European Journal of Operational Research 142 (2002) 480–496 www.elsevier.com/locate/dsw
Discrete Optimization
Infrastructure development for conversion to environmentally friendly fuel Ravi Bapna *, Lakshman S. Thakur, Suresh K. Nair Department of Operations and Information Management, University of Connecticut, 2100 Hillside Road, U-41 IM, Storrs, CT 06269-2041, USA Received 28 June 2001; accepted 9 August 2001
Abstract An important concern for any nation wishing to convert to alternate, environmentally friendly energy sources is the development of appropriate fuel distribution infrastructure. We address the problem of optimally locating gas station facilities for developing nations, like India, which are in the process of converting from leaded to unleaded fuel. Importantly, a similar approach may be used in developed countries, which are in the process of converting to automobiles using hydrogen or electrical energy. An integer-programming model with the objective of balancing the perspectives of coverage and cost is presented for this facility location problem. Given the existing network of roads, the model considers the traveling population, the location of existing facilities and the cost of, either converting these facilities to carry unleaded fuel, or of installing new facilities in an attempt to minimize cost and simultaneously maximize coverage of population. We develop a heuristic solution procedure for this problem. The methodology is applied to data sets obtained from Current et al. [J.R. Current, C.S. ReVelle, J.L. Cohon, Decision Sciences 19 (1988) 490] representing the Ohio state limited access highway network, and to the Indian national highway network. Additionally, extensive simulations are carried out in order to examine where our approach compares with the maximum weighted spanning tree approach. This work extends the Maximum Covering/Shortest Path problem (MCSPP) formulated by Current et al. [J.R. Current, C.S. ReVelle, J.L. Cohon, European Journal of Operational Research 21 (1985) 189] to accommodate multiple sources and destinations. 2002 Elsevier Science B.V. All rights reserved. Keywords: Facility location; Environmental protection; Integer programming
1. Introduction Location problems in general are spatial resource allocation problems dealing with one or
*
Corresponding author. Tel.: +1-860-486-8398; fax: +1-520395-8791. E-mail address:
[email protected] (R. Bapna).
more service facilities (servers) serving a spatially distributed set of demands (customers). The objective is to locate facilities to optimize a spatially dependent objective like the minimization of average travel time or distance between demands and servers. Our research is aimed at strategically locating gas stations in developing countries like India that are in the process of switching from leaded to unleaded fuel. The Institute of Environmental
0377-2217/02/$ - see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 0 1 ) 0 0 3 0 9 - 5
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Studies, University of Wisconsin-Madison tracks the progress of 79 countries which are in the process of this conversion. Fig. 1 depicts a global comparison of the percentage of gasoline sold that is unleaded. It is obvious that much of the mass for the developing and the former eastern block countries is below the 20% mark, with the exception of a few outliers. As the number of vehicles in these countries continues to grow, so does the problem of vehicular pollution. In India, where an estimated 70% of all air pollution can be attributed to automobiles, it has been estimated that since 1990, metropolitan vehicle population has tripled. Additionally, the number of patients with respiratory diseases and allergies has doubled and approximately 40,000 people die prematurely every year due to air pollution (Gasping for Air, 1996).
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1.1. The problem in India The four major Indian metropolitan cities (Delhi, Mumbai, Calcutta, and Chennai) which face the major brunt of this epidemic have phased in the use of unleaded fuel, an established reducer of toxic emissions from automobiles. Other smaller Indian cities and districts still offer only leaded fuel for motorists. Simultaneously, the government has ruled that all autos manufactured, after mid-1995, and sold in the four major cities, have to be fitted with a catalytic converter that requires them to run only on unleaded fuel. Thus, the country is faced with a major twofold problem: (a) Automobiles sold in the major cities cannot be used to travel outside the metropolitan area because of the lack of fuel stations that carry unleaded fuel. As a result people are unwilling to
Fig. 1. Unleaded fraction of gasoline sold.
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buy autos that cannot be used for long distance travel. This implies that the autos, which are not fitted with the catalytic converter and sold in the smaller cities, are in great demand – thus commanding a premium. This imbalance is detrimental to the fight against pollution. (b) The total countrywide reduction in toxic emissions due to the introduction of unleaded fuel is insignificant because a majority of the vehicles across the country still use leaded fuel. 1.2. The need for an objective approach The casual observer may suggest that a simple solution would be to provide all existing gas stations across the country with the capability to carry both unleaded and leaded fuel. However, for a developing nation like India, one has to consider the following significant costs involved: (a) Additional transportation costs of unleaded fuel from particular oil refineries that produce it to the gas stations. (b) Additional fixed cost of installing a separate tank, pumps and related hardware is a significant deterrent to gas station owners especially if they do not see enough current demand for unleaded fuel. Demand of course is dependent on the number of vehicles fitted with catalytic converters, which should increase if there were more unleaded gas stations. Thus, caught in this vicious circle, the countries are falling behind in the race to control vehicular pollution. 1.3. Inter city vs. local travel One of the primary objectives of our research is population coverage, i.e., we wish to locate gas stations such that they are accessible to the maximum number of people. There is also the notion that the proposed location plan should be such that, given the existing network of roads, one is able to travel from any population center to another using a vehicle that runs on unleaded fuel. Thus we have two perspectives: (a) The short distance traveler who mainly travels within a population center should not be re-
quired to travel an extraordinary distance to get unleaded fuel. This has elements of the p-Center Problem (PCP) and the Maximal Covering Location Problem (MCLP) since our primary objective is to provide coverage to populations. (b) The long distance traveler should be able to start from any population center in the country and make his way to any other center using unleaded fuel, not necessarily by the shortest path, but within some ‘‘reasonable’’ extra distance – thus a complete tour must be possible. If, for instance, we have three locations that are connected by roads to form a triangle as in Fig. 2. If the average distance an automobile can travel on a full tank of unleaded gas is j ðx 6 j 6 2xÞ, then we might choose to locate gas stations at only district A and B, and not locate another one on the arc connecting A and C, which is the shortest distance from A to C. Thus the traveler will have to travel via B to reach C from A. This may be practical if the additional distance is reasonable. This approach also introduces a distance constraint between facilities, which has as its upper bound the distance j, the average distance an automobile can travel on a full tank of gas. Little work has been done in facility location with distance constraints between facilities. The situation here is the opposite of that in Brimberg and Mehrez’s undesirable facility location problem with distance constraints (cf. Brimberg and Mehrez, 1994). Their model maximized the minimum distance between obnoxious facilities and demand points while being constrained by some k, a specified lower bound on the distance between any two new facilities. In our problem, the facilities are not obnoxious and the distance between any two facilities will be constrained by an upper bound
Fig. 2. Coverage and connectedness objectives.
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equal to an estimate of the average traveling capacity of autos with a full tank of gas. 1.4. Research objectives Thus, given the estimated traveling populations at the various cities, the existing network of highways and byways, and the existing gas station locations we need to determine the locations of new facilities, or alternatively, convert existing facilities to carry unleaded gas, in a minimal cost/maximal covering fashion. Current et al. (1985, 1988) demonstrate that many location problems are inherently multiobjective in nature. They identified four broad categories that deserve consideration. Namely: cost minimization, demand orientation, profit maximization, and environmental concern. We are concerned with all these objectives except profit maximization. They also emphasize that only three of the 45 papers that were reviewed specifically formulated environmental objectives or goals.
2. Related literature The problem then is that of facility location on a network (of roads). In order to develop an overall broad perspective on the facility location literature so that we can appropriately model our specific problem we present below a review of some major work that is related to ours. 2.1. The p-Center Problem Hakimi (1964) considered the general problem of locating p facilities on a network to minimize either the sum of distances (the p-median problem) or the maximum distance (the PCP) between facilities and demand points on a network. An example of the PCP he mentioned was the location of a police station P that serves a number of communities of different sizes that are inter-connected by a highway system so that the maximum distance from P is minimum. In doing so he generalized the concept of the ‘center’ of a graph to the ‘absolute center’. Consider a n-vertex weighted
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graph G where dðx; yÞ represents the length of the shortest path between points x; y. If vi represents vertex i then the absolute center of G can be defined as a point xo on G such that for every point x on G; max1 6 i 6 n dðvi ; xo Þ 6 max1 6 i 6 n dðvi ; xÞ. He found the absolute center by decomposing the problem into finding an absolute center for each branch of the graph G and then selecting the minimum of these. A seminal result in Hakimi’s paper was that for the p-median problem the set of all vertices forms a finite dominating set, a finite set of points to which the optimal solution must belong. This does not hold true for the PCP. Garfinkel and Hooker (1991) present an overview of the identification process of finite dominating sets in network location problems. 2.2. Maximal Covering Location Problems This model is useful for examining our problem from the local traveler’s perspective. If we factor in population as a criterion for deciding coverage, and not insist on covering all demand points, then we have the MCLP of Church et al. (1974). The goal here is to locate p facilities within distance S so as to cover the maximum population within S. They used a linear programming relaxation supplemented by occasional use of branch and bound to solve the MCLP. They also proposed several heuristics for this problem. Pirkul and Schilling (1991) generalized this idea for fire protection in which they accounted for coverage by two different types of service (engines and trucks) and both the servers and the facilities that housed these servers were to be sited simultaneously. Researchers in this stream subsequently built models for additional coverage, for example the original designated server may not be available at the time of call, and also considered probabilistic models that considered randomness in vehicle availability. Chung et al. (1983) and Current and Storbeck (1988) relaxed the underlying assumption in MCLP that the facilities being sited are uncapacitated to formulate the C-MCLP or the capacitated MCLP. Datta and Bandopadhyay (1993) apply the MCLP in order to locate primary health care centers in two Indian districts.
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2.3. The Maximum Covering/Shortest Path Problem (MCSPP) Considering our dual objectives of designing a network and locating facilities so as to (a) cover the maximum population while at the same time (b) minimize the total traveling costs of the population, the MCSPP as described by Current et al. (1985) serves as our natural base model. Their work started the literature on multiobjective network design and routing problems. Until then most formulations were single objective in nature. They extend the concept of coverage from the analysis of facility location to the analysis of network design and routing. Given a network G ¼ ðV ; EÞ, the first objective is to identify the shortest (or least costly) path through a network from a predetermined starting node, O, to a predetermined terminus node, D. The second, and often conflicting, objective is to maximize the total demand satisfied by the path for a known demand ak , (say the population) at each node k on the network. Also, the demand at a node, say k, is satisfied if the path enters node k directly, or if it enters some other node, say j, such that node j is within a pre-determined covering distance, S, from node k. It would be expected that the total path length (or cost) must increase to cover more demand. Therefore, the objective of minimizing the total path length from a starting node to a terminal node, and that of maximizing total demand covered by the path, are generally in conflict. Hence the concept of a single objective function is no longer pertinent. Instead the concept of non-inferiority (Pareto Optimality) is utilized to generate a non-inferior solution set from which the decision maker must choose depending on his or her utility function of various combinations of path length and coverage. Our research relaxes the constraint of having a single predetermined source and terminus node in Current et al.’s formulation. Instead we now connect a subset of the nodes to each other by paths, thereby making travel possible amongst them (cf. Current et al., 1985). The subset could represent a list of important demand centers or cities amongst which the decision maker choose to make travel feasible using environmentally friendly fuel. Thus
we extend the MCSPP to enable multiple source and sinks.
3. The model The formulation presented below focuses on locating unleaded gas stations to make inter-city travel possible among large, populated cities, assuming that such facilities already exist within each of these large cities. It requires only an existing network of roads and populations of demand centers as data, and as such could be used in an intra-city setting too with large neighborhoods representing demand centers. This would enable decision-makers in converting the cities themselves to the use of environmentally friendly fuel. The following sections present the formulation for inter-connecting the cities that lie on the existing road network, simultaneously considering the two conflicting objectives of minimizing cost and maximizing coverage. The problem is formulated as a Maximum Covering/Shortest Spanning Subgraph Problem, hereafter referred to as the MC3SP. 3.1. Notation and system parameters Consider a directed graph G ¼ ðV ; EÞ which consists of a finite set, V ¼ f1; 2; . . . ; ng, where N is the maximum number of nodes representing population centers, and a finite set E NXN , of m arcs representing the existing network of roads that connect these population centers. Let each arc be denoted by an ordered pair ði; jÞ where i 2 V and j 2 V , and let ak be the demand (or population) at node k for all k 2 V . Let Rkl represent the set of paths from node k to node l. At least one of these paths must be in the solution for every inter-city pair. Let pi;j P 0, be the population covered by each arc ði; jÞ 2 E. pi;j is the sum of the populations of nodes i; j, and of all other nodes which lie within distance S, the covering distance, from any point on the arc. For instance in Fig. 3 node l is not connected to arc ði; jÞ but is within a short distance from it and node k is connected by a small road to it. Thus we assume that populations in nodes like l
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Fig. 3. Coverage determination.
and k will be able to travel onto the highway. Hence pi;j ¼ ai þ aj þ ak þ al . Let ci;j be the cost of enabling arc ði; jÞ. By enabling an arc we mean locating the required number of fuel stations on that arc so that it can be used for travel. The following three parameters influence ci;j : (a) di;j the non-negative distance associated with each arc ði; jÞ 2 E. We assume di;j ¼ dj;i for all i; j 2 E. (b) fi;j , the fixed cost associated with enabling the arc connecting node i and j for travel purposes. fi;j in turn is dependent upon the following: • Number of stations that are needed to cover the arc, which in turn is proportional to the length of the arc di;j . Assuming nodes i and j themselves have unleaded gasoline facilities then, mathematically, we need at most ððdi;j 1Þ div DÞ where the div operator returns the quotient of the division. For example if D ¼ 250 then we need 1 station to cover a 500 mile road and 2 stations to cover a 750 mile road. • Locations of the existing facilities. Thus on a 1000 mile road if we have existing facilities at mile 200 and mile 800 then we need new facilities for only the 600 remaining miles i.e., 2 facilities. Of course we still would have to incur a lower fixed cost for converting the existing facilities to cater to unleaded fuel. • Fixed and operating costs like cost of land, building, equipment, relevant licenses, and
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operator’s salaries. These can be discounted at prevailing interest rates so as to compute their present values assuming a 25-year horizon as in Swersey and Thakur (1995). (c) gi;j , the variable traveling cost of the population associated with enabling the arc connecting node i and j. An estimate of such a cost can be made if the decision makers have access to the following information: • the average number of trips per person, ti;j , made on arc ði; jÞ per planning period, and • the average dollar cost of traveling a mile, say q. Then gi;j ¼ qti;j , will give us the dollar amount spent by the traveling public per planing period. Let VO V denote the set of origin nodes, from which a path must emanate, and VD V denote the set of destination nodes into which a path must terminate. Typically these would be large cities with high populations that decision-makers would wish to connect. In most situations we would like to connect all the cities on the network, in which case the sets VO ; VD and V would be identical. However we could have a network in which some cities act only as transit points and these do not have to be a part of the spanning subgraph in the end. Thus we can denote the set VT ¼ V VO VD , as the set of nodes that are only for transit purpose. Let Ni ¼ fj j ði; jÞ 2 Eg be the set of nodes adjacent to node i that are connected by outgoing arcs emanating from i. Likewise let Mj ¼ fi j ði; jÞ 2 Eg be the set of nodes adjacent to node j that are connected by incoming arcs from the various i. 3.2. Decision variables Let 1 Xi;j ¼ 0
if arc ði; jÞ is on the path; if otherwise:
3.3. Base model formulation Depending on the amount of information available to the decision maker the cost coefficient ci;j can be operationalized in different ways. If we represent ci;j by just the length of the arc di;j and
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assume that we have no transit nodes, that is, VT is empty and VO ¼ VD ¼ V we have the following formulation for MC3SP(1). n XX pi;j Xi;j Maximize Z~ ¼ Wk i2V
j¼2
ð1 Wk Þ
n XX i2V
subject to X XO;j P 1
ci;j Xi;j
ð1Þ
j¼2
ð8O 2 V Þ;
ð2Þ
ð8D 2 VÞ;
ð3Þ
j2NO
X
Xi;D P 1
i2MD
Xi;j þ Xj;i 6 1 ð8ði; jÞ 2 EÞ; jRkl j P 1 ð8k; l 2 V Þ;
ð4Þ ð5Þ
Xi;j 2 ð0; 1Þ ð8ði; jÞ 2 EÞ:
ð6Þ
Eq. (1) represents the weighted multiple objective function. Constraint (2) ensures that for each of the originating nodes belonging to the set V there is at least one outgoing arc that emanates or in other words that is enabled. This distinguishes it from the typical single source single destination MCSPP. Likewise, constraint (3) ensures that for each element of the set of destination nodes there is at least one incoming arc that is enabled. In this base model we deliberately make Xi;j 6¼ Xj;i for all ði; jÞ in E (constraint (4)) thereby forcing the model to look ahead at every node j and enable a different outgoing arc, say Xj;k rather than enabling the reverse directed equivalent counterpart Xj;i . Constraint (5) ensures that at least one of possible paths is enabled for every inter-city pair and is designed to make sure that the resultant subgraph actually spans the network thereby connecting all the significant cities to one another. Finally, constraint (6) ensures that Xi;j are 0,1 variables. To deal with our multiobjective problem we generate an approximation of the non-inferior set by combining the two objective functions by means of appropriately scaled weights Wk . This is recommended because it does not alter the structure of the constraint set (ReVelle, 1989). Thus the general form of (1) is Maximize
ZðWk Þ ¼ Wk Z1 ð1 Wk ÞZ2 :
ð1AÞ
Regarding the directed nature of our graph, even though our arcs representing highways are inherently undirected, we represent them as two directional arcs ði; jÞ and ðj; iÞ for modeling purposes. We would assume that any time either one is enabled its reverse counterpart is also enabled. However in this base model we deliberately make Xi;j 6¼ Xj;i for all ði; jÞ in E (constraint (4)). The benefit of such an approach is that it increases the overall connectivity of the model, i.e. it reduces the disjointedness of the solution obtained. This will be further illustrated in the ‘extensions section’ when we compare MC3SP(1) with another of its variations MC3SP(2) wherein we actually enforce Xi;j ¼ Xj;i for all ði; jÞ in E. It should be noted that we assume that each node has at least two arcs incident on it. Such an assumption almost always holds true when nodes represent large cities. If this assumption is violated by some extremely isolated node to which only a single road is incident, then we have to relax constraint (4) for the arc that connects such a city to the rest of the network. In reality for constraint (5) we would need an exponential number of constraints, dependent on the number of nodes, but instead we use the underlying nature of the problem to relax this constraint and deal with the issue if and when it arises. Because our approach is inherently optimistic we assume that fair amount of strong connectivity will arise out of constraints (2)–(4) for a given value of Wk . However as Wk becomes close to zero, Z2 , that is the cost, dominates Z1 and we get more and more disjoint cycles and (2)–(4) alone cannot guarantee connectivity. We resolve this constraint violation by means of a heuristic approach outlined in the following section.
4. Heuristic solution procedure We modify Kruskal’s classical algorithm for obtaining a minimum weight spanning tree as the basis for obtaining connectivity in our MC3SP (cf. Kruskal, 1956). We start by relaxing constraint (5), the connectivity constraint and solving the remaining MC3SP to obtain Z~, an intermediate solution to MC3SP(1) in which (5) is violated.
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Fig. 4. Solution procedure flowchart.
Fig. 4 pictorially depicts the steps of our solution procedure. Step 0: We define a set LIST containing the remaining arcs ði; jÞ not in Z~. For each element in LIST we compute a weight vij ¼ pij =cij for the arc representing the ratio of its population covered and the cost of enabling the arc. Thus vij gives us the population covered per unit cost. We sort LIST in non-increasing order of vij . 0 Step 1: Now to Z~ we add the arc ði; jÞ with the 0 highest weight vij and run a breadth-first forward search algorithm (Ahuja et al., 1993) with any arbitrary node as the source node that determines whether or not there exist directed paths from the source node to every other node. If yes, then we run a similar backward search algorithm to de-
termine whether there exist directed paths from every other node to the source node. If yes, then we have a strongly connected graph and we can terminate, else we go to Step 2. Step 2: If LIST is not empty, then we remove ði; jÞ0 from LIST and return to Step 1. Else, we conclude that the problem is infeasible and hence cannot be strongly connected. Note that both the forward and backward search algorithms have a worst case complexity of the order OðmÞ and hence our heuristic could at worst examine all the possible arcs and have a worst case complexity of Oð2mÞ. Regarding the quality of the heuristic solution procedure in all but one of the twelve experiments carried out (discussed later) the IP solution, Z~, was
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indeed connected and hence we did not have the necessity to apply the modified Kruskal’s MST procedure.
5. Numerical example To illustrate the applicability of the MC3SP model we consider the following network (Fig. 5). Assume that we have no transit nodes, that is the VT is empty and VO ¼ VD ¼ V . Table 1 gives (in thousands) the population covered by the arcs ði; jÞ and the cost of enabling these arcs which is assumed to be equal to the length of these arcs. Let us first examine the case where MC3SP is applied to the above data with different values of Wk . Fig. 6 shows the results obtained prior to applying the heuristic described in Section 4 for ensuring connectivity. Clearly the heuristic has to be applied in the case when Wk ¼ 0 and it works out that there are three candidate arcs (2,4), (3,4) or (3,5) that could
connect the two disjoint cycles. Their respective weights are 2.7, 2.84, and 0.79. Hence X34 is set to 1 to get the revised Zð0Þ ¼ 617. The dotted line indicates the application of the heuristic. One must be reminded that even though Fig. 6 displays only uni-directional arcs in reality both Xi;j and Xj;i will be enabled as indicated by the respective Z values. The objective function tradeoffs of the four non-inferior solutions to this problem are shown in Fig. 7. The two extreme points Zð0Þ and Zð1Þ are generated by solving the problem separately with the individual objective functions or alternatively making Wk in ð1AÞ ¼ 1 and 0, respectively. Non-inferior solutions Zð0:33Þ, Zð0:5Þ were obtained by maximizing weighted combinations of Z1 and Z2 with Wk ¼ 0:33, 0.5, respectively. Thus the decision-maker can see how much he or she has to sacrifice one of the objectives in order to obtain improvement in the other. It is clear that the Zð0Þ and Zð1Þ represent extreme scenarios and Zð0:33Þ; Zð0:5Þ represent compromise solutions amongst which the decision maker may choose depending on his or her utility function. 5.1. Model evaluation
Fig. 5. Illustrative network.
Table 1 Cost and population covered by arcs Xij
Cij
Pij
Xij
Cij
Pij
X12 X13 X21 X23 X24 X31 X32 X34 X35
100 150 100 50 75 150 50 75 150
140 99 140 137 205 99 137 213 119
X42 X43 X45 X46 X53 X54 X56 X64 X65
75 75 150 150 150 150 150 150 150
205 213 189 109 119 189 63 109 63
We compare our model with two alternative approaches. Firstly, a maximum coverage/unitcost weighted spanning tree can be used to come up with a connected graph that ensures inter-city travel. Additionally, we tested our solution procedure with another heuristic approach which involved the repeated application of the integer program described in Section 3 without constraint (5). After each application we aggregate the population covered by each disjoint sub-graph that is enabled and treat it as a single node. The internode distances are simply the shortest connections between any two points of such aggregated nodes. This we call the Repeated IP heuristic. Fig. 8 compares the minimum cost spanning tree for our network with the solution derived from using MC3SP with Wk ¼ 0:5. It is clear that although the spanning tree does connect every inter-city pair by a unique path it does very poorly in terms of coverage. That is the reason we do not restrict ourselves by searching for a spanning tree to begin with and
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Fig. 6. Solution procedure applied with different values of weightage, Wk .
Fig. 7. Non-inferior solutions.
then fixing it by means of some heuristic. Rather, cycles are welcome as long as they contribute to coverage at a reasonable cost. In addition if we examine the spanning tree below for the traveler going from node 6 to node 5 we observe that the path is 365 units long as opposed to 150 obtained by MC3SP.
Table 2, compares the various solution procedures viz. three parameters. They are: • population covered per unit cost; • aggregated extra commuting dollars in comparison with the all-pairs shortest path solution, assuming 141 trips per planning horizon – which
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Fig. 8. Spanning tree approach vs. MC3SP.
Table 2 Comparison of solution approaches
lation values are reliably available. The following equation resulted from the regression:
Solution methodology
Pop. covered/ unit cost
Extra commuting expense
MC3SP Spanning tree Repeated IP heuristic
1.3 1.2 0.9
$4388 $17,552 n.a.
is the average population covered by the arcs – and a standard commuting cost of $0.31 per mile. Thus it is evident that the MC3SP performs significantly better in both counts. The reduced extra commuting distance parameter has important implications with respect to the corresponding reduction in vehicular emissions, in addition to the readily computable dollar value listed above.
6. Application to the Indian highway network We were able to obtain data regarding the automobile populations in 22 major cities in India. Subsequently we examined the national highway network and derived a network that connected these cities. The resultant network had approximately 30 cities all of which were in close proximity to the highway. In order to obtain the coverage attribute of each arc we estimated the automobile population of those cities for which we did not have the data. This was done using simple linear regression for the 22 cities for which data was available with the automobile population as the independent variable and the actual population as the dependent variable, since actual popu-
Auto: Population ¼ 229431 þ 0:14 Actual population: Further we assumed that the cost attribute for each arc can be represented by the length of the arc. This was estimated using the actual published driving distance between the cities on our network. Fig. 9(a) shows a map of India and Fig. 9(b) shows the entire network representation of the problem under consideration. Fig. 9(b) includes all major economic and population centers. Fig. 10 shows the resultant spanning subgraph for two extreme cases where cost and coverage dominate. For lack of reliable inter-city driving distances for a few cities, the nodes that are not filled represent towns and cities that are in the formulation only for the coverage they provide. The model does not attempt to connect them. Fig. 11 displays the resultant network for different values of Wk . In case of a compromise solution where Wk ¼ 0:75 a traveler from Mangalore to Mumbai will have to go via Hubli and Belgaum, rather then take the direct route via Panaji. Finally, Fig. 12 plots the efficient frontier of the non-inferior solutions which the decision makers in India can use to maximize their total utility balancing the cost and coverage factors. 6.1. Analysis of results In order to provide a point of reference to our approach we carried out extensive simulations to
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Fig. 9. (a) India map. (b) Indian national highway network.
Fig. 10. Spanning subgraph when cost dominates (left), and when coverage dominates (right).
analyze how it compares against another major approach that a decision maker is most likely to consider, namely the maximum weighted spanning tree approach. Fig. 13 demonstrates the application of the maximum weighted spanning tree approach, where to model both the factors of cost
and coverage, we consider population covered per unit cost and construct a maximum weight spanning tree. The data set used here is the same as the representation of the Ohio network by Current et al. (1988). It should be observed that the solution of this approach is still worse off in terms of
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Fig. 11. Resultant sub-graphs (clockwise from top left) for Wk ¼ 0:25, 0.5 and 0.75.
Fig. 12. Efficient frontier for India data.
inter-city travel distances when compared to a typical ðWk ¼ 0:5Þ non-inferior solution generated using the MC3SP heuristic. For instance consider the case of the traveler going from node 14 to 15. In both cases the most direct route is not enabled but the MC3SP solution is far better than the spanning tree.
Table 3 shows the results obtained from a number of randomly generated road networks with Wk ¼ 0:25, 0.5 and 0.75, respectively. The important parameter that were varied was problem size viz. number of nodes and arcs. Cardinality of each vertex was randomly drawn between [1,5]. The integer program was solved
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Fig. 13. Spanning tree vs. MC3SP for Ohio data set.
Table 3 Simulated road networks, a comparison Problem size (nodes, arcs)
20, 104 40, 194 60, 358 80, 450 100, 556 120, 716 140, 868 160, 932 180, 984 200, 1108
Max wt spanning tree
Wk ¼ 0:25
Wk ¼ 0:5
Wk ¼ 0:75
Coverage
Cost
Coverage
Cost
Coverage
Cost
Coverage
Cost
1418 2714 3651 5080 6791 8123 9081 10,492 12,675 12,349
771 1201 1482 1966 2800 2947 3206 4245 4678 5132
10,932 3672 4232 4456 3576 3420 3528 3624 2768 2916
2937 679 345 271 541 271 289 351 271 444
6344 7352 9536 8760 5872 8064 7000 7740 7176 8356
2159 1330 1220 1009 912 1178 843 1026 942 1313
3812 10,496 11,420 11,096 9220 10,228 10,152 11,308 9728 10,472
1331 2507 1918 2040 2024 1941 2029 2364 1875 2064
using MS-Excel’s premium solver for large scale problems. Table 3 presents a typical Markowitz style riskreward trade-off scenario. Take the case of a typical problem (shaded row 7) with 100 nodes and 556 arcs. The spanning tree solution presents a low-risk low-reward kind of scenario with a coverage of 6791 at a cost of $2800. For approximately the same cost the Wk ¼ 0:75 solution gives a much higher coverage. Thus the decision-maker has to fit their utility curve along the efficient frontier provided. It should be re-emphasized that the concept of a single optimal solution is not relevant in problems of this nature and hence an approximation of the efficient frontier is provided with Wk values ranging from 0.25 to 0.75. Interestingly, as Wk increases the population covered
per unit cost decreases; a result that draws our attention to the following nuance. As the model tries to cover a larger population it starts including a greater number of arcs thus while the total coverage goes up, the ratio of coverage to cost goes down, and the solution quality suffers. This has to do with the density of population in the area being covered. Additional coverage in high density are as would be obtained with little additional cost, whereas additional coverage in sparsely populated areas would be very expensive. As problem size increases it is observed in the application of the MC3SP heuristic that connectivity becomes increasingly pervasive thereby giving us optimal solutions at the first stage of the solution procedure itself, i.e., the stage where Z~ is computed as described in Section 4. Lastly Table 4 presents a
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Table 4 Application to real data Problem size (nodes, arcs)
Max wt spanning tree
Wk ¼ 0:25
Coverage
Cost
Coverage
Cost
Coverage
Cost
Coverage
Cost
Ohio (17, 58) India (30, 86)
2456 25,033
498 8523
13,390 26,165
2816 9927
13,850 32,149
2839 14,065
18,660 33,855
3239 16,628
similar comparison for the two real world data sets we used, namely the Ohio data and the India data sets.
7. Extensions In this section we examine different extensions to our base model described in Section 3.1. Another operationalization of the cost coefficient in Z2 in (1A) could be if we just consider fi;j , the fixed cost as described in Section 3.1. This would take into account parameters like the locations of existing facilities, fixed costs like cost of land, building, equipment, relevant licenses, and operator’s salaries. However, because length of the arcs would be modeled only indirectly by the number of new facilities needed we could end up with a solution that would be bad for the traveling public from the inter-city traveling distance perspective. Thus MC3SP(1) would have Z2 ¼
X
n X
i2V
j¼2
fi;j Xi;j :
A third, and richer operationalization of Z2 would consider the joint effects of the fixed cost parameter fi;j and the traveling population’s perspective as represented by gi;j; , the variable traveling cost as defined in Section 3.1. Thus MC3SP(1) would have Z2 ¼
n XX i2V
ðfi;j þ gi;j ÞXi;j :
j¼2
Another source of variation in the formulation can come from constraints set ((2)–(4)). Assuming that we have no transit nodes, that is the VT is empty and VO ¼ VD ¼ V we can have the following formulation MC3SP(2) that exploits the essentially undirected nature of this problem. In this
Wk ¼ 0:5
Wk ¼ 0:75
formulation we drop constraint (4) and replace (3) by making Xi;j ¼ Xj;i for all ði; jÞ in E. In addition we have to adjust the coefficients in the objective functions for Xi;j by multiplying them with 0.5 so as to avoid double counting. Thus the resultant math program for MC3SP(2) is Maximize
Wk
n XX i2V
0:5pi;j Xi:j
j¼2
ð1 Wk Þ
n XX i2V
subject to X XO;j P 1
ð7Þ 0:5ci;j Xi:j
j¼2
ð8O 2 V Þ;
ð8Þ
j2NO
Xi;j ¼ Xj;i
ð8ði; jÞ 2 EÞ;
ð9Þ
constraints ð5Þ and ð6Þ: Thus constraint (9) actually makes the constraint (3) redundant because the moment we enable an outgoing arc say, Xk;l to satisfy (8) its incoming counterpart Xl;k would be enabled by constraint (9). We can implement (9) because in reality it is the same road and hence it would make sense to enable both the directions or none. However with such a formulation it becomes increasingly difficult to maintain connectivity. For any given node we now have some outgoing arcs in some incoming arcs. If all these arcs are very expensive, that is the cost is dominant over coverage then a constraint of the form Xi;j ¼ Xj;i for all ði; jÞ in E would lead to many disjoint cycles of the size 2. Thus Xi;j ¼ 1 would satisfy the outgoing constraint for i and the incoming constraint for j. It would also enable Xj;i which in turn would satisfy the outgoing constraint for j and the incoming constraint for i. The following section compares MC3SP with MC3SP(2) for our example.
R. Bapna et al. / European Journal of Operational Research 142 (2002) 480–496
495
Fig. 14. MC3SP with MC3SP(2).
7.1. MC3SP with MC3SP(2) Although the formulation for MC3SP(2) appears to be more intuitive in terms of the undirected nature of our problem the following illustration shows the problem of too many disjoint cycles when Wk is low, i.e. close to 0. Fig. 14 shows the resultant subgraphs obtained from the two models. 7.2. Comprehensive model In the general case when we have transit nodes, that is VT is not empty and the utility function of the decision maker is not strictly the weighted combination of the two individual objective functions then we have the following representation for the general maximum covering/spanning subgraph problem. Maximize Z ¼ U ðZ1 ðX^ Þ; Z2 ðX^ ÞÞ ð10Þ subject to X XO;j P 1
ð8O 2 VO Þ;
ð11Þ
ð8D 2 VD Þ;
ð12Þ
j2NO
X
Xi;D P 1
i2MD
X
Xi;j
i2Mj
X
Xj;k ¼ 0
ð8j 2 VT Þ;
k2Nj
constraints ð5Þ and ð6Þ; P Pn where Z1 ¼ i2V j¼2 pi;j Xi;j , and Z2 ¼
n XX i2V
j¼2
ci;j Xi;j :
ð13Þ
Henig (1985) presents a good discussion of generating the non-dominated paths for a single source single destination problem under different types of utility functions. He suggests that an improvement in efficiency can be obtained when quasi-concave or quasi-convex utility functions are assumed. Constraints (11) and (12) are similar to (2) and (3), respectively. Constraint (13) ensures that for all the remaining transit nodes (non-source, nondestination) if any incoming arc is enabled then a corresponding outgoing arc must be enabled so as to form a contiguous path.
8. Concluding remarks and further research issues A solution methodology is proposed to solve the important problem of objectively locating facilities for developing countries trying to promote the use of environmentally friendly fuel like unleaded gasoline. The methodology is applied to data sets obtained from Current et al. (1988) representing the Ohio state limited access highway network, and to the Indian national highway network. Additionally, extensive simulations are carried out in order to compare our approach with the maximum weighted spanning tree approach. This work extends the MCSPP formulated by Current et al. (1985) to accommodate multiple sources and destinations. Care is taken that model and subsequent solution procedures are driven by real aspects of the problem as described by decision makers. The results obtained apply as well to
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developed countries which are looking to promote the use of automobiles using natural gas or electric batteries. As problem size increases it would be interesting to decompose a large country wide problem into smaller problems say on the basis of regional or state boundaries. Subsequently it would be interesting to examine a suitable heuristic that would help us merge the smaller problems together and give us some idea about the distance of the merged solution from the optimal solution. This approach would be natural for countries that do not rely solely on centralized decision making, and where states have their own variations in matters of public policy. Also the present model does not consider the transportation cost of the fuel from the refineries. Consideration of transportation costs would bring in elements of warehouse location problems.
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Church, R., ReVelle, C., 1974. White and Case. The Maximal Covering Location Problem. Papers of Regional Science Association. Current, J., Storbeck, J., 1988. Capacitated covering models. Environment and Planning 15, 153–164. Current, J.R., ReVelle, C.S., Cohon, J.L., 1985. The maximum covering/shortest path problem: A multiobjective network design and routing formulation. European Journal of Operational Research 21, 189–199. Current, J.R., ReVelle, C.S., Cohon, J.L., 1988. The minimum covering/shortest path problem. Decision Sciences 19, 490– 502. Datta, S., Bandopadhyay, R., 1993. An application of O.R. in micro-level planning in India. Computers and Operations Research 20 (2), 121–132. Garfinkel, R.S., Hooker, J.N., 1991. Finite dominating sets for network location problems. Operations Research 39 (Jan-Feb), 100–118. Gasping for Air, 1996. India Today. December 15 1996, pp. 38– 48. Hakimi, S.L., 1964. Optimal locations of switching centers in a communications network and some related graph theoretic problems. Operations Research 12, 450–459. Henig, M.I., 1985. The shortest path problem with two objective functions. European Journal of Operational Research 25, 281–291. Kruskal, J.B., 1956. On the shortest spanning tree of graph and the traveling salesman problem. Proceedings of the American Mathematical Society 7, 48–50. Pirkul, H., Schilling, D.A., 1991. The maximal covering location problem with capacities on total workload. Management Science 37 (2), 233–248. ReVelle, C., 1989. Review, extension and prediction in emergency service siting models. European Journal of Operational Research 40, 58–69. Swersey, A., Thakur, L.S., 1995. An integer programming model for locating vehicle emission testing stations. Management Science 41 (3), 496–512.
