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Minimum cost-reliability ratio path problem
Comput. Opns Res. Vol. 15, No. 1, pp. 83-89,1988
0305-0548/88 53.00+0.00
Printed in Great Britain. All rights reserved
MINIMUM
Copyright 0 1988 Pergamon Journals Ltd
COST-RELIABILITY
RATIO PATH PROBLEM
R. K. AHUJA* Sloan School of Management, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. (Received December 1985; reoisedJuly 1986)
Scope and Purpose-In the shortest path problem, thecost and reliability are two important criteria; we wish to minimize the cost and maximize reliability. The minimum cost-reliability ratio path problem attempts to achieve both simultaneously by yielding a path with low cost and high reliability. In the paper, we develop an algorithm for solving this problem. Computational results are presented which indicate that the algorithm can be used on the real-time basis. Abstract-The minimum ratio path problem in a network is known to be a strong NP-complete problem. However its variant, the minimum cost-reliability ratio path problem, is shown to possess an entirely different structure. We observe that the optimum solution of the minimum cost-reliability ratio path problem is an efficient extreme solution ofa bicriteria path problem. We employ parametric programming to enumerate these efficient extreme solutions and a sufficiency condition is used to cut down the enumeration substantially. The algorithm is shown to be pseudo-polynomial. Computational results of the algorithm on grid as well as random networks are also presented. The algorithm is able to solve problems with several thousand nodes and arcs in a few seconds.
INTRODUCTION
Let G = (N, A) be a directed network in which cij denotes the cost of traversing an arc (i, j) E A and 0 < rij < 1 denotes its reliability. The Ci, are assumed to be nonnegative integers. We consider the problem of finding a directed path P from a source node s to a sink node t for which
is minimum among all such paths. We refer to this problem as the Minimum Cost-Reliability Ratio Path Problem (MCRRPP). The minimum cost-reliability ratio path problems are likely to arise in practice when cost as well as reliability are the criteria to be considered by the decision maker. A least cost path may have poor reliability and the most reliable path may be too expensive to travel. The minimum cost to reliability ratio path problem attempts to incorporate both the criteria into a single objective by yielding a path with low cost and high reliability. Combinatorial optimization problems related to the MCRRPP have been studied and solved by several authors including Chandrasekaran [l], Dantzig et al. [2], Fox [3], Gustield [4], Lawler [S], and Meggido [6]. These approaches are not applicable to the minimum ratio path problem which has been shown to be a strong NP-complete problem by Ahuja et al. [7]. This rules out the possibility of the pseudo-polynomial algorithm unless P = NP [S]. The MCRRPP, however, is shown to possess an entirely different structure in this paper by developing a pseudo-polynomial algorithm. The approach adopted by us is similar to those of Chandrasekaran et al. [9] for the minimum costreliability ratio spanning tree problem, and of Aneja and Nair [lo] for the minimum ratio dynamic programs. In this paper, we show that the optimum solution ofthe MCRRPP is an efficient extreme point ofa b&it&a path problem. These paths are enumerated by the parametric analysis of the shortest path problem. We further establish sufficiency conditions which reduce the path enumeration *Dr R. IL Ahuja is an Assistant Professor in Industrial and Management Engineering Programme at Indian Institute of Technology, Kanpur. He is currently a visiting scientist at the Sloan School of Management, M.I.T., Cambridge. He holds a B.Tech., M.Tech. and Ph.D. from Indian Institute of Technology, Kanpur. His major research interests are networks flows, combinatorial optimization and linear programming. His papers have appeared in Naoal Research Logistics Quarterly, Operations Research Letters, European_Journal of Operational Research and others. 83
R. K. AHUIA
84
substantially. The computational
complexity of the algorithm is shown to be
where D = max{Clj}. Computational results of the algorithm on grid as well as (&&a random networks are presented. There are certain advantages ofthe approach adopted. The parametric analysis ofthe shortest path problem is fairly standard and requires reoptimization of the current tree of shortest paths. Further, the use of the tree data structures and binary heaps result in an extremely fast implementation of the algorithm. Indeed, the computational investigations indicate that the problems with several thousand nodes and arcs can be solved in a few seconds. O(mnD log m),
PROPERTIES
OF
THE
OPTIMUM
SOLUTION
Let S be the set of all directed paths in G. For each P E S define
(1) (2)
(3) where d, = -In rij. Note that dij 2 0, V (i, j) E A, and R(P) = eeD”‘, VP ES. The MCRRPP is then to Minimize z(P) = C(P)/R(P).
(4)
PES
This problem is related to the following bicriteria path problem: Min$ize
[C(P), D(P)].
(5)
By definition, a path P E S is an eflcient solution of (5) if and only if there does not exist any path P E S such that C(P) < C(P) and D(P) < D(F) with a strict inequality in at least one case. The bicriteria space of (5) may be defined as a plane in which each path PES is mapped into a point with coordinates (C(P), D(P)). Let H denote the convex hull of all the paths in the bicriteria space. An efficient solution of (5) is an efficient extreme solution if it is mapped into an extreme point of H. For simplicity of presentation, we use P and its image (C(P), D(P)) interchangeably.
The following theorem establishes a relationship between the MCRRPP and the bicriteria path problem. Theorem 1
An optimum solution P* of the MCRRPP is an efficient extreme point of (5). Proof Let P* not be an efficient extreme point of (5). Then P* can be represented as a convex combination of h paths P,, P,, . . ., P,, such that C(P*) = CF= 1 aiC(Pi), D(P*) = CF= 1 aiD( CL 1 ai = 1 and O
Vi=l,...,h.
(6)
Minimum cost-reliability ratio path problem
85
C(P)
Fig. 1
From the convexity of eeX it follows that h
,-~:=~a,W’i) < 1 i=
ai
,-WI).
1
(7)
Substituting (6) in (7) yields
e-x:=@(Pf)< 1
i i=
~iqpi)/z(pl).
1
(8)
Hence
which contradicts the optimality of P*. In view of Theorem 1, we can restrict our search for the optimum path P* among the efficient extreme points of (5). Let Pi, . . ., Pk, Pk+ I, . . ., P, be the list ofall the efficient extreme points of (5) in the increasing order of their D(P,) value. These paths can be enumerated by performing parametric analysis of the shortest path problem with arc lengths as d, + clcij and increasing ,Ufrom 0 to a large number [ 111. The parametric analysis also enumerates some of the boundary paths which can, of course, be ignored. It was observed during our computational investigations that, in most problem instances, z(P& first decreases and then keeps on increasing for increasing values of k. In such cases, the search can be terminated whenever z(P,.) starts increasing. But it cannot be guaranteed for all problem instances due to the nonlinear nature of the function z(P,.). Hence a more sophisticated termination criterion covering all problem instances is required. We now develop one such criterion. It is well known that the efficient frontier obtained by joining the points Pk_l to Pk for all k=2, . . . . w, is a piecewise linear convex function and typically is of the form as shown in Fig. 1. Let C,, = C(P,) and C,,,,, = C(P,). Further, let L, denote the line passing through Pk_l and Pk. The equation of L, is given by y = uk - bkx, where bk = (@PA - NP,- N(C(P,- A - C(Pd) and a, = D(P,J + b,C(P,.). For any point (x, y) EL,, define gk(x) = xeY= xe’k-bbx. It is easy to see that g&) is a unimodular function and achieves its maximum at x* = l/b,. Let xI = C(P,), V i = 1, . . ., w. Further, let zk* = min
{z(P,)},
l
and P* be the path for which this minimum is attained. Theorem 2
If m(C,J
2 zt, then P* is an optimum solution of the MCRRP’
R. K. AHUJA
86
Proof
Since the efficient frontier is a piecewise linear convex function, it follows that uk - &xi < D(P,), Vi=k+l,...,w.Accordingly gk(Xi)
= Xi
eak- brxi< C(P,) eocpJ = z(pJ,
Vi=k+l,...,w.
(9)
If xk < l/b,, then in view of the nature of gk(x)
zk*G
gk(Gin)
G gktXi)< z(pi),
Vi=k+l,...,w,
(IO)
and the theorem follows. Ifx, > l/b,, then let Xkc l/b, denote the x-coordinate ofthe unique point on L, for which g,&.) = gk(xk). Clearly
for all x, < Xi < Xk. For Xi < i,, (10) applies and the proof is complete. The condition stated in Theorem 2 can be used when the paths are enumerated in the order p,, P,, . . ., P,. However, these paths can also be enumerated in the order P,, P,_ 1, . . ., P,, P, by the parametric analysis. In that case, the following condition can be used as a termination condition. Let .?t = mink.iGw {Z(Pi)} and B* b e t h e path for which this minimum is attained. Theorem 3
If gk+ r(C,,,,,) a.?:, then P* is an optimum solution of the MCRRPP. Proof
The proof is similar to that of Theorem 2 and therefore omitted. THE
ALGORITHM
A formal description of the MCRRPP algorithm is given below. Step 1
Solve the shortest path problem with cij as arc lengths. Let Cminbe the length of the shortest path from s to t. Now, solve the shortest path problem with dij as arc lengths. Let Tl be the resulting tree of shortest paths and T1 = A - T. Compute the dual variables rrj and rri as follows: nf = 0
and
K; - 7~f=+
K!= 0
and
~4 -$
= dij,
V (i,j)E I’,, V (i, j) E Tl.
Set k = 1 and z* = M (a large number). Step 2
For each (i, j) E T,, define the relative cost coefficients as Cij= cij + nf - 4 and (7j = d, + nf - ni. Let F = {(i, j) E ?$: Cij< 0). If F is empty, then go to Step 4; otherwise identify an arc (p, q) E F for which -~JZ,,, = min(,,naF { -&,/Eij} and perform the pivot operation. The pivot operation is performed by adding (p, q) to Tk and dropping the unique basic arc (r, q) incident on node q. Dropping (p, q) from Tk forms two subtrees Ti and Ti, containing and not containing the nodes respectively. Update the dual variables $(rc‘$ by adding C,&&) to $(n$ for each j E Tz. Set k = k + 1. Let Tk be the new tree of shortest paths and Pk be the unique directed path in Tk from s to t. If d, > 0 and (p, q) is incident to the path Pk_ 1, then go to Step 3; otherwise repeat this step. Step 3
Compute z(PJ = C(PJ/D(P,). If z(Pk)
then set z* = z(P& and P* = Pk. If gk(Cmin)2 z*,
Minimumcost-reliabilityratio path problem
87
Step 4
The P* is an optimum path with z* as the objective function value. STOP. It may be pointed out that every iteration of Step 2 in the above algorithm does not yield a new efficient path. The basis of the shortest path problem is highly degenerate and consequently pivot operations may change the tree of shortest paths, but may not change the path from s to t. It can be easily seen that Pk from s to t changes if and only if the entering arc is incident on a node belonging to Pk. Also, to obtain the first efficient path, the algorithm selects arcs with zij = 0, if they exist, and performs the pivot operation. Theorem 4
The computational
complexity of the MCRRPP algorithm is O(mnD log m).
Proof Step 1 requires O(n’) operations. The value ofat least one A$reduces in each execution of Step 2 by at least one unit. Since nD is an upper bound on each $, Step 2 is performed atmost n2D times. The effort involved in updating A; and 7~4is also O(n’D) if thread indices [ 123 are used to store the basis tree. The arcs in the set F can be ranked in the nondecreasing order of -Jij/tij value and stored in a heap [13]. The size of the heap is atmost m and, accordingly, the selection of entering variable requires O(n2D log m) operations in all. At each iteration, the -iij/Zij value changes only for those arcs which lie in the cutset separating Ti and Ti. The algorithm therefore, examines arcs incident to and incident from the nodes in Tz, recomputes -~j.lC, and updates the heap. Since an arc (i,j) is examined only when either n; or I$ decreases, each arc is examined no more than 2nD times and consequently the operations needed to update the heap are bounded by O(mnD log m). Finally, Step 3 requires O(nD) computations. The overall complexity of the algorithm is, therefore, O(mnD log m).
COMPUTATIONAL RESULTS The MCRRPP algorithm was coded in FORTRAN-IV and tested on grid as well as random networks of different sizes. (The program listing can be obtained from the author.) The program was run on DEC 10 in multiprogramming environment. The main aim was to estimate the number of(i) pivot operations, (ii) efficient extreme paths enumerated by the algorithm, (iii) efficient extreme paths present in the network, and (iv) to assess the computational time requirement of the algorithm. The network was stored in the forward and reverse star representation. The algorithm was implemented using the augmented threaded index method [12]. The program required 9n + 8m storage out of which 3m was exclusively used to store the heap in a manner suitable for efficient updating. To obtain the shortest paths we used Pape, d’Esopo and Moore’s algorithm as given in Syslo, Deo and Kowalik [14]. A grid network of size p consists of p2 + 2 nodes and 3p2 - p arcs. The topological structure of the grid network for p = 3 is shown in Fig. 2. The C,jwere generated randomly in the range (1,10) for arcs pointing downward or upward, and in the range (5,200) for other arcs. The rij were drawn randomly from the uniform distribution in the range (0.9, 1.0). We selected grid networks with these ranges based on our observation that such networks have generally a large number ofeflicient extreme paths and the optimum path lies in the middle region of the efficient frontier. A random network of a given size was generated along similar lines as suggested in Ahuja et al. [ 151. First a skeleton network of length and width equal to the size of the network is generated. The additional arcs are then added by generating the head and tail nodes randomly until the required number of arcs is reached. The cost and reliability data of arcs is generated in a more complex manner but keeping in mind the similar considerations as in the case of grid networks. We considered ten network sizes for grid networks and live network sizes for random networks. For each network size, five different problems were solved and the number of iterations, the number of eficient extreme paths enumerated, and computational times were noted. The total number of
R. K. AHUJA
88
Fig. 2
Table 1. Computational
results of the MCRRPP algorithm (times are in seconds on DEC 10 system) Efficient extreme paths Computational Percent time enumeration
Problem type
Problem size
Number of nodes
Number of arcs
Number of iterations
1 2 3 4 5 6 7 8 9 10
Grid Grid Grid Grid Grid Grid Grid Grid Grid Grid
5 10 13 15 18 20 23 25 28 30
21 102 171 221 326 402 531 627 786 902
70 290 494 660 954 1180 1564 1850 2324 2670
20 87 121 169 265 286 351 404 493 452
4 8 8 9 11 8 8 9 8 11
5 10 12 14 19 17 20 21 23 28
80 80 67 64 57 47 40 43 3.5 39
0.02 0.06 0.10 0.14 0.22 0.26 0.34 0.42 0.53 0.60
11 12 13 14 15
Random Random Random Random Random
10 15 20 25 30
102 227 402 627 902
500 loo0 20xJ 3000 4000
154 180 220 265 301
4 5 I 1 10
8 9 13 15 16
50 55 53 41 63
0.14 0.17 0.24 0.38 0.51
Problem set
Enumerated
Total
efficient extreme paths were also obtained for each problem. The averages of these values for each network size are given in Table 1. The following conclusions can be drawn from the table: (i)
The algorithm is fairly efficient in solving problems ofvarious sizes. It is able to solve problems with several thousand arcs in less than a second. The algorithm, therefore, can be used in real time systems. (ii) The total number of efficient extreme paths in the network are small even for large sized problems, in spite of the fact that the number of paths in a network grow exponentially with the network size. Thus, the path enumeration in the MCRRPP algorithm is not computationally taxing. (iii) The implementation of Theorem 2 is very effective in reducing the path enumeration. For larger sized problems, less than 50 % of the total efficient extreme paths are enumerated. For real life networks, we expect the optimum path to lie around Pi or P,, in which case either Theorem 2 or Theorem 3 can be used to terminate the enumeration. (iv) The number ofpivot iterations performed by the algorithm are much less than the worst case number n2D anticipated in Theorem 4. Consequently, we find the average complexity of the algorithm to be signiticantly different from the worst case complexity of O(mnD log m). (v) The random networks appear to be easier to solve than the grid networks since they have lesser number of extreme eficient solutions.
It is true that these observations are data dependent, but we believe that these will hold good for real life network problems too.
Minimum cost-reliability ratio path problem
89
REFERENCES 1. R. Chandrasekaran, Minimum ratio spanning trees. Networks 7, 335-342 (1977). 2. G. B. Dan&g, W. Blattner and M. R. Rao, Finding a cycle in a graph with minimum cost to time ratio with applications to a ship routing problem, In Theory of Graphs (Editedby P. Rosenthiel). Gordon & Breach, New York (1967). 3. B. L. Fox, Finding minimum cost-time ratio circuits. Opns Res. 17, 546-551 (1969). 4. D. Gusfield, Sensitiuity Analysis for Combinatorial Optimization, Research Report, Electronics Research Laboratory, University of California, Berkeley (1980). 5. E. L. Lawler, Optimal cycles in doubly weighted directed linear graphs. In Theory of Graphs (Edited by P. Rosenthiel). Gordon & Breach, New York (1967). 6. N. Meggido, Combinatorial optimization with rational objective functions. Mathemat. 0pn.s Res. 4, 414-424 (1979). 7. R. K. Ahuja, J. L. Batra and S. K. Gupta, Combinatorial optimization with rational objective functions: a communication. Ma&mat. Opns Res. 8, 314 (1983). 8. C. H. Papadimitriou and K. Steiglitz, Combinatorial Optimizarion: Algorithms and Complexity. Prentice-Hall, Englewood Cliffs, N.J. (1982). 9. R. Chandrasekaran, Y. P. Aneja and K. P. K. Nair, Minimum cost-reliability ratio spanning tree. In Studies on Graphs and Discrete Programming (Edited by P. Hansen). North-Holland, Amsterdam (1981). 10. Y. P. Aneja and K. P. K. Nair, Ratio dynamic programs. Opns Res. Lett. 3, 167-172 (1984). 11. M. Zeleny, Linear Multiobjective Programming. Springer, New York (1974). 12. F. Glover, D. Klingman and D. Stutz, The augmented threaded index method. INFOR 12, 293-298 (1974). 13. E. Horowitz and S. Sahani, Fundamenlals of Data Structures. Computer Science Press, Maryland (1983). 14. M. J. Syslo, N. Deo and J. S. Kowalik, Discrete Optimization Algorithms. PrenticeHall, Englewood Cliffs, NJ. (1983). 15. R. K. Ahuja, J. L. Batra and S. K. Gupta, Optimal expansion ofcapacitated transshipment networks. Nav. Res. logist. Q. In press.
0305-0548/88 53.00+0.00
Printed in Great Britain. All rights reserved
MINIMUM
Copyright 0 1988 Pergamon Journals Ltd
COST-RELIABILITY
RATIO PATH PROBLEM
R. K. AHUJA* Sloan School of Management, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. (Received December 1985; reoisedJuly 1986)
Scope and Purpose-In the shortest path problem, thecost and reliability are two important criteria; we wish to minimize the cost and maximize reliability. The minimum cost-reliability ratio path problem attempts to achieve both simultaneously by yielding a path with low cost and high reliability. In the paper, we develop an algorithm for solving this problem. Computational results are presented which indicate that the algorithm can be used on the real-time basis. Abstract-The minimum ratio path problem in a network is known to be a strong NP-complete problem. However its variant, the minimum cost-reliability ratio path problem, is shown to possess an entirely different structure. We observe that the optimum solution of the minimum cost-reliability ratio path problem is an efficient extreme solution ofa bicriteria path problem. We employ parametric programming to enumerate these efficient extreme solutions and a sufficiency condition is used to cut down the enumeration substantially. The algorithm is shown to be pseudo-polynomial. Computational results of the algorithm on grid as well as random networks are also presented. The algorithm is able to solve problems with several thousand nodes and arcs in a few seconds.
INTRODUCTION
Let G = (N, A) be a directed network in which cij denotes the cost of traversing an arc (i, j) E A and 0 < rij < 1 denotes its reliability. The Ci, are assumed to be nonnegative integers. We consider the problem of finding a directed path P from a source node s to a sink node t for which
is minimum among all such paths. We refer to this problem as the Minimum Cost-Reliability Ratio Path Problem (MCRRPP). The minimum cost-reliability ratio path problems are likely to arise in practice when cost as well as reliability are the criteria to be considered by the decision maker. A least cost path may have poor reliability and the most reliable path may be too expensive to travel. The minimum cost to reliability ratio path problem attempts to incorporate both the criteria into a single objective by yielding a path with low cost and high reliability. Combinatorial optimization problems related to the MCRRPP have been studied and solved by several authors including Chandrasekaran [l], Dantzig et al. [2], Fox [3], Gustield [4], Lawler [S], and Meggido [6]. These approaches are not applicable to the minimum ratio path problem which has been shown to be a strong NP-complete problem by Ahuja et al. [7]. This rules out the possibility of the pseudo-polynomial algorithm unless P = NP [S]. The MCRRPP, however, is shown to possess an entirely different structure in this paper by developing a pseudo-polynomial algorithm. The approach adopted by us is similar to those of Chandrasekaran et al. [9] for the minimum costreliability ratio spanning tree problem, and of Aneja and Nair [lo] for the minimum ratio dynamic programs. In this paper, we show that the optimum solution ofthe MCRRPP is an efficient extreme point ofa b&it&a path problem. These paths are enumerated by the parametric analysis of the shortest path problem. We further establish sufficiency conditions which reduce the path enumeration *Dr R. IL Ahuja is an Assistant Professor in Industrial and Management Engineering Programme at Indian Institute of Technology, Kanpur. He is currently a visiting scientist at the Sloan School of Management, M.I.T., Cambridge. He holds a B.Tech., M.Tech. and Ph.D. from Indian Institute of Technology, Kanpur. His major research interests are networks flows, combinatorial optimization and linear programming. His papers have appeared in Naoal Research Logistics Quarterly, Operations Research Letters, European_Journal of Operational Research and others. 83
R. K. AHUIA
84
substantially. The computational
complexity of the algorithm is shown to be
where D = max{Clj}. Computational results of the algorithm on grid as well as (&&a random networks are presented. There are certain advantages ofthe approach adopted. The parametric analysis ofthe shortest path problem is fairly standard and requires reoptimization of the current tree of shortest paths. Further, the use of the tree data structures and binary heaps result in an extremely fast implementation of the algorithm. Indeed, the computational investigations indicate that the problems with several thousand nodes and arcs can be solved in a few seconds. O(mnD log m),
PROPERTIES
OF
THE
OPTIMUM
SOLUTION
Let S be the set of all directed paths in G. For each P E S define
(1) (2)
(3) where d, = -In rij. Note that dij 2 0, V (i, j) E A, and R(P) = eeD”‘, VP ES. The MCRRPP is then to Minimize z(P) = C(P)/R(P).
(4)
PES
This problem is related to the following bicriteria path problem: Min$ize
[C(P), D(P)].
(5)
By definition, a path P E S is an eflcient solution of (5) if and only if there does not exist any path P E S such that C(P) < C(P) and D(P) < D(F) with a strict inequality in at least one case. The bicriteria space of (5) may be defined as a plane in which each path PES is mapped into a point with coordinates (C(P), D(P)). Let H denote the convex hull of all the paths in the bicriteria space. An efficient solution of (5) is an efficient extreme solution if it is mapped into an extreme point of H. For simplicity of presentation, we use P and its image (C(P), D(P)) interchangeably.
The following theorem establishes a relationship between the MCRRPP and the bicriteria path problem. Theorem 1
An optimum solution P* of the MCRRPP is an efficient extreme point of (5). Proof Let P* not be an efficient extreme point of (5). Then P* can be represented as a convex combination of h paths P,, P,, . . ., P,, such that C(P*) = CF= 1 aiC(Pi), D(P*) = CF= 1 aiD( CL 1 ai = 1 and O
Vi=l,...,h.
(6)
Minimum cost-reliability ratio path problem
85
C(P)
Fig. 1
From the convexity of eeX it follows that h
,-~:=~a,W’i) < 1 i=
ai
,-WI).
1
(7)
Substituting (6) in (7) yields
e-x:=@(Pf)< 1
i i=
~iqpi)/z(pl).
1
(8)
Hence
which contradicts the optimality of P*. In view of Theorem 1, we can restrict our search for the optimum path P* among the efficient extreme points of (5). Let Pi, . . ., Pk, Pk+ I, . . ., P, be the list ofall the efficient extreme points of (5) in the increasing order of their D(P,) value. These paths can be enumerated by performing parametric analysis of the shortest path problem with arc lengths as d, + clcij and increasing ,Ufrom 0 to a large number [ 111. The parametric analysis also enumerates some of the boundary paths which can, of course, be ignored. It was observed during our computational investigations that, in most problem instances, z(P& first decreases and then keeps on increasing for increasing values of k. In such cases, the search can be terminated whenever z(P,.) starts increasing. But it cannot be guaranteed for all problem instances due to the nonlinear nature of the function z(P,.). Hence a more sophisticated termination criterion covering all problem instances is required. We now develop one such criterion. It is well known that the efficient frontier obtained by joining the points Pk_l to Pk for all k=2, . . . . w, is a piecewise linear convex function and typically is of the form as shown in Fig. 1. Let C,, = C(P,) and C,,,,, = C(P,). Further, let L, denote the line passing through Pk_l and Pk. The equation of L, is given by y = uk - bkx, where bk = (@PA - NP,- N(C(P,- A - C(Pd) and a, = D(P,J + b,C(P,.). For any point (x, y) EL,, define gk(x) = xeY= xe’k-bbx. It is easy to see that g&) is a unimodular function and achieves its maximum at x* = l/b,. Let xI = C(P,), V i = 1, . . ., w. Further, let zk* = min
{z(P,)},
l
and P* be the path for which this minimum is attained. Theorem 2
If m(C,J
2 zt, then P* is an optimum solution of the MCRRP’
R. K. AHUJA
86
Proof
Since the efficient frontier is a piecewise linear convex function, it follows that uk - &xi < D(P,), Vi=k+l,...,w.Accordingly gk(Xi)
= Xi
eak- brxi< C(P,) eocpJ = z(pJ,
Vi=k+l,...,w.
(9)
If xk < l/b,, then in view of the nature of gk(x)
zk*G
gk(Gin)
G gktXi)< z(pi),
Vi=k+l,...,w,
(IO)
and the theorem follows. Ifx, > l/b,, then let Xkc l/b, denote the x-coordinate ofthe unique point on L, for which g,&.) = gk(xk). Clearly
for all x, < Xi < Xk. For Xi < i,, (10) applies and the proof is complete. The condition stated in Theorem 2 can be used when the paths are enumerated in the order p,, P,, . . ., P,. However, these paths can also be enumerated in the order P,, P,_ 1, . . ., P,, P, by the parametric analysis. In that case, the following condition can be used as a termination condition. Let .?t = mink.iGw {Z(Pi)} and B* b e t h e path for which this minimum is attained. Theorem 3
If gk+ r(C,,,,,) a.?:, then P* is an optimum solution of the MCRRPP. Proof
The proof is similar to that of Theorem 2 and therefore omitted. THE
ALGORITHM
A formal description of the MCRRPP algorithm is given below. Step 1
Solve the shortest path problem with cij as arc lengths. Let Cminbe the length of the shortest path from s to t. Now, solve the shortest path problem with dij as arc lengths. Let Tl be the resulting tree of shortest paths and T1 = A - T. Compute the dual variables rrj and rri as follows: nf = 0
and
K; - 7~f=+
K!= 0
and
~4 -$
= dij,
V (i,j)E I’,, V (i, j) E Tl.
Set k = 1 and z* = M (a large number). Step 2
For each (i, j) E T,, define the relative cost coefficients as Cij= cij + nf - 4 and (7j = d, + nf - ni. Let F = {(i, j) E ?$: Cij< 0). If F is empty, then go to Step 4; otherwise identify an arc (p, q) E F for which -~JZ,,, = min(,,naF { -&,/Eij} and perform the pivot operation. The pivot operation is performed by adding (p, q) to Tk and dropping the unique basic arc (r, q) incident on node q. Dropping (p, q) from Tk forms two subtrees Ti and Ti, containing and not containing the nodes respectively. Update the dual variables $(rc‘$ by adding C,&&) to $(n$ for each j E Tz. Set k = k + 1. Let Tk be the new tree of shortest paths and Pk be the unique directed path in Tk from s to t. If d, > 0 and (p, q) is incident to the path Pk_ 1, then go to Step 3; otherwise repeat this step. Step 3
Compute z(PJ = C(PJ/D(P,). If z(Pk)
then set z* = z(P& and P* = Pk. If gk(Cmin)2 z*,
Minimumcost-reliabilityratio path problem
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Step 4
The P* is an optimum path with z* as the objective function value. STOP. It may be pointed out that every iteration of Step 2 in the above algorithm does not yield a new efficient path. The basis of the shortest path problem is highly degenerate and consequently pivot operations may change the tree of shortest paths, but may not change the path from s to t. It can be easily seen that Pk from s to t changes if and only if the entering arc is incident on a node belonging to Pk. Also, to obtain the first efficient path, the algorithm selects arcs with zij = 0, if they exist, and performs the pivot operation. Theorem 4
The computational
complexity of the MCRRPP algorithm is O(mnD log m).
Proof Step 1 requires O(n’) operations. The value ofat least one A$reduces in each execution of Step 2 by at least one unit. Since nD is an upper bound on each $, Step 2 is performed atmost n2D times. The effort involved in updating A; and 7~4is also O(n’D) if thread indices [ 123 are used to store the basis tree. The arcs in the set F can be ranked in the nondecreasing order of -Jij/tij value and stored in a heap [13]. The size of the heap is atmost m and, accordingly, the selection of entering variable requires O(n2D log m) operations in all. At each iteration, the -iij/Zij value changes only for those arcs which lie in the cutset separating Ti and Ti. The algorithm therefore, examines arcs incident to and incident from the nodes in Tz, recomputes -~j.lC, and updates the heap. Since an arc (i,j) is examined only when either n; or I$ decreases, each arc is examined no more than 2nD times and consequently the operations needed to update the heap are bounded by O(mnD log m). Finally, Step 3 requires O(nD) computations. The overall complexity of the algorithm is, therefore, O(mnD log m).
COMPUTATIONAL RESULTS The MCRRPP algorithm was coded in FORTRAN-IV and tested on grid as well as random networks of different sizes. (The program listing can be obtained from the author.) The program was run on DEC 10 in multiprogramming environment. The main aim was to estimate the number of(i) pivot operations, (ii) efficient extreme paths enumerated by the algorithm, (iii) efficient extreme paths present in the network, and (iv) to assess the computational time requirement of the algorithm. The network was stored in the forward and reverse star representation. The algorithm was implemented using the augmented threaded index method [12]. The program required 9n + 8m storage out of which 3m was exclusively used to store the heap in a manner suitable for efficient updating. To obtain the shortest paths we used Pape, d’Esopo and Moore’s algorithm as given in Syslo, Deo and Kowalik [14]. A grid network of size p consists of p2 + 2 nodes and 3p2 - p arcs. The topological structure of the grid network for p = 3 is shown in Fig. 2. The C,jwere generated randomly in the range (1,10) for arcs pointing downward or upward, and in the range (5,200) for other arcs. The rij were drawn randomly from the uniform distribution in the range (0.9, 1.0). We selected grid networks with these ranges based on our observation that such networks have generally a large number ofeflicient extreme paths and the optimum path lies in the middle region of the efficient frontier. A random network of a given size was generated along similar lines as suggested in Ahuja et al. [ 151. First a skeleton network of length and width equal to the size of the network is generated. The additional arcs are then added by generating the head and tail nodes randomly until the required number of arcs is reached. The cost and reliability data of arcs is generated in a more complex manner but keeping in mind the similar considerations as in the case of grid networks. We considered ten network sizes for grid networks and live network sizes for random networks. For each network size, five different problems were solved and the number of iterations, the number of eficient extreme paths enumerated, and computational times were noted. The total number of
R. K. AHUJA
88
Fig. 2
Table 1. Computational
results of the MCRRPP algorithm (times are in seconds on DEC 10 system) Efficient extreme paths Computational Percent time enumeration
Problem type
Problem size
Number of nodes
Number of arcs
Number of iterations
1 2 3 4 5 6 7 8 9 10
Grid Grid Grid Grid Grid Grid Grid Grid Grid Grid
5 10 13 15 18 20 23 25 28 30
21 102 171 221 326 402 531 627 786 902
70 290 494 660 954 1180 1564 1850 2324 2670
20 87 121 169 265 286 351 404 493 452
4 8 8 9 11 8 8 9 8 11
5 10 12 14 19 17 20 21 23 28
80 80 67 64 57 47 40 43 3.5 39
0.02 0.06 0.10 0.14 0.22 0.26 0.34 0.42 0.53 0.60
11 12 13 14 15
Random Random Random Random Random
10 15 20 25 30
102 227 402 627 902
500 loo0 20xJ 3000 4000
154 180 220 265 301
4 5 I 1 10
8 9 13 15 16
50 55 53 41 63
0.14 0.17 0.24 0.38 0.51
Problem set
Enumerated
Total
efficient extreme paths were also obtained for each problem. The averages of these values for each network size are given in Table 1. The following conclusions can be drawn from the table: (i)
The algorithm is fairly efficient in solving problems ofvarious sizes. It is able to solve problems with several thousand arcs in less than a second. The algorithm, therefore, can be used in real time systems. (ii) The total number of efficient extreme paths in the network are small even for large sized problems, in spite of the fact that the number of paths in a network grow exponentially with the network size. Thus, the path enumeration in the MCRRPP algorithm is not computationally taxing. (iii) The implementation of Theorem 2 is very effective in reducing the path enumeration. For larger sized problems, less than 50 % of the total efficient extreme paths are enumerated. For real life networks, we expect the optimum path to lie around Pi or P,, in which case either Theorem 2 or Theorem 3 can be used to terminate the enumeration. (iv) The number ofpivot iterations performed by the algorithm are much less than the worst case number n2D anticipated in Theorem 4. Consequently, we find the average complexity of the algorithm to be signiticantly different from the worst case complexity of O(mnD log m). (v) The random networks appear to be easier to solve than the grid networks since they have lesser number of extreme eficient solutions.
It is true that these observations are data dependent, but we believe that these will hold good for real life network problems too.
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