A heuristic algorithm for multiple trip vehicle routing problems with time window constraint and outside carrier selection

1. Introduction

Vehicle routing problem is to find optimal routes for multiple vehicles visiting a set of locations by minimizing the total travel length. A variety of vehicle routing problems has been studied in the literature to address different practical situations. Typically, different vehicle routing problems address different practical situations. Vehicle routing problem with time windows (VRPTW) is an important and practical problem for logistics managers. In many sectors of the economy, transportation costs amount for a fifth or even a quarter (lumber, wood, petroleum, stone, clay and glass products) of the average sales amount (Schneider, 1985). Thus, appropriately identifying and modeling the problems and developing algorithms to solve them have been the continuing research effort in the last several decades.

Our motivation for this study stems from observations of a local logistics company. This company owns different types of private trucks, and its main business is delivering food and beverages to wholesalers. The logistics company promises the customer that a shipment received during business hours will be delivered to the destination within 5 h, so the delivery time window is a major concern. Furthermore, the company is facing fluctuations in demand from its customers. When the customer demands are greater than the total capacity of owned trucks during the peak season, the company has three strategies to use: using overtime, outsourcing vehicles and using outside carriers. As the overtime cost and the rents of outsourcing vehicles are much higher than that of using an outside carrier, sometimes using an outside carrier is a more attractive option.

Regarding the outside carrier selection, a logistics manager can make a choice between a truckload (a private truck with multiple trips) and a less-than-truckload carrier (an outside carrier). A private truck allows a company to consolidate several shipments, going to different destinations, and in a single truck with multiple trips. A less-than-truckload carrier usually assumes the responsibility for routing each shipment from the origin to the destination. The freight charged by a less-than-truckload carrier is typically much higher than the cost of a private truck. Choosing the right customers to be served by outside carriers may yield significant cost savings to the company.

In this paper, we address the problem of routing a fixed number of private trucks with limited capacity from a central warehouse to customers with known demand and time windows. Furthermore, each vehicle can perform several trips during the working day. The objective of this paper is to develop a heuristic algorithm to route the private trucks with time windows and to make a selection of less-than-truckload carriers by minimizing a total cost function.

The literature of VRPTW can be classified into two categories, the exact method and heuristic algorithm. Although there are some exact methods (Laporte, 1992; Laporte and Nobert, 1998; Azi et al., 2007), their application is limited because the solution time is exponentially increasing with the number of customers. Clearly, a heuristic algorithm remains a viable alternative for larger instances. Heuristic algorithms can be broadly classified into two categories: classical heuristics and metaheuristics.

Classical heuristics include construction and improvement approaches. Construction heuristics build a feasible route by iteratively inserting a customer into a current route based on maximum savings or minimum additional distance. Some examples of construction heuristics are Solomon (1987), Potvin and Rousseau (1993), Bramel and Simchi-Levi (1996) and Ioannou et al. (2001). Improvement heuristics modify the current solution iteratively by performing local searches for better solutions. Some examples of improvement heuristics are Potvin and Rousseau (1995), Russell (1995), Cordone and Wolfler Calvo (2001) and Bräysy et al. (2004).

Metaheuristics are general solution procedures exploring the solution space to identify good solutions and incorporating some classical heuristics. In contract to classical heuristics, metaheuristics allow infeasible and deteriorating solutions during the search process to escape from local optimum. So far, the Tabu Search and Genetic Algorithm have shown the best performance for vehicle routing problems (Mester and Bräysy, 2005). Tabu Search is a local search metaheuristic introduced by Glover (1986). Details about Tabu Search can also be found (Glover, 1989; Glover, 1990; Glover and Laguna, 1997). Rochat and Taillard (1995), Taillard et al. (1997), Chiang and Russell (1997), Schulze and Fahle (1999) and Cordeau et al. (2001) have successfully applied Tabu Search to the VRPTW. The ideas involved in Genetic Algorithm were originally developed by Holland (1975). A Genetic Algorithm begins with a pool of the population of chromosomes that these chromosomes undergo crossovers and mutation to generate some children. Although these children are different from parents, they inherit some characteristics from their parents. This process continues until no further improvement in the solution appears possible. Gendreau and Tarantilis (2010) reported good results with genetic algorithms. This year, Demir et al. (2019) published a book, and in Chapter 8, discussed the basic principles of vehicle routing to provide readers with a complete introductory resource.

Little research has examined the problem of choosing between a less-than-truckload and truckload carrier. Ball et al. (1985) considered a fleet planning problem for long-haul deliveries with fixed delivery locations and an option to use an outside carrier. Agarwal (1985) studied the static problem with a fixed fleet size and an option to use an outside carrier. Klincewicz et al. (1990) developed a methodology to address the fleet size planning and to route limited trucks from a central warehouse to customers with random daily demands. Chu (2005) introduced a heuristic to simultaneously select customers to be served by external transportation providers and to route a limited number of owned heterogeneous trucks. A carrier collaboration problem for less-than-truckload carriers was studied by Hernández and Peeta (2014). They considered a single-carrier collaboration problem in which a less-than-truckload carrier of interest seeks to collaborate with other carriers by acquiring the capacity to service excess demand. Recently, Wu et al. (2017) extended Chu’s work (2005) by incorporating time windows constraint into a vehicle routing problem while less-than-truckload carriers are available.

In general, our research described here differs from the previous one on fleet planning or vehicle routing in that it modifies the Clarke and Wright method by shifting the performance measure from a distance to cost and also incorporates the fixed cost of different types of private trucks into the model. In addition, we simultaneously consider the multiple trip vehicle routing problems with time windows and the selection of less-than-truckload carriers that is an integrated scenario of real-world application. Furthermore, a mathematical model is also proposed to represent and solve the problem. To sum up, our problem is a generalization of the VRPTW, the multi-trip vehicle routing problem with time windows (MTVRPTW) and the heterogeneous vehicle routing problem (HVRP) with external carriers (following the abbreviation in literature, Koc, et al. (2016)). To the best of our knowledge, this scenario has not been considered in the literature. The rest of the paper is organized as follows. The next section formulates the mathematical model for our problem. Section 3 presents the heuristic algorithm. Computational results are reported in Section 4. Finally, concluding remarks and suggestions for future research are provided in Section 5.

2. Mathematical model

We formulate our mathematical model based on the following assumptions:

• A single warehouse system is considered; all trucks start at the warehouse and return to the warehouse.

• We restrict ourselves to delivery only.

• The requirements of all the customers are known, and each customer’s requirement cannot exceed the truck capacity.

• The time window of each customer is known.

• Multiple trips are allowed on a truck.

• A maximum driving time is imposed on each route.

• Each customer is served by one truck (either by the private truck or the less-than-truckload carrier), and all customers’ requirements must be met.

• The cost of operating the truck fleet consists of a fixed cost and a variable cost. The principal items in the fixed cost include personnel, insurance and truck depreciation. The main component of the variable cost is fuel, which is usually proportional to the traveled distance.

The relevant notations and integer programming model are as follows:

• 1: the warehouse (initial point).

• D: the warehouse (terminal points).

• N: the set of demand nodes.

• TN: the set of all nodes°TN∈ N∪ {1, D}.

• R: the set of all trips.

• K: the set of all trucks.

• i: {i = 1,2,…,n}, the index set of customers including the warehouse.

• j: {j = 1,2,…,n}, the index set of customers including the warehouse.

• k: {k = 1,2,…,m}, the index set of trucks.

• r: {r=1,2,…,R}, the index set of trips.

• n: the number of customers.

• m: the number of trucks.

• R: the maximum number of allowed trips for a vehicle.

• FC_k: the fixed cost of private truck k.

• C_ijk: the cost of truck k traveling from customer i to customer j.

• CL_i: the cost charged by the less-than-truckload carrier for serving customer i.

• q_i: the delivery of customer i.

• Q_k: the capacity of private truck k.

• s_i: the required service time of customer i.

• t_ij: the travel time from customer i to customer j.

• e_i: the earliest time to start service at customer i.

• l_i: the latest time to start service at customer i.

• M: a big number.

• T_max: the maximum route time allowed for a vehicle.

Decision variables:

T_ikr = the start service time at customer i by the k-th truck at its r-th trip.

LQ_ij = the number of loaded shipments on the car while traveling from customer i to customer j.

Subject to

(5) ∑i∈Nqi*Yikr≤ Qk     ∀ k∈K, r∈R 

The objective function is to route the private trucks and to make a selection of less-than-truckload carriers by minimizing a total cost function. The first and second terms are the fixed cost and variable costs of the private trucks; the last term is the cost from the less-than-truckload carrier.

Constraint (1) ensures that at most m trucks are used.

Constraint (2) defines that each customer is served either by a trip of a private truck or a less-than-truckload carrier.

Constraints (3) and (4) guarantee that a truck arrives at a customer and also leaves that location.

Constraint (5) ensures that the loaded shipments cannot exceed the loading capacity of each truck.

Constraints (6)-(8) assure the feasibility of the schedule.

Constraint (9) guarantees that a truck leaves the initial point and arrives at the terminal point for each trip. The difference between the number of leaving the initial point and the number of arriving at the terminal point is equal to one.

Constraint (10) ensures that the loaded shipments on the truck should equal LQ_ij minus q_j after the truck leaving customer j.

Constraint (11) assures that the total loaded shipments at the warehouse is equal to the sum of the required shipment from all demand nodes.

Constraint (12) guarantees that the loaded shipments on the truck should be equal to or greater than zero; the loaded shipments on the truck cannot exceed the truck capacity.

Constraint (13) ensures that there exits the r-th trip of the k-th truck, then there is a possibility of (r + 1)-th trip.

Constraint (14) assures that the arriving time at the warehouse of the k-th truck at its r-th trip is equal to the leaving time at the warehouse of the k-th truck at its (r + 1)-th trip.

3. The heuristic algorithm

In this section, we describe our algorithm, called MTVRPTW-LTL, for solving the multiple trip vehicle routing problems with time windows, and the selection of less-than-truckload carriers. The heuristic algorithm can be decomposed into four main steps. In the following, we describe this algorithm by examining the main steps separately.

4. Computational results

As our scenario has not been explored in the literature, there are no standard instances available for our problem. As mentioned in the introduction, Wu et al. (2017) consider a vehicle routing problem with time constraints while less-than-truckload carriers are available. However, only a single trip is considered in the work of Wu et al.

In this study, we simultaneously consider the multiple trip vehicle routing problems with time windows while less-than-truckload carriers are available. Since each vehicle can perform several trips during the working day in our study, the problem studied by Wu et al. is a special case of our problem. Our results are supposed to be better than those of Wu’s work. Hence, we use the test problems proposed by Wu et al. (2017) to evaluate the efficiency and accuracy of our algorithm.

The vehicle capacities and relevant costs for test problems are shown in Tables I and II, and the detailed coordinates, demands and time windows of customers are given in the Appendix. The solutions produced by the heuristic algorithm are compared to the optimal results from the mathematical model mentioned in Section 2. The heuristic algorithm was written in FORTRAN language, and the mathematical model was solved using the software LINGO version 16.0. Both of them were implemented on a PC with an Intel Core i5 2.7 GHz processor. Computational results on test problems are reported in Tables III-VI, respectively.

Table III summarizes the results for five customers. Our heuristic algorithm obtains optimal solutions. As shown in Table III, both the mathematical model and the heuristic algorithm yield the same total cost in 20 instances. By comparing our results with the results from a single trip, we can find that our heuristic algorithm outperforms the single trip in 16 instances except for 1-9, 1-10 and 2-9, 2-10. The average percentage of savings from our algorithm is about 27.41 per cent. Combining the total cost in Table III and the vehicle capacity in Table I, we plot Figures 8 and 9 for problems 1-1-1-10 and 2-1-2-10, respectively. From Figure 8, we can find that there is a negative relationship between total costs and vehicle capacities. It means that the higher the vehicle capacity, the lower the total cost. It makes sense since the freight charged by a less-than-truckload carrier is usually much higher than the cost of a private truck.

From Figure 9, we can find that there is a negative relationship between total costs and the vehicle capacities from the range 30 to 60. With similar reasoning in Figure 8, it makes sense since the freight charged by a less-than-truckload carrier is usually much higher than the cost of a private truck. There is a positive relationship between total costs and vehicle capacities from the range 70 to 120. The reason for this is that in this range, the increase in fixed cost is much higher than that of the cost savings from increasing capacity.

Table IV summarizes the results for ten customers. For problems 3-1∼3-10, our heuristic algorithm obtains the optimal solutions in 10 instances. As shown in Table VI, both the mathematical model and the heuristic algorithm yield the same total cost and the same route sequence in 10 instances. As to problems 4-1 and 4-10, our heuristic algorithm also obtains the optimal solutions in five instances. From the computational experiments, we found that the selection of customers served by the LTL carriers, and the initial solution have a great impact on whether an optimal solution can be reached.

Table IV shows that the solution time for the mathematical model increased dramatically with the size of the problem. In general, the solution time for problem 4-1∼4-10 is range from 2 to 6 h. It takes more than 6 h to solve the problem 4-4. Notice that the execution time reported here doesn’t include the time for sub- tour breaking. Computationally, exact algorithms for the VRP are restricted to solving problems of only up to about 50 customers (Desaulniers, 2010). Even though the Branch-and-price-and-cut is used for solving the problem, it is still difficult to find the optimal solution in reasonable computing time. On the other side, our heuristic algorithm requires little time to solve the problem. The solving time of a problem is ranged from 0.5 to 11.4 s. The CPU time of test problems is not very sensitive to the problem size. From Tables III and IV, we find that the heuristic algorithm obtains the optimal or near-optimal solutions. The worst case of deviation from the optimum is only 1.5 per cent, and the average percentage deviation from the optimum for the forty test problems is 0.119 per cent and the execution time for all test problems is less than 12 s. It is an encouraging result in terms of both time and accuracy.

5. Conclusions

Vehicle routing plays a central role in logistics management. In this paper, we introduced a real world new problem, the MTVRPTW and the possible use of a less-than-truckload carrier to satisfy customer demands. Our new problem is a generalization of the VRPTW and the MTVRPTW and the HVRP with external carriers. To the best of our knowledge, the scenario has never been explored in the literature. We developed both the mathematical model and the heuristic algorithm. In all, 40 test problems developed by Wu et al. were examined with our heuristics. The results obtained from our mathematical model and heuristic algorithm outperform the results from the exact solution method by Wu et al. The results are encouraging as our algorithm obtains the optimal or near-optimal solutions in an efficient way in terms of time and accuracy.

As for future research, a wide range of test instances should be built and tested. Furthermore, it would be interesting to see if other intelligent optimization techniques, such as Tabu Search, Genetic Algorithms, Ants Colony, Simulated Annealing and Neural Networks, can be used to solve this problem and even provide better results.