We consider stochastic vehicle routing problems on a network with random travel and

Transcription

1 with Random Travel Times Astrid S. Kenyon David P. Morton Institute of Mathematics B, Technical University Graz, A-8010 Graz, Austria Graduate Program in Operations Research, The University of Texas at Austin, Austin, Texas We consider stochastic vehicle routing problems on a network with random travel and service times. A fleet of one or more vehicles is available to be routed through the network to service each node. Two versions of the model are developed based on alternative objective functions. We provide bounds on optimal objective function values and conditions under which reductions to simpler models can be made. Our solution method embeds a branch-and-cut scheme within a Monte Carlo sampling-based procedure. Introduction We consider a stochastic vehicle routing problem (SVRP) that consists of planning optimal vehicle routes to service a number of locations in the presence of random travel and service times. The system is modeled using a network whose arcs have nonnegative random travel times and whose nodes have nonnegative random service times, with distributions assumed to be known. Vehicles are uncapacitated and routes for each vehicle begin and end at a specific depot node. A route is defined as the set of arcs followed by a vehicle and the set of nodes it services. These routes are selected before knowing the random travel and service times and so that each node will be serviced. After the routes have been planned and realizations of the random travel and service times become known, the actual time to complete each route can be computed. The vehicles must follow their a priori routes; no route reoptimizations are permitted. The time at which the final vehicle returns to the depot, after all nodes have been serviced, is called the completion time. We consider two models with different objectives: The first minimizes the expected completion time and the second maximizes the probability that the operation is complete on or before a prespecified target time, T. Applications of the SVRP occur in a variety of fields. For example, Lambert et al. (1993) design routes for vehicles to collect deposits from bank branches and deliver them to a central office. In this paper, we study properties of the SVRP and present solution methods. In the remainder of this section, we briefly review related work. Section 2 provides a formal problem statement. In 3, we study properties of the SVRP: relating the models under the alternative objective functions, developing bounds, and providing conditions under which reductions to simpler deterministic models can be made. Solution methods and computational results are discussed in 4, and the paper is summarized in 5. Jaillet (1985, 1988) considers a probabilistic traveling salesman problem (TSP) with random demand. An a priori tour is constructed that includes all potential customers, and after observing the subset that requires service, the other customers are skipped. The goal is to find a tour of minimum expected length. Bertsimas and Howell (1993) further explore this problem, and Bertsimas (1992) generalizes it to a capacitated single-vehicle routing problem with random demands. For more on a priori optimization, see Bertsimas et al. (1990). Laporte et al. (1994) formulate and solve probabilistic TSPs as stochastic integer programs with so-called simple recourse (i.e., no second-stage posterior optimization is performed). Stochastic programming formulations have been developed for SVRPs in which the demand at each /03/3701/0069$ electronic ISSN Transportation Science 2003 INFORMS Vol. 37, No. 1, February 2003 pp

2 node is a random variable. In these models, the assumption is that each node must be visited, and first-stage (a priori) routes for each vehicle are to be determined. The models in the literature differ primarily in how they deal with route failure (i.e., when demand on a route exceeds the vehicle s capacity). Stewart and Golden (1983), Laporte et al. (1989), and Bastian and Rinnooy Kan (1992) consider models that constrain the probability of route failure to be at most a prespecified level. Stewart and Golden (1983), Dror and Trudeau (1986), Dror et al. (1989), Laporte and Louveaux (1990), and Bastian and Rinnooy Kan (1992) formulate models in which the expected value of a recourse function is optimized. Laporte and Louveaux (1990) use recourse decisions that, after observing the demand, optimally select points in the first-stage route when the vehicle should return to the depot. The other models are stochastic programs with simple recourse. Dynamic route-reoptimization strategies are allowed in the multistage stochastic programming formulation of Dror (1993) and the Markov decision process models of Dror et al. (1989) and Dror (1993), but the authors indicated that computational solution of these models was not possible. The stochastic component of the models described previously lies in the customer demands. For incapacitated vehicles, random demands are modeled via random node service times. The arc travel times may also be random. Leipälä (1978) studies the expected length of posterior tours of TSPs with random arc lengths. Berman and Simchi-Levi (1989) consider a variant of the probabilistic TSP in which the location of the salesman s home is to be optimized on a network with a random subset of customers requiring service and random travel times. Kao (1978), Sniedovich (1981), and Carraway et al. (1989) consider a stochastic TSP in which the objective is to maximize the probability of completing the tour by a deadline when the arcs have independent and normally distributed travel times. Laporte et al. (1992) introduce the SVRP with stochastic travel and service times. In their approach, the vehicles are incapacitated, and each node must be serviced. Each vehicle has a target time by which its route should be complete: A chance-constrained formulation ensures this with prespecified probability levels, whereas a stochastic program with simple recourse penalizes the expected value by which route travel times exceed the respective targets. Laporte et al. use a branch-and-cut approach to solve instances of the SVRP on networks with 10 to 20 nodes and 2 to 5 scenarios. Lambert et al. (1993) approximately optimize collection routes through bank branches in a network with stochastic travel times by adapting the heuristic due to Clarke and Wright (1964). As in Laporte et al. (1992), the model incorporates target route completion times. They present results for networks with 28 and 44 nodes, in which random arcs access 3 or 4 of the nodes, respectively, and can take two values (all travel times are long or all are short). We use a variant of their 28-node problem in our computational work. Like most of the work described herein, our model is static in nature (i.e., we select vehicle routes before realizing the random parameters and do not subsequently reoptimize the routes). Each node in the network must be visited by a vehicle, and arc travel times and node service times are stochastic. Even though the vehicles in the models of Laporte et al. (1992) and Lambert et al. (1993) are incapacitated, they are called SVRPs instead of stochastic multiple salesmen TSPs (perhaps to avoid confusion with probabilistic TSPs). Of the models in the literature, ours is closest to those of Laporte et al. and Lambert et al.; so, we follow their naming convention. The objective functions in most of the stochastic vehicle routing problem (VRP) and TSP models depend on the total travel costs although, as noted previously, some incorporate target completion times for each vehicle. In contrast, our models objective functions depend on the length of the longest route. Such an objective function would be most appropriate when the goal is to minimize completion time of the project. 1. Problem Statement Let G = N A be a directed graph, where N is the set of nodes and A is the set of arcs. The fleet of vehicles, indexed by k K, is based at the depot (labeled node 1), and all the routes must start and end at the depot. Each node in N \ 1 is a location to be serviced. The graph does not need to be complete, but 70 Transportation Science/Vol. 37, No. 1, February 2003

3 we assume it is feasible to visit every node and return to the depot. Let c = c ijk, i j A k K, b e the stochastic travel-time vector, and = ik i N k K, be the stochastic service-time vector. The random vector = c contains all of the random data elements in the model. Travel and service times are nonnegative random variables, and the components of may be dependent. Note that c is not assumed to be symmetric (i.e., in general, c ijk may differ from c jik because these times can depend on the direction traveled). We use to denote the support of and a realization of is denoted. We use the same type of notation to distinguish other random elements from their realizations. For example, c is a realization of the vector of random travel times c. The objective is to optimize the expectation of a function of the completion time. The operation is complete when all the vehicles have returned to the depot after finishing their routes. The completion time, denoted h x u, is thus the maximum of K random variables, namely the individual vehicles completion times, where x and u are the routing and servicing decisions ( h x u = max c ijk x ijk + ) ik u ik k K i j A i N The binary decision variable x ijk i j A k K, takes value 1 if the arc i j is part of vehicle k s route and zero otherwise. Similarly, the binary variable u ik i N k K, takes value 1 if node i is serviced by vehicle k and zero otherwise. We consider two models that differ only in their objective functions. The first model minimizes the expected completion time, Eh x u. The second model maximizes the probability of completing the operation by a given deadline T P h x u T = EI h x u T. Here, the indicator function, I, takes value 1 if its argument is true, and zero otherwise. In both models, the routes must be selected before knowing the values of the random travel and service times. Sometimes it may be desirable (or even necessary) for a vehicle to traverse a physical link in a particular direction more than once. With an obvious generalization of the notation, this is made possible by adding parallel arcs for such links. Both models are subject to the same set of constraints x ijk = x jik j N k K (1a) i j RS j i j i j A u ik j i FS j u ik = 1 i N \ 1 (1b) k K 1 i FS 1 x ijk 1 A i j FS i x ijk i N \ 1 k K (1c) x 1ik 1 k K (1d) i j A x ijk N \ 1 2 N 2 k K (1e) x ijk 0 1 i j A k K (1f) u ik 0 1 i N k K (1g) Here, FS i and RS i represent the set of arcs directed out of and into node i N, respectively (see, e.g., Ahuja et al. 1993), A = i j i j A i j, and is the complement of the index set. Constraint (1a) conserves the flow of each vehicle at each node. Constraint (1b) ensures that each node is serviced exactly once. Constraint (1c) states that a node can be serviced by a vehicle only if the node is visited on that vehicle s route. Constraint (1d) forces all the routes to include (and hence in practice start) at the depot. Constraint (1e) eliminates subtours that are isolated from the depot. We define XU to be the set of decision vectors x u that satisfy constraints (1a) (1g). The two models we study in this paper can be stated as E z E = min Eh x u s.t. x u XU and P z P = max P h x u T s.t. x u XU An example of when using completion time (instead of, e.g., total travel time) in the objective function is appropriate arises when the vehicles are leased, and returned on project completion, as a group. When leasing costs are the predominant costs, are proportional to completion time and many similar projects Transportation Science/Vol. 37, No. 1, February

4 are carried out, (E) may be a reasonable model to adopt. The objective function in (P) involves the sign, but not the magnitude of completion time less T. Sometimes this characteristic is a shortcoming, but it can also occur naturally. For instance, suppose the project is sure to be completed within one or two days and vehicle leasing costs are charged in daily increments. Then, maximizing the probability of avoiding the second cost increment may be a suitable goal. The following deterministic routing model, (M), minimizes completion time, under the assumption that the arc travel times and node service times take their mean values. Model (M) will play an important role in the next section. (M) z M = min s.t. h x u E x u XU 2. Model Properties In this section, we provide some characterizations of the models (E) and (P) developed in 1. The following proposition shows that (E) is not interesting, from a stochastic perspective, when there is only one vehicle. It states a convexity result for the objective function of (E). It relates the optimal values of (E) and (M), as well as (E) and (P). Proposition 1. (a) If K =1, an optimal solution to (E) may be obtained by solving(m). (b) H x u = Eh x u is a convex function on co(xu), where co(xu) is the convex hull of XU. (c) z M z E. (d) If T>0, then z P 1 z E T. Proof. (a) Since K =1, we have h x u = i j Ac ij1 x ij1 + i N i1 u i1. The result follows immediately from the linearity of h x u in. (b) The function h is the maximum of a collection of linear functions, and thus is convex, i.e., h x x 2 u u 2 h x 1 u h x 2 u 2 w.p.1, (2) where 0 1 and x 1 u 1 x 2 u 2 co(xu). Taking expectations on both sides of (2) yields the desired result. (c) For fixed x u XU h x u is the maximum of a collection of linear functions on co( ), the convex hull of. Thus, h x u is convex on co( ) and so Jensen s (1906) inequality implies z M = min x u XU h x u E Eh x u x u XU Minimizing the right-hand side of this expression over x u XU yields the desired result. (d) Let xe u E and x P u P denote optimal solutions to (E) and (P), respectively. A simple variant of Markov s inequality (e.g., Loéve 1963, 9) yields P ( h x E u E T ) 1 Eh x E u E T Since xp u P solves (P) and x E u E is feasible to (P), we have P h xp u P T P h xe u E T and hence the desired result. From Part (a), we see that the stochastic program (E) with one vehicle reduces to the deterministic TSP obtained by replacing the random travel and service times by their population means (i.e., model (M) with K =1). When more than one vehicle is used, this reduction is not valid because h x u is the maximum route length over all the vehicles, and hence is a nonlinear function in. In our computational results of 3, we are able to solve the instances of (E) we consider directly. That said, Part (b) of Proposition 1 is important because convexity of H x u shows that cutting-plane methods (Van Slyke and Wets 1969, Wollmer 1980) can be applied to solve (E). These methods would facilitate solving larger instances of (E). As we will see later, it does not appear that similar conclusions are available for (P). One commonly used heuristic to solve stochastic programs, such as the SVRP, is to replace the random elements with their population means and solve the corresponding deterministic model (M). An optimal solution to (M), denoted xm u M, is feasible to (E), but in general is suboptimal for (E). If we can compute or bound from above (or statistically estimate) Eh xm u M, then we can use Part (c) of Proposition 1 to bound (or develop a confidence interval for) Eh xm u M z E. Birge (1982) calls this difference the value of the stochastic solution (VSS), because 72 Transportation Science/Vol. 37, No. 1, February 2003

5 it is the magnitude by which we improve the quality of the solution by solving the stochastic program (E) instead of the mean-based deterministic approximation (M). If this difference happens to be sufficiently small, then we may deem xm u M to be an adequate solution to (E). Otherwise, it may be worthwhile to solve (E). Part (a) of Proposition 1 guarantees that the value of the stochastic solution is zero in the special case of K =1. As we will show in 3, however, this difference can be large (i.e., the quality of xm u M can be poor, for multivehicle problems). If is a discrete random variable with having a manageable number of realizations, then we can compute Eh xm u M exactly: For realization, computing h xm u M simply requires summing the travel and service times on each vehicle s route and then taking the largest value over all vehicles. If is too large for exact computation, or if is continuous, then we can apply deterministically valid upper bounds that exploit convexity of h xm u M. These include the Edmundson-Madansky bound (Edmundson 1956, Madansky 1959) and more recent extensions and alternatives (see, e.g., Edirisinghe and Ziemba 1992 and Birge and Louveaux 1997, 9). The effort required to compute such bounds can grow exponentially in the dimension of. As a result, for problems with many stochastic parameters and a large number of realizations, estimating Eh xm u M via Monte Carlo sampling may be the only viable option. We pursue this issue further in 3. The final result in Proposition 1 provides a nonvacuous lower bound on the optimal objective function value of model (P) when E h xe u E <T. Like any distribution-free result (the bound depends only on the first moment of h x u ), we can expect that the bound will be crude. In general, H x u = P h x u T is not a concave function and so, unlike (E), standard cuttingplane methods cannot be applied to solve (P). (It may be possible to develop a procedure with specialpurpose cuts as done in Laporte and Louveaux (1993), but we do not pursue this here.) However, a convexity result arises when only one vehicle is used under the assumption that the travel and service times are normally distributed. Strictly speaking, assuming is normally distributed is inconsistent with nonnegative travel and service times. However, the normal distribution may provide an appropriate probability model if the likelihood of negative times is negligible. Proposition 2. Consider the special case of (P) with K =1, where = c is distributed as a multivariate normal with mean = E and positive definite covariance matrix V. Let y = x u. Then, solving (P) is equivalent to finding the largest Rfor which the following system is feasible y XU T y T + y T Vy 0 (3a) (3b) Proof. Because h is a linear combination of normal random variables, h x u = c ij1 x ij1 + i1 u i1 = T y i N i j A is normally distributed with mean T y and variance y T Vy. Thus, is equivalent to max y XU P max P T y T y XU ( N 0 1 T T y yt Vy ) (4) where N 0 1 is a normal random variable with zero mean and unit variance. Solving (4) is equivalent to solving max y XU [ = T T y yt Vy ] (5) which may be restated as: Find the largest Rsuch that (3) is feasible. Proposition 2 is related to convexity results for chance-constrained stochastic programs: Kataoka (1963) shows that y T Vy 1/2 is a convex function, and hence shows the set of y s satisfying (3b) is convex provided 0 (i.e., provided we have at least a 50% chance of meeting the deadline). This condition is relatively easy to check. First, we solve the deterministic model (M) to find z M. By examining (5), it is clear that we can achieve 0 for some Transportation Science/Vol. 37, No. 1, February

6 x u XU if and only if z M T. Thus, the significance of Proposition 2 is that we can solve (P), when K =1, N V, and z M T, by testing the feasibility of (3) for a sequence of values of (using, for example, a bisection search) that finds the largest for which this system is feasible. Testing feasibility of each system is equivalent to solving a deterministic nonlinear integer program whose continuous relaxation is a convex program. See Geoffrion (1989) and Henig (1990) for solution approaches of this type, and see Morton and Wood (1998) for a scheme that, under certain independence and integrality assumptions, eliminates the need to have z M T. Stewart and Golden (1983) perform related reductions for capacitated SVRPs with random demands. For more results on chance-constrained programs, see Prekopa (1995). Generalizing the result of Proposition 2 to the multivehicle case does not appear possible. The objective function P h x u T, with obvious generalizations of the notation developed in Proposition 2, can be expressed P ( h x u T ) ( ) = P max T y k T k K ( = P T y 1 T ) T y K T Proposition 2 shows that when K =1, the set { ( ) } y = y 1 y K P max T y k T k K p is convex, provided p 1/2. It is easy to develop counterexamples to this statement for K Solution Methods We develop two solution methods, both centered on a branch-and-cut approach. The first method, called DESVRP, solves the deterministic equivalent of the stochastic problem and is applicable when the cardinality of the sample space is small. The second method, MCSVRP, embeds the branch-andcut scheme in a Monte Carlo sampling-based solution procedure to address SVRPs with larger sample spaces or continuous random parameters. We present results for a 9-node network and for an adaptation of a 28-node network of bank branches in Belgium (Lambert et al. 1993) Method to Solve Instances of the SVRP with Small Sample Space When the number of realizations of the travel- and service-time vector is finite, we can express (E) and (P) as large-scale integer linear programs. Let the realizations of be indexed by so that they can be enumerated = c,, with probability mass function p = P =,. Then, (E) may be written z E = min p (6a) x u s t ik u ik c ijk x ijk + i j A i N k K x u XU (6b) (6c) The continuous decision variable represents the length of the longest route under sample point. Similarly, model (P) may be written z P = max p v (7a) x u v s t ik u ik T + M k 1 v i j A c ijk x ijk + i N x u XU v 0 1 k K (7b) (7c) (7d) The binary variable v takes value one if the length of the longest route under sample point is at most T. Parameter M k is sufficiently large so that when v = 0 constraint (7b) is vacuous. Excessively large values of M k can make the integer program (7) more difficult to solve, and this is why we allow the value to depend on the scenario and vehicle. Although tighter bounds may be achieved, a value for M k that is ensured to be sufficiently large is i j A c ijk + i N ik T. Models (6) and (7) are called deterministic equivalent models. If we were to explicitly enumerate all of the subtour elimination constraints (1e) in the definition of XU, then we could, in principle, solve (6) and (7) using a branch-and-bound integer programming algorithm. However, such an approach is not practical because there are an exponential number of 74 Transportation Science/Vol. 37, No. 1, February 2003

7 constraints in (1e). Instead, we solve a sequence of relaxations of (6) and (7), identifying subtours that are isolated from the depot and appending subtour elimination constraints on an as-needed basis. This type of solution procedure is known as branch-and-cut, and began with the work of Dantzig et al. (1954) for the traveling salesman problem. For related work in vehicle routing, see, for example, Laporte et al. (1985), and for a branch-and-cut procedure for a variant of our SVRP, see Laporte et al. (1992). The proposed method, called DESVRP, is summarized below. We state the algorithm for (E); but, with minor changes, it is also applicable to (P). Step 0: Select >0 and let z =. Form (6 ) with feasible region XU, the relaxation of (6), which removes all constraints of type (1e) from XU. Step 1. Solve (6 ) to obtain solution x R u R and objective function value z. Step 2. Check x R for subtours that are isolated from the depot. If no such subtours exist, then stop: x R u R is an optimal solution to (6). Step 3. Let u F = u R. Use a heuristic that joins isolated subtours of x R to construct a solution x F so that x F u F XU. Let z = p h x F u F. Step 4. If z < z, then let z = z and x F u F = x F u F. Step 5. If z z /z, then stop: x F u F is a solution with objective function value z within 100 % of z E. Otherwise, update (6 ) by adding constraints of type (1e) to XU that eliminate infeasible subtours of x R u R. Go to Step 1. We obtain z in Step 1 by solving (6 ), a relaxation of (6) in which only a subset of the subtour elimination constraints are present, and hence, z z E. The heuristic in Step 3 constructs a feasible solution to (6) and, hence, its objective function value is an upper bound on z E. Step 4 keeps track of the smallest upper bound and the corresponding solution. The only changes required to apply the method to (P) are: (i) replace references to (6) and (6 ) with references to (7) and the analogous relaxation (7 ); (ii) in Step 3 let z = p I h x F u F T ; and (iii) recognize that solving the relaxation of a maximization problem yields an upper bound (labeled z) instep1onz P, so that, we want to track the greatest lower bound (labeled z) in Step 4 and modify the termination criterion to z z / z in Step 5. We note that more sophisticated implementations of branch-and-cut more fully integrate the addition of cuts and the use of heuristics during the branchand-bound process. See, for example, Crowder et al. (1983), Hoffman and Padberg (1991), and Padberg and Rinaldi (1991). We do this in only a very limited manner. In particular, it is unnecessary to solve (6 ) in Step 1 to within a tight error tolerance in the early iterations of DESVRP. So, these integer programs are (approximately) solved with larger tolerance values that gradually decrease so that the specified relative tolerance can be achieved. The heuristic used in Step 3 to construct a feasible solution works as follows: Each vehicle has a main (sub)tour that contains the depot node. If a vehicle also has subtours isolated from the depot, then they are sequentially processed as follows. We attempt to join the isolated subtour with the main tour by deleting one arc from each subtour and finding two arcs to add to construct a single tour. We consider all possible direct connections of this form and choose the cheapest one, assuming that arc travel times take their mean values. It is possible that no such direct connections exist. In this case, for each node in the isolated subtour, we compute the shortest path to and from the depot node, again assuming deterministic travel times. After eliminating an isolated subtour in this manner, the definition of the main tour is updated, and we process the next isolated subtour. At each iteration in Step 1, we must solve a stochastic integer program with A + N K + decision variables and more than 2 N + +1 K + N structural constraints. As a result, DESVRP can be applied only if is small enough so that we can actually solve (6 ). We have implemented DESVRP in C using the CPLEX 6.0 library (1997). When solving (6 ) in Step 1, we improve the performance of the integerprogramming solver by using the best candidate solution, x F u F, from previous iterations and pruning the branch-and-bound tree when the solution value of the current linear programming relaxation exceeds the upper bound z. We reduce the total number of iterations required in DESVRP by first solving the Transportation Science/Vol. 37, No. 1, February

8 smaller deterministic problem (M) via a branch-andcut scheme and then initiating the solution of the stochastic problem by including in XU the subtour elimination constraints generated when solving (M). In our computational work, branching first on the servicing decision variables and setting binary variables to 1 first, sped up Step 1. To provide some insight into the nature of solutions to the SVRP, we begin by considering a small problem with a fleet of two vehicles on the nine-node network shown in Figure 1. The travel time on each stochastic arc in the figure can take one of two values: The nominal value multiplied by 1, with probability p, and the nominal value multiplied by 2, with probability 1 p. The random travel time on arc i j is the same as (i.e., is perfectly correlated with) that of arc j i, the travel times on the five different random segments are independent, and the value of p is the same for each arc. Two vehicles are available for use, and the travel time on each arc is identical for both vehicles. We consider instances of this nine-node problem under four different probability distributions detailed in Table 1. As Table 1 indicates, the marginal distributions for P 1 are symmetric about the nominal travel Table 1 Travel-Time Distributions in the Nine-Node Network Distribution 1 2 p Skewness P 1 1/2 3/ P 2 1/ P 3 1/ P 4 4/ Note. This table shows four distributions P 1 P 2 P 3 and P 4, characterized by different values of 1 2 p. The random travel time on arc i j takes value 1 times that arc s nominal travel time with probability p and takes value 2 times the nominal travel time with probability 1 p. The nominal (and mean) travel time between two adjacent nodes on vertical and horizontal segments is one unit of time, whereas that of the diagonal arcs is 2 units of time. time, whereas P 2, P 3, and P 4 are distributions with increasingly long right-hand tails. By construction, (M) for each of the four distributions in Table 1 is the same. This problem has 55 distinct optimal solutions with z M = Under P 1,2of the 55 optimal solutions to (M) are also optimal to (E) and z E P 1 = E P 1 h x M u M ranges from to for the different solutions xm u M, meaning that the value of the stochastic solution under P 1 (i.e., E P 1 h x M u M z E P 1 ), can be as large as 6% of z E P 1. The routing solution displayed in Figure 2 is optimal to (E) under each of the three distributions P 2, P 3 and P 4, with an optimal objective function value z E P i = 11 i= This solution has the property Figure 1 Nine-Node Network Note. Node 1 is the depot. The service time at the depot node is zero, and every other node has a deterministic service time of one unit. A line segment between two nodes i and j represents two directed arcs, i j and j i. The nominal travel time between two adjacent nodes on vertical or horizontal lines is 1 unit of time and is 2 units of time on a diagonal arc. Solid-line segments represent arcs with deterministic travel times (i.e., the travel time is simply the nominal value). Dashed lines indicate arcs with random travel times with two realizations: the nominal time multiplied by 1 and by 2 (see Table 1) Figure 2 Optimal Solution for E with K =2 Vehicles Under P 2, P 3, and P 4 Note. The routing and servicing decisions for Vehicle 1 (2) are represented with continuous (dashed) lines and black (white) nodes. Because of the symmetry in the travel times, the routes can be traversed in a clockwise or counterclockwise manner. Also, the vehicles are identical and hence can swap routes. Thus, in terms of the decision variables used in (E), the figure actually corresponds to eight solutions with identical objective function values. 76 Transportation Science/Vol. 37, No. 1, February 2003

9 Table 2 Performance of the 55 Optimal Solutions to (M) According to the Objective Function of the Stochastic Problem (E) Distance Range Mean Variance VSS (%) P 1 [10.88, 11.54] [0.0, 6.1] P 2 [11.03, 11.96] [0.3, 8.7] P 3 [11.37, 12.52] [3.4, 13.8] P 4 [11.33, 12.74] [3.0, 15.8] Note. The range of values for E P h x M u M is given in the first column. The mean expected completion time and its variance for these 55 solutions are given in the next two columns. Finally, the range of values for VSS, as a percentage of z E P i, is given in the last column. that the completion time of Vehicle 1 is the same as that of Vehicle 2 under each scenario. As a result, the expected completion time will be the same (value 11) for any travel-time distribution in which the first moments equal the nominal values. None of the 55 optimal solutions to (M) are optimal to (E) under P 2, P 3 or P 4. To get a feel for how (E) changes with increasing skewness of the underlying distribution, we computed E P i h x M u M, i = 1 4 for the 55 optimal solutions to (M). As Table 2 indicates, the average (over the 55 solutions) expected completion time, its variance, and the value of the stochastic solution tend to increase with increased skewness of the travel-time distributions. Table 3 gives a similar, but briefer, summary for (P) for two target completion times T = 12 and T = 13. As Table 3 indicates, the value of the stochastic solution can be significantly larger for (P) than for (E). VSS, as a percentage of z P P i, isaslarge as 50% for P 1 with T = 12 and 61% for P 4 with T = 12. Table 3 Performance of the 55 Optimal Solutions to M According to the Objective Function of the Stochastic Problem P T = 12 T = 13 Distance z P P i P h x M u M T z P P i P h x M u M T P [0.5, 1.0] 1.0 [0.87, 1.0] P [0.5, 0.84] 1.0 [0.58, 1.0] P [0.59, 0.81] 1.0 [0.64, 0.91] P [0.31, 0.91] 1.0 [0.59, 0.81] Note. The columns z P P i give the probability of completing the operation by target time T for an optimal solution to (P ) for T = 12 and T = 13 for distributions P 1 P 4. The other two columns give the range of values for this probability when using the 55 solutions to (M) Method to Solve Larger Instances of the SVRP When the number of scenarios is too large, or when the distributions of the random travel times are continuous, the deterministic equivalent of the SVRP cannot be solved exactly. In this case, we are forced to resort to approximation or estimation techniques. In this section, we couple DESVRP and a Monte Carlo sampling-based approach in a method we call MCSVRP. Unlike the deterministic procedure described in 4.1, MCSVRP does not, in general, find an optimal solution to (E) or (P). Instead, it finds a solution ˆx û XU and constructs a (random) confidence interval width such that, for (E), P (Eh ˆx û ) z E + 1 (8) for 0 < <1. So, although we cannot ensure that ˆx û is an optimal solution to (E), we are confident, at a 1 level, that ˆx û has an objective function value that is at most worse than the optimal value z E. The confidence interval (8) arises from upperbound and lower-bound estimators on z E. The upper bound is simply the standard sample-mean estimator of Eh ˆx û, i.e., U n u = 1 n u n u i=1 h ˆx û i where 1 n u are independent and identically distributed (i.i.d.) as. Because ˆx û is feasible but, in general, suboptimal, we have E U n u = Eh ˆx û z E, and from the central limit theorem for i.i.d. random variables we know that U n u is asymptotically normal. The lower-bound estimator is rooted in solving (E) under an empirical measure: L = min x u XU 1 n n h x u i (9) Mak et al. (1999) show that if the objective function in (9) is an unbiased estimator of Eh x u, then E L z E. In general, L is not asymptotically normal. We circumvent this difficulty by generating n l i.i.d. observations of L and call the associated sample mean L n l. Let su 2 n u and s 2 l n l denote sample variances i=1 Transportation Science/Vol. 37, No. 1, February

10 of h ˆx û and L, t m bea1 quantile from a t distribution with m degrees of freedom, u = t n u 1 /2s u n u and l = t n l 1 /2s l n l nu nl Then, { P L n l l E L z E Eh ˆx û } U n u + u { } 1 P L n l l >E L { P U n u + u <Eh ˆx û } 1 (10) From (10), we infer the desired confidence interval (8) with = U n u L n l + + u + l, where y + = max y 0. The inequality in (10) follows from the Boole-Bonferroni inequality, which is used because the two events that the probabilistic upper and lower bounds are valid need not be independent. The in (10) arises because the statement is exact only as n l and n u grow large. We call the resulting method MCSVRP and summarize it below. Step 0. Select the sample size for the upper bound n u, and the batch size n and number of batches n l for the lower bound. Select 0 < <1. Step 1. For each sample i, i = 1 n l : (a) Generate i1 in i.i.d. as. (b) Calculate L i 1 n = min h x u ij x u XU. n j=1 Step 2. Form L n l = 1 nl L i and s 2 n l i=1 l n l = 1 nl L i L n n l 1 i=1 l 2. Step 3. Select a candidate solution ˆx û by choosing one of the solutions obtained in Step 1b. Step 4. Generate n u i.i.d. observations, i, i = 1 n u. Compute U n u = 1 nu h ˆx û i and n u i=1 s 2 u n u = 1 n u 1 nu h ˆx û i U n i=1 u 2. Step 5. To evaluate the quality of the solution ˆx û, form the confidence interval 0 U n u L n l + + u + l, in which u = s u n u t nu 1 /2/ n u, l = s l n l t nl 1 /2/ n l. The output of MCSVRP is a feasible solution ˆx û and a one-sided 1 -level confidence interval on the quality of that solution (i.e., on the optimality gap, Eh ˆx û z E ). The confidence interval width = U n u L n l + + u + l has contributions due to: (i) the difference in the upper- and lower-bound point estimates, (ii) sampling error in the upper-bound estimator, and (iii) sampling error in the lower-bound estimator. The sample size n l is typically fixed (at, say, 30), and we control the confidence interval width by selecting the sample size parameter n u and the batch size n. Because upper-bound estimation is so inexpensive for the SVRPs under consideration, n u can be selected large enough that contributions due to (ii) are dominated by those of (i) and (iii). Increasing the batch size n increases E L (Mak et al. 1999) and tends to produce better solutions ˆx û (for convergence results see, e.g., Kleywegt et al. 1999). Increasing n also decreases the variance of the objective function in (9). Thus, increasing n tends to decrease both (i) and (iii). The observations generated in Step 1 are not only independent within a batch, but also between batches. And, independent streams of observations of are used to construct the upper- and lower-bound estimators. Mak et al. (1999) discuss the merits of using common random number streams for the upper and lower bounds. However, our computational work suggests that, for the SVRP, the large values of n u that we can select more than offset any variance reduction achieved by using common random numbers. In Step 3, we must select a feasible solution, ˆx û that we will use as our solution to the problem. In principle, this solution can be selected in any manner we like; but, of course, if we pick a poor quality solution, then the width of the confidence interval constructed in Step 5 will be large. In executing Step 1b, we generate up to n l distinct feasible solutions to (E), say ˆx û 1 ˆx û n l. Our implementation of Step 3 generates a stream of i.i.d. observations 1 m for some large m and constructs the sample-mean estimator 1 m m i=1 h ˆxj û j i, for j = 1 n l. Note that common random numbers are used in each of these sample means. We select the candidate solution ˆx û to be the solution that gives the smallest point estimate. We then use an independent stream of i.i.d. observations of to (re)estimate Eh ˆx û in Step 4. Virtually all of the computational effort of MCSVRP is in the optimization performed in Step 1b. This step requires that we solve n l separate SVRPs each of the form of (6) with the n scenarios being the observations 78 Transportation Science/Vol. 37, No. 1, February 2003

11 generated in Step 1a by i.i.d. sampling and with p = 1/n, = 1 n. To solve each of these problems, we use DESVRP, with certain enhancements. In Step 2 of the DESVRP procedure, we add subtour elimination constraints. When n is sufficiently large, the different versions of (6) that we must solve have similar solutions and hence the same subtour elimination constraints are often generated. Thus, continually growing the set of subtour elimination constraints from one batch to the next decreases the number of iterations required in DESVRP and speeds the execution of Step 1b. A second computational enhancement to speed the optimization of Step 1bin the MCSVRP method concerns the computation of upper bounds on L i. As described in 3.1, tighter upper bounds can be used to accelerate solution of the integer program in Step 1 of DESVRP by aiding in pruning the branch-andbound tree. We use two methods to obtain an upper bound on L i. The first is as described in 3.1 (i.e., using feasible solutions generated by the heuristic in Step 3 of DESVRP). Second, when solving (6) under the ith batch, we have optimal solutions from batches 1 i 1, and we also compute the objective function values of these feasible solutions under the observations in batch i and use the smallest upper bound. Note that in using DESVRP in Step 1b, we do not compute L i exactly, and because the solution value is used in a lower-bounding procedure, we use z in place of L i. Choosing the tolerance in DESVRP commensurate with the relative width of the confidence interval is computationally prudent. We have implemented MCSVRP in C using the DESVRP implementation described in 3.1, with the modifications outlined previously, to solve the stochastic integer programs in Step 1b. We tested MCSVRP on the 28-node network of Lambert et al. (1993) for model (E). Figure 3 gives the underlying network for the problem. Each line segment in the figure represents two directed arcs, i j and j i, and there are a total of 276 arcs in this network. We again use K =2 vehicles so the model has 552 routing decision variables, x ijk, and 56 servicing decision variables u ik. Intercity distances, d ij, are symmetric and are shown in Table 4; these values were estimated from a map of the area. We call the city labeled Figure 3 Bank Branches Network in Belgium Note. The 28-node network represents the depot and branches of a bank in the French-speaking part of Belgium. Node 1 is the depot. and the thirteen cities above it in Figure 3 Group 1 cities and the other 14 cities Group 2 cities. Arcs terminating in Group 1 (2) cities are labeled Group 1 (2) arcs. The nominal travel speed is 0 = 25 km/hr, so that the nominal travel time on arc i j is d ij / 0. However, travel times on all the arcs are assumed to be random via c ij = d ij / 0 U, where U is a continuous uniform random variable on 1/4 7/4. The random travel times on arcs within each group are perfectly correlated. However, travel times on arcs between the two groups are negatively correlated; in particular, if c ij = d ij / 0 U for Group 1 arcs, then c i j = d i j / 0 2 U for Group 2 arcs. The service times at each node are set to five minutes, except for the depot (Node 1) that does not require service. In running MCSVRP, for the lower-bound computation, we chose a batch size of n = 30 with n l = Transportation Science/Vol. 37, No. 1, February

12 Table 4 Intercity Distances Between the Branches of the 28-Node Network Note. Because the distances are symmetric, only the upper part of the distance matrix is listed. These distances were estimated from a map of the area. 30 batches. As indicated previously, computing the upper bound is not expensive, and we chose n u = 20,000. The solution we obtained is given in Figure 4. We found L n l = 6 507, s l n l = 0 048, U n u = 6 532, s u n u = 0 195, and thus l = 0 018, u = , and a 95% confidence interval of Recall that this is a confidence interval on the optimality gap, Eh ˆx û z E, in which ˆx û is the solution shown in Figure 4. The confidence interval width is about 0.7% of (the point estimate for) z E. When the random travel times are replaced by their mean values, we found z M = 6 087, which is, by Proposition 1, a lower bound on z E. Estimating Eh x M u M via the standard sample-mean estimator with 20,000 observations gave a 95% confidence interval of 7 41 ± Thus, our point estsimate of Eh xm u M Eh ˆx û is 0 882, and the value of the stochastic solution is about 13% of z E. Note that we are able to obtain a solution ˆx û and ascertain its quality (estimated to be within 1% of optimal) by solving instances of the problem with only 30 scenarios. 4. Conclusion In this paper, we have considered an SVRP with random travel and service times. The objective function depends on the completion time of the project (i.e., the time at which the final vehicle returns to the depot after all nodes have been serviced). We developed two stochastic programming models. The first model minimizes the expected completion time, whereas the second model maximizes the probability of completing the project by a prespecified deadline. 80 Transportation Science/Vol. 37, No. 1, February 2003

13 16 21 grateful to two referees and an associate editor who helped improve the paper Figure 4 Decision Vector Selected by MCSVRP for Model (E) Note. The route and service of vehicle 1 (2) are represented in continuous (dashed) lines and black (white) nodes. Node 1 is the depot and does not require service. We have provided conditions under which these models reduce to simpler deterministic optimization models and have given bounds on the optimal solution values. A branch-and-cut algorithm is embedded within a Monte Carlo solution procedure for solving the most general forms of our SVRPs. A small nine-node SVRP indicates that solutions to the stochastic model can be significantly better than solutions obtained by solving the associated mean-value model (i.e., the deterministic vehicle routing problem in which all random parameters are replaced with their population means). Solution of a 28-node model with two vehicles and continuous random parameters suggests that modest-sized SVPRs may be solved using our procedure. Acknowledgments This research was supported by Schlumberger and by the National Science Foundation through Grant DMI The authors are References Ahuja, R. K., T. L. Magnanti, J. B. Orlin Network Flows. Prentice Hall, Upper Saddle River, NJ. Bastian, C., A. H. G. Rinnooy Kan The stochastic vehicle routing problem revisited. Eur. J. Oper. Res Berman, O., D. Simchi-Levi The traveling salesman location problem on stochastic networks. Transportation Sci Bertsimas, D. J A vehicle routing problem with stochastic demand. Oper. Res , L. H. Howell Further results on the probabilistic traveling salesman problem. Eur. J. Oper. Res , P. Jaillet, A. R. Odoni A priori optimization. Oper. Res Birge, J. R The value of the stochastic solution in stochastic linear programs with fixed recourse. Math. Programming , F. Louveaux Introduction to Stochastic Programming. Springer-Verlag, New York. Carraway, R. L., T. L. Morin, H. Moskowitz Generalized dynamic programming for stochastic combinatorial optimization. Oper. Res Clarke. G., J. W. Wright Scheduling of vehicles from a central depot to a number of delivery points. Oper. Res CPLEX CPLEX Manual, Version 6.0: Usingthe CPLEX Callable Library Includingthe CPLEX Base System with CPLEX Barrier and Mixed Integer Solver Options. CPLEX Division, ILOG, Inc., Incline Village, NV. Crowder, H., E. L. Johnson, M. Padberg Solving large-scale zero-one linear programming problems. Oper. Res Dantzig, G. B., D. R. Fulkerson, S. M. Johnson Solution of a large-scale traveling-salesman problem. Oper. Res Dror, M Modeling vehicle routing with uncertain demands as a stochastic program: Properties of the corresponding solution. Eur. J. Oper. Res , P. Trudeau Stochastic vehicle routing with modified savings algorithm. Eur. J. Oper. Res , G. Laporte P. Trudeau Vehicle routing with stochastic demands: Properties and solution frameworks. Transportation Sci Edirisinghe, N. C. P., W. T. Ziemba Tight bounds for stochastic convex programs. Oper. Res Edmundson, H. P Bounds on the expectation of a convex function of a random variable. Technical Report, The Rand Corporation Paper 982, Santa Monica, CA. Geoffrion, A. M Computer-based modeling environment. Eur. J. Oper. Res Henig, M. I Risk criteria in a stochastic knapsack problem. Oper. Res Hoffman, K. L., M. Padberg Improving LP-representations of zero-one linear programs for branch-and-cut. ORSA J. Comput Transportation Science/Vol. 37, No. 1, February

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