paper describes a network-based decision support system for a general form of ... It is an integrated optimization system that generates the model, optimizea.
Mathl. Compsi. Modelling Vol. 15, No. 8, pp. 71-76, Printed in Great Britain. All rights reserved
A DECISION PARKING
1991 Copyright@
0895-7177/91 $3.99 + 0.09 1991 Pergamon Press plc
SUPPORT SYSTEM FOR SPACE ASSIGNMENT
M. A. VENKATARAMANAN Decision and Information
AND MARC
Indiana University, Bloomington, (Received
BORNSTEIN
Systems, School of Business
December
IN 47405
1990)
Abstract-This paper describes a network-based decision support system for a general form of assigning parking spaces. It is an integrated optimization system that generates the model, optimizea it, and produces reports. The problem is modeled as a pure assignment network, which minimizes the preference values. The preference values are penalty costs, which are a function of walking distance, cost and priority. The methodology presented here can be applied to a variety of assignment situations Since the approach is based on a network algorithm, it is where there are competing objectives. capable of solving very large systems in an efficient way. A description of how the model was used to assign parking spaces for Dunhill Apartments in Bloomington, Indiana is provided. The proposed system can be used for any multi-building and multi-lot situation.
1. INTRODUCTION Assigning parking spaces according to preference and availability is a common problem. This paper presents a network-based decision support system (DSS) that can be successfully used to improve the overall preference of the parking assignments. Phillips [l] suggested a multicriteria optimization approach to assigning university personnel to parking lots. Dinkel, Mote and Venkataramanan [23, Dyer and Mulvey [3], Dyer, Feinberg and Geoffrion [4] and Klingman, Mote and Phillips [5] used network based decision support The network based models provide integer solutions due systems for a variety of situations. The efficiency of network algorithms is well to the unimodularity of the constraint matrix. documented [6]. The use of this decision support system provides an effective method for dealing with a large, complex and time consuming process, while improving the performance. Due to the ease of solving the model, it is possible to allow changes in priorities and to present alternate solutions effectively. Thus the system allows the decision maker to maintain control of the process. The structure of the parking lot assignment problem lends itself to be modeled as a network problem. The proposed network combines the objectives of priority, cost and distance by weighting factors. With very little effort, the underlying model can be modified to serve a wide variety of assignment environments. In the following section the problem is explained through the parking lot assignment model for Dunhill Apartments in Bloomington, Indiana. The model is presented in Section 3 and the decision support system is described in Section 4. 2. BACKGROUND
OF PROBLEM
Dunhill Apartments has 11 apartment buildings labeled “A” through “K” and 232 parking spaces (refer to Figure 1). The parking spaces are scattered over the apartment property. A shortage of parking spaces occurred because nonresidents were occupying some of the spaces. To solve this problem, the management planned to issue parking permits and tow cars without permits. Once the decision to issue permits was made, an efficient way of distributing them was needed. This led to the analysis of the problem and the development of the decision support system. Another reason for the parking shortage at Dunhill is that many residents are college students and often each student has a car. Thus, it is possible for the residents of a two bedroom apartment, Typeset by &S-QX 71 HCH 15:8-F
M.A.
VENKATARAMANAN, M. BORNSTEIN Dunn
Street
ia th Grant
St.
St.
Figure 1. Dnnhill Apartments.
for example, to require as many as four permits. The model allows the user to differentiate between the priority of the permits requested. For example, the first permit requested from an apartment is a priority one permit, the second permit requested from the same apartment is a priority two, and so on. In this example for the “A” building, nine apartments requested at least one permit [A1=9], six apartments requested at least two permits [A2=6], one apartment requested at least three permits [A3=1], and one apartment requested at least four permits [A4=1] (see Figure 2). None of the apartments requested more than four permits, but the model could have accommodated additional requests. If the users do not want to differentiate between the number of requests per apartment building, they simply set all priorities equal to an average value. Since the apartment complex is not charging residents for the permits, the cost is currently set to zero. If the apartment management wished to charge for parking permits, they could increase the prices of permits to various levels and raise or lower the weighting factors to accommodate the reduction of walking distance proportional to the increase in money spent for the permit. 3. THE
MODEL
The problem is modeled as a pure network problem (refer to Figure 2). There are two classes of nodes or constraints in this network model. The buildings are the supply nodes and the parking lots are the demand nodes. Buildings with four doors are counted as two demand nodes to make a total of 15 nodes in this case. For example, the whole “B” building is broken up with B representing apartments 1-12 and BB representing apartments 13-24 (refer to Figure 2). The
Decision support system
73
0 0 0
1
A3
1
A4
10
Bi 0 a
0
12
BBl 0 0 0
0 K4
2
Value on an
Figure
nodes are replicated inthenetwork(i=1,2,3
2. Parking
for each priority ,..., I).
arc is the composite
a&gnment
cost
network model.
level. Thus the four priority
level gives sixty demand
nodes
The 232 spaces are grouped into 42 segments containing four to eight parking spaces. They form the 42 demand nodes (j = 1,2,. . . , J). To balance the problem an additional dummy demand node is added as there were three more parking spaces than the number of total requests. The arcs or variables associated with the nodes are denoted as Xij. The arcs connect a supply node i to a demand node j. The flow on these arcs is the number of parking permits issued for building i to lot j under a priority. The objective function coefficient is the weighted sum of priority, cost and walking distance. Priority, cost and walking distance can be viewed as competing objectives. The assignments can be altered by weighting these objectives differently. For example, if we weight priority and cost at a zero level, we will minimize the walking distance. In the case where priority and walking distance are weighted at a zero level, we will minimize the cost of obtaining permits. In order to deal with the objectives, we impose relative weights on the various objective function components. Let CYLe = (1,2,3) be the relative weight assigned to the objective function component e. Mathematically
the network
mm {
model is given as:
a,~~p,Xij+rr~~~Ci,Xii+u3~~~ijXil i=l j=l i=lj=l
(1) i=lj=l
1
or I
min
C
J C[ol
i=l j=l
pij +
a2
Cij + 03Dij]
, Xij
(2)
M.A. VENKATAFLAMANAN, M. BORNSTEIN
74
subject
to
c = c I
Xij
di,
fori=1,2
,...,
I,
(3)
Xij = Sj,
forj=1,2
,...,
J,
(4)
Xij
for all i, j,
j=I J
i=l
Where
QL is relative
> 0,
weight on objective
(5)
e,
Pij is the priority of a permit, Xij is the arc connecting building i to lot j at a priority level, Cij is the cost of parking in lot j from building i, Dij is the distance between building i and lot j, di is the permits demanded at a particular priority level in a building, Sj is the number of parking spaces in lot j.
and
Constraint set (3) ensures that the number of parking spaces allotted to a building at a given priority level is equal to the demand at that priority level. Constraint set (4) ensures that the number of parking assignments to a lot will equal the number of spaces available. Constraint set (5) includes the non-negativity constraints. The priority Pij can be given a very high numerical value for higher priority residents to give them closer parking spaces. The lot capacity data, permit request, and priority information were provided by Dunhill Apartments. The distance from each lot to each building was measured in feet. Since the apartment complex does not charge the residents, the cost of a permit is set to zero. The objectives were weighted equally for the example. 4. DECISION
SUPPORT
SYSTEM
The primary software components include a problem generator, a primal optimizer and a report writer. The system is illustrated in Figure 3.
simplex
network
Walking ICost. Distance. 8 I Lot Data’
‘r’
Figure 3. Parking assignment decision support system.
The first component is a problem generator which creates a problem file from two input files in IBM-SHARE format for the network optimizer. One input file has cost data involving walking distance and available space in each lot, while the other consists of the number of requests under each priority. The next component, the primal simplex network optimizer, reads the problem file and weighting factor and solves the resulting problem to optimality. The third component,
Decision support system
75
the report writer, takes the output from the network optimizer and presents it in usable form. A report is generated for each building giving the assignments under each priority to various parking lots. The weighting scheme generates a pareto-optimal solution. To provide the decision maker with the tradeoffs involved in the three objectives, a payoff table is developed. For each of the three objectives, the weights are assigned such that one objective is given a high weight, and the other two a minimal weight. The three solutions obtained by each of these three weighted combinations yield a payoff table, exhibited in Table 1. The attainments for each objective are presented to the decision maker, providing a compact demonstration of the tradeoffs involved. Table 1. Payoff table.
The diagonal elements are minimal nondominated attainments for each of the three objective functions. Z,? = Attainment of 2, while minimizing Zi.
Then the problem is solved using the weights provided by the decision maker. Prior to implementation, the decision maker can review the report to assess the quality of the solution. If a different solution is desired, the decision maker can change the weighting scheme, alter the priority data or lock-in certain parking spaces. When the decision maker is comfortable with assignments, final reports can be generated. It only takes a few seconds to run the model, allowing the decision maker to make multiple “what if” runs. Other “what if” runs can involve changing the cost of parking or acquiring additional parking spaces. In the latter case, building a parking garage may be necessary. The location of the garage can be investigated as well, given a possible set of locations. The problem for Dunhill Apartments has 103 nodes and 2688 arcs. The model took one second to generate the problem, solve it and report the results. The software is written in FORTRAN 77 and implemented in an IBM-4381 computer. Once the database for a particular setting is created, it is simple to maintain the system. The apartment complex would have to update the supply only if they gain or lose parking spaces. Demand will change each September when new residents move in. The cost, priority, and weighting factors can be altered at any time to view many different scenarios. The model can also accommodate special situations like a handicapped person or a maintenance vehicle by including only one arc with a zero cost on it. The report gives the assignment for each priority level of each building to parking lots. The decision maker assigns individual permits to the occupants or the building based on the solution, thus maintaining further control over the process. 5. CONCLUSIONS The model takes little time to solve and provides in depth information. Instead of asking for an “optimal solution,” the model can be used to analyze pareto-optimal solutions and related trade offs. The model and the solution method are robust and can be modified to serve a wide variety of assignment environments with competing objectives. REFERENCES 1. N.V. Phillips, An application of multicriteria optimization to assigning university personnel to pa&kg lots, ORSA/TlMS National Conference, Atlanta (1985). 2. J.J. Dinlcel, J. Mote and M.A. Venl&aramanan, An efficient DSS for academic course scheduling, Operations Research 37, 85.3-864 (1989).
76
M.A. 3.
VENKATARAMANAN, M.BORNSTEIN
J.S. Dyer and J.M. Mulvey, An integrated optimization/information system for academic departmental planning, McmcrgemenlScience 22, 1332-1341 (1976). 4. J.S. Dyer, A. Feinberg and A. Geoffrion, An interactive approach for multicriterion optimization with an application to the operating of au academic department, Management Science 19,357-368 (1972). 5. D. Klingman, J. Mote and N.V. Phillips, A logistics planning system at W.R. Grace, Operations Rereunh 36,811~822 (1988). 6. F. Glover, D. Karney and D. Klingmau, Implementation and computational comparisons of primal, dual and primal-dual computer codes for minimum cost network flow problems, Netwottr 4,191-212 (1974).
