Site Selection Using Optimization Techniques

International Journal of Scientific & Engineering Research, Volume 4, Issue 4, April-2013 1687

ISSN 2229-5518

Vandana Bagla, Anjana Gupta

Abstract— The process of selection of sites for commercial activities involves myriad of qualitative, quantitative and financial factor s. In general, there are multiple diversified factors. Due to the human tendency to depend more on emotions than reasons, there is every chance of reaching an irrational conclusion. This paper presents a formal system to evaluate comparative ranks of available sites for commercial activities and thereby to determine the long termed profitable decision using Lexicographic Approach. A versatile solution is provided to given problem using Weighted Penalty Method.

Index Terms— Analytic Hierarchy Process (AHP), Consistency, Eigen value, Eigen vector, Lexicographic Approach, Mixed Integer

Programming, Weighted Penality Method .

1 INTRODUCTION

——————————  ——————————
lmost all investment decisions involve multiple, diverse and complex set of social and financial factors which are
quite hard to be overcome by mere intuition. The site selection process is an exemplification of an investment decision which involves the evaluation of attributes for maximization of profit and minimization of cost which are the most important re- quirements for the successful functioning of a particular busi- ness activity. Qualitative factors must also be considered while selection of a site for an activity.
A number of allocation problems have been solved in the re- cent past; for ready reference see Carlsson & Fuller[1], Ig- nizio[4], Ignizio & Cavalier [5], Serfini[7] and Azarm[8].
In this paper, we envision a site selection model for commer- cial activities which efficiently explores a multi-criteria deci- sion-making model involving three objectives. Maximization of profit and minimization of set-up cost are the major objec- tives which are taken up as first and second objectives alterna- tively . Last but not the least objective is to rank various attrib- utes such as capacity, neighborhood, connectivity, transport availability and proximity which are considerably important factors for a flourishing business.
The solution procedure consists of two phases. In the first phase, weights are allocated to various available sites based on various important aspects such as neighboring locality, broad or narrow connecting roads, area/capacity of the available sites, proximity factors such as competitive business rivals in

————————————————

 Vandana Bagla is currently pursuing her Doctoral program in Opera- tions Research and is working as Assistant Professor in Maharaja Agresen Institute of technology(MAIT), Delhi, India.

E-mail: vandana_6928e@yahoo.com

 Anjana Gupta is working as an Associate Professor in Delhi Technologi- cal University (DTU), Delhi, India.

E-mail: guptaanjana2003@yahoo.co.in

the nearby areas, transport availability such as metro or other public transports. To accomplish this, an approach of Analytic Hierarchy Process (AHP) given by Saaty [10] is used.
In the second phase, the proposed problem seeking to maxim-
ize the profit, minimize the set-up cost and maximize the allot- ted weights, is modeled as mixed integer programming prob- lem. Since the objective is to maximize the profit and weights. Also at the same time to minimize the cost, reversed costs are being taken after normalization. The problem is solved using hierarchical optimization method. The said problem is also be solved using Weighted Penalty Method developed by Prakash and Gupta[9].
The paper is organized as follows. Section 1 is introductory.
Section 2 explains the Analytic Hierarchy Process (AHP). Sec- tion 3 describes the problem mathematically. Section 4 ex- plains the proposed methodology to find the set of solutions. Section 5 illustrates the method via an example. The paper concludes in Section 6 for further application in real estate and many other important fields

2 ANALYTIC HIERARCHY PROCESS (AHP)

The analytical hierarchy process (AHP) is a decision mak- ing approach designed to aid in the solution of complex multiple criteria problems in number of application domains. The outcome of AHP is a prioritized weighting of each deci- sion alternative. The first step in the analytical hierarchy pro- cess is to model the problem as a hierarchy. The hierarchy is a structured mean of describing the problem at hand. It consists of an overall goal at the top level, a group of options or alter- natives for reaching the goal and a group of factors or criteria that relate the alternatives to the goal. In most cases the crite- ria are further broken down into sub criteria, sub-sub criteria and so on in many levels as per the requirement of the prob- lem. Once the hierarchy has been constructed, the participants use the AHP to establish priorities for all its nodes. In this, the
elements of a problem are compared in pairs with respect to

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their relative impact on a property they share in common. The pair wise comparison is quantified in a matrix form by using the scale of Relative Importance given in Saaty [10] as shown in Table 1.
This scale has been validated for effectiveness, not only in many applications by a number of people, but also through
theoretical comparison with a large number of other scales.
then the derived priority vector w satisfies wi / wj = aij , i < j (2)
Any reciprocal matrix satisfying (1) is called consistent. How- ever in practice, the priority matrix seldom satisfies (1), there- by making it more important to define some relax measuring of consistency check, Saaty [10] introduced the concept of
Consistency Index (CI) of a reciprocal matrix as the ratio
During the elicitation process, a positive reciprocal matrix is   n
formed in which (i,j)th element aij is filled by the corresponding

max

n  1

where λmax and n , respectively stand for the max-
number from the Table 1.
TABLE 1

Analytic Hierarchy Measurement Scale

imum eigen value and order of the reciprocal matrix.
The obtained CI value is compared with the Random Index (RI) given in Table 2. The table 2 had been calculated as an average of CI’s of many thousands matrices of the same order whose entries were generated randomly from the scale 1 to 9 with reciprocal force. The simulation results of RI for matrices of size 1 to 10 had been developed by Saaty [10] and are given
in Table 2.

TABLE 2
RANDOM INDEX (RI)

The number is chosen according to the following criterion.
The ratio of CI and RI for the same order matrix is called the
Consistency Ratio (CR).

3 PROBLEM DESCRIPTION AND MATHEMATICAL FORMULATION

Suppose a corporate body/M.N.C. deals in m different busi- ness/outlets and there are n available sites.
Problem is to allocate a suitable site out of the available sites to each business/outlet. While doing this, the two main objec- tives of the company are to maximize the overall profit and to minimize the overall cost. At the same time, the company
wants to prioritize the sites carrying more weights.

Let pij

(i=1, …,m; j=1,…,n) denote the expected profit, when

aij , if xi dominates xj

i th business is set up on

j th site. Also let c be the overall cost

th j

1/aij , if xj dominates xi
and w j
be the weight of
j site (j=1, …,n) calculated by

1, if xi and xj do not dominate over one another
AHP. Let  j
denote the normalized cost of the j site. Then

(1-  j )=

r j denotes the reversed cost of the j site.

The matrix so formed is called the reciprocal matrix. This re- ciprocal matrix is used to calculate the local priority weight of each criterion. The local priority weight (w) is the normalized eigen vector of the priority matrix corresponding to the maxi- mum eigen value of the matrix. For detailed reasoning of this account we refer to Lunging [2], Bal & Srinivasan[3] , Bryson &
Mobolurin[6] and Saaty[10].
It is to be noted that we have reversed the cost as our objective
is to minimize the cost and on the contrary, we are dealing with a maximization problem.
Then the above described model is formulated as the follow-
ing three objective problem.
Maximize Z( x )= (P( x ), R( x ), W( x ))

Where P( x ) =
An interesting property of the priority matrix is that if in addi-
tion its elements are such that
aij ajk = aik , i ≤ j ≤ k (1)



i 1

 pij

j 1

xij

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R( x ) = 

i 1

 rj

j 1

xij

programming in the first iteration.
Find the optimal solution of reconstructed single objective problem with R( x ) as the objective function and an added constraint P( x ) ≥ k1 ,with the original constraint equations.
Let R( x ) = k2 be the optimal solution obtained in the second
W( x ) =
Subject to



i 1

 w j j 1

xij

iteration.
Finally find optimal solution to the given problem in final iter- ation by reforming the problem as:
Maximize W( x )
Subject to



j 1

xij

= 1, i=1, …,m (1)
P( x ) ≥ k1
R( x ) ≥ k2



i1

xij

 1, j =1, …,n (2)



j1

xij

=1, i=1, …,m (1)
xij  {0,1}, i=1 …m; j=1, …,n (3)
The constraint (1) ensures that each kind of outlet is allotted with a site. It is clear in constraint (2) that same site is not allot-



i1

xij  1, j =1, …,n (2)
ted for more than one outlet. In constraint (3) the value of
xij  {0,1}, i=1 …m; j=1, …,n (3)

xij

is one if i th outlet is allotted with is zero.

j th activity, otherwise

The above stated procedure provides us with an optimal solu- tion to the given three objective programming problem with prioritized objectives. A similar methodology may be adopted
while considering cost as the first priority objective.

4 SOLUTION PROCEDURE

4.1 Allotting Weights Using AHP

In the site selection model, we construct hierarchy of attributes which are most important in decision making using AHP. To evaluate the hierarchy, various surveys are conducted to rate each attribute to others at the same level in a series of pair wise comparisons using a scale from 1 to 9 (Table 1).We rank each of the available site in the final set by evaluating the site with respect to upper level attributes separately as an illus- trated in Table 4 and Table 5. The evaluation process finally generates the global weights for each available site of interest, as shown in Table 6 of the illustration.
But the above stated model has its limitations. It does not pro- vide alternatives to the aspirant which suits best to his pocket.

4.3 Procedure to obtain a set of efficient solutions us- ing Weighted Penalty Method

Based on the feedback provided by the investor/corporate body/M.N.C, priorities are assigned to each of the three objec- tives. Here we have taken maximization of overall profit and minimization of cost as first and second priority objectives alternatively. Also maximization of qualitative ranks (weights) is assigned third priority.
This three-objective problem is reduced to an equivalent sin-
gle-objective integer programming problem following the pro-

4.2 Procedure to obtain optimal solution using Lexi- cographic Approach

cedure developed by Prakash and Gupta [9]. Here

pij

de-
Consider the three- objective linear programming problem
notes the expected annual profit if i th

business activity is set
Maximize Z( x )= (P( x ), R( x ), W( x )) subject to given con-
up at

j site. Now we partition the set

straints.
The method requires that the objective functions are to be pri-
{ pij : i=1, …,m ; j=1,…,n} into the subsets Lk (k = -1, 1, …,q)
in the following way.
oritized in decreasing order of importance. Let P( x ) be the most prioritized and W( x ) is the least prioritized objective.
L-1 consists of those

p for which i th business activity can not

Then the method consists of following procedure.
be set up at

jth site. For example a petrol pump can not be set

Optimize the single objective problem consisting of P( x ) as the objective function subject to given constraints. All other
up in a multistoried shopping complex. Consequently we
block allocation in that particular ( i, j )th cell. Afterwards we
objectives are ignored.
Let P( x ) = k1 be the optimal solution obtained using integer
follow the lexicographic arrangement of

pij ’s among the re-

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maining pij . Let L1 consists of those numerical value.

pij

having the largest
The optimal solution for the above problem yields the first efficient solution. Estimated annual profit of the business ac-
tivities at the selected site is determined by adding the profits
L2 consists of

pij

having the next largest numerical value.
in the allocated cells. Minimum cost is calculated by adding
Continuing in this way, finally Lq consists of

pij

having the
the costs corresponding to allocated sites. Also corresponding total weights can be found out in the similar manner. Now to
smallest numerical value.
Now to deal with the cost function simultaneously, we calcu-
obtain second efficient solution, associate a cost M-1
(zero)
late normalized cost

 j , for each potential site and conse-

with each of the variables

xij for which

pij is maximum and

quently respective reversed cost rj = (1-  j ) for each available site.
rest of the problem remains unchanged. This will somehow
reduce the profit but at the same time reduce the cost born by the corporate body/ M.N.C. in general. Solve the resultant
Weights w j
have already been calculated for each site S1,
problem by adopting the same procedure.
The third and subsequent efficient solutions for the problem
S2,…Sn via AHP, as explained in section 4.1. Now Since the
profit function P( x ) is the first priority factor followed by cost

function R( x ) and then weight factor W( x ). Assigning posi- tive priorities M1,……,Mq, Mc , Mw , M-1 to each of the sum

are obtained by repeatedly modifying the objective function and proceeding in the similar manner described above. This will provide a variety of solutions to the investor which suits
best to his pocket.

 xij ,...,  xij ,

m n

  rj

m n

xij ,  

w j xij ,  xij

Remark:- The proposed methodology provides an alternative

L1 Lq

i 1

j 1

i 1

j 1

L1

to goal programming. As evident in goal programming, se-
respectively. Here  xij

is the sum of

xij ’s corresponding to pij’s belong-

cond and third priorities may be partially fulfilled.

5 ILLUSTRATION

We will now illustrate the proposed methodology via an ex-
ing to Lk. Following points should be observed while allotting
the priorities.
(i) No allocation can be made in set L-1.
(ii) Cost factor has been reversed as our objective is max- imization of the three given factors.
Now the priority weights assigned are

M1 > > M2 > > M3 …>> Mq > > MC >> Mw >> M-1

The symbol a >> b indicates a is arbitrarily large compared to b. Having done this, the problem with maximization of P( x ) as the first priority objective, maximization of R ( x ) (minimi-
zation of C(x)) as the second priority objective and maximiza- tion of W( x ) as the third priority objective, is reduced to a
single objective integer programming problem.
ample. Suppose a multi national company is seeking to invest in its different proposals like glossary stores, apparel outlets, gold outlets and petrol pumps. Each proposal requires a suit- able site to be established. As a sample survey, Table 3 shows a set of available sites.
TABLE 3
BUISINESS PROPOSALS VERSES AVILABLE SITES



i 1

 w j

j 1

q m n

xij + M-1  xij

L1

Subject to

 xij  1,

j 1

i=1,…,m

Here S2 and S7 are sites on upper levels of a multistoried shopping complex. Last row shows the costs of the available

 xij  1 , j=1,…,n

i1

sites in requisite units. Intercellular costs show expected an-
nual profit. For example the cell (1,1) shows that if B1 proposal

xij  {0,1}, i=1, …,m; j=1, …,n

is setup on S1
site, then annual expected profit is 3.5 units. The

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cells (4,2) and (4,7) are left blank because a petrol pump can not be set up on the upper levels of a multistoried building. The company wants an optimal set of solutions which should maximize the overall profit at the minimum cost which suits its pocket. At the same time the company does not want to compromise from quality point of view for the sake of its rep- utation and long term profitable business. To meet the re- quirement given by the M.N.C., all available sites were ranked as explained in section 4.1 by taking into account all important affecting attributes given in Fig. 1.
Fig. 1
Hierarchy for Site Selection
For this purpose a survey on thirty two people was conducted and reciprocal matrices were generated by taking mean of all values of matrices (having C.R < 0.1 ) for further calculations. The important attributes were neighborhood, connecting roads, capacity, transport availability and Proxima. Neighbor- hood takes into account standard of the surrounding localities. Broad Connecting roads were given preferences. Capaci- ty/Area of the available site also plays an important role. Availability of transport (Metro/Bus) is an important attribute for middle class people. Proxima plays a very important role in running of a successful business activity. Based on various surveys conducted on the available sites, weights were calcu- lated for each attribute at each level. As an example Table 4 shows the comparison matrix and calculated weights at level two and Table 5 shows the comparison matrix for the neigh- borhood. Further more the global weights are calculated in Table 6.
TABLE 4
RANKING OF ATTRIBUTES

max = 5.23748 C.I.= 0.0593688 C.R.=.0539164
TABLE 5
COMPARISON MATRIX FOR NEIGHBORHOOD

max = 7.44384 C.I.= 0.0739736 C.R.=.056041
TABLE 6
CALCULATION OF GLOBAL WEIGHTS FOR SITES

Table 7 gives the reversed cost and calculated normalized weights for each site.

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TABLE 7

CALCULATION OF REVERSED COST FOR SITES

First phase of the solution procedure has been accomplished us- ing AHP.

5.1 Solution Using Lexicographic Approach

Now to solve the given problem using Lexicographic Ap- proachexplained in section 4.2, taking maximization of profit as the single objective function subject to given constraints, the problem reduces to following integer programming problem. Maximize P(x) = 3.5x11 + 3x12 + ….+ 1.5x46
Subject to

 xij  1, i=1,2,3,4

j 1

 xij  1 , j=1,…,7

i 1

x13=1 , x22=1 , x37=1 , x41=1 yielding P(x) = 14.5 , R(x) =3.3243 and W(x)=0.6554.

Table 8 shows the efficient solution by taking maximization of profit as first priority objective.

TABLE 8

PRIORITIZING MAXIMIZATION OF PROFIT

Therefore the required optimal solution using Lexicographic Ap- proachis given by

[Optimal profit = 14.5 units, Optimal cost = 250 units and

Optimal weight = 0.6554]

Similarly the said problem is solved by the explained methodolo-

gy taking minimization of cost as the first priority objective, max-

imization of profit as the second and maximization of weights as the third priority objectives. The efficient solution is given in

Table 9.

TABLE 9

PRIORITIZING MINIMIZATION OF COST

xij

 {0,1}; i=1,…,4 ; j=1, …,7

Solving the above linear programming problem using integer programming, we get
x13=1 , x22=1 , x37=1 , x41=1 yielding P(x) = 14.5.
Reconstructing the above problem in the second iteration as

Maximize R(x) = 0.7838x11 + 0.8108x12 + ….+ 0.9181x46

Subject to

P(x) ≥ 14.5 and all above constraints in iteration 1.

Again solving the problem using integer programming, we get

x13=1 , x22=1 , x37=1 , x41=1 yielding P(x) = 14.5

and R(x) =3.3243

Reforming the above problem in the third iteration as shown:

Maximize P(x) = 0.0880x11 + 0.1611x12 + ….+ 0.631x46

Subject to

R(x) ≥ 3.3243 and all above constraints in iteration 2.

Solving above using integer programming

[Optimal cost = 160 units, Optimal profit = 10 units and Optimal weight = 0.4421]

5.2 Solution Using Weighted Penalty Method

Now to solve the above problem using Weighted Penalty Method, the reversed costs and normalized weights are calculated by adopting the similar procedure as explained before in section 4.1. we now provide priorities to respective profits viz. M1 to 4 units, M2 to 3.5 units, M3 to 3 units, M4 to 2.5 units, M5 to 2 units, M6

to 1.5 units and M7 to 1 unit as maximization of profit is the first

priority objective. Also assigning Mc to respective reversed costs and Mw to normalized weights such that M1 >> M2 >> …>>M7

>>Mc >> Mw and adopting the procedure obtained in section 4.3, following set of values are obtained for various cells as shown in Table 10.

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TABLE 10
Assigning Priorities Using Weighted Penalty Method
Similarly a set of efficient solutions is obtained by taking max- imization of reversed cost (minimization of cost) as the first priority objective, maximization of profit as the second and maximization of weights as the third priority objective. Table
12 shows the set of efficient solutions by prioritizing minimi- zation of cost.

TABLE 12
PRIORITIZING MINIMIZATION OF COST

The given integer programming problem reduces to MaximizeZ(x)=(M2+0.7838Mc+0.0880Mw)x11+ (M3+0.8108Mc+0.1611Mw)x12
+…….+(M6+0.9181Mc+.0631Mw) x46
Subject to

5.3 Comparison of the Two Methodologies

Lexicographic Approach provides an optimal solution to the given problem according to the priorities assigned to objec- tives. But it does not provide choices to the aspirant which suit best to his pocket. Whereas using Weighted Penalty Method, a set of efficient solutions is obtained and investor has a wider choice. As evident from Tables 8 and 11, the optimal solution given by Lexicographic Approachby prioritizing maximiza- tion of profit is same as the first efficient solution obtained by Weighted Penalty Method. Same is the trend shown in Tables
9 and 12, while considering minimization of cost as the first priority objective. Also one may notice that the sixth efficient

 xij  1,

j 1

i  1, 2, 3, 4

solution in Table 11 shows a profit of 5 units at a set up cost of
160 units and qualitative weights 0.4421. While at the same
cost and weight, first efficient solution in Table 12 gains a

 xij  1, j  1,, 7

i 1

profit of 10 crores. So the investor is provided with a lucrative option at the same set up cost.

xij

 {0,1}; i=1,…,4 ; j=1, …,7

6 CONCLUSION

By solving the above problem, the set of efficient solutions
prioritizing maximum profit is obtained as shown in Table 11.
TABLE 11

PRIORITIZING MAXIMIZATION OF PROFIT

While strategically managing our location decisions, we need solutions that address multiple logistics and economic factors involving real estate and our customers. Maximization of prof- it and minimization of set-up cost are the two major objectives. Third objective is to rank the sites qualitatively, for a flourish- ing business. The method also suggested that once starting up with a low profit/low cost model, high futuristic growth may be expected if qualitative ranks are high. For examplefifth effi- cient solution given in Table 11 and second efficient solution given in Table 12 gained highest ranks in spite of low cost and low profit, but the project may earn higher profits in near fu- ture in spite of low one time set up cost due to qualitative as- pects. Present paper can be solved alternately by changing the level of priorities. For example high ranks may be prioritized to low cost etc. The proposed methodology may prove to be a powerful tool for efficient decision making. The model pre- sented has potential application in the area of real estate, portfolio management, investment theory etc.

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