J Glob Optim (2015) 63:149–163 DOI 10.1007/s10898-015-0280-9
Constrained shortest path with uncertain transit times Shaghayegh Mokarami · S. Mehdi Hashemi
Received: 10 February 2014 / Accepted: 5 February 2015 / Published online: 21 February 2015 © Springer Science+Business Media New York 2015
Abstract This paper is concerned with the constrained shortest path (CSP) problem, where in addition to the arc cost, a transit time is associated to each arc. The presence of uncertainty in transit times is a critical issue in a wide variety of world applications, such as telecommunication, traffic, and transportation. To capture this issue, we present tractable approaches for solving the CSP problem with uncertain transit times from the viewpoint of robust and stochastic optimization. To study robust CSP problem, two different uncertainty sets, Γ -scenario and ellipsoidal, are considered. We show that the robust counterpart of the CSP problem under both uncertainty sets, can be efficiently solved. We further consider the CSP problem with random transit times and show that the problem can be solved by solving robust constrained shortest path problem under ellipsoidal uncertainty set. We present extensive computational results on a set of randomly generated networks. Our results demonstrate that with a reasonable extra cost, the robust optimal path preserves feasibility, in almost all scenarios under Γ -scenario uncertainty set. The results also show that, in the most cases, the robust CSP problem under ellipsoidal uncertainty set is feasible. Keywords Constrained shortest path · Robust optimization · Stochastic optimization · Uncertain optimization · Approximation algorithm
1 Introduction Nowadays, route guidance systems are more essential than ever due to a rapid growth in population, traffic, and urban and suburban trips. One important reason for using route guidance systems is to find a path with minimum cost or length or maximum safety to reach the
S. Mokarami · S. M. Hashemi (B) Faculty of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Ave., Tehran, Iran e-mail:
[email protected] S. Mokarami e-mail:
[email protected]
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destination on time. For example, this issue is of importance for emergency vehicles such as ambulances and fire engines that should arrive at the scene and transfer the injured people to hospital within a pre-defined time interval. Another example is when a person wants to find the best route to get access to public transportation, such as airports when on-time arrival is highly important. For this purpose, the Constrained Shortest Path (CSP) problem can be applied to design route guidance systems. The aim of the CSP is to find a path from the origin to the destination with minimum cost in such a way that the total transit time of path does not exceed from the given time horizon in a network in which each link has a transit time as well as a cost. One simple way is to assign a fixed value to the transit time of links based on the data gathered in the past in the same place and time of day or week. In this method, which is called priori guidance, the best route is found before starting the trip, and usually the proposed direction does not change during the trip. But there are a lot of factors, such as bad weather, incidents and accidents, festivals and exhibitions, rush hours, and varying demand for transportation on different days of the week, namely working days and weekends, which make it difficult to determine a fixed value as transit time of links. Therefore, assigning a single value as transit time of links in the network reduces the accuracy of routing. Hence, a reliable route guidance system should take account of uncertainty in transit time of links. Applying online data is a way to overcome the problem of changing traffic conditions. Although this method takes into account all changes in traffic, there are computational challenges in largescale networks due to the huge amount of data. On the other hand, the systems that operate based on online data, such as Variable Message Signs (VMS) and radio or cell phone-based systems are faced with time and space constraints. Furthermore, the provided information is usually general and users should analyze the information based on their own experience and knowledge about the network. The user who uses online route guidance systems also needs to make a decision to choose his route several times, which may cause distraction. To overcome these drawbacks and make a reliable route guidance system, a system which suggests directions a priori but based on uncertain data is proposed. Here is a place where uncertain optimization for solving the CSP can facilitate designing a reliable route guidance system. A common approach to model uncertainty is using stochastic programming, which was introduced by Dantzig [9]. This approach needs to know a probability distribution over the uncertain parameters, but finding the distribution of data is a difficult task, especially in real world applications. On the other hand, for solving stochastic programming models, the size of model usually increases with respect to the number of scenarios; this matter causes computational challenges, so finding the best solution, particularly in large-scale networks, becomes more complicated. In addition, stochastic programming models find a solution which works well based on mean values; this solution may not be suitable in some applications like guiding services. Robust optimization is another approach to deal with uncertainty (see, e.g., [7,8]). This approach assumes a deterministic uncertainty set, rather than a probability distribution, and attempts to find a solution which is robust against all possible scenarios. In transportation networks, due to complicated supply and demands in origin-destination travels, accidents and special events and same things else, assigning an interval as a set into transit time of links without distribution is more convenient and applicable. Moreover, sometimes the value of routing guidance systems strongly depends on arriving on time even in the worst conditions. So, with respect to the properties of robust optimization, it seems reasonable to employ it for solving CSP problem which can be applicable in routing guidance systems.
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It is worth mentioning that the CSP problem has many real-world applications in other areas, such as telecommunications, in addition to the stated application in route guidance systems. In this paper, we study the CSP problem with uncertain transit times. This problem is a NPhard optimization problem; so it is important to propose an efficient algorithm which works well in large-scale networks. In the rest of this section, a brief review of related problems are mentioned. The CSP problem can be solved by using a dynamic programming method. For the case where the time horizon is small enough (polynomially bounded in the input size of the problem), this leads to a polynomially-time algorithm. But in general, the problem is NPcomplete as shown by Garey and Johnson [12]. However, Joksch [15] proposed a pseudopolynomial algorithm based on dynamic programming for solving the CSP problem, so the CSP problem is NP-Complete problem in weak sense. Handler and Zang [13] proposed two approaches to solve the CSP problem in which one of them is based on a k-th shortest path algorithm, and another one utilizes a Lagrangian relaxation method. Based on dynamic programming, Hassin [14] presented FPTAS for the CSP problem. Using a scaling and rounding technique similar to that of Hassin [14], he improved running time of FPTAS for acyclic graphs. But if the time horizon is not polynomially bounded in the input size of the problem, this yields a pseudo-polynomial time algorithm. However, one can scale the transit times and present an FPTAS for solving the CSP problem. A number of researchers have studied shortest path problems with uncertain costs. In particular, there are a number of papers on the stochastic shortest path problem, where arc costs are given as random variables. In literature, there are different criteria for stochastic shortest path problem. The most common criteria is to find a path with minimum expected cost (a path whose cost is minimum in average) (for example [19]). However, this objective function leads to simple structure, but the variety of path’s cost is not considered. To overcome this drawback, the combination of mean and variance of path is considered as objective function. This model has a quadratic form and some strategies including bi-criteria optimization [20] and Lagrangian relaxation [22] were proposed to solve it. Another criteria for stochastic shortest path problem is to find a path such that the probability that the cost of path is less than the desired value, is maximized. Nikolova [18], studied this kind of stochastic shortest path problem when the arc costs distribution are Normal, Exponential or Bernoulli. Kosuch and Lisser [16] considered constrained shortest path in which the arc costs are deterministic and the arc transit times are random variables. In their studies, for each unit of delay, the penalty occurs and the aim of problem is to find a path minimizing the sum of path cost and total penalty delay. The same work was studied by Verweij et al. [21] in which the discrete distribution was assumed for the transit times. In stochastic optimization, one would need to know exact distribution for the data and the size of optimization model as a function of the number of scenarios is very large. Here is where robust optimization comes into play to overcome these difficulties in uncertain environments. In the robust optimization, the worst case scenarios are considered and optimization is done against them that usually is used a min-max objective function. Kouvelis and Yu [17] proposed a framework for robust discrete optimization which seeks to find a solution that minimizes the worst case performance under a set of scenarios for the data. A related objective is the min-max regret approach which seeks to minimize the worst case loss in objective value that may occur. In both approaches the robust version of many polynomially solvable discrete optimization problems becomes NP-hard. Averbakh [1] showed that when each cost coefficient can vary within an interval, polynomial time solvability of some specific discrete optimization problem
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is preserved; however the approach does not seem to generalize to other discrete optimization problems. Bertsimas and Sim [7,8] proposed a powerful modeling approach in robust optimization to control the level of conservatism in the solution. The main advantage of their model is that the robust counterpart of trackable problems remain trackable. For 0–1 integer problem, they [7] showed that the robust counterpart problem can be solved efficiently if the original problem is efficiently solvable. They also showed that there exist approximation algorithms for the case in which the original problem can be approximated efficiently. Our contributions. In this paper, we study the Constrained Shortest Path problem, in which the transit times are subject to uncertainty. We use the robust optimization methodology and stochastic optimization techniques to deal with this situation. We first assume that the uncertain transit times lie within a deterministic uncertainty set and seek a solution which remains feasible under any possible scenario. For this purpose, two different kind of uncertainty sets are considered, Γ -scenario model and ellipsoidal model. We show that the resulting problem reduces to solving m instances of the CSP problem, where m is the number of the arcs, in Γ -scenario model and in ellipsoidal model, we show that robust CSP problem can be solved thorough a number of CSP problems. We then suppose that the transit times are given by independent random variables. For the case where the random variables have the same standard deviation, We only need to solve m instance of CSP problem. Finally, we report extensive computational experiments to evaluate the performance of the optimal path of robust CSP on random networks.
2 Robust CSP problem We consider a directed graph G = (V, E) with a set V of n nodes and a set E ⊆ V × V of m arcs. Each arc e ∈ E is associated with a cost (or length) ce and a transit time τe . More precisely, ce denotes the cost for traversing arc e, and τe gives the required time to pass the arc. A path P from node v to node w is a sequence of nodes as P := v1 , v2 , . . . , v such that v1 = v, (vi , vi+1 ) ∈ E for i = 1, . . . , − 1 and v = w. The cost and transit time of the path P are defined as the sum of transit costs and transit times of the arcs in the path, respectively, i.e., c(P) :=
−1
c(vi ,vi+1 ) ,
i=1
τ (P) :=
−1
τ(vi ,vi+1 ) .
i=1
There are two specific nodes; a source node s ∈ V and a sink node t ∈ V . Given a parameter T ≥ 0, the constrained shortest path (CSP) problem is to find a path from the source s to the sink t with minimum cost so that its transit time is no more than T . This problem has been well-studied in the literature. Traditionally, it is assumed that the cost and transit times are fixed and deterministic in advance. This is not a realistic assumption as in many practical situations, transit times are subject to uncertain. We address this issue in this section from the robust optimization prospective. In particular, we assume that the transit times are uncertain, but lie within a deterministic uncertainty set.
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We suppose that the transit time of arc e can increase by δe from its nominal value τe . A path is called robust if its transit time is no more than T under any scenario. It follows from this definition that a path P is robust if and only if: τe + max δe ≤ T, e∈P
δ∈U
e∈P
Where δ = (δ1 , δ2 , . . . , δm ) and U denotes the uncertain set contains all possible increases in the transit time coefficients. The robust CSP problem seeks a robust path with minimal cost. Then the robust CSP problem can be formulated as follows: min x∈X
s.t.
cx
τe xe + max
δ∈U e∈E
e∈E
δe xe ≤ T,
(1)
here and subsequently, X ⊆ {0, 1}m is the set of all solutions satisfying the following constraints: ⎧ ⎨ 1, if v = s, ∀v ∈ V, xe − xe = −1, if v = t, ⎩ (2) 0, otherwise, e∈A(v) e∈B(v) xe ∈ {0, 1}, ∀e ∈ E, where, for each node v ∈ V , A(v) and B(v) are the set of outgoing and incoming arcs in node v, respectively. In continue of this section, we consider two special cases of uncertainty set U and solve the robust CSP problem with respect to them. 2.1 Robust CSP problem under Γ -scenario uncertainty set It is unlikely that all coefficients take their upper bounds. To control the level of uncertainty, a parameter Γ ∈ {0, . . . , m} is introduced, allowing the number of transit times to deviate from their nominal values, as proposed by Bertsimas and Sim [7]. So, the uncertainty set U is defined as follows and called Γ -scenario uncertainty set, U := {d = (de1 , . . . , dem ) : 0 ≤ d ≤ δ and dei > 0 for at most Γ arcs}.
Then the robust CSP problem under Γ -scenario uncertainty set can be formulated as follows: min x∈X
s.t.
cx
τe xe + max
e∈E
d∈U e∈E
de xe ≤ T,
(3)
Lemma 1 Problem (3) is equivalent to the following problem min x∈X
s.t.
cx e∈E
τe xe + Γ α +
α ≥ 0.
max{0, δe − α}xe ≤ T,
(4)
e∈E
Proof By applying strong duality, one can show the correctness of Lemma 1.
Problem (4) is a nonlinear 0–1 integer program. However, for a fixed value of α, this problem reduces to an instance of the CSP problem. We next show that an optimal value for α can be determined by restricting our attention to a finite set of m elements.
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Fig. 1 Robust CSP under Γ -scenario uncertainty set algorithm
Algorithm 1 for all e ∈ E, solve CSP from s to t with cost c, transit time τ + max{0, δ − δe } and time horizon T − e select the best solution as optimal solution of robust CSP.
Theorem 1 If Problem (3) has a feasible solution, then there exists an optimal solution (x ∗ , α ∗ ) for Problem (3) such that α ∗ ∈ {δ1 , . . . , δm }. Proof We use the equivalency of Problems (3) and (4). Suppose that (x, ˜ α) ˜ is an optimal solution of Problem (4). For a fixed α, we define Z (α) as τe x˜e + Γ α + max{0, δe − α}x˜e . (5) Z (α) := e∈E
e∈E
It is clear that min Z (α) ≤ Z (α) ˜ ≤ T. α≥0
(6)
By similar argument as that of Theorem 3 in Bertsimas and Sim [7], we can show that the problem minα≥0 Z (α) has an optimal solution in the set {δ1 , . . . , δm }, say α ∗ . This shows that (x, ˜ α ∗ ) is an optimal solution for problem (4). We can now conclude the following results. Corollary 1 The robust counterpart of the CSP problem under Γ -scenario uncertainty set can be solved by at most m CSP problems. In particular, it admits a fully-polynomial time approximation algorithm. It is known that the CSP can be solved in polynomial-time by using a dynamic programming approach, if the transit times are polynomially-bounded in the input size of the problem. This remains true for the robust counterpart of CSP problem under Γ -scenario uncertainty set. Corollary 2 If the CSP problem is polynomially solvable, then so is the robust counterpart of problem. Figure 1 gives a formal description of robust CSP algorithm based on Theorem 1. 2.2 Robust CSP problem under ellipsoidal uncertainty set The ellipsoidal uncertainty set U , defined as U = {δ : Σ −1/2 δ 2 ≤ Ω}, and proposed by Ben-Tal and Nemirovski [3–5] and El-Ghaoui et al. [10,11]. In this definition Σ is the covariance matrix of the transit times.Suppose that the transit√times are uncorrelated, i.e., Σ = diag(d1 , . . . , dm ), then maxδ∈U e∈E δe xe is equal to Ω d x. Therefore the following problem models the robust CSP problem under ellipsoidal uncertainty set, min x∈X
s.t.
cx
√ τ x + Ω d x ≤ T.
(7)
Notice that Problem (7) is a non-linear model. But, we show that it can be written as a parametric linear program.
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Fig. 2 Robust CSP under ellipsoidal uncertainty set algorithm
Algorithm 2 for all θ ∈ {1, 2, . . . , mdmax }, solve CSP from s to t with cost c, dx transit time τ + 2 √ θ √ and time horizon T − 2 dθ select the best solution as optimal solution of robust CSP.
Theorem 2 Problem (7) is equivalent to the following problem min
cx
s.t.
min
x∈X
√ Ω 2
θ >0
θ + τx +
dx Ω √ 2 θ
≤ T.
(8)
Proof Let Z ∗ and W ∗ be the optimal values of Problems (7) and (8), respectively. We show that Z ∗ = W ∗ . Let x ∗ be an optimal solution for Problem (7). It is clear that for θ = d x ∗ , x ∗ is a feasible solution for Problem (8). So, W ∗ ≤ cx ∗ = Z ∗ . Next we show that √ Z ∗ ≤ W ∗ . Assume that x is an optimal solution for Problem (8). Since Ω d x = arg minθ >0 { 2 θ + τ x + Ω2 d√x }, we can conclude that x is a feasible solution for θ Problem (7). It means Z ∗ ≤ cx = W ∗ . Problem (8) is a parametric linear programming problem for solving (7). This problem has the advantage that it reduces to the classical CSP problem for fixed θ . We next show that one can solve Problem (8) by solving a number of CSP problems. Theorem 3 Let Z (θ ) be the optimal value of the following problem min
cx
s.t.
Ω 2
x∈X
√ θ + τx +
Ω √ dx 2 θ
≤ T,
(9)
then the optimal value of Problem (8) is equal to minθ ∈{1,2,...,mdmax } Z (θ ) where dmax = max{de : e ∈ E}. Furthermore, if de = d for all e ∈ E, then the optimal value of Problem (8) is equal to minθ ∈{1,2,...,m} Z d (θ ), where Z d (θ ) be the optimal value of the following problem min x∈X
cx
√
(10) dθ + τ x + Ω2 √d x ≤ T. dθ √ dx Proof It is clear that for fixed x ∈ X , arg min{ Ω2 θ + τ x + Ω2 √ } occurs at θ = d x. Hence, θ the set {1, 2, . . . , mdmax } contains all possible values for θ , since x is a binary vector. This prove the first part of the theorem. Similarly, we can prove the second part. s.t.
Ω 2
Corollary 3 The robust counterpart of the CSP problem under ellipsoidal uncertainty set by solving the number of CSP problems. Figure 2 gives a formal description of robust CSP under ellipsoidal uncertainty set algorithm based on Theorem 3.
3 Stochastic CSP problem So far we have confined our attention to robust counterpart of CSP problem in which the uncertain transit times lie within a deterministic uncertainty set and we optimize against the
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worst case scenarios. In this section, we assume that the transit times are random variables and look for a path with minimal cost whose transit time is no more than T with high probability. Some researchers are using the advantages of robust optimization to deal with tractable methods for stochastic optimization. In particular, they design uncertainty sets for different kinds of probabilistic distributions, which the robust solution has a desirable probabilistic guarantee. For instance, Bertsimas et. al. [6] applied hypothesis test to construct proper uncertainty sets. Also, Bandi and Bertsimas [2] presented tractable approach for some stochastic problems via robust optimization in high dimensions. Here, we show that CSP problem with uncorrelated random transit times can be treated with robust counterpart of CSP problem under ellipsoidal uncertainty set. Suppose that the transit time of each arc e is given by a random variable τ˜e . We assume that the transit times are independent random variables. Then the transit time of a path P is a random variable given by τ˜ (P) := e∈P τ˜e . The stochastic CSP problem is to find a path P with minimum cost so that the transit time of P is no more than T with a probability of at least 1 − for some given > 0. This problem is formulated as follows:
min
cx
s.t.
Pr [τ˜ x ≤ T ] ≥ 1 − .
x∈X
(11)
Here > 0 shows the probability of violation of feasibility. The smaller yields the more reliable solution. The stochastic CSP problem is an instance of chance-constrained problems. These kinds of problems have received a great deal of attention, both from practical and theoretical points of view. In contrast to robust optimization, these problems arise whenever we are interested in finding solutions which perform well in average, rather than the worst case scenarios. However, one can use the robust optimization model to get a reasonable solution for the stochastic version. In particular, Bertsimas and Sim [8] showed that the robust counterpart problems generate solutions which are feasible with high probability for the stochastic variants of the problems.
Lemma 2 (Bertsimas and Sim [8]) Suppose that the random variables τ˜e√ , e ∈ E, are independent and symmetrically distributed. Let ε ∈ (0, 1), if Γ ≥ 1/2(m + −2m ln(ε)), then the robust CSP problem yields a feasible solution for Problem (11).
Lemma 2 expresses that robust approach is an useful method for solving chanceconstrained problems. However, it does not provide an optimal solution for Problem (11) and the obtained solution might be conservative. Therefore, we aims to develop a method to compute an optimal solution for the stochastic CSP problem. Suppose that τe and δe are the mean and the standard deviation of the random variable τ˜e . Then we can write
Pr [τ˜ x ≤ T ] =Pr
123
τ˜ x − τ x T − τx T − τx
=Φ ≤ √ . √ √ δx δx δx
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where Φ is the cumulative distribution function of the standard normal distribution. Since Φ is an increasing function, we have T − τx Pr [τ˜ x ≤ T ] ≥ 1 − ⇐⇒ Φ ≥1−
√ δx T − τx ≥ Φ −1 (1 − ) ⇐⇒ √ δx √ ⇐⇒ τ x + Φ −1 (1 − ) δx ≤ T. In what follows, we let Ω := Φ −1 (1 − ). Then, Problem (11) can be written as follows: min x∈X
s.t.
cx
√ τ x + Ω δx ≤ T.
(12)
We can interpret Model (12) as finding a path from source to sink with minimum cost such that the sum of mean and variance of transit times is no more than the given time horizon T . So, this problem can be solved through robust counterpart of CSP problem under ellipsoidal uncertainty set. Corollary 4 There exists a pseudo-polynomial time algorithm to solve stochastic CSP problem. Corollary 5 Suppose that the standard deviation of all transit times of arcs are equal, then stochastic CSP problem admits a FPTAS.
4 Experimental results In this section, we present some computational results to evaluate the performance of the proposed algorithms. The algorithms are implemented by C++ and all the tests are carried out on a computer equipped with a Intel(R) Core(TM) i5, 2.8 GHz processor and 4G of memory. The networks with different number of nodes and arcs are generated randomly, in fact, two families of relatively sparse and dense networks are considered. The number of nodes and arcs of each family of networks are summarized in Table 1. A number between 1 and 30 is chosen randomly as the cost of each arc, and the transit time of each arc is a random number between 1 and 40, the transit time of each arc can also increase to δ present of its nominal value, where δ is randomly selected between 10 and 20. Algorithm 1 is implemented for both dense and sparse networks for 99 times to compute robust CSP for Γ = 1, . . . , 10. In practice, it’s likely that some arcs have same δ, and the iterations of Algorithm 1 to solve the robust CSP problem under Γ -scenario uncertainty set, are fewer than the number of arcs. However, the number of iterations of Algorithm 2 depend directly to number of arcs of network and the run time of algorithm dramatically
Table 1 The number of nodes and arc of generated networks Number of nodes Number of arcs (sparse) Number of arcs (dense)
40 57 780
50 72 1225
60 87 1770
70 102 2415
80 117 3160
90 132 4005
100 147 4950
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Table 2 The average of ratio of optimal robust value to optimal nominal value for sparse networks Number of nodes Γ
40
50
60
70
80
90
100
1 2 3 4 5 6 7 8 9 10
1.693 1.639 1.559 1.444 1.542 1.472 1.300 1.588 – –
1.581 1.564 1.525 1.433 1.444 1.404 1.292 – – –
1.659 1.680 1.622 1.548 1.484 1.545 1.406 1.688 – –
1.784 1.672 1.526 1.546 1.553 1.453 1.136 – – –
1.637 1.611 1.595 1.474 1.558 1.469 1.556 2.300 – –
1.613 1.573 1.434 1.426 1.447 1.266 1.280 1.244 – –
1.653 1.517 1.458 1.367 1.329 1.254 1.246 – – –
Table 3 The average value of ratio of optimal robust value to optimal nominal value for dense networks Number of nodes Γ
40
50
60
70
80
90
100
1 2 3 4 5 6 7 8 9 10
2.544 1.869 1.712 1.656 1.460 1.555 1.595 1.348 1.000 –
2.482 2.541 2.038 1.963 1.444 1.315 1.182 1.158 1.155 1.167
2.284 1.987 1.796 1.641 1.401 1.264 1.243 1.190 1.085 1.111
1.995 2.031 1.761 1.552 1.534 1.268 1.286 1.354 1.000 –
2.368 2.137 1.779 1.527 1.387 1.203 1.184 1.112 1.067 1.110
2.070 1.898 1.597 1.475 1.437 1.274 1.206 1.104 1.273 1.091
2.131 1.873 1.764 1.736 1.651 1.421 1.201 1.220 1.107 1.091
Table 4 The number of networks with no feasible solution for sparse family Number of nodes Γ
40
50
60
70
80
90
100
1 2 3 4 5 6 7 8 9 10
16 33 47 62 81 91 96 98 99 99
11 21 36 60 83 83 96 99 99 99
16 31 55 69 80 87 97 98 99 99
14 30 46 67 79 88 97 99 99 99
8 20 37 59 75 90 96 98 99 99
9 19 43 54 74 86 94 96 99 99
8 22 35 52 68 82 95 99 99 99
increases when the number of arcs increase. So, only sparse networks are considered to evaluate Algorithm 2 which are implemented for one hundred times. The ratio of the cost of robust optimal path to cost of nominal optimal path shows how much we should pay to protect against uncertainty. Therefore, our measure to evaluate the algorithms is the average value of this ratio. Tables 2 and 3 show these ratios for sparse and
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Table 5 The number of networks with no feasible solution for dense family Γ
40
50
60
70
80
90
100
1 2 3 4 5 6 7 8 9 10
9 27 42 54 71 83 88 94 98 99
9 22 46 56 72 79 89 92 97 98
11 19 33 51 73 85 89 91 97 98
12 21 35 57 70 81 88 91 98 99
10 24 41 51 67 81 89 93 95 97
12 24 41 53 65 81 87 95 98 98
7 14 26 48 65 79 82 93 96 98
Table 6 The average value of ratio of optimal robust value to optimal nominal value under ellipsoidal uncertainty set Nodes
Ellipsoid
Γ =1
Γ =2
Γ =3
Γ =4
Γ =5
40 50 60 70 80 90 100
1.747 1.787 1.883 1.791 1.704 1.681 1.59
1.747 1.736 1.844 1.703 1.492 1.598 1.566
1.82 1.815 1.65 1.771 1.425 1.629 1.566
1.575 1.703 1.667 1.72 1.474 1.633 1.544
1.58 1.677 1.408 1.558 1.473 1.607 1.571
1.394 1.661 1.413 1.518 1.286 1.624 1.611
Table 7 The number of networks with no feasible solution for ellipsoidal uncertainty set Nodes
Ellipsoid
Γ =1
Γ =2
Γ =3
Γ =4
Γ =5
40 50 60 70 80 90 100
5 7 5 4 9 6 4
8 15 10 11 15 9 8
23 26 32 22 31 21 22
53 48 54 48 46 38 36
70 61 71 71 67 55 59
87 81 84 85 87 76 72
dense networks respectively for different values of Γ . It follows from Tables 2 and 3 that this ratio in sparse networks is less than this value in dense networks on average, for fixed value of Γ . We also see that by increasing the value of Γ , this ratio decreases. however, one should notice that by increasing Γ , the number of networks with no feasible solutions gradually increase, and these averages get from feasible solutions. Naturally, we expect that the number of infeasible problems grow when the value of Γ increases. Tables 4 and 5 confirm this dependency. As seen, the sparse networks go to infeasibility more quickly than dense networks, for fixed value of Γ . For example, let set the value of Γ to 4, the number of infeasible problems is 60.43 on average for sparse networks while this value equals to 52.86 for dense networks. We also see that for Γ greater than 5, at least 81 % of networks don’t have feasible solution, for both sparse and dense networks. Table 6 represents the ratio of robust optimal path to nominal optimal path for ellipsoidal uncertainty set. To better evaluation and comparison, Table 6 also contains the average ratio
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Table 8 The percentage of feasibility of optimal path and the ratio of optimal robust cost to optimal nominal cost for Γ = 1, . . . , 4 Iteration %F O N S %F O RS1
O RS1 ONS
%F O RS2
O RS2 ONS
%F O RS3
O RS3 ONS
%F O RS4
O RS4 ONS
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
2.67 0 1.25 1.5 1.75 2 1.33 1.5 1 1.25 1.75 3 0 2 1.4 1.75 2 4 1.75 1.5 1.167 2.5 1.5 1.25 7 0 2.33 1.75 1.2 1.25 0 0 2.6 3.5 2.25 1.4 1.5 0 5 2 5 1.33 1.2 2 1.75 1.25 2.5 1.33 1.4 2
100 0 100 100 100 100 100 100 100 100 100 100 0 100 100 100 100 100 100 100 100 100 100 0 0 0 0 100 100 100 0 0 0 100 100 0 100 0 0 100 0 100 100 100 100 100 100 100 100 100
3 0 1.25 1.5 1.75 2.33 1.67 1.5 1 1.25 1.75 3.5 0 2 1.4 1.75 2 4.5 1.75 1.5 1.167 2.5 1.5 0 0 0 0 1.75 1.2 1.25 0 0 0 3.5 2.25 0 1.5 0 0 2.2 0 1.33 1.2 2 1.75 1.25 2.5 2 1.4 2
100 0 100 0 100 100 100 100 100 100 0 100 0 0 0 100 0 100 100 100 100 100 100 0 0 0 0 0 100 100 0 0 0 100 0 0 0 0 0 100 0 0 100 100 100 100 0 100 100 100
3 0 1.25 0 1.75 2.33 2 1.5 1 1.25 0 3.5 0 0 0 1.75 0 4.5 1.75 1.5 1.167 2.5 1.5 0 0 0 0 0 1.2 1.25 0 0 0 3.5 0 0 0 0 0 2.2 0 0 1.2 2 2.25 1.25 0 2 1.4 2
100 0 0 0 0 100 100 0 100 0 0 0 0 0 0 100 0 0 0 100 100 0 100 0 0 0 0 0 100 0 0 0 0 0 0 0 0 0 0 0 0 0 100 100 100 0 0 100 0 100
3 0 0 0 0 2.33 2 0 1 0 0 0 0 0 0 2 0 0 0 1.5 1.167 0 1.5 0 0 0 0 0 1.2 0 0 0 0 0 0 0 0 0 0 0 0 0 1.2 2 2.25 0 0 2 0 2
4.06 100 7.32 8.33 4.04 7.81 0 4.05 100 4.14 0 0.01 33.9 12.1 0 0.01 4.2 3.98 8.06 0 23.36 4.15 0 8.41 13.3 40.81 33.18 8.33 0 64.32 100 24.64 33.62 4.1 4.11 9 16.36 46.27 21.02 84.05 50.02 4.14 0 0 4 8.08 25.76 0 8.07 0
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91.93 0 100 100 100 84.33 95.96 100 100 100 100 100 0 100 100 100 100 100 100 100 100 100 100 100 100 0 100 100 100 100 0 0 100 100 100 100 100 0 100 100 100 100 100 100 100 100 100 100 100 100
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Table 9 The results of simulation Γ
The percentage of Feasibility of optimal robust path
Ratio of cost robust path to nominal path
The percentage of infeasibility of robust CSP
0 1 2 3 4
18.82 99.37 100 100 100
– 2.08 1.88 1.94 1.80
0 7 14 24 37
of the cost of robust optimal path under Γ -scenario uncertainty set, to cost of nominal optimal path on the same networks that robust CSP with ellipsoidal uncertainty set were solved. As seen, these ratios for ellipsoidal uncertainty set model are close to Γ -scenario model, and even for Γ = 1, these ratios are almost equal. But, there is another important factor to compare two models, the number of cases that Γ -scenario model or ellipsoidal model have no feasible solution, which were shown in Table 7. It could be seen that the number of networks with no feasible solution in Γ -scenario model are far more than the ellipsoidal model. This value, even for Γ = 1, is about twice of ellipsoidal case, in average. One important criterion to evaluate robust optimal path is to find the probability that it remains feasible when a scenario occurs in compare to feasibility of nominal optimal path. For this purpose, The random dense networks with 100 nodes are generated for fifty times, for Γ -scenario model. After solving nominal CSP and robust CSP for Γ = 1, . . . , 4, for each network, 10,000 scenarios are generated randomly. At each scenario, a random number in interval [τe , τe + δe ] is assigned to transit time of arc e for all e ∈ E. Then the optimal path of nominal CSP and robust CSP problems are tested to verify that they remain feasible or not. Table 8 illustrates the results of this experiment. The notations %F O N S, %F O R Si, i = 1, . . . , 4, denote the percentage of times which optimal path remains feasible for nominal and robust problems respectively, and the notations OORSi N S , i = 1, . . . , 4, show the ratio of the cost of optimal robust solution to the cost of optimal nominal solution. It can be seen from Table 8 that the optimal nominal path at only 18.82 % of scenarios remains feasible while the robust optimal path with Γ = 1, at 99.37 % of cases preserve feasibility, if the robust solution exists. To reach this high rate of feasibility, one should pay only 2.08 times of the cost of nominal optimal path on average. Table 9 summarizes the results of this experiment as average value. To determine the cost of preserve feasibility under ellipsoidal uncertainty set, one hundred sparse random networks with 40 and 100 nodes were considered, separately. After solving robust CSP problem using Algorithm 2, one thousand scenarios, based on the ellipsoidal uncertainty set, were generated randomly. Table 10 represents the percent of number of scenarios in which the nominal optimal path remains feasible that denoted by %F O N S, and the ratio of cost of optimal robust path to cost of optimal nominal path which denoted by O E RS O N S . In sparse networks with 40 nodes, with costing 1.76 more times, the feasibility of all cases are preserved instead of 49.62 %. In the sparse networks with 100 nodes, this cost equals to 1.85 more times against 77.46 % of feasibility.
5 Conclusion In this paper, we have considered the constrained shortest path problem in uncertain environment, in which the transit times of arcs are expressed as inexact values. We have examined
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Table 10 The cost of preserve feasibility under ellipsoidal uncertainty set Iteration
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
40 Nodes
100 Nodes
Iteration
%F O N S
O E RS ONS
%F O N S
O E RS ONS
20 19 6 57 6 68 37 100 40 25 6 100 25 100 100 87 50 57 100 100 37 25 20 100 6 43 100 50 68 100 68 100 46 94 68 46 68 35 6 20 6 57 6 25 68 50 100 100 100 6
1.36 2.91 1.74 1.25 1.26 0 1.44 2 1.13 2.29 4.4 1 1.4 1 1.9 1.29 1.33 1.11 1 1 1.11 1.04 1.39 1.28 4.83 1.88 1 2 1.88 1 1.08 2.09 1.22 1.64 0 1.08 1.68 1.4 3.89 2.64 5.45 1.04 1.15 1.21 1.82 1.7 1.28 1.6 1.05 1.04
20 19 6 57 6 37 100 40 25 6 100 25 100 100 87 50 57 100 100 37 25 20 100 6 43 100 50 68 100 68 100 46 94 46 68 35 6 20 6 57 6 25 68 50 100 100 100 6 46 100
1.36 2.91 1.74 1.25 1.26 1.44 2 1.13 2.29 4.4 1 1.4 1 1.9 1.29 1.33 1.11 1 1 1.11 1.04 1.39 1.28 4.83 1.88 1 2 1.88 1 1.08 2.09 1.22 1.64 1.08 1.68 1.4 3.89 2.64 5.45 1.04 1.15 1.21 1.82 1.7 1.28 1.6 1.05 1.04 1.28 2.28
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51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
40 Nodes
100 Nodes
%F O N S
O E RS ONS
%F O N S
O E RS ONS
46 100 100 37 7 6 68 100 19 20 25 46 68 40 46 57 6 100 19 68 45 43 20 46 6 7 68 2 100 2 43 6 2 37 94 100 57 50 37 19 94 68 7 25 35 68 37 7 94 57
1.28 2.28 2.38 1.04 5.4 2.25 1.11 4.79 1.11 3.71 1.18 1.67 2 2.08 1.62 1.33 1.09 0 1.13 1.8 1.63 3.09 1.47 1.4 1.07 1.6 1.09 1.09 1.68 1.48 0 2.6 1.39 0 1.13 1 1.35 0 1.55 0 1.25 2.9 1.04 0 1.47 2.16 2.56 2.4 1.03 1.31
100 37 7 6 68 100 19 20 25 46 68 40 46 57 6 19 68 45 43 20 46 6 7 68 2 100 2 6 2 94 100 57 37 94 68 7 35 68 37 7 94 57 43 68 68 100 37 50 19 25
2.38 1.04 5.4 2.25 1.11 4.79 1.11 3.71 1.18 1.67 2 2.08 1.62 1.33 1.09 1.13 1.8 1.63 3.09 1.47 1.4 1.07 1.6 1.09 1.09 1.68 1.48 2.6 1.39 1.13 1 1.35 1.55 1.25 2.9 1.04 1.47 2.16 2.56 2.4 1.03 1.31 0 0 0 0 0 0 0 0
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this problem in both robust (with two different kinds of uncertainty set) and stochastic setting and established structural properties and computational results. In particular, we showed that the robust counterpart of the CSP problem is no harder than the CSP problem as it admits a FPTAS, for both uncertainty sets. We also showed that the stochastic version can be solved through solving a number of CSP problems, where the transit times are independent random variables. In addition, if the standard deviations of transit times are equal, there exists a FPTAS algorithm.
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