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Location and size of day nurseries— a multiple goal approach
Location and size of day nurseries a multiple goal approach 2. The problem and the model
M.A. B E N I T O A L O N S O and P. D E V A U X Public Investment Research Center, Facult~ Universitaire Catholique de Mons, B- 7000 Mons, Belgium
In 1976 the local authorities of Mons, Belgium, decided to ~iry and meet the demand for pre-school facilities as estimated by [1 ]. The demand (ai) for the twelve areas of the conurbation is shown in Fig. 1. To identity optimal locations and sizes, we need to assume that local authorities will be trying to reduce both distances and costs (Section 2.1,2.2) while meeting the children's educational needs as fully as possible (Section 2.3).
Received May 1979 Revised January 1980 This paper uses a mathematical model to determine the location and size of day nurseries within a town. The solution consists of finding a compromise among three conflicting objectives. Mixed integer multiple goal programming is used to solve the problem. Actual data is analysed.
2.1. Minimizing children's travel distances
Keywords: Nursery schools, multiple goal programming
Let dii denote the distance in Km from area i to area ]. Three assumptions are made regarding tiffs distance: (a) distances within an area are equal to one, dii= 1 f o r / = / ; (b) no child will travel too far. From interviews with potential users the critical threshold was estimated to be dij = 11; (c) the distance between any two areas is equal to the distance between the centers of the two areas.
1. Introduction The growing tendency for married women to go to work increases the demand for child-cale services. Day nurseries rank first among these services. Planning the nursery network requires answers to: how many nurseries to operate, where to locate them, and at what capacities. One approach (see for example [3,8,12]) to the above questions has been to identify which areas should be given priority in a nursery building program. Besides focusing on parental demand such approach took into account the educational needs of the children. For example its so-called 'priority scores' were measured by the difference between (forecast demand x need index) and current provision. A second approach (see [4,5])applies operations research to solving the problem of where to locate day nurseries. It uses a location-allocation model which generally maximizes accessibility to potential patrons. Such treatments, although by l~o means negligible, are not comprehensive enough. On practice decision makers face conflicting targets'such as educational needs, accessibility and budget considerations.
© North-Holland Publishing Company European Journal of Operational Research 6 (1981 ) 195 - 198
Fig. 1. Demand for day-care services within each area. 195
M.A. Benito.Alonso, P. Devaux / Multiple goal approach
196
and where
2.2. Minimizing the costs o f the nursery n e t w o r k
We assume this aggregate cost to be a linear function of both the number of places and the number of day nurseries set up.
xii
si
2. 3. Satisfying the educational needs
~r~a ],
It is not enough to meet the spatially distributed demand for day nursery places but one must also consider social needs of each area in its broader context. Using experts' opinion it is possible to construct an index of educational needs which measures how important and pressing the educational needs of infant children are within each area in town. Let wi denote the value of the index for area i. We derived its estimates from study [ 1] and obtained the following: w i = 3 for i = 1, ..., 5; wi = I for i = 6, ..., 12. Minimizing the extent to which children's educational needs are unmet will thus mean maximizing the degree to which they are indeed met. The three (possibly conflicting) objectives may be expressed as follows: 12
12
Minimize i=l j=l 12
12
/=I
ZICF "I"~-I CLSj, /=l
Minimize 12
Minimize
~
wy i
i= 1
subject to the constraints~2 i= 1,..., 12,
(1)
j = 1.... ,12,
(2)
s/<~zj(O M - Din) ,
j = 1, ..., 12,
(3)
z/= 1 or 0 ,
/ = 1, ..., 12,
(4)
xi/>~O , s/>~O ; yi>~O ,
i , j = 1,..., 12,
(5)
~_i xi] + Yi = ai ,
/=!
denotes the number of under-fours living in area i and attending a day nursery located in area/, the number of extra places in addition to the minimum capacity of the day nursery within
Yi
Dm DM
the unmet demand for day nursery places from area i, the minimum size to be ceached if a day nursery is to be set up (set to be 30), the maximum size of a day nursery (set to be
50), CF
CL
the cost of building rninimum-sized day nursery (estimated to be 6.4 million BF), the building cost per place over minimum capacity (estimated to be 0.2 million BF).
The above model is a linear mixed integer multipleobjective program. Powerful technique for analyzing such models have been proposed (see for example [2,6,7,11]). We tested two strategies. One gives priority to balancing the budget, while the other reflects the opinions of those who lay greater emphasis on meeting the children's educational needs. Thus under strategy A, first priority is given to minimizing the gap between aggregate building cost and budget allocation (which assumed to be 20 million BF). Maximizing educational provision ranks second, and the distance problem comes last. Strategy B first ensures the satisfaction of educational needs, then tries to reduce the gap between spending and allocation, and the .,ainimization of journeys ranks third. In our Mons model we have 142 variables and 39 constraints. We used the BBMIP package [ 14] which is based upon Land and Doig's algorithm [ 10].
12
i= 1
x i / = D m z i + sj,
where
l
1 if a day nursery is to be built in area/,
zi = 0 otherwise
3. Results and interpretation Solving the model provides the level of each objective and the location-allocation matrix describing the solution. Column $16 in Table l provides the solution ~nder strategy A: two nurseries are opened;one in area 8 caters for areas 1~ and part of 8, the second in area 6 covers areas 5 - 6 and part of 7. The other areas are not served. Under strategy B, column Sl in Table l, 5 nurseries are built: one with 32 places in area 2 and four with 50 places each in areas 6, 9,
11 and 12.
M.A. Benito-Alonso, P. Devaux / Multiple goal approach
r--
e~
197
These strategies provide extreme solutions, which are but the untempered embodiment of conflicting viewpoint within a group of decision-makers. Each strategy makes it possible to arrive at an optimum solution corresponding to one point of view, and then measure the consequent relaxation of the other objectives. Those solutions are use~'ul in that they help determine the feasibility boundaries within which negotiation is to take place. To help decisionmakers with the negotiation, we studied the efficient frontier by applying the method of artificial constraints [9]. The resulting set of Pareto solutions is shown in Table 1. The best compromise will be reached by analysts and decision-makers engaging in successive trade-offs among the objectives proposed in Table 1. However, since the latter does not enumerate all possible Pareto efficient solutions, more refined rest,arch in the region of the compromise solution might prove worthwhile.
~er~
References C) ¢q 0
¢.q
O0
~D
0
[1 ] L. Bodart, P. Devaux and L. lieuskin, Etude du d6veloppement de l'infrastructu~'e sociale en mati~re de garde des epfants en bas age du grand mons (1976). [2] A. Charnes and W.W. Cooper, Goal programming, and multiple objective optimizations, Part I, European J. Operational Res. 1 (1977) 39- 54. [3] J. Gamble, Priority areas for the provision of day care facilities for the under-fives, County of South Glamorgan (1977). [4] M.J. Hodgson and P. Doyle, The location of public services considering the mode of travel, Socio-l-conomic Planning Sci. 12 (1978) 4 9 - 5 4 . [5 ] J. Holmes, F.B. Williams and L.A. Brown, Facihty location under maximum travel restrictions: an example using day care facilities, Geograph. Anal. 4 (1972) 258- 266. [6] J.P. Ignizio, Goal Programming and Extensions (Lexington, 1976). 17] J.P. lgnizio, A review of goal programming: a tool for multiobjective analysis, J. Operational Res. Soc. 29 (1978) 1109-1119. [8] G.K. Jones and S. Roper, Planning a nursery programme, Greater London Intelligence Quarterly 27 t 1974) 51--61. [9] R.L. Keeney and H. Raiffa, Decisions with Multiple Objectives: Preferences and Value Tradc-offs i Wiley, New York, 1976). 1101 A.H. Land and A.G. Doig, An automatic method of solving discrete programming problems, Ecol~ometrica 28 (1960) 497-520. 1111 S.M. Lee, Goal Programming for Decision Analysis Auerbach, Princeton, 1972). I121 London Borough of Greenwich, Programme Planning
"98
M.A. Benito.Alonso, P. Devaux / Multiple goal approach
Section, Priority locations of pre-school facilities in Greenwich, London Borough of Greenwich (1973). [ 131 A. Ostanello-Borreani and P. Capellaro, Efficient solutions of a multiple-objective programming model for a problem of school allocation and dimensioning, Second
European Congress on Operations Research, Stockholm, Sweden (1976). [ 14] R. Shareshian, Branch and bound mixed-integer programming, IBM Corporation, New York (1967).
M.A. B E N I T O A L O N S O and P. D E V A U X Public Investment Research Center, Facult~ Universitaire Catholique de Mons, B- 7000 Mons, Belgium
In 1976 the local authorities of Mons, Belgium, decided to ~iry and meet the demand for pre-school facilities as estimated by [1 ]. The demand (ai) for the twelve areas of the conurbation is shown in Fig. 1. To identity optimal locations and sizes, we need to assume that local authorities will be trying to reduce both distances and costs (Section 2.1,2.2) while meeting the children's educational needs as fully as possible (Section 2.3).
Received May 1979 Revised January 1980 This paper uses a mathematical model to determine the location and size of day nurseries within a town. The solution consists of finding a compromise among three conflicting objectives. Mixed integer multiple goal programming is used to solve the problem. Actual data is analysed.
2.1. Minimizing children's travel distances
Keywords: Nursery schools, multiple goal programming
Let dii denote the distance in Km from area i to area ]. Three assumptions are made regarding tiffs distance: (a) distances within an area are equal to one, dii= 1 f o r / = / ; (b) no child will travel too far. From interviews with potential users the critical threshold was estimated to be dij = 11; (c) the distance between any two areas is equal to the distance between the centers of the two areas.
1. Introduction The growing tendency for married women to go to work increases the demand for child-cale services. Day nurseries rank first among these services. Planning the nursery network requires answers to: how many nurseries to operate, where to locate them, and at what capacities. One approach (see for example [3,8,12]) to the above questions has been to identify which areas should be given priority in a nursery building program. Besides focusing on parental demand such approach took into account the educational needs of the children. For example its so-called 'priority scores' were measured by the difference between (forecast demand x need index) and current provision. A second approach (see [4,5])applies operations research to solving the problem of where to locate day nurseries. It uses a location-allocation model which generally maximizes accessibility to potential patrons. Such treatments, although by l~o means negligible, are not comprehensive enough. On practice decision makers face conflicting targets'such as educational needs, accessibility and budget considerations.
© North-Holland Publishing Company European Journal of Operational Research 6 (1981 ) 195 - 198
Fig. 1. Demand for day-care services within each area. 195
M.A. Benito.Alonso, P. Devaux / Multiple goal approach
196
and where
2.2. Minimizing the costs o f the nursery n e t w o r k
We assume this aggregate cost to be a linear function of both the number of places and the number of day nurseries set up.
xii
si
2. 3. Satisfying the educational needs
~r~a ],
It is not enough to meet the spatially distributed demand for day nursery places but one must also consider social needs of each area in its broader context. Using experts' opinion it is possible to construct an index of educational needs which measures how important and pressing the educational needs of infant children are within each area in town. Let wi denote the value of the index for area i. We derived its estimates from study [ 1] and obtained the following: w i = 3 for i = 1, ..., 5; wi = I for i = 6, ..., 12. Minimizing the extent to which children's educational needs are unmet will thus mean maximizing the degree to which they are indeed met. The three (possibly conflicting) objectives may be expressed as follows: 12
12
Minimize i=l j=l 12
12
/=I
ZICF "I"~-I CLSj, /=l
Minimize 12
Minimize
~
wy i
i= 1
subject to the constraints~2 i= 1,..., 12,
(1)
j = 1.... ,12,
(2)
s/<~zj(O M - Din) ,
j = 1, ..., 12,
(3)
z/= 1 or 0 ,
/ = 1, ..., 12,
(4)
xi/>~O , s/>~O ; yi>~O ,
i , j = 1,..., 12,
(5)
~_i xi] + Yi = ai ,
/=!
denotes the number of under-fours living in area i and attending a day nursery located in area/, the number of extra places in addition to the minimum capacity of the day nursery within
Yi
Dm DM
the unmet demand for day nursery places from area i, the minimum size to be ceached if a day nursery is to be set up (set to be 30), the maximum size of a day nursery (set to be
50), CF
CL
the cost of building rninimum-sized day nursery (estimated to be 6.4 million BF), the building cost per place over minimum capacity (estimated to be 0.2 million BF).
The above model is a linear mixed integer multipleobjective program. Powerful technique for analyzing such models have been proposed (see for example [2,6,7,11]). We tested two strategies. One gives priority to balancing the budget, while the other reflects the opinions of those who lay greater emphasis on meeting the children's educational needs. Thus under strategy A, first priority is given to minimizing the gap between aggregate building cost and budget allocation (which assumed to be 20 million BF). Maximizing educational provision ranks second, and the distance problem comes last. Strategy B first ensures the satisfaction of educational needs, then tries to reduce the gap between spending and allocation, and the .,ainimization of journeys ranks third. In our Mons model we have 142 variables and 39 constraints. We used the BBMIP package [ 14] which is based upon Land and Doig's algorithm [ 10].
12
i= 1
x i / = D m z i + sj,
where
l
1 if a day nursery is to be built in area/,
zi = 0 otherwise
3. Results and interpretation Solving the model provides the level of each objective and the location-allocation matrix describing the solution. Column $16 in Table l provides the solution ~nder strategy A: two nurseries are opened;one in area 8 caters for areas 1~ and part of 8, the second in area 6 covers areas 5 - 6 and part of 7. The other areas are not served. Under strategy B, column Sl in Table l, 5 nurseries are built: one with 32 places in area 2 and four with 50 places each in areas 6, 9,
11 and 12.
M.A. Benito-Alonso, P. Devaux / Multiple goal approach
r--
e~
197
These strategies provide extreme solutions, which are but the untempered embodiment of conflicting viewpoint within a group of decision-makers. Each strategy makes it possible to arrive at an optimum solution corresponding to one point of view, and then measure the consequent relaxation of the other objectives. Those solutions are use~'ul in that they help determine the feasibility boundaries within which negotiation is to take place. To help decisionmakers with the negotiation, we studied the efficient frontier by applying the method of artificial constraints [9]. The resulting set of Pareto solutions is shown in Table 1. The best compromise will be reached by analysts and decision-makers engaging in successive trade-offs among the objectives proposed in Table 1. However, since the latter does not enumerate all possible Pareto efficient solutions, more refined rest,arch in the region of the compromise solution might prove worthwhile.
~er~
References C) ¢q 0
¢.q
O0
~D
0
[1 ] L. Bodart, P. Devaux and L. lieuskin, Etude du d6veloppement de l'infrastructu~'e sociale en mati~re de garde des epfants en bas age du grand mons (1976). [2] A. Charnes and W.W. Cooper, Goal programming, and multiple objective optimizations, Part I, European J. Operational Res. 1 (1977) 39- 54. [3] J. Gamble, Priority areas for the provision of day care facilities for the under-fives, County of South Glamorgan (1977). [4] M.J. Hodgson and P. Doyle, The location of public services considering the mode of travel, Socio-l-conomic Planning Sci. 12 (1978) 4 9 - 5 4 . [5 ] J. Holmes, F.B. Williams and L.A. Brown, Facihty location under maximum travel restrictions: an example using day care facilities, Geograph. Anal. 4 (1972) 258- 266. [6] J.P. Ignizio, Goal Programming and Extensions (Lexington, 1976). 17] J.P. lgnizio, A review of goal programming: a tool for multiobjective analysis, J. Operational Res. Soc. 29 (1978) 1109-1119. [8] G.K. Jones and S. Roper, Planning a nursery programme, Greater London Intelligence Quarterly 27 t 1974) 51--61. [9] R.L. Keeney and H. Raiffa, Decisions with Multiple Objectives: Preferences and Value Tradc-offs i Wiley, New York, 1976). 1101 A.H. Land and A.G. Doig, An automatic method of solving discrete programming problems, Ecol~ometrica 28 (1960) 497-520. 1111 S.M. Lee, Goal Programming for Decision Analysis Auerbach, Princeton, 1972). I121 London Borough of Greenwich, Programme Planning
"98
M.A. Benito.Alonso, P. Devaux / Multiple goal approach
Section, Priority locations of pre-school facilities in Greenwich, London Borough of Greenwich (1973). [ 131 A. Ostanello-Borreani and P. Capellaro, Efficient solutions of a multiple-objective programming model for a problem of school allocation and dimensioning, Second
European Congress on Operations Research, Stockholm, Sweden (1976). [ 14] R. Shareshian, Branch and bound mixed-integer programming, IBM Corporation, New York (1967).