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A goal programming model for facility location planning
0038-0121/87 $3.00 + 0.00 Copyright 0 1987 Pergamon Journals Ltd
So&-Econ. Plann. Sci. Vol. 21, No. 4, PP. 251-255, 1987 Printed in Great Britain. All rights reserved
A GOAL PROGRAMMING MODEL FOR FACILITY LOCATION PLANNING S. Department
B. SINHA
of Mathematics, (Received
Indian
and
S. V. C. SASTRY
Institute
of Technology,
Kharagpur
16 September 1986; in revised form 30 December
721 302, India 1986)
Abstract-The paper presents a goal programming model for facility location planning. Often the location-decision is coupled with multiple objectives, at times conflicting among themselves. Specifically the model considers the four major objectives: (i) necessary locations, (ii) maximum number of locations, (ii) capacity restrictions and (iv) transportation cost/walking distance minimization, simultaneously and proposes optimal locations in conjunction with the existing facilities in the region. The model developed has been illustrated with an example considering the location of community storage facilities in a specified region.
1. INTRODUCTION
field of enquiry in which the Operations Research studies are very suitable is facility location analysis. Facility location problems are strategic in nature. Any technique for calculating the optimum (minimum) transportation cost from its origin to the location of a new/main facility(s) on a network, is not only mathematically elegant but also provides a powerful tool for solving many real-life problems in location analysis. In the recent past, there have been several studies aiming at the single objective of minimization of transportation cost subject to the environmental conditions of the problem and they find an optimal solution making use of general assignment algorithms and/or set covering approaches. In this direction the works of Ross and Soland [l], Gleason [2] and Toregas et al. [3] are worth mentioning. Charles et al. [4] presented a detailed discussion of private and public sector location models, which they broadly classified into location problems on a plane and location problems on a network. They also distinguished the private and public sector models and compared as to structure, criteria and constraints in formulating such problems. Re Velle and Swain [5] considered a public sector location problem explicitly and formulated it as a LP model. In any real-life decision problem, the management must weigh multiple objective simultaneously. Quite often, due to the conflicting and incommensurable nature of the multiple objectives, infeasibility arises in solving the problem with the standard LP technique 161. To overcome the drawbacks inherent in this technique and to consider the multi-dimensionality of the problem a relatively new technique, popularly known as, “Goal Programming” (GP) has been employed in this paper [7,8,9]. For an extensive study of GP methods and applications one can refer to the works of Kornbluth [lo], Hwang and Masud [ll] and Soyibo 1121. The main aim of the present study is to investigate the potentiality and applicability of GP technique in facility location analysis with multiple objectives. The latter sections of the A major
251
paper present a general mathematical the problem.
2. FORMULATION
OF THE
modelling of
MODEL
The problem considered for the development of a model is related to the location of community storage facilities for foodgrains. However, as it is a community storage facility location-decision, the management will have to deal with multiple objectives/goals simultaneously, in order to fulfil the needs of the society. They are described as follows: (i) Necessary locations: it is highly necessary to keep the existing facility OT to locate a new facility at particular places due to their increasing demand. The term, necessary also may relate to the political and social influences of the region and/or the interest of the management, (ii) Number of new locations: this goal is also equally important due to the limitations on the budget in a specific planning period, which constrains the number of new locations to be proposed for this facility. (iii) Capacity of each facility: optimization of the capacity of each storage facility is also needed for its smooth running. Otherwise, it may have adverse affects on the maintenance and management of such facilities. (iv) Transportation cost/walking distance goal: similarly, the optimization of transportation cost/ walking distance for a customer from his/her settlement (village) to travel to the proposed new location is rather an important goal to be considered in a location-decision. Otherwise, the main aim of social utility of a facility may not be achieved. Specifically with these objective factors, the present study proposes to develop a zero-one goal programming model to select the location of new facilities to augment the existing facilities in a specified region. A zero-one GP model is a sub-class of GP models with an additional constraint on the decision variables to take into account of logical restrictions.
252
S. B. SINHA and S.
V. C. SASTRY
Before an optimal solution is considered, it should ascertain that (i) the facility is provided to all the customers, and (ii) all the customers use this facility.
Converting it to a goal constraint form, we have II xx,j+d,--d:=S, t=m+l. (5)
The variables
Capacity of each facilitv: the maximum canacitv of each new storage facility-depends on the prod&on of that region, and also on the interaction from the nearby locations. Also, the capacity of each such facility should be within the limits of the Government policy. Mathematically, these can be expressed as:
j=/
Before proceeding to the formulation of the problem, the present section defines the designation of the variables in the models as follows. M
T 4 ci
41"i
Xij
xjj
Ni Qj S
d: Id, m n
set of the numbers of necessary locations the total production of foodgrains at the ith location the maximum expected amount of produce that will arrive at the ith storage facility location capacity of a storage facility required at an ith location the lower/upper bounds on the capacity of ith location represents the interaction from the ith location to the jth new facility (or the percentage ratio of foodgrains that will go from ith location to the jth new facility) represents a zero-one variable; it takes a value “1” (one) if the existing/the new facility at the jth location is a proposed location and takes a value “0” (zero) otherwise the set of all location-numbers that are within the acceptable walking distance from the ith location the distance between the ith location and the jth new facility the maximum number of new facilities to be located in the region represents the positive/negative deviational variable associated with the tth goal number of new facilities proposed to be set up in the region number of existing facilities in the region
The functional relationships Based on the above stated goals, the model functional relationships can now be formulated. The necessary locations: out of a given total number of “n” locations of the region, only some of them can be made as essential locations to meet various objectives. That is mathematically, x,~= 1 for all j EM
xjla 1 for all j EM
(2) form, we have
xi, + d; - d: = 1 for all j EM
(3)
where t=l,2,..., m for each j of M sequentially. The maximum number of locations: the total number of new locations to be identified for a specified region should be less than a specific number. It can be mathematically expressed, as i j-1
xjl
<
s.
(4)
forall
i=l,2
,...,
n,
forall
i=l,2
,...,
n.
(6)
(7) Converting these into goal constraints form, we have 1 Ajxji+d;
-d:
= Ci
j,Ni
where t=m+2,m+3,...m+n+l 2, . . , n respectively.
for
i=l,
Ci+d;-d:=I,
(9)
wheret=m+n+2,m+n+3,...,m+2n+lfor i=l, 2,..., n respectively and Cj+d;-d:
=ui
(10) where t=m+2n+2, m+2n+3,..., m+3n+l for i = 1,2, . . . , n respectively. Transportation goal: the total transportation cost or alternatively the total walking distance for each new storage facility from the existing ones should be minimized. Mathematically, it can be expressed as:
(11) Converting
it to a goal constraint
form we have
i c AixuDu+d, -d: =0 (12) i=l jEN, where t =m f3n +2. The system constraints: the rigid goals of the study would form the system constraints of the problem. They are (i) all the customers should be provided with the storage facility and (ii) all the customers should use these facilities. Mathematically, these can be stated as: xxii=1 ieX
(1)
since xjj is a zero-one variable, the above constraints may be written as:
converting these into goal constraint
c A,x,,
forall
i=1,2 ,...,
n.
(13)
and x,/>xi/
forall
i=l,2
,...,
n,jENi,i#j.
(14)
The other basic assumptions underlying the model are all the decision and deviational variables must be non-negative and also, the product of the negative and positive deviational variables of each goal should be zero. Mathematically, x,2 0 d:, d; >O
for all
and
i
d2.d;
3. THE ACHIEVEMENT
and =0
j for all
(15) t. (16)
FUNCTION
In GP techniques, the model’s objective function is termed as achievement function, which is composed
253
Facility location planning Table 1. Preliminary sites for the location based on Central Place Theory No. of dependent
Sl.
NO. 1. 2. 3. 4. 5. 6. I. 8. 9. 10.
11.
Name of village
Production in MTs
villages
Guddigudem Gopalapuram Ponguturu Yarrampeta Devarapalli Duddukuru Yamagudem Malakapalli Tallaoudi
4 5 6 4 2 4 4 6 1
12,698 7893 12,636 8344 2206 7714 11,360 4614
Annidevarapeta Gutala
3 3
3999 10,794
48
88,188
Total
Block
of deviational variables only. These deviations are minimized subject to a pre-emptive priority ranking of the various goals stated by the management. The pre-emptive priority factors have a relationship as follows: Pj> > > P,+,,
forall
j=1,2
,...,
k.
(17)
which means that P,, the jth priority factor, always takes priority over Pj+,, the (j + 1)th priority factor. In other words, the lower priority goals are considered only after the higher priority goals are completely achieved or reached to a point beyond which no further improvement can be made. Moreover the appearance of a negative, a positive or both the deviational variables at the respective priority level depends on the analysis of each goal in terms of their achievements. In this way for the present study the achievement function is formulated by assigning priority levels for each goal. However, in GP techniques the system constraints will have the highest priority during the course of optimization. The first priority is given to the goal of necessary locations followed by the goal related to the specific number of facilities to be located. The third priority is given for the optimization capacity of each such facilities and other related constraints on the maximum capacity goal. The fourth priority is assigned to the objective of minimization of the total walking distance for each new facility or alternatively minimization of total transportation cost. Consequently the achievement function can be seen as:
studied under Integrated Rural Development Plan thoroughly [13]. The major crop of the region is paddy and estimated production of this crop for the year 1985 is 88200 MTs. Now, our problem is to find out optimal locations for these new storage facilities to take care of the entire produce of the region. One can identify preliminary locations for these facilities by Central Place Theory or other Spatial Location Models. According to these studies the centrality of a settlement (village) is judged on the basis of the functional importance of that settlement over the other surrounding settlements and also depends on the socio-economic factors of those settlements. Based on the functional hierarchy of the various settlements in a region, the management can identify a specific number of locations to be considered as preliminary locations. To decrease the dimensionality of the present model, we assumed the eleven preliminary loca$ons for these facilities. The names of the places, the number of villages that each can serve and estimated crop production in each group is serially given in Table 1. But the location of a facility at each place is further constrained by several other socio-economic factors of the region. Thus, only Guddigudam, Ponguturu, Devarapalli and Gutala are considered as necessary locations. Although the management desired to provide storage facility for the entire produce, it is expected that only 7080% of the produce will arrive at these centres for processing. Accordingly, the management wished to identify only five potential locations for the establishment of the facilities. Each new facility should take care of the 70% of the produce from its own group of villages and remaining 30% of the produce from the nearby other group of villages, depending on the existing facilities in those places. In order to minimize the total transportation cost or walking distance objective, it is presumed that no customer of the centre would travel more than 10 km from his settlement to the nearest new facility. Based on the order of importance, these goals have been prioritized as follows in the decreasing order of importance: P, for necessary locations, P, for maximum number of locations, P, for capacity restricti-Pns and P4 for minimization of transportation objective. Consequently, the model can be stated as follows: Minimize: P,(d;+d;+d,+d,-)+P,(d:)
Minimize:
+P,(d,+d;+...+d,)+Pq(d:7)
P,(d;+d;+...+d,)+P*(d,f+,) +P3(d,++z+d:+3+
d,f+,,+,+ G,,,,, + . + f’ddm++3n+d.
+
4. APPLICATION
subject to goal constraints:
. ..+d.+,+,
+ d,+,+z + dm+n+3+ .
(19)
xu + d; - d: = 1
+ d,+z,+,
(20)
where t = 1 to 4 for i = 1, 3, 5, 11 respectively.
+ G,,,,,,)
$x,+d,-d,+=5
(18)
1 A, xji + jENi
AND SOLUTION
The model developed in the above sections has been employed to identify potential locations for the establishment of rural community storage facilities to augment the existing ones in the Gopalapuram block of West Godavari district in A.P. The block has been
(21)
j=l
d; - d,+ = C,
(22)
where t = 6-16 for i = 1-l 1 respectively coefficient Aj=0.7(Tj) and
for
i=j
and
joNi
and the
254
S.
B.
SINHA
and S. V. C.
Table 2. Priority-wise information Priority level
1.
I
Definition of the problem
0.0
Completely satisfied the environmental conditions of the problem.
2.
2
Necessary locations
0.0
Fully achieved i.e. all the essential locations are established.
3.
3
Maximum number of locations
0.0
Exactly satisfied this goal by additionally locating a new facility at Malakapalli also.
4.
4
Capacity restrictions
3.4
Due to interaction from the nearby locations the capacity goals are not satisfactory.
5.
5
Transportation/walking
for
i#j
Achievement Goals
and
g 1 Aix,L$+d;-d: i=l /EN!
kVC?l
distance
jENi
=0
(23)
and system constraints: C xi/= 1 for all iEN, x,~-x@>O forall i,j~N~
i
(24)
and
i#j.
(25)
and x,20 d:, d; > 0
forall and
i
and
d: .d; = 0
j
(26)
for all t.
(27)
The model has been run on WIPRO-Z650 computer system and the solution obtained is as follows: Xn=1,x33=1,X55=l,X*8=lrX,,,1,=l 1,x43= 1,x,,=
X 1o,8 = 1
on various goals
Sl. No.
A,=O.3(7;)
x,,=
SASTRY
1,x,5= 1,x98= 1
and all other decision variables are zero.
It implies that the management has to establish the new facilities at Guddigudem, Ponguturu, Devarapalli, Malakapalli and Gutala to serve the entire block in such a way that the customers of Gopalapuram are served by Guddigudem; Yerrampeta are served by Ponguturu; Duddukuru and Yarnagudem are served by Devarapalli; Tallapudi and An-
422
Description
The satisfaction of higher priority goals led to the under-achievement of this goal.
nadevarapeta are served by Malakapalli storage locations and Gutala have an individual storage for its customers, The achievement levels of various goals kept at different priority levels and their description is given in Table 2. All the higher priority goals have been fully achieved. The under-achievement in capacity goals of the storage centre provide the necessary information in order to increase the capacity so that the total production of the region is taken care of. Finally, the total transportation or walking distance is minimized to 422 km in the region. The sequence of ranking of multiple goals as presented in the above section need not be unique and may vary from the decision of the management. Accordingly the reordering of priority levels of the goals, changes in target levels of different objectives etc. have been considered to study the influence of such changes on the solution. Changing the necessary locations, from four to three and keeping the priority system as: P, for necessary locations, P, for minimization of transportation factor, P3 for maximum number of location and P4 for capacity requirements, the following solution was obtained: x11= 1,x33= 1,x55= l,xgg= 1,x,,=
Xl,,,1 = 1, x*, = 1, x43= 1, X65= 1, x,* = 1 X 1o,9 =
1 and all other decision variables are zero.
Table 3. Priority-wise information on various goals from the final solution Sl. No.
Priority level
Achievement level
Goals
Interpretation
1.
1
Definition of the problem
0
Completely achieved.
2.
2
Necessary locations
0
The goals at this priority level are fully achieved. That is, all the necessary locations are identified in the solution.
3.
3
Transportation/walking minimization goal
4.
4
5.
5
398.5
The total distance traversed by all the customers in the region is minimized.
Maximum number of locations
1
Represents under-achievement of the goal i.e. the total number of locations identified in the final solution is exceeded by “one” additional location in the region.
Capacity requirements
0
The goal is completely achieved i.e. the capacity requirement of the entire region is met.
distance
1
255
Facility location planning
It is observed from this solution, the management of the centre has to establish storage centres at Guddigudem, Ponguturu, Devarapalli, Malakapalli, Tallapidi and Gutala. The total distance traversed by all the customers in the region is reduced to 398.5 km just by exceeding the limit on the maximum number of locations by one unit. It is due to the transportation goal at the higher priority level. The achievement levels of various goals and the interpretation is presented in Table 3. In this way the management can have a great deal of information from the sensitivity analysis of the model before reaching a plan of action.
5. CONCLUSIONS The paper presented a zero-one linear goal programming model and demonstrated its application potential in facility location problems with multiple
objectives. The reordering of the priorities and/or incorporation of some more goals and constraints, if any, further provide useful information for the management in decision-making process. The model developed in this paper is not only limited to the location of these storage facilities but also applicable to a wide range of location problems related to emergency facilities etc. to augment the existing facilities of any other region, with or without any modifications.
REFERENCES 1. G. T. Ross and R. M. Soland, Modelling facility location problems as generalized assignment problems. Mgmt Sci. 24, 345-351 (1977).
2. M. Gleason, A set covering approach to bus stop location. Omega 3, 605-608 (19i$. 3. C. Toreaas. R. Swain. C. Re Velle and L. Bereman. The location- of emergency service facilities. 0~: Resl 19, 1363-1371 (1971). 4. C. Re Velle, D. Marks and J. C. Liebman, An analysis of private and public sector location models. Mgmt Sci. 16(11), 692-707 (1970). 5. C. Revelle and R. Swain, Central facilities location. Geogr. Analysis 2(l), (1970). 6. A. J. Hughes and D. E. Grawiog, Linear Programming: An Emphasis on Decision Making. Addison-Wesley, Reading, Mass. (1972). I. A. Charnes and W. W. Cooper, Management Models and Industrial Applications of Linear Programming. Vol. I. Wiley, New York (1961). 8. S. M. Lee, Goal Programming for Decision Analysis. Auerbech, Philadelphia, Pa (1972). 9. J. P. Ignizio, Goal Programming and Extensions. Lexington Books, Lexinsrton. Mass. (1976). 10. J. S. H. Kornbluth, A survey of goal programming. Omega 1, 193-205 (1975). 11. C. L. Hwang and A. S. Md. Masud, MODM: Methodr and Application, Lecture notes in Economics and Mathematical Systems. Vol. 164. Springer-Verlag, Berlin
(1979). 12. A. Soyibo, Goal programming methods and applications. A survey. Optimization Sci. 6, 247-264 (1985). 13. Technical Report on Integrated Rural Development Plan, Gopalpuram Block, West Godavari District, Andhra Pradesh (1981).
So&-Econ. Plann. Sci. Vol. 21, No. 4, PP. 251-255, 1987 Printed in Great Britain. All rights reserved
A GOAL PROGRAMMING MODEL FOR FACILITY LOCATION PLANNING S. Department
B. SINHA
of Mathematics, (Received
Indian
and
S. V. C. SASTRY
Institute
of Technology,
Kharagpur
16 September 1986; in revised form 30 December
721 302, India 1986)
Abstract-The paper presents a goal programming model for facility location planning. Often the location-decision is coupled with multiple objectives, at times conflicting among themselves. Specifically the model considers the four major objectives: (i) necessary locations, (ii) maximum number of locations, (ii) capacity restrictions and (iv) transportation cost/walking distance minimization, simultaneously and proposes optimal locations in conjunction with the existing facilities in the region. The model developed has been illustrated with an example considering the location of community storage facilities in a specified region.
1. INTRODUCTION
field of enquiry in which the Operations Research studies are very suitable is facility location analysis. Facility location problems are strategic in nature. Any technique for calculating the optimum (minimum) transportation cost from its origin to the location of a new/main facility(s) on a network, is not only mathematically elegant but also provides a powerful tool for solving many real-life problems in location analysis. In the recent past, there have been several studies aiming at the single objective of minimization of transportation cost subject to the environmental conditions of the problem and they find an optimal solution making use of general assignment algorithms and/or set covering approaches. In this direction the works of Ross and Soland [l], Gleason [2] and Toregas et al. [3] are worth mentioning. Charles et al. [4] presented a detailed discussion of private and public sector location models, which they broadly classified into location problems on a plane and location problems on a network. They also distinguished the private and public sector models and compared as to structure, criteria and constraints in formulating such problems. Re Velle and Swain [5] considered a public sector location problem explicitly and formulated it as a LP model. In any real-life decision problem, the management must weigh multiple objective simultaneously. Quite often, due to the conflicting and incommensurable nature of the multiple objectives, infeasibility arises in solving the problem with the standard LP technique 161. To overcome the drawbacks inherent in this technique and to consider the multi-dimensionality of the problem a relatively new technique, popularly known as, “Goal Programming” (GP) has been employed in this paper [7,8,9]. For an extensive study of GP methods and applications one can refer to the works of Kornbluth [lo], Hwang and Masud [ll] and Soyibo 1121. The main aim of the present study is to investigate the potentiality and applicability of GP technique in facility location analysis with multiple objectives. The latter sections of the A major
251
paper present a general mathematical the problem.
2. FORMULATION
OF THE
modelling of
MODEL
The problem considered for the development of a model is related to the location of community storage facilities for foodgrains. However, as it is a community storage facility location-decision, the management will have to deal with multiple objectives/goals simultaneously, in order to fulfil the needs of the society. They are described as follows: (i) Necessary locations: it is highly necessary to keep the existing facility OT to locate a new facility at particular places due to their increasing demand. The term, necessary also may relate to the political and social influences of the region and/or the interest of the management, (ii) Number of new locations: this goal is also equally important due to the limitations on the budget in a specific planning period, which constrains the number of new locations to be proposed for this facility. (iii) Capacity of each facility: optimization of the capacity of each storage facility is also needed for its smooth running. Otherwise, it may have adverse affects on the maintenance and management of such facilities. (iv) Transportation cost/walking distance goal: similarly, the optimization of transportation cost/ walking distance for a customer from his/her settlement (village) to travel to the proposed new location is rather an important goal to be considered in a location-decision. Otherwise, the main aim of social utility of a facility may not be achieved. Specifically with these objective factors, the present study proposes to develop a zero-one goal programming model to select the location of new facilities to augment the existing facilities in a specified region. A zero-one GP model is a sub-class of GP models with an additional constraint on the decision variables to take into account of logical restrictions.
252
S. B. SINHA and S.
V. C. SASTRY
Before an optimal solution is considered, it should ascertain that (i) the facility is provided to all the customers, and (ii) all the customers use this facility.
Converting it to a goal constraint form, we have II xx,j+d,--d:=S, t=m+l. (5)
The variables
Capacity of each facilitv: the maximum canacitv of each new storage facility-depends on the prod&on of that region, and also on the interaction from the nearby locations. Also, the capacity of each such facility should be within the limits of the Government policy. Mathematically, these can be expressed as:
j=/
Before proceeding to the formulation of the problem, the present section defines the designation of the variables in the models as follows. M
T 4 ci
41"i
Xij
xjj
Ni Qj S
d: Id, m n
set of the numbers of necessary locations the total production of foodgrains at the ith location the maximum expected amount of produce that will arrive at the ith storage facility location capacity of a storage facility required at an ith location the lower/upper bounds on the capacity of ith location represents the interaction from the ith location to the jth new facility (or the percentage ratio of foodgrains that will go from ith location to the jth new facility) represents a zero-one variable; it takes a value “1” (one) if the existing/the new facility at the jth location is a proposed location and takes a value “0” (zero) otherwise the set of all location-numbers that are within the acceptable walking distance from the ith location the distance between the ith location and the jth new facility the maximum number of new facilities to be located in the region represents the positive/negative deviational variable associated with the tth goal number of new facilities proposed to be set up in the region number of existing facilities in the region
The functional relationships Based on the above stated goals, the model functional relationships can now be formulated. The necessary locations: out of a given total number of “n” locations of the region, only some of them can be made as essential locations to meet various objectives. That is mathematically, x,~= 1 for all j EM
xjla 1 for all j EM
(2) form, we have
xi, + d; - d: = 1 for all j EM
(3)
where t=l,2,..., m for each j of M sequentially. The maximum number of locations: the total number of new locations to be identified for a specified region should be less than a specific number. It can be mathematically expressed, as i j-1
xjl
<
s.
(4)
forall
i=l,2
,...,
n,
forall
i=l,2
,...,
n.
(6)
(7) Converting these into goal constraints form, we have 1 Ajxji+d;
-d:
= Ci
j,Ni
where t=m+2,m+3,...m+n+l 2, . . , n respectively.
for
i=l,
Ci+d;-d:=I,
(9)
wheret=m+n+2,m+n+3,...,m+2n+lfor i=l, 2,..., n respectively and Cj+d;-d:
=ui
(10) where t=m+2n+2, m+2n+3,..., m+3n+l for i = 1,2, . . . , n respectively. Transportation goal: the total transportation cost or alternatively the total walking distance for each new storage facility from the existing ones should be minimized. Mathematically, it can be expressed as:
(11) Converting
it to a goal constraint
form we have
i c AixuDu+d, -d: =0 (12) i=l jEN, where t =m f3n +2. The system constraints: the rigid goals of the study would form the system constraints of the problem. They are (i) all the customers should be provided with the storage facility and (ii) all the customers should use these facilities. Mathematically, these can be stated as: xxii=1 ieX
(1)
since xjj is a zero-one variable, the above constraints may be written as:
converting these into goal constraint
c A,x,,
forall
i=1,2 ,...,
n.
(13)
and x,/>xi/
forall
i=l,2
,...,
n,jENi,i#j.
(14)
The other basic assumptions underlying the model are all the decision and deviational variables must be non-negative and also, the product of the negative and positive deviational variables of each goal should be zero. Mathematically, x,2 0 d:, d; >O
for all
and
i
d2.d;
3. THE ACHIEVEMENT
and =0
j for all
(15) t. (16)
FUNCTION
In GP techniques, the model’s objective function is termed as achievement function, which is composed
253
Facility location planning Table 1. Preliminary sites for the location based on Central Place Theory No. of dependent
Sl.
NO. 1. 2. 3. 4. 5. 6. I. 8. 9. 10.
11.
Name of village
Production in MTs
villages
Guddigudem Gopalapuram Ponguturu Yarrampeta Devarapalli Duddukuru Yamagudem Malakapalli Tallaoudi
4 5 6 4 2 4 4 6 1
12,698 7893 12,636 8344 2206 7714 11,360 4614
Annidevarapeta Gutala
3 3
3999 10,794
48
88,188
Total
Block
of deviational variables only. These deviations are minimized subject to a pre-emptive priority ranking of the various goals stated by the management. The pre-emptive priority factors have a relationship as follows: Pj> > > P,+,,
forall
j=1,2
,...,
k.
(17)
which means that P,, the jth priority factor, always takes priority over Pj+,, the (j + 1)th priority factor. In other words, the lower priority goals are considered only after the higher priority goals are completely achieved or reached to a point beyond which no further improvement can be made. Moreover the appearance of a negative, a positive or both the deviational variables at the respective priority level depends on the analysis of each goal in terms of their achievements. In this way for the present study the achievement function is formulated by assigning priority levels for each goal. However, in GP techniques the system constraints will have the highest priority during the course of optimization. The first priority is given to the goal of necessary locations followed by the goal related to the specific number of facilities to be located. The third priority is given for the optimization capacity of each such facilities and other related constraints on the maximum capacity goal. The fourth priority is assigned to the objective of minimization of the total walking distance for each new facility or alternatively minimization of total transportation cost. Consequently the achievement function can be seen as:
studied under Integrated Rural Development Plan thoroughly [13]. The major crop of the region is paddy and estimated production of this crop for the year 1985 is 88200 MTs. Now, our problem is to find out optimal locations for these new storage facilities to take care of the entire produce of the region. One can identify preliminary locations for these facilities by Central Place Theory or other Spatial Location Models. According to these studies the centrality of a settlement (village) is judged on the basis of the functional importance of that settlement over the other surrounding settlements and also depends on the socio-economic factors of those settlements. Based on the functional hierarchy of the various settlements in a region, the management can identify a specific number of locations to be considered as preliminary locations. To decrease the dimensionality of the present model, we assumed the eleven preliminary loca$ons for these facilities. The names of the places, the number of villages that each can serve and estimated crop production in each group is serially given in Table 1. But the location of a facility at each place is further constrained by several other socio-economic factors of the region. Thus, only Guddigudam, Ponguturu, Devarapalli and Gutala are considered as necessary locations. Although the management desired to provide storage facility for the entire produce, it is expected that only 7080% of the produce will arrive at these centres for processing. Accordingly, the management wished to identify only five potential locations for the establishment of the facilities. Each new facility should take care of the 70% of the produce from its own group of villages and remaining 30% of the produce from the nearby other group of villages, depending on the existing facilities in those places. In order to minimize the total transportation cost or walking distance objective, it is presumed that no customer of the centre would travel more than 10 km from his settlement to the nearest new facility. Based on the order of importance, these goals have been prioritized as follows in the decreasing order of importance: P, for necessary locations, P, for maximum number of locations, P, for capacity restricti-Pns and P4 for minimization of transportation objective. Consequently, the model can be stated as follows: Minimize: P,(d;+d;+d,+d,-)+P,(d:)
Minimize:
+P,(d,+d;+...+d,)+Pq(d:7)
P,(d;+d;+...+d,)+P*(d,f+,) +P3(d,++z+d:+3+
d,f+,,+,+ G,,,,, + . + f’ddm++3n+d.
+
4. APPLICATION
subject to goal constraints:
. ..+d.+,+,
+ d,+,+z + dm+n+3+ .
(19)
xu + d; - d: = 1
+ d,+z,+,
(20)
where t = 1 to 4 for i = 1, 3, 5, 11 respectively.
+ G,,,,,,)
$x,+d,-d,+=5
(18)
1 A, xji + jENi
AND SOLUTION
The model developed in the above sections has been employed to identify potential locations for the establishment of rural community storage facilities to augment the existing ones in the Gopalapuram block of West Godavari district in A.P. The block has been
(21)
j=l
d; - d,+ = C,
(22)
where t = 6-16 for i = 1-l 1 respectively coefficient Aj=0.7(Tj) and
for
i=j
and
joNi
and the
254
S.
B.
SINHA
and S. V. C.
Table 2. Priority-wise information Priority level
1.
I
Definition of the problem
0.0
Completely satisfied the environmental conditions of the problem.
2.
2
Necessary locations
0.0
Fully achieved i.e. all the essential locations are established.
3.
3
Maximum number of locations
0.0
Exactly satisfied this goal by additionally locating a new facility at Malakapalli also.
4.
4
Capacity restrictions
3.4
Due to interaction from the nearby locations the capacity goals are not satisfactory.
5.
5
Transportation/walking
for
i#j
Achievement Goals
and
g 1 Aix,L$+d;-d: i=l /EN!
kVC?l
distance
jENi
=0
(23)
and system constraints: C xi/= 1 for all iEN, x,~-x@>O forall i,j~N~
i
(24)
and
i#j.
(25)
and x,20 d:, d; > 0
forall and
i
and
d: .d; = 0
j
(26)
for all t.
(27)
The model has been run on WIPRO-Z650 computer system and the solution obtained is as follows: Xn=1,x33=1,X55=l,X*8=lrX,,,1,=l 1,x43= 1,x,,=
X 1o,8 = 1
on various goals
Sl. No.
A,=O.3(7;)
x,,=
SASTRY
1,x,5= 1,x98= 1
and all other decision variables are zero.
It implies that the management has to establish the new facilities at Guddigudem, Ponguturu, Devarapalli, Malakapalli and Gutala to serve the entire block in such a way that the customers of Gopalapuram are served by Guddigudem; Yerrampeta are served by Ponguturu; Duddukuru and Yarnagudem are served by Devarapalli; Tallapudi and An-
422
Description
The satisfaction of higher priority goals led to the under-achievement of this goal.
nadevarapeta are served by Malakapalli storage locations and Gutala have an individual storage for its customers, The achievement levels of various goals kept at different priority levels and their description is given in Table 2. All the higher priority goals have been fully achieved. The under-achievement in capacity goals of the storage centre provide the necessary information in order to increase the capacity so that the total production of the region is taken care of. Finally, the total transportation or walking distance is minimized to 422 km in the region. The sequence of ranking of multiple goals as presented in the above section need not be unique and may vary from the decision of the management. Accordingly the reordering of priority levels of the goals, changes in target levels of different objectives etc. have been considered to study the influence of such changes on the solution. Changing the necessary locations, from four to three and keeping the priority system as: P, for necessary locations, P, for minimization of transportation factor, P3 for maximum number of location and P4 for capacity requirements, the following solution was obtained: x11= 1,x33= 1,x55= l,xgg= 1,x,,=
Xl,,,1 = 1, x*, = 1, x43= 1, X65= 1, x,* = 1 X 1o,9 =
1 and all other decision variables are zero.
Table 3. Priority-wise information on various goals from the final solution Sl. No.
Priority level
Achievement level
Goals
Interpretation
1.
1
Definition of the problem
0
Completely achieved.
2.
2
Necessary locations
0
The goals at this priority level are fully achieved. That is, all the necessary locations are identified in the solution.
3.
3
Transportation/walking minimization goal
4.
4
5.
5
398.5
The total distance traversed by all the customers in the region is minimized.
Maximum number of locations
1
Represents under-achievement of the goal i.e. the total number of locations identified in the final solution is exceeded by “one” additional location in the region.
Capacity requirements
0
The goal is completely achieved i.e. the capacity requirement of the entire region is met.
distance
1
255
Facility location planning
It is observed from this solution, the management of the centre has to establish storage centres at Guddigudem, Ponguturu, Devarapalli, Malakapalli, Tallapidi and Gutala. The total distance traversed by all the customers in the region is reduced to 398.5 km just by exceeding the limit on the maximum number of locations by one unit. It is due to the transportation goal at the higher priority level. The achievement levels of various goals and the interpretation is presented in Table 3. In this way the management can have a great deal of information from the sensitivity analysis of the model before reaching a plan of action.
5. CONCLUSIONS The paper presented a zero-one linear goal programming model and demonstrated its application potential in facility location problems with multiple
objectives. The reordering of the priorities and/or incorporation of some more goals and constraints, if any, further provide useful information for the management in decision-making process. The model developed in this paper is not only limited to the location of these storage facilities but also applicable to a wide range of location problems related to emergency facilities etc. to augment the existing facilities of any other region, with or without any modifications.
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