A Hybrid Meta-Heuristic Approach for Design and Solving a Location Routing Problem Considering the Time Window

محورهای موضوعی : نشریه بین‌المللی هوش تصمیم

Mohammad Amin Rahmani ¹ (School of Industrial Engineering, University of Tehran, Tehran, Iran)
Ahamd Mirzaei ² (School of Computer Faculty, Islamic Azad University, electronic campus, Tehran, Iran)
Milad Hamzehzadeh Aghbelagh ³ (School of Faculty of Information Technology Engineering, Shiraz University of Technology, Shiraz,Iran)

تاریخ دریافت : 1402/07/22 تاریخ پذیرش : 1402/07/26 تاریخ انتشار : 1403/07/01

کلید واژه: Time Window, The problem of routing - locating, Meta-heuristic algorithm of gray wolf optimization, Meta-heuristic algorithm of locust optimization, Hybrid meta-heuristic algorithm,

چکیده مقاله :

The supply chain requires a distribution network between customers and suppliers. This distribution network can be multifaceted. Combining these two problems into a single problem increases the efficiency of the distribution network and ultimately increases the efficiency of the supply chain. Establishing a window of time to deliver goods to customers also increases their satisfaction and, as a result, more profitability in the long run. Therefore, in this research, an attempt has been made to present a routing-location problem in the multimodal transportation network. A time window is also included in this model. To solve such a model, especially in large dimensions, exact solution methods cannot be used. Based on this, a combined meta-heuristic algorithm (genetic optimization algorithm and neural network) has been proposed to solve the model, and the result has been compared with two gray wolf optimization algorithms and grasshopper optimization algorithms. The presented results indicate the effectiveness of the proposed algorithm.

چکیده انگلیسی:

The supply chain requires a distribution network between customers and suppliers. This distribution network can be multifaceted. Combining these two problems into a single problem increases the efficiency of the distribution network and ultimately increases the efficiency of the supply chain. Establishing a window of time to deliver goods to customers also increases their satisfaction and, as a result, more profitability in the long run. Therefore, in this research, an attempt has been made to present a routing-location problem in the multimodal transportation network. A time window is also included in this model. To solve such a model, especially in large dimensions, exact solution methods cannot be used. Based on this, a combined meta-heuristic algorithm (genetic optimization algorithm and neural network) has been proposed to solve the model, and the result has been compared with two gray wolf optimization algorithms and grasshopper optimization algorithms. The presented results indicate the effectiveness of the proposed algorithm.

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International Journal of Decision Inelligence

Vol 1, Issue 1, Winter 2024 , 49-61

 

 

 

 

 

Developing a Decision Model as Budget Assignment Method for Locating Industrial Facilities: Real Case Study

Parsa Fallah Sheikhlaria, Seyed Habib A.Rahmatib,*

aJawaharlal Nehru Technological University Hyderabad, India

bDepartment of Mechanical Engineering, Qazvin Branch, Islamic Azad University, Qazvin, Iran

Received 19 October  2022; Accepted 28  December  2022

Abstract

 

In today's world, due to the existence of several criteria in every decision, budgeting has become a complex issue. Applying decision-making methods significantly helps to make optimal decisions. Multiple criteria decision making (MCDM) is a branch of operational research dealing with finding optimal results in complex scenarios including various indicators, conflicting objectives, criteria, and various indicators. The paper presents a location-allocation decision-making problem resulting in the selection of the most desirable location for the consulting service center in the Qazvin state of Iran. In the first stage, different location criteria are determined. Then, the decision-making matrix preparation is completed based on the criteria dimension and expert opinions. The decision problem is formulated as a multiple criteria ranking problem (MCDM). In the second stage, the decision-making is performed by using all three models of the PROMETHEE method. Finally, considered locations are ranked from the best choice to the worst one with the application of the PROMETHEE MCDM/A method.

 

Keywords: Locational alocation, multicriteria decision making, PROMETHE, location problems.

 

1.Introduction

In today's economy, characterized by a dynamic and volatile environment, many researchers stress the significance of location factors(Kaboli & et. al., 2007). Location allocation decisions are made in both private and public sectors. For example, governments need to determine the locations for emergency bases highway patrol vehicles, fire bases, ambulances, television antennas, and exploratory oil wells. In all cases, poor locations can increase the likelihood of property damage and cost life. In private sectors, locations of warehouses and distribution centers, production and assembly facilities, offices, and retail outlets must be onsidered. Facility location applications are concerned with the location of one or more facilities in such a way that a certain objective such as minimizing transportation cost, providing equitable service to consumers, capturing the largest market share, and etc. Facility location problems may rise challenging geometrical and combinational problems. The research on facility location problems spans many research fields such as operations research/management science, industrial engineering, geography, economics, computer science, mathematics

 

and marketing (Kim & et.al, 1999). Location theory was first introduced by Weber (Tabari & et. al., 2008), who considered the problem of locating a single warehouse in order to minimize the total travel distance between the warehouse and a set of spatially distributed costumers. In fact, he proposed a material index for selecting the location in which if this index is grater than one, the warehouse should be installed in the vicinity of the source of raw material; or otherwise, it should be close to the market. Smithies and Stevens (Stevens & et. al., 1961) extended the Hotelling's problem later. Hakimi (Hakimi, 1964) considered a general problem to locate one or more facilities on a network by minimizing the sum of the distances and the maximum distance between facilities and points on a network. Considerable research and theoretical interest in the location problem have been carried out after this seminal paper. Brown and Gibson (Tabari & et. al., 2008), and Buffa and Sarin proposed a facility location model for a multidimensional location problem based on critical factors, objective factors, and subjective factors. Fortenberry and Mitra (Fortenberry & Mitra, 1986) resented a model for the location-allocation

* Corresponding Author. Email: [e-mail]

problems considering both qualitative and quantitative factors. Kahne (Kahne, 1975) considered 29 attributes and

used a weighting model to determine the relative mportance with uncertainty in attributes. Kirkwood (Kirkwood, 1982) discussed a multi-disciplinary study conducted to select a site for a nuclear power facility. Linares and Romrero (Liang, 2000) roposed a methodology that combined several multi-criteria methods to address electricity planning problems. Several MCDM methods for the location selection are used such as Liang and Wang ( Liang & Wang, 1991) who proposed an algorithm for a site selection based on the concepts of the fuzzy set theory. Lianares (2000) proposed a holistic MCDM model for the facility location selection. Montajabiha and Wang (2016) proposed a fuzzy TOPSIS approach to select the best facility location under linguistic environment. MCGDM methods if decision makers (DMs) are not able to treat precise data in order to define their preferences, the intuitionistic fuzzy set (IFS) theory enables them  (Montajabiha and Wang, 2016). The ranking of options is done by comparing their pairs in each index. The ROMETHEE method provides six generalized criteria to define the superiority function for the decision maker (Figueroa, 2019).The structure of this paper is as follows: First, the PROMETHE model is introduced. Second, the case study is described in detail. Analysis of a case study is then discussed in order to verify the practicability and effectiveness of the proposed

model in the facility location problem. Finally, this paper concludes with gained results.

 

 

Pj: Preference Function of Criteria j sc: Sign of Criteria (1×n vector) DC: Differences on Criteria (Deviation between alternatives over jth criteria) PFC: Preference Function on Criteria PFV: Preference Function Value PI: Preference Index Value WPI: Weighted Preference Index pfv : Preference Function Value Vector  PI: Preference Index

 

 

2. PROMETHEE Decision-Making Process

 

2.1

Before describing the detailed process of the algorithm some notations are summarized for more simplification.

The PROMETHEE method is well-known as a multi-criteria decision-making method in which alternatives ranking is obtained based on preference function, criteria, and their criteria. For this purpose, determined preference function Pj() has been used to show preference of alternative a1 over alternative a2 for specific criteria, fj. In other words, Pj() is a type of function that can convert alternative differences on jth criteria = ±1*(x1 – x2) to find required values for further calculation. Sign of criteria (cj) can be stored in sc1×n vector (sc=[±1 ±1 …±1] ) and criteria weights are demonstrated in weight wector (w). The PROMETHEE method encompasses six types of preference function for criteria as mentioned in table 1.

 

 

 

 

 

Table 1.

Preference function types in the PROMETHEE method

Type I (Usual criterion)	Type II (Quasi-criterion)	Type III (V-sharp criterion)
Type IV (Level-criterion)	Type V (Linear criterion)	Type VI (Gaussian criterion)

 

2.1.1

PROMETHEE as a powerful decision-making tool can calculate alternatives ranking by various criteria consideration, in 2 phases. In the method, w j is the weight of the jth criteria and normalized weights aggregation should be equal to on ().

Phase 1 started by gathering and preparing data in the shape of a matrix, which introduces alternatives and their criteria values in every row as xij. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 2.

Decision matrix (alternatives and criteria)

 

By considering the criteria sign vector (sc), the elements () of matrix DC can be interpreted as the deviation values between alternatives and all pairwise comparisons over jth criteria (size of DC matrix m2×n). In table 3, 

 

= ±1*(x13 - x23) illustrates mentioned comparison between alternative a1 and a2 on 3rd criteria considering cj sign for instance.

 

 

Table 3.

Matrix of differences on criteria

		±c1	±c2	…	±cn
	a1	x11	x12	…	x1n
	a2	x21	x22	…	x2n
D=
	am	xm1	xm2	…	xmn

 

 

2.1.2

 

Preference function value and weighted preference index show a relation between alternatives. Based on criteria features, elements of DC have to adjust with the

 

 

preference function of jth criteria. Here preference functions convert DC matrix to PFC which is a fundamental matrix for further calculation, including PFV

, PI and finally alternative ranking can be obtained.

 

 

Table 4.

 Matrix of preference function on DC

		±c1		±c2	…	±cn
	a1a1				…
	a1a2				…
DC=
	amam-1				…
	amam				…

 

 

 

2.1.3 Preference Function Value

 

 

Horizontal summation of PFC expresses pfv vector of all pair alternatives within m2×1 vector which could be elaborately reshaped to matrix PFV m×m as table 5

 

 

Table 5.

 Matrix of preference function value

		±c1	±c2	…	±cn
	a1a1			…
	a1a2			…
PFC=
	amam-1			…
	amam			…

 

 

 

2.1.4.Weighted Preference Index

 

By applying criteria weights on the PFC matrix, needed matrixes like WPI and PI are obtainable. According to Eq.1, the PI matrix is obtained from the multiplication of wj to cj value of PFC. In this study, wj values of all criteria follow the normalization condition which means criteria weights summation is equal to 1 ().

 Horizontal summation on WPI matrix results in pi vector with the size of m2×1. As each element of pi vector expresses a relation of two alternatives, it could be reshaped to PI matrix m×m as it is shown in table 6.

 

 

Table 6.

 preference index Matrix

		a1	a2	…	am
	a1	∑	∑	…	∑
	a2	∑	∑	…	∑
PFV =
	am	∑	∑	…	∑

 

 

 

 

2.2.

 

 

In the second step, based on PROMETHEE method, pertinent calculations on alternatives obtain alternatives out ranking.

 

 

 

 

 

 

 

 

 

 

 

 

 

2.2.1 PROMETHEE I

 

As PROMETHEE basis is on a relation between alternatives,the entering and leaving flows are determined through Eq.

lternatives flow calculations based on PROMETHEE I are determined through Eq.4 to Eq. 7.(2) and Eq.(3).

 

 

 

		a1	a2	…	am
	a1	( a1 a1)	( a1 a2)	…	( a1 am)
	a2	( a2 a1)	( a2 a2)	…	( a2 am)
PI =
	am	( an a1)	( an a2)	…	( amam)

(4)

 

(5)

 

(6)

 

(7)

 

 

In this method, the a1 alternative is superior to alternative a2 , as Eq.8..

 

:              (8)

In addition, the alternatives a1 and a2 are indifferent to each other under the condition of Eq.9.

 

(9)

 

 

2.2.2. PROMETHEE II

 

In this method, net flow of alternatives is obtained from Eq. 10 and then a comparison between pairs of alternatives is accomplished by Eq.11.The a1 alternative is better than the alternative a2 under the condition of Eq.9 and Eq. 10.

 

 

10)

 

 

(11)

 

Furthermore, two alternatives are indifferent to each other under the condition of Eq.12.

 

(12)

 

2.2.3 PROMETHEE III

 

 

In 3rd method, for every alternative ai an interval value  is determined, and alternative a1 is superior to alternative a2 , considering conditions of Eq.13:

 

(13)

 

And, they are indifferent to each other under conditions of Eq.14.

 

(14)

 

 

In this technique, final alternatives ranking is calculated by using interval values. The interval values are obtained from Eqs.15. and Eq.16.

 

 

 

 

 

 

3. Problem Definition and Decision-Making Process

Industrial Estates Management Organization of Ghazvin province tends to establish a consulting service department in one of five main industrial estates. This department provides companies with engineering and financial consulting. All cities are located in the province with different specifications which are defined as their

 

criteria. Included cities or alternatives in this decision-making case are Abeyek (a1), Arasanj (a2), Heydariyeh (a3), Khorram Dasht (a4), Lia (a5).

Problem features include 17 criteria which are defined by experts after a deep field survey as mentioned in table 7. By conducting meetings with informants and converting qualitative elements to quantitative, matrix (D) values have been prepared as mentioned in table 8.

 

Table 7

 criteria of the location budget assignment case study

c1:	General conditions of land
c2:	Subside supporting
c3:	Welfare and healthcare facilities
c4:	Public infrastructure facilities
c5:	Number of industrial units that can be installed in the state
c6:	Counselors willingness to settle in the center
c7:	Special features and infrastructure
c8:	Employable people population in the state
c9:	Cultural and social contexts in the state
c10:	Recruiting potential of the state
c11:	Distance to the capital of the province
c12:	Distance to roads and highways
c13:	Distance to the railway
c14:	Distance to the airport
c15:	Distance to research centers and universities
c16:	Distance to other industrial states
c17:	Distance to administrative centers

 

 

Table 8.

 Decision matrix for location- budget assignment

		c1	c2	c3	c4	c5	c6	c7	c8	c9	c10	c11	c12	c13	c14	c15	c16	c17
	a1	85	77	88	104	91	92	92	85	76	99	72	103	93	67	70	78	75
	a2	110	77	89	115	93	80	81	87	74	89	74	106	100	83	82	92	81
D=	a3	112	70	69	100	84	58	69	76	65	74	96	122	97	104	99	104	108
	a4	113	72	68	104	85	57	74	85	65	74	132	124	103	104	105	119	91
	a5	121	88	116	126	123	112	101	120	92	108	69	100	79	78	74	77	74

 

 

Fig.1. Alternatives of the research in Ghazvin province case study.

 

Criteria sign vector is introduced based on criteria characteristics as shown in the table.9. The positive signs express the fact that larger amounts are more favorable in the criteria, and in the negative criteria smaller values play this role.

 

 

 

 

Table 9 

 

Sign of criteria

 

cj	c1	c2	c3	c4	c5	c6	c7	c8	c9	c10	c11	c12	c13	c14	c15	c16	c17
Sign	1	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	-1

 

 

With the respect to criteria features, 17 different values were gathered for every alternative by expert informant questionnaire and field survey as shown in table.3. So, alternatives comparisons and calculations have been accomplished based on thier D matrix.

 

As mentioned in 2.1.1 deviation between alternatives over jth criteria is demonstrated in DC matrix in table 10. The values of the matrix are obtained based on a pairwise comparison of alternatives in all criteria. Obviously, aii values of the matrix are zero as there is no difference between one alternative with itself.

 

 

 

 

 

 

 

Table 10

 DC matrix

a1a1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a1a2	0	0	0	0	0	0.165	0.236	0	0.014	0.118	0	0	0	0.5	0.5	0.5	0
a1a3	0	0.092	0	0	0	0.764	0.691	0.063	0.343	0.542	0.5	0.5	0	0.5	0.5	0.5	0.5
a1a4	0	0.066	0	0	0	0.784	0.513	0	0.343	0.542	0.5	0.5	0	0.5	0.5	0.5	0.5
a1a5	0	0	0	0	0	0	0	0	0	0	0	0	0	0.5	0	0	0
a2a1	0.227	0	0	0	0	0	0	0.003	0	0	0	0	0	0	0	0	0
a2a2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a2a3	0	0.092	0	0	0	0.454	0.274	0.092	0.245	0.245	0.5	0.5	0	0.5	0.5	0	0.5
a2a4	0	0.066	0	0	0	0.484	0.103	0.003	0.245	0.245	0.5	0.5	0	0.5	0.5	0.5	0
a2a5	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a3a1	0.245	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a3a2	0.018	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a3a3	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a3a4	0	0	0	0	0	0.001	0	0	0	0	0.5	0	0	0	0	0.5	0
a3a5	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a4a1	0.255	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a4a2	0.027	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a4a3	0.009	0.026	0	0	0	0	0.054	0.063	0	0	0	0	0	0	0	0	0.5
a4a4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a4a5	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a5a1	0.327	0.145	1	1	0	0.393	0.165	0.625	0.589	0.096	0	0	0.5	0	0	0	0
a5a2	0.1	0.145	1	0	0	0.722	0.589	0.582	0.675	0.363	0	0	0.5	0	0	0.5	0
a5a3	0.082	0.237	1	1	1	0.974	0.897	0.787	0.92	0.764	0.5	0.5	0.5	0.5	0.5	0.5	0.5
a5a4	0.073	0.211	1	1	1	0.977	0.802	0.625	0.92	0.764	0.5	0.5	0.5	0.5	0.5	0.5	0.5
a5a5	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0

 

 

 

 

According to 2.1.2, applying appropriate preference function on DC matrix can transform it into PFC matrix.

In this case study, preference function type 2 has been used for c1 and c2 and type1 for criteria c3, c4, and c5.

Criteria c6 to c10 meet the type 6 preference function. The remaining criteria from c11 to c17 were compatible with preference type 4 and their result is demonstrated in table 11.

 

 

 

 

 

 

 

 

 

 

Table 11

PFC matrix

a1a1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a1a2	0	0	0	0	0	0.165	0.236	0	0.014	0.118	0	0	0	0.5	0.5	0.5	0
a1a3	0	0.092	0	0	0	0.764	0.691	0.063	0.343	0.542	0.5	0.5	0	0.5	0.5	0.5	0.5
a1a4	0	0.066	0	0	0	0.784	0.513	0	0.343	0.542	0.5	0.5	0	0.5	0.5	0.5	0.5
a1a5	0	0	0	0	0	0	0	0	0	0	0	0	0	0.5	0	0	0
a2a1	0.227	0	0	0	0	0	0	0.003	0	0	0	0	0	0	0	0	0
a2a2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a2a3	0	0.092	0	0	0	0.454	0.274	0.092	0.245	0.245	0.5	0.5	0	0.5	0.5	0	0.5
a2a4	0	0.066	0	0	0	0.484	0.103	0.003	0.245	0.245	0.5	0.5	0	0.5	0.5	0.5	0
a2a5	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a3a1	0.245	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a3a2	0.018	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a3a3	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a3a4	0	0	0	0	0	0.001	0	0	0	0	0.5	0	0	0	0	0.5	0
a3a5	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a4a1	0.255	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a4a2	0.027	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a4a3	0.009	0.026	0	0	0	0	0.054	0.063	0	0	0	0	0	0	0	0	0.5
a4a4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a4a5	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a5a1	0.327	0.145	1	1	0	0.393	0.165	0.625	0.589	0.096	0	0	0.5	0	0	0	0
a5a2	0.1	0.145	1	0	0	0.722	0.589	0.582	0.675	0.363	0	0	0.5	0	0	0.5	0
a5a3	0.082	0.237	1	1	1	0.974	0.897	0.787	0.92	0.764	0.5	0.5	0.5	0.5	0.5	0.5	0.5
a5a4	0.073	0.211	1	1	1	0.977	0.802	0.625	0.92	0.764	0.5	0.5	0.5	0.5	0.5	0.5	0.5
a5a5	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0

 

 

As it is explained in 2.1.3, horizontal aggregation on PFC matrix for bilateral elements results pfv vector with m2

rows. By putting these elements in m×m matrix respectively, PFV matrix has resulted as mentioned in table12. So, preference function values of every two alternative are depicted.

 

 

Table 12

PFV matrix

	a1	a2	a3	a4	a5
a1	0	2.032	5.496	5.248	0.500
a2	0.230	0	3.902	3.646	0
a3	0.245	0.018	0	1.0012	0
a4	0.255	0.027	0.652	0	0
a5	4.840	5.176	11.162	10.872	0

 

 

By multiplying normalized weights vector into PFC matrix, WPI matrix is calculated which is shown in table 13. If weight vector elements aggregation was equal to one. These weights are called normalized weights. By using this matrix, every weighted pairwise amount of the main matrix is obtained.

 

 

 Table 13

 WPI matrix

a1a1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a1a2	0	0.001	0.007	0	0	0	0.029	0.029	0.029	0	0	0.001	0.007	0	0	0	0.029
a1a3	0.004	0.020	0.032	0.029	0.029	0	0.029	0.029	0.029	0.029	0.004	0.020	0.032	0.029	0.029	0	0.029
a1a4	0	0.020	0.032	0.029	0.029	0	0.029	0.029	0.029	0.029	0	0.020	0.032	0.029	0.029	0	0.029
a1a5	0	0	0	0	0	0	0.029	0	0	0	0	0	0	0	0	0	0.029
a2a1	0.000	0	0	0	0	0	0	0	0	0	0.000	0	0	0	0	0	0
a2a2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a2a3	0.005	0.014	0.014	0.029	0.029	0	0.029	0.029	0	0.029	0.005	0.014	0.014	0.029	0.029	0	0.029
a2a4	0.000	0.014	0.014	0.029	0.029	0	0.029	0.029	0.029	0	0.000	0.014	0.014	0.029	0.029	0	0.029
a2a5	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a3a1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a3a2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a3a3	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a3a4	0	0	0	0.029	0	0	0	0	0.029	0	0	0	0	0.029	0	0	0
a3a5	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a4a1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a4a2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a4a3	0.004	0	0	0	0	0	0	0	0	0.029	0.004	0	0	0	0	0	0
a4a4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a4a5	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a5a1	0.037	0.035	0.006	0	0	0.029	0	0	0	0	0.037	0.035	0.006	0	0	0.029	0
a5a2	0.034	0.040	0.021	0	0	0.029	0	0	0.029	0	0.034	0.040	0.021	0	0	0.029	0
a5a3	0.046	0.054	0.045	0.029	0.029	0.029	0.029	0.029	0.029	0.029	0.046	0.054	0.045	0.029	0.029	0.029	0.029
a5a4	0.037	0.054	0.045	0.029	0.029	0.029	0.029	0.029	0.029	0.029	0.037	0.054	0.045	0.029	0.029	0.029	0.029
a5a5	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0

 

 

Based on Eq.1, horizontal summation of every row of the WPI matrix leads to making a pi vector with m2 elements. Then by putting this vector values in an m×m matrix, a preference index value with the name of PI is created as depicted in Table14.

 

 

 

Table 14

PI matrix

	a1	a2		a3	a4	a5
a1	0	0.014		0.014	0.015	0.285
a2	0.120	0		0.001	0.002	0.304
a3	0.323	0.230		0	0.038	0.657
a4	0.309	0.214		0.059	0	0.640
a5	0.029	0		0	0	0

 

In the first method of PROMETHEE, the priority of alternatives is determined based on the leaving and entering flow based on Eq.8. Alternatives leaving and entering flows are depicted in table 13. So, the fifth city

(a5) with the greatest leaving values is the best alternative among available 5 alternatives and the ranking of alternatives is mentioned in Fig.2.

 

        

 

 

 

 

 

Table 15

 Leaving& entering flow of alternatives

	a1	a2	a3	a4	a5
Lev_flow	0.195	0.114	0.019	0.014	0.471
Ent_flow	0.082	0.107	0.312	0.305	0.007

 

 

 

 

Fig.2. PROMETHEE I

 

In PROMETHEE II method, regarding Eq.11, alternatives net flow is computed. These gained results have been depicted in table 14. Based on alternative net-flows, the last alternative with the biggest net-flows is the best option for the budget assignment to establish an industrial facilities center. The alternatives ranking are shown in Fig.3.

 

Table 16

 Net flow of alternatives

a1	0.113
a2	0.008
a3	-0.293
a4	-0.291
a5	0.463

 

Fig.3. PROMETHEE II

In PROMETHEE III, alternatives’ intervals lower and upper bound are obtained based on Eq.15. As it is mentioned before, if alternative lower bound Xai was greater than another alternative upper bound Yaj, alternative ai is superior to aj. Alternatives interval bounds are shown in table 17 which is used for alternatives ranking.

 

Table 17

Interval values of alternatives

	X	Y
a1	0.057	0.124
a2	-0.026	0.038
a3	-0.274	-0.195
a4	-0.270	-0.196
a5	0.331	0.411

For better demonstration, alternatives intervals are illustrated in Fig.4. Considering mentioned explanation, a5 is superior to other alternatives and its interval is greater than others. Alternatives a1 and a2 are in second and third positions but for a3 and a4 no difference can be expressed. So, alternatives priority is depicted in Fig.5.

 

YA5

XA5

YA1

XA1

YA2

XA2

YA3

YA4

XA4

XA3

Fig.4. Interval of alternatives

 

Fig.5. PROMETHEE III

All above, point out that fifth alternative is the best alternative clearly among nominated cities in Qazvin province for a consulting service department establishment.

 

 

 

 

 

 

4.Conclusion

In comparative analysis, the ranking results derived by all types of PROMETHEE illustrates that Lia city ( a5) was the superior than other alternatives. In another word PROMETHEE methods results were to some extent alike. The simplicity of the PROMETHEE methodology as a source for comparing to the other outranking techniques can be considered as one of the main advantages of the PROMETHEE. So based on the method and mentioned nominated city in the province Qazvin, fifthe alternative accuaired the most desirability to allocate the location for consulting service center. The paper presents a location-allocation decision-making problem resulting in the selection of the most desirable location for the consulting service center in the Qazvin state of Iran.

 

 

 

 

 

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