A TSSP + 1 decomposition strategy for the vehicle routing problem

524

European Journal of Operational Research 79 (1994) 524-536 North-Holland

Theory and Methodology

A TSSP + 1 decomposition strategy for the vehicle routing problem C h a r l e s E. N o o n

Management Science Program, The University of Tennessee, Knoxville, TN 37996, USA

John Mittenthal

Department of Decision Sciences and Engineering Systems, Rensselaer Polytechnic Institute, Troy, NY 12180, USA

R e k h a Pillai

Energy and Transportation Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Received June 1992; revised October 1992

Abstract: The basic, capacity-constrained vehicle routing problem (VRP) is to determine a set of capacity-feasible vehicle routes with minimum total travel cost so that all customers are visited by exactly one vehicle. In this paper, we introduce a new decomposition strategy for the VRP which separates the decisions of a dispatcher from those of the individual drivers. The dispatcher is responsible for assigning a reward value to each of the customer cities so that each customer will be visited by exactly one vehicle. Based on the assigned reward values together with the given travel costs, each vehicle driver is responsible for choosing which customers to visit and determining an individual feasible route. The driver's problem is modeled as a Traveling Salesman Subset-tour Problem with one additional constraint (TSSP + 1). The TSSP + 1 decomposition is based on a Lagrangian relaxation and is capable of producing valid lower bounds for the VRP. The best bound attainable is shown to be at least as good as the bound obtained by solving the linear programming relaxation of the classic set partitioning formulation of the VRP. To test the practical value of the decomposition, we present a decompositionbased heuristic and examine its performance on a collection of problems from the literature.

Keywords: Combinatorial optimization; Vehicle routing; Lagrangian relaxation

Correspondence to: Dr. C.E. Noon, Management Science Program, The University of Tennessee, Knoxville, TN 37996, USA.

0377-2217/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0 3 7 7 - 2 2 1 7 ( 9 3 ) E 0 0 5 2 - Y

C.E. Noon et al. / A TTSP + 1 decomposition strategy for the VRP

Introduction In the capacity-constrained vehicle routing problem (VRP), V vehicles are based at a single depot and must be used to make deliveries of a single product to an indexed set of customers, X = {1, 2 . . . . . n}. Each customer i must be supplied with a particular amount of product, qi, while each vehicle v has a specific limit on the total amount of product it can supply, Q~,. The problem is to determine a set of capacity-feasible vehicle routes with minimum total travel cost so that each customer is serviced by exactly one vehicle. Numerous methods have been proposed to solve this well known, difficult and practical problem (see [3] for an overview). These methods may be categorized as heuristic, exact/mathematical programming, and combination approaches. In the heuristic arena, vehicle routes have been constructed based on savings/insertion criteria [7], a cluster first-route second philosophy [14], a route first-cluster second philosophy [2], an e x c h a n g e / improvement philosophy [18,27] and a tabu search methodology [12]. Exact approaches have also been developed relying on branch and bound [6] and set partitioning [1], but their success has been limited to problems involving fewer customers. Combination approaches are usually based on an exact methodology but rely on heuristics to solve certain subproblems. These methods have been successful in producing good solutions in reasonable time without excessive use of computer storage space. Foster and Ryan [10] formulate and solve the VRP as a set partitioning problem where each column represents a possible vehicle route. Due to the huge number of possible routes, the set partitioning problem is solved over only a heuristically generated subset of columns. Fisher and Jaikumar [9] formulate and solve the customer-to-vehicle assignment problem as a generalized assignment problem (GAP) and then sequence the individual routes using a TSP algorithm. Stewart and Golden [28] formulated the VRP as an m-TSP with additional constraints corresponding to vehicle capacities. In their solution, these additional constraints are relaxed using Lagrangian relaxation and the resulting relaxed problem is solved using an arc exchange heuristic. In this paper, we present a decomposition

525

strategy for the VRP and test its effectiveness through the development of a new combination approach. Under this strategy, the challenge of the dispatcher is to assign a reward value to each of the customer cities so that each customer will be visited by exactly one vehicle. Based on the assigned reward values together with the given travel costs, each vehicle driver is responsible for choosing which customers to visit and determining an individual feasible route. The decomposition is based on a Lagrangian relaxation and is capable of producing valid lower bounds for the VRP. We show that for the case of identical vehicles, this relaxation is capable of producing lower bounds that are superior to the bound produced by the linear programming relaxation of the set partitioning formulation for the VRP. Details of the formulation and decomposition are provided in the next section along with the results on lower bounding. In Section 2, a new combination approach based on our decomposition strategy is presented for the capacity-constrained VRP. Section 3 summarizes the performance of the heuristic on a collection of problems from the literature. Section 4 contains our summary and conclusions.

1. Problem formulation and decomposition Let 5~' represent the set of possible travel arcs among the customers (i = 1, . . . , n) and depot (i = 0). We assume that traveling from any customer i to any other customer j is permitted. In effect, ~¢= { ( i , j ) 1 i = 0 ,

1, 2, . . . , n; j - 0 ,

1, 2, . . . , n;

i~j}. Let ciL) be the cost for vehicle v (v = 1, 2, . . . , V) to travel from customer i to customer j for all (i,j) ~e,¢. It is assumed that I CI~ [ < + ~ for all (i,j) ~.~" and v = 1, 2 . . . . . V, and c~) need not equal cyr Let the decision variable, x~), be defined for all vehicles v and for all (i,j) ~.ae such that xy~- is 1 if vehicle v travels directly from customer i to customer j and 0 otherwise. The VRP formulation we present below is more general than the basic capacity-constrained VRP. For each vehicle, we allow the set of feasible routes to be restricted by a single general

C.E. Noon et al. / A TTSP + 1 decomposition strategy for the VRP

526

constraint defined over the travel arcs. The general constraint can be used to enforce a vehicle capacity limitation, as 'in the capacity-constrained VRP, or a limit on some other route characteristic such as time, length, or number of stops. The general constraint is given such that aij~' is the constraint requirement for vehicle v (v = 1, 2 . . . . . V) to travel from customer i to customer j for all (i,j) ~a¢, and b,, is vehicle v's route constraint limitation. Our formulation (P) is similar to vehicle flow based formulations given in [17] and [23].

(P) v

Min ~

~

ci~xi~)

v= l (i,j)~A s.t.

E

v = l i=0

~+

i~j

~ =2

E

for all j ~ ./V,

v = l k=O

k~j

(1) n

E x;j = 1,

(2 /

sure that each vehicle leaves the depot, visits each customer j at most once, and returns to the depot. Constraint set (5) enforces flow conservation for each vehicle tour. Subtour elimination constraints are given in (6) and the general route limitation constraints are given in (7). Note that a capacity-constrained VRP can be derived from (P) by assigning b v equal to the vehicle capacity Qv and setting airy = l ( q i + qj) for all v where qi is defined as the amount of product to be delivered to customer i and q0 is defined to be 0. Finally, the binary nature of the variables is described in (8). In this formulation, constraint set (1) is the only set which contains constraints that involve decision variables corresponding to more than one vehicle. As was observed in [23], when these constraints are relaxed, the problem decomposes into V smaller, specially structured problems one for each vehicle. Our decomposition strategy is based on exactly this Lagrangian relaxation of problem (P). Let (P,~) be the relaxation obtained by taking constraint set (1) into the objective function via unrestricted Lagrange multipliers ~(assume ~-0 is fixed at 0). Define (P,~) as follows.

j=l

~xi~)< 1 for all j ~ V ,

(3)

i=0 i ~j

Min E E (¢~i--71"i--Trj)x~j + 2 E "lTi v=l (i,j)~" i¢X

XYo= 1, i=l ~

(4 /

n Xi~j -- E Xjtk = 0

i=o i =~j

[~[

for all

j ~ v¢/',

(5)

1

for all subsets S~'___./V,

jei

Is"l >2, E

a ivj x ivj _<

(6)

by,

(7)

(i,j) ~

Xij - a

0

or 1 for all

s.t.

(2)-(8)

k=0 k #j

E E l , x
u

v

(i,j) ~s¢,

(8)

where, in addition, constraint sets (2)-(8) are constructed over v = 1, 2, . . . , V. The objective function minimizes the total cost for servicing all customers. The constraint set (1) ensures that each customer is visited exactly once. Constraint sets (2), (3) and (4), respectively en-

over v = 1, . . . , V.

Problem (P~) can be solved as V independent vehicle problems. Each of these independent vehicle problems is a Traveling Salesman Subset-tour Problem with one additional constraint (TSSP + 1). The Traveling Salesman Subset-tour Problem (TSSP) is a relaxed variant of the TSP in which a feasible tour need not visit every customer. The TSSP differs from the well-known TSP in that the salesman is not required to visit every customer, but rather the salesman chooses which customers to visit. Many problems, such as the prize collecting TSP [8], the orienteering problem [16,1921,29], and the time constrained TSP [13,15], can be viewed as TSSP's with one additional constraint (TSSP + 1). See [24] for more details. In the dispatcher/driver analogy, the dispatcher controls the values of the multipliers

C.E. Noon et al. / A TTSP + 1 decomposition strategy for the VRP

(rewards) corresponding to each customer. The goal of the dispatcher is to assign a reward value to each of the customer cities so that each customer will be visited by exactly one vehicle. Based on the assigned reward values together with the given travel costs, each vehicle driver then is responsible for choosing which customers to visit and determining an individual feasible route, i.e., each driver solves a TSSP + 1 problem over the Lagrangian adjusted objective function. There are a number of observations which can be made between our strategy for decomposition and that of Fisher and Jaikumar [9]. Both solution approaches decompose the VRP into a dispatcher level (or master) problem and a collection of driver level problems (or subproblems). Fisher and Jaikumar decompose the problem so that the dispatcher is responsible for making capacity-feasible customer-to-vehicle assignments and the drivers are responsible for sequencing their individual routes. The dispatcher level problem is modeled as a GAP while the resulting dispatcher level problems are modeled as TSPs defined over the dispatcher-assigned customers. There are several limitations to this approach. First, the GAP decomposition will only be valid when the VRP is defined with respect to vehicle capacity constraints. Problems with vehicle route limitations based on distance or time cannot be handled easily within the GAP framework. Second, Fisher and Jaikumar point out that there is no easy way to analytically characterize the GAP objective function. This means that valid lower bounds cannot be easily computed using their decomposition. In contrast, our problem decomposition places a greater share of the decision making on the individual vehicle drivers. Specifically, each driver will independently route the customers that h e / s h e has chosen to service while the dispatcher is simply responsible for setting the 'rewards' associated with each customer. There are a number of advantages to viewing the VRP in this way. First, additional routing constraints, such as the length of the routes (in time or distance) or a limit on the number of customers that can be serviced, may be included into the driver level subproblems. Note that the resulting subproblems are TSSP's with one or more additional constraints. Hence, maintaining our general approach for more complex problems is possible as long as a method exists for solving the

527

resulting subproblems. Second, since each subproblem is independent of the others, applying a solution method to the subproblems may naturally be implemented in a parallel fashion. Last, methods for obtaining lower bounds on each TSSP + 1 subproblem exist, see [8], [20], [21] or [25]. These individual TSSP + 1 bounds could be used for bounding the VRP since any lower bound on u(P~) will be a lower bound on v(P), where v(.) is the optimal objective function value of problem (-). In the next section, we show analytically that the lower bounds attainable through (P~) can be quite strong.

Relationship to the set partitioning formulation The problem (P~) provides a means for computing a valid lower bound for the heterogeneous-vehicle VRP. In general, given a set of multipliers we need to solve one TSSP + 1 problem for each vehicle. However, in the special case that the cij. . .s,. . .aij s, and b~.'s are the same across vehicles, we only need to solve one TSSP + 1 problem since the subproblems are identical. In this case, v(P~) may be computed by multiplying the TSSP + 1 subproblem's optimal objective value by V and adding the t e r m 2]~ i ~ .W."Wi.In this section, we take a closer look at this special case and show that t h e lower bound attainable using (P=) is at least as good as the lower bound obtained by solving the linear programming relaxation of a set partitioning representation of the problem. In the set partitioning formulation for the VRP, a 0-1 variable is associated with each feasible vehicle route, ket Yr be the 0-1 variable associated with route r for r = 1. . . . . R and let 3 r represent the set of arcs traversed in route r. The cost associated with route r can be easily computed as dr=

E ciL) • (i,j) ~ 3-r

Also, define eir = 1 if customer i is visited along route r and zero if not. The basic set partitioning formulation for the VRP, denoted as (Pp), is given below. (Pp) R

Min ~ d~y r r=l

C.E. Noon et al. / A TTSP + 1 decomposition strategy for the VRP

528 Soto

R

eirY r = 1 for all i ~ X ,

show that the TSSP + 1 subproblem solved over the Lagrangian adjusted costs,

E

r=l

y~=0orl

forr=l,

...,R.

In the above formulation, it is assumed that the number of vehicles is free. The linear programming relaxation of (Pp), denoted (Pp), often provides a tight lower bound on v(Pp), see [1]. A major difficulty, however, in using set partitioning approaches for the VRP is that the number of columns grows very quickly as the number of customers increases. Due to the large number of columns, explicit problem representation for even medium-sized problems is not possible. Column generation has been used to implicitly represent the problem but has been successful on only very small capacity-constrained VRP's (see [1]). In the following theorem, we establish the relationships among the problems mentioned so far. Theorem 1. For problems (P) and (P,~) with identical vehicles, and problems (Pp) and (Pp) as defined, the following are true: a) v ( P p ) _< v ( P p ) __< v(P); l-b) v(P~)= v(Pp) with "~i ----- ~tz i where -fii is the optimal dual variable associated with the i-th (i X ) constraint of (Pp); and C) /-+(Pp) __~max+~ v(P+~) _< v(P). Proof. For part a), v(Pp) is the LP relaxation of

v(Pp) so the first relation holds. Since the number of vehicles in (Pp) is free, this problem is a relaxation of (P), where the number of vehicles is fixed, and so v(Pp) < v(P). For part b), observe that icX

since at LP optimality the primal and dual objectives must be equal. By definition of ~'i, we have v(Pp) = 2 E ~'ii~

Therefore, the objective function of (P~_) may be written as v

min ~

Y'~ (ci~)-~'i-~'j)x~) + v(Pp).

v = 1 (i,j) ~ . ~

To establish the desired result, we need to

(i,j) E~¢ will have an objective value of zero. For any route r which is feasible for the TSSP + 1 subproblem, there exists a corresponding column in (Pp). Further, the Lagrangian adjusted cost of route r may be given as E ( c i V j ' - - ~ i - - ~ j ) - - d r - 2 ~ eir~'i (i,j) ~ ~ i=1 =d r-

~ eir-~i, i=1

i.e., the LP reduced cost of the column corresponding to route r in an optimal solution of (Pp). Therefore, there cannot exist a negative cost solution to the TSSP + 1 subproblem since this would contradict the assumption of LP optimality. Using the same relationship, any column in the final basis of (Pp) will correspond to a zero cost feasible solution to the TSSP + 1 with Lagrangian adjusted costs. Therefore, the optimal objective value of the TSSP + 1 subproblem will necessarily be equal to zero. For part c), the first inequality holds since ~may not necessarily be the set of multipliers which results in the largest lower bound, while the second holds from Lagrangian relaxation theory. [] From this theorem, we see that the TSSP + 1 decomposition is capable of producing a lower bound that is superior to the bound obtained by solving the LP relaxation of the full statement of (Pp). This bound can be obtained by choosing an appropriate set of multipliers and optimally solving a TSSP + 1 over the Lagrangian adjusted objective function. It should also be pointed out that the TSSP + 1 subproblem can be used within a set partitioning approach to either find a negative reduced cost column or to establish that none exist.

2. A heuristic for the capacity-constrained VRP with identical vehicles

In order to examine the practical value of our strategy, we developed and tested a heuristic

C.E. Noon et al. / A TTSP + 1 decomposition strategy for the VRP

based on the TSSP + 1 decomposition. As was noted earlier, for the case of identical vehicles, (P~) can be solved for a given vector of ~- by solving only one of the TSSP + 1 subproblems. Although this appears attractive because we need to solve only one subproblem as opposed to V, the resulting solution is not likely to provide a feasible u p p e r bound. In our heuristic, we overcome this issue by preassigning one customer to each vehicle. These preassignments result in each subproblem being distinct, and therefore contribute to finding feasible u p p e r bound solutions. Define ( P ' ) as the problem (P) with customer j assigned to vehicle j for j = 1 . . . . . V. Problem ( P ' ) may then be formulated as (P) with the additional set of constraints n

E x[j + i=0

i4=j

xJ1, = 2

for j = 1, 2 . . . . .

V,

(9)

k=0

k4=j

which simply enforces vehicle j to service cust o m e r j for j = 1 . . . . . V. Although these preassignments restrict the solution space of the original problem (P) they can be useful in developing results applicable to (P). Note that if x, an optimal solution for (P), is feasible for ( P ' ) then y, an optimal solution for (P'), is also optimal for (P). As a result, in this instance, bounds and solutions developed for ( P ' ) will be valid for (P). For many problems it is possible to identify customers that are not likely to be serviced by the same vehicle based on their geographical proximity or the weight of their delivery demands. Assuming we have preassigned V such customers, then (P'), the related Lagrangian relaxation of (P'), can be solved by solving V distinct TSSP + 1 problems. Note that our preassignment of customers to vehicles serves a very different purpose than the selection of 'seed' points in [9]. In that work, seed points were chosen so as to provide good approximations to the distance function used within the GAP. In our approach, we seek to identify customers who cannot be serviced by the same vehicle, and then preassign these customers to distinct vehicles. This ensures that each vehicle's initial tour contains at least one customer, and possibly that this preassignment may be charac-

529

teristic of an optimal solution to (P). The end result is that a vehicle is now committed to serving its preassigned customer and the job of the dispatcher is simplified to altering the rewards (multipliers) of the non-preassigned customers so that each is serviced by a vehicle. The decomposition heuristic begins by preassigning V customers and then employs an iterative subgradient algorithm for computing multipliers and for finding a feasible solution. If the subgradient algorithm converges to a feasible solution, a new set of preassignment customers is determined and the procedure is repeated. The process continues in this manner until either the subgradient algorithm fails to converge or converges to an inferior solution. In each iteration of the subgradient algorithm, ( P ' ) is solved heuristically and its solution is used to alter the n - V multipliers corresponding to the remaining customers not preassigned. The heuristic we use to solve each of the V TSSP + 1 problems when solving ( P ' ) is a combination of 3-opt for the symmetric TSP [22] and an i n s e r t / delete heuristic described in [24]. The basic premise for this heuristic is that a customer may be inserted into, or deleted from, a route as long as route feasibility is maintained or the amount of infeasibility with respect to the one additional constraint of the TSSP + 1 problem is decreased. The TSSP + 1 Decomposition Heuristic. Step O. Determine preassignments for V customers. Set UB = + oo. Step 1. Index the customers so that customer i is preassigned to vehicle i for i = 1. . . . . V. Initialize 7rj = 0 for j = V + 1 . . . . . n and l = 0. Set max = 2000. Step 2. Set I = l + 1 and heuristically solve (P'). Let ~ be the resulting solution to (P'). Let w be a vector of n components whose j-th component is

2- E v=l

E i=0

i~j

v=l

k=0

k4~j

If ~ is feasible for (P'), i.e., each customer is visited by exactly one vehicle and so w = 0, go to Step 4. Otherwise, set

A t = 50/(1°.711wll )

C.E.Noonetal. / A TTSP+I decompositionstrategyfor theVRP

530

and for j = V + 1. . . . . according to

n recompute multipliers

{ V n

makes these customers more attractive in the next iteration. For customers visited by more than one vehicle, the corresponding multipliers are decreased, making them less attractive. If a set of multipliers is selected so that each customer is visited by exactly one vehicle, a set of feasible routes has been achieved and the subgradient algorithm returns the feasible solution. The initial preassigned customers are replaced by a new set and the whole process is repeated. This overall process continues until the subgradient algorithm fails to find a feasible solution or finds one that is inferior to the previous solution. The preassignment of customers mentioned in Step 0 may be accomplished in one of two ways. As the first alternative, it is possible to look at the distribution of customer locations and simply choose V of them to be the route preassignments. This 'eyeball' method works well when the problem instance involves clustered customer locations. In the other alternative, we first identify the convex hull of customer locations, remove and save these locations until we have collected at least 3V of them. Then we solve the problem of selecting V customers from this set such that the minimum distance between any two of these customers is maximized. This problem can be solved in O ( n 3) when certain conditions on the convex hull of points are satisfied. For details, see [26].

V~)

v=l i=0

v=l k=0

i*j

k*j

and go to Step 3. Step 3. If 1 --- max, stop. Otherwise, go to Step 2. Step 4. If c$ < UB, select as new preassignmerits those customers that are visited immediately after the previous preassignments on the routes defined by $. Set UB = c~ and go to Step 1. Intuitively, the algorithm works as follows. The dispatcher preassigns one customer to each vehicle and sets the 'customer reward values' (the ~-'s) to zero for the remaining n - V customers. For cost values reflecting Euclidean distance, each driver will simply want to leave the depot, visit h i s / h e r preassigned customer, and then return to the depot. This completes the first subgradient iteration. In subsequent iterations, the dispatcher looks at the number of times a customer was visited and adjusts the 'customer reward values' accordingly. The multipliers for customers who are visited by exactly one vehicle are not changed. For customers not visited by any vehicle, the corresponding multipliers are increased which

2

O

O O

1~

O0

0

0 o

0

0

0

0 0

• DEPOT

Oo

o

Oo 0 0

@

•

O

3 Figure 1. A 3-vehicle 32-customer problem with numbered preassignments selected from the shaded points

531

C.E. Noon et al. / A TTSP + 1 decomposition strategy for the I/RP 2

11~-~l~ .............

0-

......... . . . . .

0

il ..--" o

oo" .....-'°",0 1

I

s S sJ"

1

"'.

I~s ~ I

ss S

f

I

~

I

I

0

3

i¢ Figure 2. Set of routes half-way through the subgradient algorithm

Figures 1, 2 and 3 display a 3-vehicle 32customer problem at various stages of the overall approach. Data for the problem are given in the Appendix. Figure 1 shows how the customer preassignments are determined. The seven customer locations shaded black represent the convex hull of locations. Since seven is less than 31/, these locations are removed and the convex hull of the remaining points is determined to be the four gray shaded points. The number of convex hull points now totals eleven which is greater than 31/.

The three numbered points represent the preassignments chosen from the eleven convex hull points based on maximizing the minimum interpoint distance. Figure 2 displays a set of routes reached approximately half-way through the subgradient algorithm. The points shaded black are those visited by exactly one vehicle. The grayshaded point is visited by two vehicles and the unshaded points are not visited by any of the vehicles. Figure 3 displays a final set of feasible routes for the problem.

O

I I

I

,I ,

/ / I

,

jJ ;

j"

'I

s Sr

II . ~

It Figure 3. Final set of feasible routes

"-.

/ -., / "Oa

C.E. Noon et al. / A TTSP + I decomposition strategy for the VRP

532

In any subgradient iteration, the heuristic solution to ( P ' ) requires the use of a TSSP + 1 solver for each vehicle. This solver begins by setting the initial route to be the result from the previous iteration. In the first iteration, this is simply the route from the depot to the preassigned customer and back to the depot. The costs for this problem are adjusted for the impact of the dispatcher's updating of rewards and then this initial route is checked for improvements via a 3-opt heuristic. Three routes are generated from this initial, possibly improved, route. The first is the result of the i n s e r t / d e l e t e heuristic. The second is obtained by setting the vehicle capacity to 60% of its original value, applying the i n s e r t / d e l e t e heuristic, and then resetting the capacity to its original value and resolving the problem. The effect of this procedure is to force expensive customers out of the route so that more cost effective customers may be included. The third route is obtained by setting the vehicle capacity to 140% of its original value, applying the i n s e r t / d e l e t e heuristic, and then resetting the capacity to its original value and resolving the problem. The effect of this procedure is to enlarge the set of customers on the route so that the heuristic may then weed out the more expensive customers. Finally, the costs of these three routes are compared and the route with the least cost is selected.

3. Computational results We tested our approach on a set of problems available in the literature, and compared our solutions with those reported by other researchers. The set of problems attempted using the decomposition heuristic is described in Table 1 according to customer and vehicle data source in the literature, and two algorithm performance measures. Included as part of the customer and vehicle data is a column, Capacity utilization, which represents the percentage of total vehicle capacity required to satisfy all the customer demands. The two measures of algorithm performance are CPU time, measured in seconds on a VAX 9000, and the number of passes, i.e., the number of times preassignments were established, required until the algorithm terminated. If the number of subgradient iterations within the first pass reached 2000, e.g., as in problems 6 and 10, then the algorithm terminated without ever having achieved a feasible set of routes. The problems in Table 1 can be further categorized by the distribution of the customer locations over the delivery region. Problems 5-11 and 14 are instances where customer locations are somewhat evenly distibuted over the region; problems 12 and 13 are instances where the customer locations are primarily clustered into groups; and problems 1-4 are instances which share aspects

Table 1 Test problem characteristics Problem

N u m b e r of customers

Vehicle capacity

No. of vehicles

Capacity utilization

Source

Decomp. C P U sec. a

Heuristic passes

1 2 3 4 5 6 7 8 9 10 11 12 13 14

22 29 32 32 50 75 75 75 75 100 100 100 120 150

4 500 4 500 8 000 38 000 160 100 140 180 220 112 200 200 200 200

3 3 4 3 5 14 10 8 7 14 8 10 7 12

75.5 94.4 91.8 86.5 97.1 97.4 97.4 94.7 88.6 93.0 91.1 90.5 98.2 93.1

[ 11 ] [11] [11 ] Table 2 [4] [12] [4] [12] [12] [12] [4] [5] [5] [5]

1.02 4.93 26.35 34.78 101.44 451.40 143.47 25.78 573.91 52.89 1484.79 2705.89

4 2 3 5 6 1 2 4 2 1 2 2 3 5

a V A X 9000 seconds.

C.E. Noon et al. / A TTSP + 1 decomposition strategy for the I/RP

of both these two extremes. In addition, the data for problem 4 occur in a real-world application and are provided in Table 2. Also, problems 1-3 represent relaxations of their original problem descriptions in the literature, where a constraint related to the maximum distance traveled on a route is ignored. Table 3 lists known solution values of the best performing heuristics on this set of test problems. All of the solution approaches have been mentioned earlier, but it should be noted that the values reported for Christofides, Mingozzi and Toth correspond to the best of their tree and 2-phase algorithm solutions. In addition, in all of these solutions, except where noted, the number of vehicles used is equal to the value reported in Table 1.

We report two columns of objective values for the decomposition heuristic. The column labeled Rounded represents final route costs calculated using cost coefficients, cir., that reflect Euclidean distances rounded to the nearest integer, while the column labeled Real represents the cost of the same routes calculated using the actual Euclidean distances. The inclusion of the latter column allows the future comparison of results on the basis of 'actual' distances traveled, free of bias introduced from cost rounding or truncation. With respect to algorithmic performance, observe that the TSSP + 1 decomposition heuristic yields best known solutions in six of the 14 problems and performs competitively, i.e., the solution value is within 3% of the best known solution, in the remainder of the problems for which it con-

Table 2 Data for problem 4. Depot coordinates are (250,200). There are three available vehicles, each with capacity 38000 Customer number 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

533

x

y

10 313 467 205 275 269 293 333 304 286 288 295 50 484 447 215 22 267 391 399 363 355 378 458 383 240 273 278 352 324 249 65

260 382 67 254 34 262 269 212 202 207 191 235 249 179 189 204 255 316 196 122 187 236 203 218 181 326 349 374 271 295 250 248

Demand 3500 25705 713 2267 447 1847 1437 3720 1115 273 5494 1944 250 1500 3585 140 629 479 17456 1143 1919 826 3264 1570 2215 1239 580 5000 100 201 6747 1260

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C.E. Noon et al. / A TTSP + 1 decomposition strategy for the VRP

ve r g ed . A m o n g t h e p r o b l e m s w h e r e the c u s t o m e r locations are evenly d is t r ib u t e d o v e r t h e delivery region, t h r e e best k n o w n solutions w e r e o b t a i n e d , t h r e e solutions w e r e c o m p e t i t i v e a n d in two problems the a l g o r i t h m did n o t c o n v e r g e ( D N C ) . A m o n g t h e two c l u s t e r e d c u s t o m e r l o c a t i o n p r o b lems, two c o m p e t i t i v e solutions w e r e achieved. A m o n g t h e four m i x e d c u s t o m e r location p r o b lems, which are p e r h a p s m o r e r e p r e s e n t a t i v e o f real p r o b l e m s , t h r e e best k n o w n solutions a n d o n e c o m p e t i t i v e solution w e r e o b t a i n e d . O f course, t h es e are t h e p r o b l e m s with w h i c h we have at m o s t o n e solution a p p r o a c h to c o m p a r e . T h e two p ro b l em s , n u m b e r e d 6 a n d 10, for which o u r a l g o r i t h m did not c o n v e r g e are worthy o f discussion. F o r b o t h p r o b l e m s , the c u s t o m e r s are s o m e w h a t evenly d is t r ib u t e d o v e r a s q u a r e r e g i o n an d t h e n u m b e r o f vehicles is at its greatest level, 14. T h e s e characteristics t o g e t h e r m a k e it difficult to preassign c u s t o m e r s that are 'sufficiently' apart. T h e result is that virtually all o f th e n o n - p r e a s s i g n e d c u s t o m e r s could be r e a s o n a b l y serviced by m o r e t h a n o n e vehicle. In such a situation, o u r a l g o r i t h m w o u l d p e r h a p s r e q u i r e an e n o r m o u s n u m b e r o f iterations to fine t u n e t h e costs so that e a c h c u s t o m e r w o u l d be visited by exactly o n e vehicle. It is reassuring, however, that the type of p r o b l e m for w h ic h the a l g o r i t h m did not c o n v e r g e is p e r h a p s least r e p r e s e n t a t i v e of real-world p r o b l e m s . F u r t h e r m o r e , it w o u l d be

s t r a i g h t f o r w a r d to i n c o r p o r a t e an e n d - p r o c e d u r e which, af t er t h e a l g o r i t h m has r e a c h e d its maxim u m n u m b e r o f iterations, w o u l d first e n s u r e that all c u s t o m e r s are serviced by at least o n e v eh i cl e an d t h e n for e a c h multiple-visited cust o m e r d e t e r m i n e t h e most a p p r o p r i a t e r o u t e to r e t a i n that cu st o m er . A s a final c o m m e n t , n o t e that a l t h o u g h the C P U t i m es for m a n y o f t h e s e test p r o b l e m s are q u i t e large, s p e e d - u p s are possible if efforts w e r e to be m a d e to o p t i m i z e t h e a l g o r i t h m ' s coding. W e m a d e no such effort since o u r p r i m a r y interest is in establishing t h e ef f ect i v en ess o f t h e dec o m p o s i t i o n strategy. A d d i t i o n a l s p e e d - u p s are, o f course, possible by p a r a l l e l i z i n g t h e p r o c e d u r e to solve the individual s u b p r o b l e m s .

4. Summary

and

conclusions

In this p a p e r , we i n t r o d u c e d a V R P f o r m u l a tion w h i ch i n c o r p o r a t e s a set o f V T S S P + 1 p r o b l e m constraints an d d e m o n s t r a t e d how t h e special s t r u c t u r e of this f o r m u l a t i o n gives rise to a n ew d e c o m p o s i t i o n strategy for solving t h e V R P . This n e w strategy has a n a t u r a l i n t e r p r e t a t i o n w h e r e the d i s p a t c h e r sets a r e w a r d associated with e a c h c u s t o m e r while t h e v e h i c l e drivers r o u t e the c u s t o m e r s that they have selected. T h e dec o m p o s i t i o n allows for the inclusion o f f u r t h e r

Table 3 Comparison of algorithms based on solution values Problem

1

2 3 4 5 6 7 8 9 10 11 12 13 14

Number of customers (vehicles) 22 (3)

29 (3) 32 (4) 32 (3) 50 (5) 75 (14) 75 (10) 75 (8) 75 (7) 100(14) 100 (8) 100 (10) 120 (7) 150 (12)

Foster & Ryan -

521 b 1081 852 760 692 1116 825 -

Fisher & Jaikumar -

524 857 833 824 1014 b

Christofides, Mingozzi & Toth a -

534 871 c 851 816 b 1066 1064

Stewart & Golden

Gendreau, Hertz & Laporte

-521 b 1058 c 847 751 692 1117 829

521 b -832 b -815 b 824 1035 b 1024

-

--

a Best solution value from either their tree or 2-phase algorithm. b Best known solution. c No. of vehicles used is 1 more than the number reported in Table 1.

Harche & Raghavan

TSSP + 1 decomp. heuristic Rounded Real

569 b 534 b 492 b 2005 b 521 b 1042 b 847 751 695 1113 b 825 1045 1070

569 b 545 492 b 2005 b 521 b DNC 845 739 b 690 b DNC 831 824 1037 1041

568.56 548.87 494.55 2006.34 524.93 852.22 746.00 695.94 841.56 824.05 1050.08 1057.30

C.E. Noon et al. / A TTSP + 1 decomposition strategy for the VRP

constraints without affecting the structure of the solution procedure. The only requirement is that there exists a solver for the new, corresponding driver subproblems. This issue will be taken up in future work. The decomposition is based on a Lagrangian relaxation which is capable of producing valid lower bounds for the VRP. For the case of identical vehicles, it was shown that the relaxed problem is capable of producing a lower bound which is greater than or equal to the lower bound obtained by solving the linear programming relaxation of a set partitioning representation of the problem. We developed and tested a decompositionbased heuristic for the capacity-constrained VRP with identical vehicles. From the comparison of heuristic solutions in Table 3, it is clear that the TSSP + 1 decomposition approach is effective across the three categories of problem types. In six of the 14 problems, the heuristic achieved best known solutions, while in the rest of the problems for which it converged solutions close to the best known were obtained.

Acknowledgment The authors wish to thank the associate editor and several anonymous referees for their helpful comments which resulted in a much improved paper.

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