A study of vehicle routing problem via trade-off ranking method

Vehicle routing defines selecting the minimum cost, distance, and/or time path from a depot to several alternatives for a goods or service to reach its destination. The objective of most routing problem is to minimize the total cost of providing the service. But other objectives also may come into play, particularly in the public sector. For emergency services, such as ambulance, police, and fire engine, minimizing the response time to an incident is of primary importance. A few routing algorithms do not use a deterministic algorithm to find the “best” route for a goods to get from its original source to its destination. Instead, to avoid congestion, a few algorithms use a randomized algorithm that routes a path to a randomly picked intermediate destination, and from there to its true destination. In this paper, the trade-off ranking method is used to solve for the vehicle routing treated as a conflicting multi-criteria problem. The integration of the trade-off ranking method into the vehicle routing problem gives another perspective on how to solve the problem, hence broadened the decision support system for the vehicle routing problem.


Introduction
The first model of vehicle routing problem (VRP) was solved by Dantzig and Ramser [1]. They develop a mathematical programming to solve gas delivery to service stations. From then, the VRP evolved and extensively studied. The VRP is considered as NP-hard problem. Thus, the problem is usually solved using heuristics or meta-heuristics algorithm. Using such algorithm, the optimal solution is not guaranteed, however it is the best method to solve the VRP in real-life problem, due to their highperformance. The real-life problem usually involves several criteria which are conflicting between each other. The most common objective of VRP is to minimize the total distance of vehicle routes. However, for delivery service, maximize the fulfillment of the customers' demand may be the one of the objective. But then, to minimize the total distance travel by a vehicle while maximizing the demand of the customer can be conflicting, moreover during peak hours. The Decision Maker (DM), where in the delivery service would be the rider, may have to choose between the two conflicting objective. In making a decision, there is no right or wrong. As making decision A may be the best for one, taking B would be the best for others. Hence the main contribution of this paper is a study of VRP via trade-off ranking method, which is a method used to solve multi-criteria decision making problem with conflicting criteria [2]. The method aims to give the best solution that reflects the DM's preference and minimize the trade-off between the conflicting criteria.
The rest of the paper is organized as follows. The literature on types of VRP and methods used to solve the problem is presented in section 2. The trade-off ranking method is briefly discussed in section 3. In section 4, the test case for VRP is solved via trade-off ranking method. Lastly, the conclusion is given in section 5.

2.
Vehicle routing problem and solution methods Many methods have been used to solve abundance VRP problems. Jeon et al. [3] considered a hybrid genetic algorithm solving a VRP problem with heterogeneous vehicles, double trips, and multiple depots. Akbar and Aurachmana [4] considered a problem of capacitated vehicle routing problem with time windows (CVRPTW), with pick-up and delivery for mineral water company distributor. They used a hybrid genetic-tabu search algorithm to solve the CVRPTW. Pan et al. [5] also considered a capacitated vehicle routing problem (CVRP). They implemented the multi-group grasshopper optimisation algorithm (MGOA), which is an algorithm mimicking the behaviour of grasshopper in nature. Carić and Fosin [6] used historical data to solve the time dependent vehicle routing problem (TDVRP). They derived a zone of congestion during certain times and applied it to an optimisation algorithm to find the solution. Ng et al. [7] solved the CVRP using artificial colony algorithm with multiple colonies. They suggested the re-routing strategy to enhance the routing flexibility. Wang et al. [8] addressed a twoechelon capacitated routing problem (2E-CVRP). They designed a metaheuristic method based on variable neighbourhood search and integer programming to solve the 2E-CVRP. Wei et al. [9] used simulated annealing algorithm to solve the capacitated vehicle routing problem with two-dimensional loading constraints (2L-CVRP). Santos et al. [10] proposed a branch-cut-and-price algorithm. They suggested the use of intermediate depots called satellite to solve 2E-CVRP [10].
Sitek et al. [11] considered the CVRP with alternative delivery, pick-up, and time windows. They solved the problem using a hybrid of Constraint Programming, Genetic Algorithm and Mathematical Programming. Zhen et al. [12] used hybrid particle swarm optimization and hybrid genetic algorithm to solve the problem of designing a set of tris for the fleet of vehicles supplied by different depots. Zulvia e al. [13] proposed green VRP considering time windows, working hours, and different travelling time. The proposed model is solved using a many-objective gradient evolution (MOGE). Zhang et al. [14] presented a novel model of fuzzy electric vehicle routing problem with time windows and recharging stations (FEVRPTW). They used fuzzy numbers to denote the uncertainties in the model parameter. The model is solved using an adaptive large neighbourhood search (ALNS) algorithm.
Many other methods have been widely used in solving variants of VRP model. In this paper, we are using the trade-off ranking method [2] to solve the capacitated VRP with conflicting criteria. In a capacitated VRP model, the vehicle has limited capacity, thus it can only reach to a certain place, and not all. Consider a model of n customers, in which due to some restrictions, the vehicle can only approach some of the customers. Thus, choosing a set of customers to be served may depends on the objective of the travelling. Moreover, with the pandemic crisis, travelling is limited to a certain place and for a certain reason only. With the current pandemic as the motivation, we are looking to solve the VRP as a conflicting multi-criteria problem, where the objectives of the travelling are treated as criteria and the customers are regarded as the alternatives. Now, given a set of j alternatives (customers), the DM (traveller) must choose which alternative is the best to fulfil the i criteria (objectives). In the next subsection, we briefly discussed the trade-off ranking algorithm, used to solve the conflicting criteria VRP.

3.
Trade-off ranking method algorithm The trade-off ranking method has been used to solve various multi-criteria decision-making problems, moreover in a conflicting environment, including a general test cases, a car selection problem [2], and a personnel selection problem [15]. The algorithm of the trade-off ranking method used to solve the VRP is given as follows: [15] where ij f is the value of each criteria i and alternative j after normalization.

2.
Determine the objective of each criteria (max or min).

5.
Calculate the trade-off between each customer by formula: where j w is the weight for criteria j.

6.
Rank the customer accordingly. Customer with the least trade-off value, A DT  is ranked at the top.
In the next subsection, a test case is solved using the trade-off ranking algorithm.

4.
Experimental study Consider a VRP with single depot, 1 vehicle and 18 customers. The data for the location of each customer (given as latitude and longitude), distance from the depot (in km), demand per customer and profit (RM) are shown as table 1. As can be seen from table 1, the data for profit reflected the value of customers' demand. Thus, in the dataset provided, we only consider the distance travelled by the vehicle and the profit gained. The latitude and longitude are used for mapping purpose, as shown in figures 1-3.
The distance and profit criteria are conflicting criteria. The most profitable customer is C2 with RM150 profit. However, it is not the closest to the depot. The closest one would be C7 with distance value of 5.4km and gives RM75 of profit, half-less than C2. Note that, the distance is the cost criteria, which we want to minimize, while the profit is the benefit criteria to be maximized.
Let say, in certain times, the DM can only travel to half of the customers. Thus, looking roughly at the data, one DM may choose to travel to C7 first if the DM's aims is to minimize the distance. On the other hand, other DM may want to travel to C2 to gain as much profit as possible, regardless the distance.
By implementing the trade-off ranking algorithm to the problem via equations (1)-(4), table 3 shows the result of the ranking. In this case, we investigate a problem of constraint resources where the DM can only cater 10 customers to be served at one time. The VRP is considered in three objectives; (i) to minimize distance, (ii) to maximize profit; (iii) trade-off in between distance and profit.
In objective (i), we set the weightage for distance criteria as 1, and profit criteria as 0. Meanwhile for objective (ii), it is vice-versa. In objective (iii), both criteria are considered equally important, hence the weights are set to 0.5 each.
Since the data in table 1 gives the distance of each customer from the depot only, the distances in between customers are approximated using the Pythagoras formula. In the case of uncertainty, approximation is considered adequate as a reliable information. From table 2, the trade-off ranking method proposes a route for objective (i) with minimum approximate total distance at 16.95km. For objective (ii), since the goal is to maximize profit, then the trade-off ranking method proposes a solution with total profit of RM525, regardless of distance, where in this objective, it took the longest distance to travel. In a win-win situation, i.e., minimize the distance travel while maximize the profit, the trade-off ranking method gives a route solution with least compromise of both criteria. Figures 1-3 show the route taken for each objective.

Conclusion
In this paper, a trade-off ranking method has been used to solve the capacitated VRP by treating the problem as a conflicting multi-criteria problem. In such a problem, the solution is not unique. It depends on the DM's preference towards one or more criteria. From the experimental objective, the trade-off ranking method proposes a route with least compromise value in between the distance and the profit. Further research of testing the trade-off ranking method with VRP would be done in the future considering other variants of VRP, including with time windows, multi-depots, and multi-vehicles.