Computers Opns Res. Vol. 17, No. 3, pp. 297-307, 1990 0305-OS48/90 83.00 + 0.M) Printed in Gnat Britain. All rights reserved Copyright 6 1990 Perpmon Press plc

THE EQUITY CONSTRAINED SHORTEST PATH PROBLEM

RAM GOPALAN,‘** RAJAN BA’ITA’*t*$ and MARK H. KARWAN~*$

‘Operations Research Center, Massachusetts Institute of Technology, Cambridge, MA 02139 and 2Department of Industrial Engineering, State University of New York at Buffalo, 342 Lawrence D. Bell Hall,

Buffalo, NY 14260, U.S.A.

(Receioed November 1988; revised July 1989)

Scope and Purpose-This article examines the problem of linding a shortest path on a network subject to “equity” constraints. Suggested applications for this equity constrained shortest path problem include routing of a vehicle carrying hazardous materials, routing a police car through a city, and planning a politician’s campaign tour. The equity constraints ensure that the route generated is fair to all the concerned parties. Of course, this fairness is achieved by sacrificing the quality of the route in terms of its cost. However, we find that a good degree of fairness can be achieved with an accompanying fairly modest increase in the route’s cost.

Abstract-This paper examines the problem of finding the shortest path on a network subject to “equity” constraints. A Lagrangean dual bounding approach is utilized, which relaxes the “complicating constraints” of the problem. After solving the Lagrangean dual, the duality gap is closed by finding the r shortest paths with respect to the Lagrangean function. Both looping and loopless paths are considered. A quick-and-dirty heuristic procedure is also suggested. We report a sampling of our computational experiences with the model.

1. INTRODUCTION

Equity is often an important issue in routing problems. Examples include the routing of a vehicle carrying hazardous materials, a police car patrolling neighborhoods in a city, and a politician planning a campaign tour through their constituency. For the hazardous materials vehicle, the objective might be to transport waste on the shortest risk path-the path on which the risk of exposure to the population is minimized, where risk is measured by an appropriate quantity, like the expected number of fatalities, for example-subject to constraints on equitably distributing the risk among zones of the transportation network. The police car may wish to maximize the coverage received by citizens while equitably distributing its time among the communities for which it is responsible. On the other hand, a politician might want to maximize his/her voting potential in a geographical area while simultaneously maintaining equity in the time that they spend among the cities and towns in their constituency. The equity constrained shortest path problem that we examine in this paper is motivated by these application areas.

Equity as a concept in problem solutions is not new. For example, Federal agencies are running into monolithic opposition whenever they attempt to locate a new hazardous materials dump-site in the vicinity of a particular community. Morrell [l J has proposed an innovative alternative to overcome this problem; namely siting numerous dump-sites in various counties simultaneously, with capacities in proportion to each county’s waste generation. Thus, no community feels “singled out” for the dubious distinction of accepting a dump-site in its neighborhood. Kenney [2,3]

*Ram GopaIan is a PhD candidate at the Operations Research Center at M.I.T. He has an MS degree from the Department of Industrial Engineering at SUNY at BuiTalo, and a B. Tech. degree from the Department of Mechanical Engineering at I.I.T. Madras, India. His interests include the applications of operations nsearch to problems in urban systems and production systems. This article is based upon his MS thesis, which he completed under the joint supervision of Prokurors Batta and Karwan.

t Rajan Ratta is an Assistant Professor in the Department of Industrial Engineering at SUNY at Buffalo. He has a PhD in operations research from M.I.T., and a B. Tech. degree in mechanical engineering from I.I.T. Delhi, India. His interests include the applications of operations research to problems in urban systems and production systems. He has published extensively on these topics.

$To whom correapondena should be addressed. #Mark H. Karwan is Professor and Chairman of the Department of Industrial Engineering at SUNY at ButYalo. He has

a PhD from the Department of Industrial and Systems En-g at Georgia Tech., and BBS and MSE degrees from the Mathematical Sciences Department at Johns Hopkins. His interests include multiple criteria decision making and mathematical programming and its applications in a variety of problem areas. He has published extensively on these topics.

298 RAM &PALAN et al.

discusses issues involved in the equitable distribution of risk among groups of individuals. He holds the global risk constant while making a comparison between different alternative distributions of this risk among individuals. A case for equitable solutions has also been made by Larson [4], though in the context of queuing. He contends that a lengthy, but orderly wait in a queue does not bother people as much as the occasional violation of the first-in-first-out discipline. Larson calls for amelioration of those attributes that capture “social justice”, in lieu of optimization of the traditional parameters that measure “goodness” of a queuing system.

The rest of this paper is’organized as follows. Section 2 presents two slightly different versions of the equity constrained shortest path problem. Sections 3 and 4 are devoted to the solution of each version of the problem. Section 5 describes a quick-and-dirty heuristic solution procedure. Section 6 provides a sampling of our computational experience. Finally, Section 7 contains a summary, conclusions, and directions for future research.

2. EQUITY CONSTRAINED SHORTEST PATH PROBLEM

We are supplied with an undirected transportation network defined by a node set N, INI = n, and a link set A. An origin (0) and a destination (D) are specified for the equity constrained shortest path problem. The network is divided into K mutually disjoint zones, labeled Z,, . . . , 2,. For each link (i, j)oA and ko{l,. . . , K}, Cij Z 0 represents the cost associated with travel on link (i, j), and x&i, j) z 0 represents the cost to zone Z, associated with travel on link (i, j) such that

A path is said to be equitable if the difference in costs between any two arbitrary zones is under a threshold p for travel on it. p is called the equity parameter. Later, we show how to obtain reasonable estimates for this parameter.

The objective of the equity constrained shortest path problem is to find the shortest equitable path from 0 to D. Mathematically, it is stated as follows:

P) MIN C C CijXdj i j

ST: (I) c c bz,(i, i) - Q9 i)Kj G P forall a,bE{l,...,K); f I

1 ifi=O,

(II) C Xi, - c Xii = 0 ifi # 0,i # D, I I -1 if i=D;

(III) {Xl 1 Xij 2 0 and integer} OR Xij ( Xij = 0 or 1,x Xii < 1 . I

We note that, strictly speaking, sub-tour elimination constraints should be added to prevent solutions that include a path from 0 to D and loops outside this path.

Xi, is a decision variable that represents the number of times that link (i, j) is used during travel from 0 to D. The problem formulation is classically alluded to in the literature as the constrained shortest path problem-see Handler and Zang [SJ.

Constraint set (I) has to be repeated for every node pair ordering (Z,, Z,) rather than just some selection of a pair of zones. This is because we do not know Q priori which pair of zones is going to sustain greater cost for travel on an arbitrary path. Given K zones, there are (K’ - K) constraints of type (I). Constraint set (I) ensures that, for any path vector x = {Xi,}, the absolute value of the difference in cost between any two arbitrary zones is under the required threshold p. Any transportation planner should solve the problem for a range of equity parameter p, in order to pre-determine the “equity level” that should be enforced on transporters. We note here as an aside that the smallest p for which a feasible solution exists can be determined by solving the following

The equity constrained shortest path problem

mathematical programming problem:

MIN (6 -0)

ST: cc xz.(i, j)X, 2 0 foralla= 1,. . . , K;

$ + nz.(iy jKj G 6 foralla=l,...,K;

Constraint sets (II) and (III).

In the above mathematical program, we have equity as the only criterion, ignoring the total transportation cost. Also note that, in our original problem formulation, constraint set (II) requires that our solution vector x describe a path from 0 to D.

There are two fundamentally different ways by which we can achieve equity. One approach is to follow more or less the shortest cost path and occasionally deviate or “loop” in order to “spread” the cost among the zones. This corresponds to the constraint (Xii 1 Xij 3 0 and integer} in set (III). We refer to this as the “looping-path” problem. Another line of attack is to ensure equity within the framework of a loopless path. This corresponds to the constraint (X,l 1 X, = 0 or 1, &X, d 1) in (III). We refer to this problem as the “loopless-path” problem. We note that it may be sensible in some applications to consider looping paths (e.g. routing a police car through a city), while it may be more reasonable in other applications to restrict our attention to loopless paths (for example, routing a vehicle carrying hazardous materials).

3. THE LOOPING-PATH PROBLEM

Our objective here is to look for a path, that may contain one or more loops, that minimizes cost of travel while satisfying the equity constraints. This corresponds to (X, 1 Xii 2 0 and integer} in constraint set (III). Constraint sets (II) and (III) just correspond to finding a path, a relatively easy problem in the absence of constraint set (I). A strategy that immediately suggests itself is to relax constraint set (I) and form a less constrained optimization problem. This can be done by assigning Lagrange multipliers Uzezb to each constraint in constraint set (I) and forming a Lagrangean subproblem as follows:

MIN L(x, U) = C C CijXij + 2 2 uz,,z, C C (nz,(i9 j) - nzb(k j))&j - P x 1 j z. Zb i j >

ST: Constraint sets (II) and (III).

It is well known that Min,L(x, u) supplies a lower bound on the value of the actual objective value of problem (P), and that the Lagrangean dual D,, corresponding to Max,,, Min,L(x, u) supplies the best such lower bound. In our Lagrangean subproblem, Uzmzb is the Lagrange multiplier corresponding to the constraint for zone pair ordering (Z,,, 2,).

The Lagrangean objective function can also be viewed as follows:

We can use an efficient shortest path algorithm to find the solution to the subproblem-see, for example, Glover er al. [6]. For a given vector of multipliers u, p(‘&.&, UzaZb) is a constant. Hence, any given Lagrangean subproblem can be solved easily by finding a shortest path with modified link lengths

Cij = ( Cij + C C ~z.z,(~z.09 i) - +Ji, j)) - z. Zb >

One important pre-condition to such a tractable solution is that the modified link lengths satisfy

c;,>-0 for all (i, j)cA.

300 RAM GOPALAN et al.

The nz,(i, j)‘s are just numbers, and it is quite possible when computing an expression involving the difference of numbers that the answer turns out to be negative. Also, since looping paths are allowed, even if one Ch < 0 the shortest path value is unbounded since we can shoot up and down this link reducing our cost to as low as we want it to be. Hence, in solving the Langragean dual for the looping-path problem, we must restrict our multiplier search so that all modified link lengths satisfy the condition that they are nonnegative.

3.1. Subgradient search for optimal Lagrange multipliers

Since any optimal solution to the Lagrangean dual must have all modified link lengths nonnegative, it would make sense to constrain our subgradient search itself so as to ensure this condition. Viewing subgradient optimization as a linear relaxation applied to a system of inequalities [7,8] can help us in achieving this. Consider the system:

MIN cx

ST: Dx<e;

XCZE.

E represents a system that is easy to solve, for example the path and integrality constraints in the formulation of problem (P). Dx d e represents a system that complicates the solution procedure, such as the equity constraints in problem (P).

If there are a finite number of solutions obtainable from E (the number of simple paths from origin to destination is finite), say p = 1, . . . , L, we can write the dual as

V(D,) = Max Min [cxp + u(Dxfi - e)]. ub0 @=1,...,L

Let us suppose for the moment that we knew V(D,). Then applying linear relaxation [9, lo] to the following system of inequalities produces one common derivation of the subgradient search method:

cx@+u(Dx@-e)aV(D,,), fi=l,...,L;uaO.

To obtain our desired result, we simply add another constraint on u to the above system. Let Au be a matrix containing as its entires (nx,(i, i) - a,,& j)). Then nonnegative link lengths implies that C, + uA,~ > 0.

Since we do not know what Y(DJ is, we use an under-estimate z by solving L(x, u) with u = 0. As our under-estimate is approached, we increase it by a set amount-we have found that increasing it by 10% worked well in our computational experience.

Applying linear relaxation to the full system of constraints on u leads to the generalized subgradient algorithm stated below. Note that we alternately adjust u for nonnegativity and for the condition of nonnegative modified link lengths. When both have been satisfied we solve the next Lagrangean subproblem to choose an inequality of the form cxp + u(Dxfi -e) 3 z. That is, u is modified so as to increase cx@ + u(Dx@ - e), where x@ is the solution to our most recently solved subproblem.

3.2. Generalized subgradient algorithm

1. Set t= 1, ~‘-0, and V’= -co. t is the iteration counter, u’ is the Lagrangean vector at iteration t, and V’ is the incumbent at iteration t.

2. Solve the Lagrangean relaxation (F”), by solving a shortest path problem with modified link lengths that are given by:

Ct = C, + C C G,z,(~z.(iY j I- n& j N. z. Zb

Let an optimal solution to this problem be x’. If (Dx’- e) Q 0 and u’(Dx’ - e) = 0, stop. x’ solves the primal problem (P) and V(Pt) = V(D,) = V(P). V(P:) is the value of the Lagrangean dual at iteration t. V(D,) is the optimal objective function value of the dual problem. V(P) is the optimal objective function value of the primal problem.

3. If V(P:) > V’, set V(‘+l) = V(Pt) to obtain a new incumbent for Y(D,). Otherwise, set P(r+l) = J/r

The equity constrained shortest path problem

4. Find a new vector of multipliers by setting

%Cz - v(pt,)l C ClrZ,Ci9 j) - nZb(iv i)lxij- P #+ 1) = ut +

[

[ (i.jkA 1

C C [ C Cnz,K j) - nz,Ii, jIlxij - P-J ’ 2. Zb U,j)EA I

where z is an under-estimate for V(D,). 5. If Ug,$‘,) < 0, set U$,i’,) = 0. 6. Check if all modified link lengths satisfy

C:j = Cij + C C U j) - nZb(i, j)) 3 0. z, zb

If some C;j < 0, then set

#+l) = #+‘) _ ((Cij + u”+ “Aij)Aij) 1 IIAi,l\’ *

If any component of the vector u(‘+ l) is altered, go to 5. Else go to 2 after setting t = c + 1, provided that t does not exceed the maximum permissible number of iterations. If the maximum iterations allowed for is exceeded, proceed with the current dual solution to the stage of closing the duality gap, which is described below in the next sub-section.

Thus, by using the modified subgradient procedure, we can successfully solve the Lagrangean dual, at which point we have the “best” lower bound on the optimal objective function value. For most problems there will be a duality gap between the dual solution and the optimal objective of the primal. So the next consideration is elimination of this duality gap.

3.3. Closing the duality gap

After solving the Lagrangean dual, we have a lower bound dual solution as well as an upper bound, which is the best primal feasible solution obtained during the relaxation procedure (this may be + 00 in the event no feasible solution is located). We have to update these lower and upper bounds until we solve the problem to optimality. Handler and Zang [S) present a general procedure to close the duality gap for constrained shortest path problems. We use their procedure, which, informally stated, proceeds as follows:

1. Find the t shortest paths with respect to the Lagrangean function. 2. Update the upper bound whenever Dx G e and cx c (best upper bound available). 3. Continue this procedure until the best lower bound exceeds the best upper bound, in which

case the path associated with the best upper bound is the optimal solution.

We note that the dual does not have to be solved to optimality to undertake the above gap-closing procedure. In fact, we can undertake the above procedure for any u rather than u*, even u = 0. The latter degenerate case corresponds to finding the t shortest cost paths until we satisfy the equity constraints. As a final point, when finding the t shortest paths we are interested only in looping paths, and therefore need to use the algorithm described in Shier Ill].

4. THE LOOPLESS-PATH PROBLEM

The only difference to our problem formulation is that we now insist on (X,1 Xij = 0 or 1, &Xrl < 1) instead of {Xii 1 Xij >, 0 and integer). We can partially mimic the old solution procedure by forming a relaxed Lagrangean subproblem L(x, u) which can again be solved by finding a shortest path on our network with modified costs given by

C’;j = Cij + UAij.

We can claim again that the optimal value of the Lagrangean dual V(&) > 0, since t(x, 0) > 0 and V(D,) = Max,,, L(x, u). What we cannot claim now is that, in an optimal solution to the

302 RAM GOPALAN et ai.

dual, all modified link lengths Ch satisfy

C$=Cij+uA,aO.

This is because we are now looking for a loopless path, which implies that we cannot travel on a link (i, j) more than once. Thus it is entirely possible that an individual link weight is negative, yet the value of a path of which this link is a part could be positive. Therefore, two facts emerge, namely :

(i) We need not bother restricting the multiplier search to only those u’s that produce a nonnegative value of the modified link length C, + UAij.

(ii) After allowing for (Cii + uAj,) < 0, we should be able to find the shortest loopless path on the network.

In the absence of simple negative cycles, we can use either the dynamic programming approach proposed by Dial et al, [12] or the modified labeling approach suggested by Gopalan [13] to find the shortest simple path. Out computational experience with the modified labeling approach has been very satisfactory-the modified labeling approach is of polynomial complexity [13]. Dial et al. [12] indicate that their procedure is quite efficient from a computational standpoint. Therefore, it appears that either procedure-dynamic programming or modified labeling-can be used to find the shortest simple path with reasonable computational effort.

We enact the gap-closing procedure when the Lagrangean dual is solved to optimality, or when the number of subgradient iterations exceeds a set threshold, or when a simple negative cycle is found. Only now we need to use Yen’s algorithm [14], which finds the t shortest loopless paths on a network.

5. A QUICK-AND-DIRTY HEURISTIC

Let us assume for the moment that we are interested in the quick determination of a path that reasonably distributes the cost among the zones of a transportation network. Our line of attack is to assign to each link a weight that represents the link’s ability to equitably distribute cost among the zones. Then, by solving the shortest path problem on our network with these new weights, we might discover a path that had a “good” distribution of cost among the zones of the network. One such weight for a link is

~(i, j) = Max rrr,(i, j) - Min nzb(i, j), D b

which says, if the difference between the maximum cost to a zone and minimum cost to a zone for travel on a link (i, j) is small, then the link (i, j) spreads the cost “equitably”.

Let Y(P, Z,) = cu.l)rp a 2. (i, j) be the cost to zone 2, by travel on path p. We define the equity of path p as

C Max HP, Z,) - Min Y(P, 5) 9

11 b 1 and the most equitable path of the network as

Min C

Max r(p, Z,,) - Min y(p, Zb) .

P 0 b 1 Now consider some specific path pl. Let zone Z, sustain the maximum cost for travel on p’ and zone Zb the minimum cost. Then,

Y(P’, Z,,) G c Max Qi, j), (i.JkP’ C

and

Y(P’, zb) 2 C Min wzc(i, i), (i&p1 c

The equity constrained shortest path problem 303

which implies that

Y(P’, 2,) - Y(P’, ZtJ G c Max ~z,O, 1) - 2 Min ~f(i, j) W)cP' c (4iW c

cx [ Max n,(i, j) - Min n,Ji, j) U.ikr’ c e 1 G C rl(i, A. (i.lW

Hence, solving for the least cost path with the q-weights provides a “good” upper bound on the equity level of the “most equitable” path of the network. The heuristic clearly has polynomial complexity, since the q-weights can be computed in 0(n2) time, where n is the number of nodes, and from the fact that there exist a host of polynomial shortest path algorithms [6].

6. COMPUTATIONAL EXPERIENCE

We base our computational experience on a 50-node network derived from the city and county of Albany, New York. The county divides into 10 townships, as displayed in Fig. 1. These townships serve as the zones for the equity constrained shortest path problem. We obtain the link lengths C, and the zonal costs xz,(i, j) from the context of hazardous materials (HM) routing, one of the proposed application areas for the model. In the HM context, C, is the global risk due to travel of the vehicle carrying hazardous materials on a link (i, j), and x,(i, j) is the risk to zone Z,, due to the vehicle’s travel on link (i, j). We use the Abkowitz and Cheng [15] model to calculate the C, values. The xz,(i, j) values are got by assuming that the population is uniformly distributed within each zone. Various data sets are created by varying the O-D combinations, type of HM,

Fig. 1. W-Node, IO-zone transportation map of the county of Albany, New York.

304 RAM GOPALAN et al.

and the equity parameter. All algorithms are programmed in Fortran 77 on a VAX 11/780; computation times are quoted in CPU seconds.

We also introduce the concept of “equity profiles”. Given a path, an equity profile will tell us about the number of zone pairs that fall within a specific equity range. Thus, given a solution, we can count the number of zone pairs for which the difference in risk is say between 0 and 50. It is useful to develop equity profiles for solutions obtained from the heuristic and for paths that we come across when trying to solve infeasible problems. The equity profile is valuable for solution evaluation since it can tell us how the solution “distributes” the zone pairs among different equity ranges. In the case of infeasible problems, our equity profile can tell us exactly the number of zone pairs that are “falling out” of the required level of equity. Thus, given an infeasible solution for a problem with say 50 zone pairs, if only two zone pairs are infeasible to the required equity level, a transportation planner may still push the solution through.

Our experience with the looping-path problem has proved to be unsatisfactory in terms of our ability to obtain an optimal solution. Recall that closing the duality gap entails finding the t shortest looping paths with respect to the Langragean function. Unfortunately, the second shortest looping path is only the smallest possible “deviation” from the first shortest looping path. Hence, the lower bounds do not grow fast enough and it becomes very hard to close even small duality gaps. However, we are always able to generate “good” feasible solutions during the course of the gap-closing procedure in reasonable computation time. That is, a feasible solution is always quickly (within a few minutes) identified in the gap-closing procedure which is reasonably close (within 5%) to the current best lower bound. We therefore recommend the procedure discussed in this paper as a computationally efficient heuristic procedure, only for finding “good” solutions for the equity constrained looping-path problem.

In contrast, our experience with the loopless-path problem has proved to be quite satisfactory. Table 1 presents one representative data set. The Langrangean relaxation procedure works exceedingly well. For most feasible problems, the bound procedure discovers a value within 10% of the optimal solution, if not the optimal solution itself. The subgradient procedure is terminated if complementary slackness is satisfied or if there is less than 0.5% improvement in the bound value over the last 10 iterations. Thus, at least 11 subgradient iterations are necessary, unless complementary slackness is satisfied beforehand. Under these conditions, only 20-30 subgradient iterations are performed, on the average.

Oddly enough, as the equity level is made bigger (or the problem looser), the duality gaps seem to increase. We feel that this is due to the fact that the dual penalties are not strong enough to

Table 1. Results for the loopless-path problem, while using benzyl chloride hazardous material, with an assumed radius of spread of 1.5 miles

No. of shortest

paths solved Equity Risk

Dual gap W)

No. of subgr. iter.

Round time

Time to close gap

200 210 220 230 240 250 260 270 280 290 300 310 320 330

:: 1000

- - - - -

9716.4 3.37 716.4 3.12 716.4 4.35 716.4 5.15 716.4 1.59

t716.4 8.34 (0.33$) 716.4 8.89 (0.82f) 716.4 9.23 (1.09$) 716.4 9.39 (1.24$) 716.4 9.49 (1.33X) 716.4 9.53 (1.37:)

$648.02 0.0 648.02 0.0

- - -

50 50 50 50 9 9

12 17 44 50

:8 50 50 50 1

23 23 23 22 23 40 34 36 22 I8 14 11 11 11 11

122.1 116.25 115.74 109.6 117.8 206.66 175.9 189.44 114.46 89.6 69.6 54.4 54.0 54.3 54.3 6.6 6.5

908.23 934.89 926.05 905.92 152.76 157.85 211.32 314.69 819.59 903.53 897.56 897.5 892.59 901.4 907.91

1.73 I .7a

*Optimal node sequence is:l-2-9-6-17-21-22-28-30-50. tRepresents best feasible solution in R paths. $Represents duality gap after 50 shortest paths. §Optimal node sequence: I-2-3-4-5-49-13-35-36-37-38-44-24-26-50.

The equity constrained shortest path problem 305

Table 2. Protiles for infeasible problems encountered in solution of problems in Table I. These ~rotilcs arc described for the 5th. IO&,. . . ,5Oth shortest paths

Path No. O-50 50-100 100-150 150-200 200-250 250+

EQUITY LEVEL 200 5 28 0 1 2 6 8

10 28 0 1 3 5

:: 23 13 13 5 1 1 8 2 10 6 : 8

:: 28 28 0 0 1 1 2 3 6 5 : 35 24 4 1 2 6 40 16 9 3 7 10 : 45 28 0 1 3 5 8 50 21 1 2 1 6 8

EQUITY LEVEL 210 5 18 7 2 7 11 0

10 28 0 1 3 5 8 15 19 5 1 9 11 0 20 13 I3 1 8 10 0 25 24 4 1 2 6 8 30 16 9 3 7 10 0 35 28 0 1 2 6 8 40 25 3 1 2 6 8 45 16 10 2 7 10 0 50 24 4 1 2 6 8

EQUITY LEVEL 220 5 18 I 2 7 11 0

10 28 0 1 3 5 8 15 19 5 1 9 11 0 20 28 0 1 2 6 8 25 13 13 1 8 10 0 30 24 4 1 2 6 8 35 28 0 1 4 4 8 40 16 10 2 1 10 0 45 21 7 2 1 6 8 50 23 5 1 2 6 8

push the bound up. Our computational experience indicates that the gap increases in this fashion until, all of a sudden, it drops to zero when the problem becomes exceedingly easy.

Results that match intuition are:

The higher the duality gap, the more the number of shortest paths we need to close it. Times spent in bounding and closing the gap keep step with the number of subgradient iterations performed and shortest paths solved respectively. Equity is achieved by sacrificing the quality of the route in terms of its cost. However, our findings indicate that a sizeable amount of equity can be achieved with an accompanying modest increase in the route’s cost.

The general structure of the solutions obtained is as follows. For low levels of equity the problem is infeasible. We solve 50 shortest paths before declaring a problem infeasible, because around this point the paths we obtain are meandering and quite unacceptable as a solution. Then, at equity level 240, the problem becomes feasible with a risk value of 716.4. For a range of equity from 240 to 340, the same solution remains optimal. When we get to equity level 350, however, we obtain a better solution with a lower risk of 648.02. This solution remains optimal for equity levels above 350. This gradation of lower risk with looser equity is characteristic of almost all solutions. An examination of the equity profiles of infeasible problems (see Table 2) shows that there are always “problematic zone pairs” between whom it is very difficult to achieve equity. The identification of such zones is useful information to a transportation planner.

Sometimes, though feasible solutions to a problem can be discovered, it cannot be confirmed that these solutions are optimal by solving just 50 shortest paths. However, the lower bound value keeps increasing and, comfortingly, the final lower bound is within 2% of the best feasible solution at hand. See, for example, equity level 340 in Table 1.

The heuristic performs well (see Fig. 2) often discovering a path tenable both in terms of equity and risk. The heuristic solution is useful in indicating the range of the minimum achievable equity-240 is the minimum achievable equity, and the heuristic does not produce any zone pairs

RAM GOPALAN et al.

7 polrs of zones had tan absolute dlfhrence I” risk lncurr.?d between 100 and 150

Key:

Equity range

1. Optlmol heUrlStlC Sequence IS: l-2-9-17-21-22-26-30-50 2. Risk of heurlstlc path IS 716.14 3. Heurlstlc solution IS InfeasIble for ’ 150 equity

Fig. 2. Heuristic performance for the problem whose exact solution is displayed in Table 1.

in the 250+ range. As a final point, since the heuristic solves for the shortest path on a modified set of link lengths that always have positive length, it always generates a loopless path as its recommended solution. This loopless path solution can of course be used for a looping-path problem, as the set of loopless paths are a subset of the set of looping paths.

7. SUMMARY, CONCLUSIONS, AND FUTURE RESEARCH DIRECTIONS

The contents of this paper can be summarized as follows: in this paper, we have examined the shortest path problem with equity constraints. Applications for this equity constrained shortest path problem include routing of a vehicle carrying hazardous materials, routing a police car through a city, and planning a politician’s campaign tour. The equity constraints ensure a specified degree of fairness in the route between the various zones in the transportation network. Both looping and loopless paths were examined. For the looping-path problem, we developed a generalized subgradient algorithm to solve the Lagrangean dual obtained by relaxing the equity constraints, and proposed a procedure to close the resulting duality gap, if any, based upon finding the t shortest looping paths in a network. For the loopless-path problem, we also developed a subgradient algorithm to solve the resultant dual, and proposed a modified labeling procedure to close any remaining duality gap. In addition, a quick-and-dirty heuristic procedure was developed. Finally, we reported computational experiences based upon a hazardous materials routing scenario from Albany, New York.

The conclusions that we can draw from this research are as follows:

1. A reasonable degree of equity can be achieved with an accompanying fairly modest increase in the route’s cost.

2. The solution methodology proposed for the looping-path problem is efficient in terms of its ability to obtain good heuristic solutions within reasonable computation time, but it is not efficient in terms of its ability to obtain optimal solutions.

3. The solution methodology proposed for the loopless-path problem is efficient in terms of its ability to obtain optimal solutions.

4. The duality gap increases as the equity level is made larger, until, all of a sudden, it drops to zero when the problem becomes exceedingly easy.

5. The quick-and-dirty heuristic performs well, often discovering a path tenable both in terms of equity and risk.

An obvious and meaningful extension of this work is to consider the case where more than one route has to be taken, such that equity must be achieved at the end of the specified number of routes. Gopalan et al. [ 161 have taken a step in this direction by iteratively using the single-route

The equity constrained shortest path problem 301

procedure developed in this paper to solve the multip\e-route case. Another important extension is to examine other forms of the equity constraints. $e&fi&&y, the equity constraints in this paper are indifferent to differences in cost between zones that do not determine the extreme cost difference. For example, consider a 4-zone case in which zones 1, 2, 3, and 4 get costs of 100, 130, 170, and 200 units respectively, on a given route. An alternative route in which the costs to zones 1 and 4 remain unchanged but the costs to zones 2 and 3 are 150 units each, is clearly better in terms of equity but still has an equity level of 100 units, and hence our method of defining equity will-incorrectly-consider both routes to be equivalent in terms of their solution quality. The work of Mandell [17] may be helpful for this purpose.

Acknowledgements-The authors would like to thank two anonymous referees for their helpful comments on an earlier version of this paper. They would also like to thank the Department of Industrial Engineering at SUNY at Buffalo for providing a hospitable home for this research. This paper represents part of Research Initiative X6, “Spatial Decision Support Systems,” of the National Center for Geographic Information and Analysis, supported by a grant from the National Science Foundation (SES-88-10917); support by NSF is gratefully acknowledged.

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