. [4], Cao et al. [7], Liu et al. [16], Wu et al. [24], Zhang and Guo [25] and Dong and Cooper [26]. However, when it comes to the field of vehicles fleet, a few studies have been carried out by using these tools. Gumus [8] proposed a hybrid model wherein FAHP and TOPSIS are combined for the assessment of hazardous material transportation companies. They have used such criteria as safety, service quality, compliance with standards, and servicing time and embarked on assessing and ranking five companies active in the transportation of hazardous materials. Yeh et al. [22] applied a multi criteria analysis approach for the performance assessment of urban bus companies under uncertainty conditions. They have used fuzzy TOPSIS method and such criteria as comfort, safety, and social duty to rank 10 bus companies.

Lee [11], Morisugi [13], and Quinet [15] have proposed models based on benefit-cost analysis to assess the transportation projects in the USA, Japan, and France, respectively.

Zak & Kruszynski [23] proposed an approach based on AHP and ELECTRE III/IV for the assessment of urban transportation projects. They applied the aforementioned approach for 18 projects where the results indicated the efficiency and user friendliness of their proposed approach.

Haghshenas et al. [9] assessed urban transportation projects with regard to sustainable development policies. They made use of system dynamics approach and assessed urban transportation projects from the perspective of sustainable development in Isfahan by using the data pertaining to the past four decades.

Shen et al. [17] proposed an applied approach for the assessment of metro infrastructure projects in China. They used such criteria as population of city, length of metro system, annual ridership of metro system, and Gross Domestic Product (GDP) to assess the metro system of 17 cities in China.

Baran & Zak [5] proposed an approach based on AHP for assessing the transportation performance of agribusiness companies. They used transportation cost, delivery time, fleet modernity, transportation reliability, transportation quality, safety, environmental friendliness, and fleet utilization as criteria for the assessment of companies.

In a review paper written by Macharis & Bernardini [12], different types of papers about transportation projects have been reviewed, and, then, the methods used in these papers have been identified, among which approaches based on ANP/AHP with the weight of 33% have been used more frequently than the other methods. In Table 1, some studies in the field of transportation assessment have been shown.

Table 1
Comparison of the proposed approach with those of other studies

Author(s) Year Assessment Approach Quantitative Criterion Qualitative Criterion Uncertainty Case Study Field

Morisugi [13] 2000 Cost-Benefit Analysis Yes Yes – Yes Assessment of transportation projects in Japan

Quinet [15] 2000 Cost-Benefit Analysis Yes Yes – Yes Assessment of transportation projects in France

Talvitie [19] 2000 Cost-Benefit Analysis Yes Yes – Yes

Lee [11] 2000 Cost-Benefit Analysis Yes Yes – Yes Assessment of transportation projects in the USA

Iniestra and Gutiérrez [10] 2009 A hybrid approach, combining multi-objective programming and ELECTRE III Yes Yes – – Transportation projects portfolio

Wu et al. [21] 2011 Mixed integer programming Yes – – – Assessment of Build-operation-implementation transportation projects

Shi and Zhou [18] 2012 Nonlinear mathematical programming Yes Yes – Yes Assessment of Road projects

Haghshenas et al. [9] 2014 System Dynamics Yes Yes – Yes Isfahan Urban transportation projects

Baran and Zak [5] 2014 AHP – Yes – Yes Transportation projects of agribusiness companies

Shen et al. [17] 2015 utility efficiency evaluation index (UEEI) model Yes Yes – Yes Metro infrastructure project

Macharis and Bernardini [12] 2015 A review paper about the application of multi criteria decision making approach in assessing transportation projects – – – – –

Quadros and Nassi [14] 2015 AHP – Yes – Yes Assessment of infrastructure transportation projects

Wandelt and Sun [20] 2015 – Yes Yes – Yes Assessment of transportation infrastructure projects

Żak and Kruszyński [23] 2015 AHP and ELECTRE III/IV Yes Yes – Yes Urban transportation projects

Proposed approach FAHP and DEA Yes Yes Yes Yes Road Maintenance and Transport Organization of Iran

Author(s)	Year	Assessment Approach	Quantitative Criterion	Qualitative Criterion	Uncertainty	Case Study	Field
Morisugi [13]	2000	Cost-Benefit Analysis	Yes	Yes	–	Yes	Assessment of transportation projects in Japan
Quinet [15]	2000	Cost-Benefit Analysis	Yes	Yes	–	Yes	Assessment of transportation projects in France
Talvitie [19]	2000	Cost-Benefit Analysis	Yes	Yes	–	Yes
Lee [11]	2000	Cost-Benefit Analysis	Yes	Yes	–	Yes	Assessment of transportation projects in the USA
Iniestra and Gutiérrez [10]	2009	A hybrid approach, combining multi-objective programming and ELECTRE III	Yes	Yes	–	–	Transportation projects portfolio
Wu et al. [21]	2011	Mixed integer programming	Yes	–	–	–	Assessment of Build-operation-implementation transportation projects
Shi and Zhou [18]	2012	Nonlinear mathematical programming	Yes	Yes	–	Yes	Assessment of Road projects
Haghshenas et al. [9]	2014	System Dynamics	Yes	Yes	–	Yes	Isfahan Urban transportation projects
Baran and Zak [5]	2014	AHP	–	Yes	–	Yes	Transportation projects of agribusiness companies
Shen et al. [17]	2015	utility efficiency evaluation index (UEEI) model	Yes	Yes	–	Yes	Metro infrastructure project
Macharis and Bernardini [12]	2015	A review paper about the application of multi criteria decision making approach in assessing transportation projects	–	–	–	–	–
Quadros and Nassi [14]	2015	AHP	–	Yes	–	Yes	Assessment of infrastructure transportation projects
Wandelt and Sun [20]	2015	–	Yes	Yes	–	Yes	Assessment of transportation infrastructure projects
Żak and Kruszyński [23]	2015	AHP and ELECTRE III/IV	Yes	Yes	–	Yes	Urban transportation projects
Proposed approach	FAHP and DEA	Yes	Yes	Yes	Yes	Road Maintenance and Transport Organization of Iran

According to Table 1, a new hybrid approach based on FAHP and DEA is proposed to evaluate the relative efficiency of each province from the perspective of Road Maintenance and Transport Organization in Iran. In the proposed approach, the weights of the input and output criteria are calculated using the FAHP method, and the provinces are prioritized relative to each other in terms of road maintenance and transportation according to their weights. Then, the prioritization results are implemented as the constraints of the proposed DEA model. To the best of the authors’ knowledge, this is one of the first studies that puts its focus on the ranking of road transport areas by using a novel Decision Support System (DSS). The effectiveness and efficiency of the DSS represent its appropriate performance.

The next section of the paper is dealt with the description of the problem and the proposed approach. The third part is dedicated to the implementation of the proposed model in road Maintenance and Transport Organization of Iran. Finally, the conclusion of this paper will be presented in the fourth section.

2 Motivations

All public and private organizations require a performance assessment system for development, growth, and sustainability so that they can measure the efficiency and effectiveness of their processes, organizational programs, and also human resources. Efficient organizations not only gather and analyze data and information, but also use such data and information to improve the organization, realize the missions, and implement the strategies. In other words, they put their main focus on performance management rather than performance assessment.

The management quality and effectiveness of the administrative system are crucial factors in the realization of development and welfare programs of countries. The high organizational costs, limited financial resources, and the low efficiency of these organizations have compelled governments to assign attention to the realization of these organizations’ goals. Considering the results and the realization of goals, the continuous quality improvement of services and products, and citizens’ satisfaction, it is required to focus on performance assessment and management.

Road transport area is not an exception and it requires an assessment and ranking system for the purpose of development, growth, and sustainability in the competitive environment in order to achieve such goals as the creation of an appropriate legal platform for privatization, cost reduction, economy of activities, increased productivity, qualitative development of human resources, improvement of service quality, enhancement of business and marketing activities, use of equipment and fleet with new technologies, reduction of environmental pollution and increase of safety, expansion of multi-modal, compound and container-based transport, and establishment of a trade-off in road transport.

According to the statistics released by Road Maintenance and Transport Organization of Iran, more than 90% of products are carried by road transport fleets. To this end, more than 4,200 companies are active in the field of road transportation in the country, around 90% of which belong to the private sector (http://www.rmto.ir). In the last three years, the annual rate of 385 million tons of products have been carried through road transportation by 423000 trucks and 464000 truck drivers on average.

Based on the above information, the field of road transportation has a vital role in goods delivery, lots of industries are benefited through this field, and a large number of jobs have been influenced either directly or indirectly by this area. Hence, the presentation of a performance assessment approach can have a significant impact on the improved performance of this domain.

3 The proposed approach

This article aims to propose a hybrid approach based on FAHP and DEA to measure and rank the performance of Iranian provinces in terms of their effectiveness in the field of road transportation. The proposed approach is illustrated in Fig. 1 and presented as follows:

First step: Identifying evaluation criteria

This step determines the criteria for fleet performance evaluation to rank the performance of provinces. To the best of the authors’ knowledge, due to the absence of any criterion for performance assessment in the area of road transportation in the literature, the evaluation criteria were obtained based on the experts’ consensus and brainstorming method. These criteria are divided into two groups, namely input criteria and output criteria, as shown below:

Input criteria:

Average freight cost per ton-kilometer based on the origin province (IC1)

Number of public transportation vehicles (IC2)

Number of transportation vehicle drivers (IC4)

Number of active companies and organizations in load transportation area (IC4)

Output criteria:

Quantity of imported goods into the province (OC1)

Quantity of exported goods out of the province (OC2)

Quantity of transported goods in the province (OC3)

Average travelled distance in each travel based on the origin province (OC4)

Number of inter-provincial travels by each loading truck (OC5)

Number of out-of-province travels by each loading truck (OC6)

Number of loading trucks entered to the province (OC7)

Fig.1

Procedure of proposed approach.

Data extraction is fulfilled by a performance assessment team in the Road Organization where these data are collected annually and are included on the website of the Road Organization (www.rmto.ir).

Second step: Criteria weighting using FAHP

In this step, FAHP presented by Bozbura and Beskese [6], is used to weight the criteria so that the importance of each criterion is obtained. For this purpose, pairwise comparison questionnaires were sent to the experts and, they were asked to fill the questionnaires based on linguistic terms of Table (2). Then, the weight of each criterion is determined using Bozbura and Beskese’s [6] method and, thereafter, the constraints pertaining to the weight of each criterion will be recognized by applying data environment analysis. This process is presented as follows:

Table 2

Linguistic scale for difficulty and importance [1]

Linguistic scales for difficulty	Linguistic scales for importance	Triangular fuzzy scale	Triangular fuzzy reciprocal scale
Just equal	Just equal	(1, 1, 1)	(1, 1, 1)
Equally difficult	Equally importance	(1/2, 1, 3/2)	(2/3, 1, 2)
Weakly more difficult	Weakly more importance	(1, 3/2, 2)	(1/2, 2/3, 1)
Strongly more difficult	Strongly more importance	(3/2, 2, 5/2)	(2/5, 1/2, 2/3)
Very strongly more difficult	Very Strongly more importance	(2, 5/2, 3)	(1/3, 2/5, 1/2)
Absolutely more difficult	Absolutely more importance	(5/2, 3, 7/2)	(2/7, 1/3, 2/5)

Assume that $M_{g_{i}}^{j}$ represents the triangular fuzzy numbers located in the ith row and the jth column of the pairwise comparison matrix and the following is true:

$\begin{matrix} \sum_{j = 1}^{m} M_{g_{i}}^{j} = (\sum_{j = 1}^{m} a_{ij}, \sum_{j = 1}^{m} b_{ij}, \sum_{j = 1}^{m} c_{ij}), \\ i = 1, 2, 3, \dots, n \end{matrix}$ (1)

Where a, b and c represent the lower, middle and upper bounds of triangular fuzzy numbers, respectively. The following operation should be performed to obtain ${[\sum_{i = 1}^{n} \sum_{j = 1}^{m} M_{g_{i}}^{j}]}^{- 1}$ : $\begin{matrix} [\sum_{i = 1}^{n} \sum_{j = 1}^{m} M_{g_{i}}^{j}] \\ = (\sum_{i = 1}^{n} \sum_{j = 1}^{m} a_{ij}, \sum_{i = 1}^{n} \sum_{j = 1}^{m} b_{ij}, \sum_{i = 1}^{n} \sum_{j = 1}^{m} c_{ij}) \end{matrix}$ (2) $\begin{matrix} {[\sum_{i = 1}^{n} \sum_{j = 1}^{m} M_{g_{i}}^{j}]}^{- 1} \\ = (\frac{1}{\sum_{i = 1}^{n} \sum_{j = 1}^{m} c_{ij}}, \frac{1}{\sum_{i = 1}^{n} \sum_{j = 1}^{m} b_{ij}}, \frac{1}{\sum_{i = 1}^{n} \sum_{j = 1}^{m} a_{ij}}) \end{matrix}$

Fuzzy synthetic extent (S_i) is defined as below: $S_{i} = \sum_{j = 1}^{m} M_{g_{i}}^{j} \otimes {[\sum_{i = 1}^{n} \sum_{j = 1}^{m} M_{g_{i}}^{j}]}^{- 1}$ (3)

Then, the possibility degree must be determined. For example, the possibility degree of expression M₂ ≥ M₁ is defined as: $V (M_{2} \geq M_{1}) = Sup [Min (μ_{M_{1}} (x), μ_{M_{2}} (y))], y \geq x$ (4)

This possibility degree is calculated via the following formula:

$\begin{matrix} V (M_{2} \geq M_{1}) = hgt (M_{1} \cap M_{2}) \\ = μ_{M_{2}} (d) = {\begin{matrix} 1 \\ 0 \\ \frac{a_{1} - c_{2}}{(b_{2} - c_{2}) - (b_{1} - a_{1})} \end{matrix} \begin{matrix} b_{2} \geq b_{1} \\ a_{1} \geq c_{2} \\ O . W \end{matrix} \end{matrix}$ (5)

Where d is the maximum height between μ_{M
₁} and μ_{M
₁}. Figure 2 illustrates this concept.

Fig.2

M1 and M2 intersection diagram.

The next stage defines the possibility degree for convex fuzzy numbers:

$\begin{matrix} V (M \geq M_{1}, M_{2}, \dots, M_{k}) \\ = V [(M \geq M_{1}) & (M \geq M_{2}) & \dots & (M \geq M_{k})] \\ = Min (M \geq M_{i}) \\ i = 1, 2, \dots, k \end{matrix}$ (6)

Thus, it is assumed that $d^{'} (A_{i}) = Min V (S_{i} \geq S_{k})$ (7)

Then, the weighting vector is represented as follows: $W^{'} = (d^{'} (A_{1}), d^{'} (A_{2}), \dots, d^{'} (A_{n}))^{T}$ (8)

At this point, the obtained weight vector should be normalized: $W = (d (A_{1}), d (A_{2}), \dots, d (A_{n}))^{T}$ (9)

Thereafter, it is possible to calculate the local weight of each criterion.

After determination of the weights, the criteria are prioritized. Then, the priority between the criteria is expressed in terms of mathematical constraints so that these constraints can be added to DEA model. For example, if there are n input criteria and m output criteria, then, n-1 constraints are added to the proposed DEA model for input criteria and m-1 constraints are added to it for output criteria.

Third step: Province ranking using DEA method

In this step, the extended form of the model proposed by Azadeh et al. [3] is used to rank provinces in terms of their efficiency in the domain of road transport. For the development of DEA model, the constraints pertaining to the prioritization of criteria, which were determined in the second step, will be added to Azadeh et al.’s model [3]. The developed model has been presented below:

Indices

i	Province (DMU)	i = 1, 2, …, I
k	Evaluation criteria (input and output)	k = 1, 2, …, n + m
k ₁	Input criteria	k₁ = 1, 2, …, n
k ₂	Output criteria	k₂ = n + 1, …, n + m
k₁ ∪ k₂ = k

Parameters

inp _{ik ₁}	Performance of province i (DMU) with respect to kth input criterion
outp _{ik ₂}	ith province performance (DMU) with respect to kth output criterion
ɛ	Very small number

Variables

d _i	Deviation of ith province efficiency
w _{k ₁}	Weight of input criterion k₁
$w_{k_{2}}^{⌢}$	Weight of output criterion k₂
z	Objective function value

Mathematical model

Objective Function $Min z$ (10) $\begin{matrix} St : \\ z \geq d_{i} \forall i \end{matrix}$ (11) $\frac{\sum_{k_{2}} {outp}_{{ik}_{2}} \times w_{k_{2}}^{⌢}}{\sum_{k_{1}} {inp}_{{ik}_{1}} \times w_{k_{1}}} + d_{i} = 1 \forall i$ (12) $w_{k_{1}} \geq ɛ$ (13) $w_{k_{2}}^{⌢} \geq ɛ$ (14)

The objective function of the proposed model aims at minimizing deviation from efficiency.

Constraint (11) defines the value of the objective function as the greatest deviation from efficiency. In other words, the objective function embarks on minimizing the deviation from efficiency based on constraint (11). The total of efficiency and deviation from efficiency for each province should be equal to one where constraint (12) represents this formula.

Constraints (13) and (14) ensure that the weights obtained from input and output criteria should not be zero.

Based on the proposed model, the efficiency deviation of each province is calculated and is ranked according to the mentioned score.

4 Case study

Iranian road transport is of great importance due to such reasons as vast climate variation, expansion of road transport network, limited coverage of rail and air transportation, lack of navigable waterways, flexibility, economic nature, and potential usability of environmental and tourism attractions. This is highlighted by the fact that more than 90 percent of Iranian product transportations are done through roads.

Each year 1 , in Iranian road transport, 4299 active companies or organizations transport more than 380 million tons of goods in 27 million travels where more than 90 percent of these companies belong to the private sector.

The performance improvement of these companies depends on the assessment of their performance and efficiency so that this important goal (i.e., performance improvement) can be achieved through the reinforcement of the weaknesses and the consolidation of the strengths. The development of an evaluation approach requires the selection of an appropriate evaluation criteria and an effective and efficient model. For this purpose, a combined approach of DEA and FAHP was presented. In this section, the presented approach was implemented in Iranian Road Maintenance and Transportation Organization for its efficiency measurement and validation. The results are presented in following steps:

First step

In this step, the evaluation criteria required for the evaluation of each province performance are identified. The data related to input and output criteria are presented in Tables (3) and (4), respectively.

Table 3
Input criteria data for each province

Average freight cost per ton-kilometer on its origin province (Rial) Quantity of goods transported in each province Number of transportation vehicle drivers Number of active load transportation companies and organizations

East Azerbayejan 943 24563 29487 265

West Azerbayejan 885 15462 23802 135

Ardabil 1041 7560 11368 77

Isfahan 791 36266 48935 449

Alborz 1286 13490 8820 68

Ilam 856 2534 4362 43

Booshehr 993 2994 4369 120

Tehran 1129 31874 16180 369

Chaharmahal and Bakhtiari 850 7551 14603 47

South Khorasan 676 3316 8254 55

Khorasan Razavi 670 28882 39321 269

North Khorasan 737 3289 6944 24

Khoozestan 829 13138 18401 289

Zanjan 954 4983 6638 66

Semnan 649 3126 4758 71

Sistan and baluchestan 779 4901 8139 115

Fars 809 36614 54099 218

Qazvin 1110 9269 12084 76

Qom 637 3589 5090 39

Kordestan 623 5407 10450 67

Kerman 836 12821 16528 250

Kermanshah 1071 12948 17368 65

Kohkiluye and Boyerahmad 1441 954 1829 16

Golestan 950 7574 11361 67

Gilan 1128 4834 8283 136

Lorestan 754 9371 15301 86

Mazandaran 1267 11420 16343 116

Markazi 851 8440 11044 121

Hormozgan 968 4250 3817 154

Hamedan 796 13240 19230 91

Yazd 646 8757 13307 164

	Average freight cost per ton-kilometer on its origin province (Rial)	Quantity of goods transported in each province	Number of transportation vehicle drivers	Number of active load transportation companies and organizations
East Azerbayejan	943	24563	29487	265
West Azerbayejan	885	15462	23802	135
Ardabil	1041	7560	11368	77
Isfahan	791	36266	48935	449
Alborz	1286	13490	8820	68
Ilam	856	2534	4362	43
Booshehr	993	2994	4369	120
Tehran	1129	31874	16180	369
Chaharmahal and Bakhtiari	850	7551	14603	47
South Khorasan	676	3316	8254	55
Khorasan Razavi	670	28882	39321	269
North Khorasan	737	3289	6944	24
Khoozestan	829	13138	18401	289
Zanjan	954	4983	6638	66
Semnan	649	3126	4758	71
Sistan and baluchestan	779	4901	8139	115
Fars	809	36614	54099	218
Qazvin	1110	9269	12084	76
Qom	637	3589	5090	39
Kordestan	623	5407	10450	67
Kerman	836	12821	16528	250
Kermanshah	1071	12948	17368	65
Kohkiluye and Boyerahmad	1441	954	1829	16
Golestan	950	7574	11361	67
Gilan	1128	4834	8283	136
Lorestan	754	9371	15301	86
Mazandaran	1267	11420	16343	116
Markazi	851	8440	11044	121
Hormozgan	968	4250	3817	154
Hamedan	796	13240	19230	91
Yazd	646	8757	13307	164

Table 4

Output criteria data for each province

	Quantity of goods imported into the province (thousand tons)	Quantity of goods exported from the province (thousand tons)	Quantity of goods transported in the province (thousand tons)	Average distance travelled in each travel in the province (kilometer)	Number of inter-provincial travels by each loading truck (thousand travel)	Number of out-of-province travels by each loading truck (thousand travel)	Number of active companies and organizations in load transportation area (thousand travels)
East Azerbayejan	9673	9408	5238	461	411	793	726
West Azerbayejan	5272	4859	4360	441	338	390	437
Ardabil	2129	2293	1215	410	90	194	189
Isfahan	20846	29877	11369	476	654	1899	1460
Alborz	7818	4374	1784	291	138	439	589
Ilam	3130	1586	1148	311	79	95	217
Booshehr	4772	4740	1604	691	92	334	401
Tehran	37865	22781	5618	532	401	2565	3081
Chaharmahal and Bakhtiari	2066	2463	639	371	67	188	164
South Khorasan	3424	2651	1090	474	95	169	222
Khorasan Razavi	12575	14118	10429	576	866	1002	953
North Khorasan	1882	2532	772	407	70	160	155
Khoozestan	22874	28249	8253	678	481	1552	1350
Zanjan	3011	3505	1777	320	133	239	228
Semnan	3143	7561	1170	435	90	477	216
Sistan and baluchestan	5888	1559	4416	514	289	110	361
Fars	11623	14931	9054	464	785	1227	891
Qazvin	7299	6507	665	349	54	509	489
Qom	5118	4255	397	475	28	391	387
Kordestan	3659	3460	1594	379	121	234	304
Kerman	7666	13524	3594	625	206	749	553
Kermanshah	9029	3675	3821	309	249	249	606
Kohkiluye and Boyerahmad	1415	342	157	309	14	29	117
Golestan	4353	3984	1366	394	115	341	324
Gilan	6467	5201	2693	353	169	411	525
Lorestan	2778	3284	1446	359	97	280	249
Mazandaran	11197	6884	6130	340	442	579	874
Markazi	6965	14126	1557	391	101	965	468
Hormozgan	22312	13821	4239	934	204	779	1225
Hamedan	4442	7228	1939	362	141	474	364
Yazd	8178	15092	1555	626	80	827	523

Second step: Weighting of criteria

In this step, the questionnaires are submitted to the experts and they are requested to make pairwise comparisons using linguistic terms of Table (2). Tables (5) and (6) present the obtained results of the questionnaires.

Table 5

Pairwise comparison of input criteria

Input criteria	IC1	IC2	IC3	IC4
IC1	(1,1,1)	(1/2,1,3/2)	(1,3/2,2)	(1,3/2,2)
IC2	(2/3,1,2)	(1,1,1)	(1/2,1,3/2)	(1,3/2,2)
IC3	(1/2,2/3,1)	(2/3,1,2)	(1,1,1)	(1/2,1,3/2)
IC4	(1/2,2/3,1)	(1/2,2/3,1)	(2/3,1,2)	(1,1,1)

Table 6

Pairwise comparison of output criteria

Output criteria	OC1	OC2	OC3	OC4	OC5	OC6	OC7
OC1	(1,1,1)	(1,3/2,2)	(1,3/2,2)	(1/2,1,3/2)	(1/2,1,3/2)	(1,3/2,2)	(3/2,2,5/2)
OC2	(1/2,2/3,1)	(1,1,1)	(1,3/2,2)	(1/2,2/3,1)	(1,3/2,2)	(1,3/2,2)	(3/2,2,5/2)
OC3	(1/2,2/3,1)	(1/2,2/3,1)	(1,1,1)	(1/2,2/3,1)	(1/2,2/3,1)	(1,3/2,2)	(3/2,2,5/2)
OC4	(2/3,1,2)	(1,3/2,2)	(1,3/2,2)	(1,1,1)	(1/2,2/3,1)	(1,3/2,2)	(3/2,2,5/2)
OC5	(2/3,1,2)	(1/2,2/3,1)	(1,3/2,2)	(1,3/2,2)	(1,1,1)	(1,3/2,2)	(1,3/2,2)
OC6	(1/2,2/3,1)	(1/2,2/3,1)	(1/2,2/3,1)	(1/2,2/3,1)	(1/2,2/3,1)	(1,1,1)	(1/2,1,3/2)
OC7	(2/5,1/2,2/3)	(2/5,1/2,2/3)	(2/5,1/2,2/3)	(2/5,1/2,2/3)	(1/2,2/3,1)	(2/3,1,2)	(1,1,1)

Now, the weight of each criterion is obtained using the method developed by Bozbura and Beskese [6]. As an example, the weighting process of input criteria is presented based on Table (5) below.

According to Equation (1), the sum of fuzzy numbers in each row is calculated: $\begin{matrix} \sum_{j = 1}^{4} M_{g_{1}}^{j} = (3.5, 5, 6.5) \\ \sum_{j = 1}^{4} M_{g_{2}}^{j} = (3.167, 4.5, 6.5) \\ \sum_{j = 1}^{4} M_{g_{3}}^{j} = (2.667, 3.667, 5.5) \\ \sum_{j = 1}^{4} M_{g_{4}}^{j} = (2.667, 3.33, 5) \end{matrix}$

As per Equation (2), the reverse of the sum of triangular fuzzy numbers is obtained for all rows and columns: $\begin{matrix} {[\sum_{i = 1}^{4} \sum_{j = 1}^{4} M_{g_{i}}^{j}]}^{- 1} = (\frac{1}{23.5}, \frac{1}{16.497}, \frac{1}{12.001}) \\ {[\sum_{i = 1}^{4} \sum_{j = 1}^{4} M_{g_{i}}^{j}]}^{- 1} = (\frac{1}{23.5}, \frac{1}{16.497}, \frac{1}{12.001}) \end{matrix}$

Then, the fuzzy synthetic extent is determined by Equation (3): $\begin{matrix} S_{1} & = & (0.149, 0.303, 0.542) \\ S_{2} & = & (0.135, 0.273, 0.542) \\ S_{3} & = & (0.113, 0.222, 0.458) \\ S_{4} & = & (0.113, 0.202, 0.417) \end{matrix}$

Then, the possibility degree is obtained using Equations (4), (5) and (6): $\begin{matrix} V (S_{1} \geq S_{2}) = 1 V (S_{1} \geq S_{3}) = 1 V (S_{1} \geq S_{4}) = 1 \\ V (S_{2} \geq S_{1}) = 0.929 V (S_{2} \geq S_{3}) = 1 \\ V (S_{2} \geq S_{4}) = 1 \\ V (S_{3} \geq S_{1}) = 0.792 V (S_{3} \geq S_{2}) = 0.864 \\ V (S_{3} \geq S_{4}) = 1 \\ V (S_{4} \geq S_{1}) = 0.726 V (S_{4} \geq S_{2}) = 0.799 \\ V (S_{4} \geq S_{3}) = 0.938 \end{matrix}$

For each pairwise comparison, the minimum of the possibility degrees is obtained by Equation (7): $\begin{matrix} MinV (s_{1} \geq s_{i}) = 1 \\ MinV (s_{2} \geq s_{i}) = 0.929 \\ MinV (s_{3} \geq s_{i}) = 0.792 \\ MinV (s_{4} \geq s_{i}) = 0.726 \end{matrix}$

Thus, the non-normal weighting vector can be generated (see Equation (8)): $W^{'} = (1, 0.929, 0.792, 0.726)^{T}$

Via normalization, the importance weights of criteria are calculated as follows (see Equation (9)): $W = (0.2901, 0.2695, 0.2298, 0.2106)^{T}$

Therefore, the weights of all criteria are calculated. Table (7) presents the weights of all input and output criteria, which are calculated via FAHP.

Table 7

Weights of criteria

Criteria	IC1	IC2	IC3	IC4	OC1	OC2	OC3	OC4	OC5	OC6	OC7
Weight	0.2901	0.2695	0.2298	0.2106	0.1761	0.1661	0.137	0.1715	0.1643	0.1022	0.0828

Now, based on the results of Table (6), the constraints pertaining to weights of criteria are determined and are applied in the proposed DEA model. Based on the weights presented in Table (6) and the proposed DEA model, the following will be true: $\begin{matrix} w_{IC 1} > w_{IC 2} \\ w_{IC 2} > w_{IC 3} \\ w_{IC 3} > w_{IC 4} \\ w_{OC 1}^{⌢} > w_{OC 4}^{⌢} \\ w_{OC 4}^{⌢} > w_{OC 2}^{⌢} \end{matrix}$ (15) $\begin{matrix} w_{OC 2}^{⌢} > w_{OC 5}^{⌢} \\ w_{OC 5}^{⌢} > w_{OC 3}^{⌢} \\ w_{OC 3}^{⌢} > w_{OC 6}^{⌢} \\ w_{OC 6}^{⌢} > w_{OC 7}^{⌢} \end{matrix}$

Third step: Ranking of provinces

In this step, the proposed DEA model and the resultant constraints from previous step (FAHP) are merged and, thereby, the mathematical model is developed as follows: $Min z$ (16) $St :$ (17) $z \geq d_{i} \forall i = 1, 2, \dots, 31$ (18) $\begin{matrix} \frac{\sum_{k_{2}} {outp}_{{ik}_{2}} \times w_{k_{2}}^{⌢}}{\sum_{k_{1}} {inp}_{{ik}_{1}} \times w_{k_{1}}} + d_{i} = 1 \\ \forall i = 1, 2, \dots, 31 \end{matrix}$ (19) $w_{k_{1}} \geq ɛ \forall k_{1} = IC 1, IC 2, IC 3, IC 4$ (20) $\begin{matrix} w_{k_{2}}^{⌢} \geq ɛ \forall k_{2} = OC 1, OC 2, OC 3, OC 4, \\ OC 5, OC 6, OC 7 \end{matrix}$ (21) $w_{IC 1} > w_{IC 2}$ (22) $w_{IC 2} > w_{IC 3}$ (23) $w_{IC 3} > w_{IC 4}$ (24) $w_{OC 1}^{⌢} > w_{OC 4}^{⌢}$ (25) $w_{OC 4}^{⌢} > w_{OC 2}^{⌢}$ (26) $w_{OC 2}^{⌢} > w_{OC 5}^{⌢}$ (27) $w_{OC 5}^{⌢} > w_{OC 3}^{⌢}$ (28) $w_{OC 3}^{⌢} > w_{OC 6}^{⌢}$ (29) $w_{OC 6}^{⌢} > w_{OC 7}^{⌢}$ (30)

By running the model in GAMS24.1/BARON software, the efficiency deviation of each province is calculated. The province with the lowest level of deviation benefits form higher efficiency. The results of the model implementation are presented in Table (8):

Table 8

Provinces’ efficiency and rank

Province	d _i	1 - d_i	Rank
East Azerbayejan	0.683	0.317	8
West Azerbayejan	0.701	0.299	9
Ardabil	0.904	0.096	29
Isfahan	0.624	0.376	4
Alborz	0.865	0.135	26
Ilam	0.779	0.221	15
Booshehr	0.813	0.187	19
Tehran	0.58	0.42	3
Chaharmahal and Bakhtiari	0.909	0.091	30
South Khorasan	0.753	0.247	13
Khorasan Razavi	0.637	0.363	5
North Khorasan	0.822	0.178	20
Khoozestan	0.398	0.602	2
Zanjan	0.81	0.19	18
Semnan	0.869	0.131	27
Sistan and baluchestan	0.805	0.195	17
Fars	0.678	0.322	7
Qazvin	0.858	0.142	23
Qom	0.712	0.288	10
Kordestan	0.793	0.207	16
Kerman	0.739	0.261	12
Kermanshah	0.833	0.167	21
Kohkiluye and Boyerahmad	0.91	0.09	31
Golestan	0.838	0.162	22
Gilan	0.859	0.141	24
Lorestan	0.89	0.11	28
Mazandaran	0.733	0.267	11
Markazi	0.659	0.341	6
Hormozgan	0	1	1
Hamedan	0.754	0.246	14
Yazd	0.86	0.14	25

As shown in Table (8), Hormozgan province with the relative efficiency of 1 and Kohkiluye and Boyerahmad with the relative efficiency of 0.09 are known as the most and the least efficient provinces in road transport in Iran, respectively.

5 Conclusion

Transportation, as an important activity in servicing sectors, has an undeniable role in the growth of industries. In recent years, transportation section has experienced a high added value which is constantly increasing. For any improvement in this section, the realization of the current weaknesses is required at first. Accordingly, a hybrid method based on FAHP and DEA was proposed in this study to measure the efficiency of transportation section in each province. The proposed method includes the importance of the criteria as an input and also considers expert opinion in the form of constraints in DEA model. In order to validate the model, we applied the proposed method in Road Maintenance and Transport Organization of Iran and the results proved its effectiveness.

Footnotes

Based on statistics of 2015 ().

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