A tabu search method for the truck and trailer routing problem

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Abstract

A solution construction method and a tabu search improvement heuristic coupled with the deviation concept found in deterministic annealing is developed to solve the truck and trailer routing problem. We test our tabu search method on 21 problems that have been converted from the basic vehicle routing problem. Our construction method always solves a problem (it always finds a feasible solution) and the tabu search improvement heuristic significantly improves an initial solution.

Scope and purpose

The vehicle routing problem holds a central place in distribution management and logistics, and its practical significance has been well documented in the literature. The truck and trailer routing problem is a variant of the vehicle routing problem to take into account some real-life applications in which fleet of mk trucks and ml trailers (mkml) services a set of customers. Some customers can be serviced by a complete vehicle (that is, a truck pulling a trailer) or by a truck alone, whereas others can be serviced only a truck alone. There are three types of routes in a solution to the problem: (1) a pure truck route traveled by a truck alone, (2) a pure vehicle route without any sub-tours traveled by a complete vehicle, and (3) a complete vehicle route consisting of a main tour traveled by a complete vehicle, and one or more sub-tours traveled by a truck alone. A sub-tour begins and finishes at a customer on the main tour where the truck uncouples, parks, and re-couples its pulling trailer and continues to service the remaining customers on the sub-tour. The objective is to minimize the total distance traveled, or total cost incurred by the fleet. The problem is more difficult to solve than the basic vehicle routing problem, but it occurs in many real-life applications. The purpose of this article is to develop a solution method that generates an initial solution and improves the solution using tabu search. The tabu search procedure uses the deviation concept found in deterministic annealing to further improve the initial solution. Our heuristic solves the truck and trailer routing problem efficiently and effectively.

Introduction

In the basic vehicle routing problem (VRP), routes are constructed to dispatch a fleet of homogenous or heterogeneous vehicles to service a set of customers from a single distribution depot. Each vehicle has a fixed capacity and each customer has a known demand that must be fully satisfied. Each customer must be serviced by exactly one visit of a single vehicle and each vehicle must depart from and return to the depot. Each route has a route length constraint that limits the distance traveled by each vehicle. The objective is to provide each vehicle with a sequence of visits so that all customers are serviced and the total distance traveled by the fleet (or the total travel cost incurred by the fleet) is minimized. The basic VRP and its many variants are NP-hard optimization problems [1] and have received a great deal of attention in the literature [2], [3], [4], [5].

The truck and trailer routing problem (TTRP) extends the basic VRP to take a real-life application into account. In the TTRP, a fleet of mk trucks and ml trailers (mkml) services a set of customers from a central depot. Each truck and each trailer has a fixed capacity of Qk and Ql, respectively. A complete vehicle (a truck pulling a trailer) has a total capacity equaling of Qk+Ql. The locations of the customers vary from a city center to a mountain village and some of these locations make maneuvering a complete vehicle unlikely. Some customers, called vehicle customers (v.c.), are reachable either by a complete vehicle or by a truck alone, but others, called truck customers (t.c.), are reachable only by a truck alone. There are three types of routes in a solution to the TTRP: a pure truck route, a pure vehicle route, and a complete vehicle route. A pure truck route contains v.c. or t.c. customers serviced by a truck alone. A pure vehicle route contains only v.c. customers serviced by a complete vehicle without any sub-tours. A complete vehicle route consists of a main tour traveled by a complete vehicle and one or more sub-tours traveled by a truck alone. A sub-tour starts from and returns to a customer found on the main tour (the trailer is parked at this customer). At the parking place, the truck uncouples and parks the trailer that it is pulling, departs to service the customers on the sub-tour, returns to re-couple the trailer, and then continues to service the remaining customers on the main tour. In Fig. 1, we give two solutions to a TTRP with 38 v.c. customers and 12 t.c. customers. The solid segments, passing only v.c. customers which are depicted by circles, nodes, are edges traveled by a complete vehicle. The dashed segments, passing v.c. customers or t.c. customers that are depicted by dots, are edges traveled by a truck alone on a pure truck route or on a sub-tour. Solution 1 with an objective function value of 600.35 has two pure truck routes and three complete vehicle routes (two have one sub-tour and one has two sub-tours). Solution 2 with an objective function value of 565.02 has two pure truck routes, one pure vehicle route, and two complete vehicle routes with one sub-tour each.

In the TTRP cost can be incurred in several different ways including the different traveling costs by a complete vehicle or by a truck alone, the cost of parking a trailer, the cost of shifting demands between a truck and its pulling trailer, and the fixed cost of maintaining the fleet. For simplicity, we assume that the cost in the TTRP is proportional to the distance traveled by the fleet; therefore, the objective of the TTRP is to minimize the total distance traveled by the fleet on all three types of routes. In addition, we assume that the numbers of trucks and trailers are known in advance and all trucks and trailers are identical. Therefore, each truck is able to pull a trailer. Finally, we assume that the depot and every v.c. customer location can be selected as a trailer-parking place.

It is worth noting that the total demand loads in a pure truck route or in a sub-tour cannot exceed the truck capacity Qk; however, the summation of the demand loads of all sub-tours in a complete route can exceed Qk because shifting demand loads from a truck to its pulling trailer is allowed. The total demand loads in a pure vehicle route and a complete vehicle route, including the main tour and all its associated sub-tours, cannot exceed the sum of the capacities of the truck and its pulling trailer, that is, Qk+Ql. Just as in the basic VRP, each customer's demand must be fully satisfied and be serviced exactly once, that is, there is no splitting of demand. For every route, there is a restriction on tour length.

We now describe a network optimization formulation of the TTRP. Let V={0,1,2,…,n} be the set of customer nodes where 0 is the depot and E is the set of edges between nodes in V.G={V,E} is a complete graph in which each point iV⧹0 has a positive demand qi and an index either equaling to 1 indicating a t.c. customer that can be solely serviced by a truck alone, or 0 indicating a v.c. customer that can be serviced by either a complete vehicle or a truck alone. Each edge ijE has a symmetric, nonnegative cost cij associated with it, where cij is the Euclidean distance between node i and node j. In the TTRP, a set of routes consisting of ml pure and complete vehicle routes and mkml pure truck routes are constructed so that the total distance traveled by the fleet over all three types of routes is minimized and all constraints are satisfied.

It is important to point out that the TTRP is a multi-level optimization problem. At the first level, an allowable type of route has to be selected for each customer and no t.c. customer can appear on a pure vehicle route or on the main tour of a complete vehicle route. At the second level, three kinds of routes need to be constructed. The pure truck and the pure vehicle routes can be constructed by using a method for solving the traveling salesman problem. The complete vehicle routes are more difficult to construct since we need to decide on the number of sub-tours, the trailer-parking place for each sub-tour, and the sequence of customers on the main tour and each sub-tour. Other examples of multi-level routing problems are the multiple-depot VRP, the period VRP, and the site-dependent VRP [6], [7], [8], [9].

A problem related to the TTRP, called the Vehicle Routing Problem with Trailer (VRPWT), is due to Gerdessen [10]. Gerdessen presents two actual TTRP applications. The first is the distribution of dairy products by the Dutch dairy industry in which many customers are located in crowded cities. Maneuvering a complete vehicle is very difficult, so that the trailer is often parked while only the truck delivers the products. Another application is the delivery of compound animal feed that has to be distributed among farmers. On many roads, there are narrow roads or small bridges, so various types of vehicles are needed to distribute animal feed to farmers. One type of vehicle, called double bottoms, has a truck and a trailer. The trailer is left behind at a parking place while the truck is servicing some part of the route. Another actual application related to the TTRP appears in the article by Semet and Taillard [11]. This application occurred in a major chain store with 45 grocery stores located in the cantons of Vaud and Valais in Switzerland. The stores were serviced by a fleet of 21 trucks and 7 trailers. The site-dependent VRP [6], [7], [12], [13] is related to the TTRP. In the SDVRP, the fleet has many types of vehicles and there are vehicle–site compatibilities between customer sites and vehicle types.

The TTRP can be reduced to the VRP if there are no truck customers. Therefore, the TTRP is at least as difficult as the VRP which is NP-Hard [1]. An exact algorithm that can optimally solve a large-size TTRP problem is unlikely. In this paper, we develop and test a heuristic method for solving the TTRT that is based on tabu search and is coupled with the deviation concept from deterministic annealing. The remainder of this paper is organized as follows. In Section 2, we briefly review solution approaches to the TTRP and related problems. In Section 3, we describe a solution construction method that generates an initial solution to the TTRP. In Section 4, we develop an innovative tabu search improvement heuristic for solving the TTRP. In Section 5, we present the computational results from applying our tabu search method to 21 test problems. In Section 6, we give our conclusions.

Section snippets

Reviewing solution approaches to the TTRP and related problems

In the past three decades, the basic VRP and its variants have attracted the attention of many researchers [2], [3], [4], [5]. However, to our knowledge, only a few papers describe solution approaches for solving the TTRP and related problems. Semet [14] models a TTRP-related problem called the Partial Accessibility Constraint VRP (PACVRP). The author assumes that all available trucks are used and the number of trailers needed has to be determined. This assumption will increase the total cost

A solution construction approach for the TTRP

Our solution construction approach consists of a relaxed generalized assignment, route construction, and descent improvement. We describe each individual step in detail, and then integrate them into a construction approach that generates an initial solution to the TTRP.

The tabu search improvement heuristic for the TTRP

In this section, we develop an improvement approach based on tabu search coupled with the deviation concept from deterministic annealing to improve the initial solution generated in the construction step. Tabu search (TS), first presented by Glover [19] and also sketched by Hansen [20] in 1986, is a general improvement heuristic procedure. Although no clean proof of convergence has been presented in the literature, tabu search has been successful in a variety of problem settings like

New test problems for the TTRP

To our knowledge, there are no test problems in the OR literature for the TTRP. In order to test our heuristic, we select seven basic VRPs from the well-known test problems of Christofides, Mingozzi, and Toth (CMT) [24] and convert them into 21 TTRPs. The characteristics of the 21 TTRP problems are shown in Table 8. The 21 TTRPs are generated in the following way. For each customer i in the CMT problem, the distance between i to its nearest neighbor customer is calculated and this is denoted by

Conclusions

In this article, we developed a new tabu search heuristic for solving the truck and trailer routing problem. Our new heuristic always generated a feasible solution to the test problems (the ratio of the total demand to the total capacity of each test problem was above 90%). The construction step roughly allocated customers to routes and three types of routes were constructed by using a cheapest insertion heuristic. In the descent step, two criteria were used to convert an initial solution

Acknowledgements

The support of the National Science Council of the Republic of China in Taiwan under grant number NSC 87-2416-h-145-001 is gratefully acknowledged. I am indebted to two referees for their helpful comments and to Edward Wasil for his editing assistance.

I-Ming Chao received his M.S. from the University of New Mexico at Albuquerque and Ph.D. from the University of Maryland at College Park. He is now an associate professor and the chairman of the Department of Management Science at Chinese Military Academy in Taiwan. His research interests include network optimization, heuristic search, and applied operations research.

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