Urban pickup and delivery problem considering time-dependent fuzzy velocity

doi:10.1016/j.cie.2011.01.020

Computers & Industrial Engineering

Volume 60, Issue 4, May 2011, Pages 821-829

https://doi.org/10.1016/j.cie.2011.01.020 Get rights and content

Abstract

The velocity in vehicle routing problem is usually fuzzy and uncertain, which would influence the flexibility and feasibility of the dispatch result of vehicles. In this paper a realistic model is proposed to solve urban pickup and delivery problem with consideration of time-dependent fuzzy velocity of vehicles. Firstly, based on collected traffic data of typical roads in an urban city, the membership function of fuzzy velocity was created and simulated. Then the fuzzy arriving time to customers was calculated with Alfa-Cut Set Algorithm. Next, the urban pickup and delivery problem under time-dependent fuzzy velocity was described, and a modified Tabu Search was used to solve the problem. Finally, the paper introduces a real pickup and delivery problem with 28 customers in a logistics company to illustrate the proposed method. The results show the dispatch result considering fuzzy velocity is more feasible and robust than that with deterministic velocity, and the improved Tabu Search could be used to solve the urban pickup and delivery problem with shorter computation time.

Introduction

Since vehicle routing problem (VRP) was first introduced by Dantzig and Ramser (1959), scientists, engineers and management specialists have developed various accurate or heuristic algorithms for different VRPs. Laporte, Mercure, and Nobert (1986) established Branch and Bound Algorithm. Gillett and Miller (1974) developed Sweep Algorithm. Clarke and Wright (1964) established Saving Algorithm, Holland (1975) used Genetic Algorithm. And Glover (1986) introduced Tabu Search Algorithm. After that, several typical type of VRP was solved with Tabu Search, such as the traveling salesman problem with backhauls (Gendreau, Hertz, & Laporte, 1996), the traveling salesman problem with time-window (Gendreau, Hertz, & Laporte, 1998), VRP with stochastic demands and customers (Gendreau et al., 1996a, Gendreau et al., 1996b), the periodic vehicle routing problem (Gaudioso & Paletta, 1992). Alfa, Heragu, and Chen (1991) introduced Simulated Anneal Algorithm for VRP.

However, in most research on VRP, the information, such as customer information, vehicle parameters and traffic conditions was assumed to be known and predetermined before routing is constructed. Therefore these algorithms are only valid for VRP under deterministic conditions.

In real dispatch application, parameters in VRP are usually fuzzy (e.g. vehicle velocity) due to influence of uncertain traffic conditions and fuzzy recognition of human beings. Research on VRP with fuzzy parameters is relatively few. Perincherry and Kikuchi (1990) studied reship/reload problem under fuzzy parameters, they assumed that both the cargo supply and requirement are fuzzy, but traveling time among reship/reload locations and related costs are determined. And fuzzy linear programming was introduced to solve the reload problem with fuzzy inputs. Teodorovic and Pavkovi (1996) studied VRP with fuzzy customer requirement for single vehicle, and triangular fuzzy number was used to indicate fuzzy requirement of customers. Chen and Gen (1995) described VRP with fuzzy due-time and developed a genetic algorithm to solve it. In these work, the fuzzy parameter was assumed to have triangular fuzzy number characteristics. This assumption may be far from the reality, in particular for crowded urban areas in big cities.

In this paper, the membership function of fuzzy vehicle velocity for urban pickup and delivery problem (UPDP) is simulated with collected traffic data, which shows “pseudo normal distribution”. A model is proposed to describe the urban pickup and delivery problem with fuzzy velocity (UPDPFV). Using Alfa-Cut Set Algorithm, the fuzzy arriving time to each customer was calculated. Since Tabu Search is a local-search based optimization method that has been successfully applied to the solution of many difficult combinatorial optimization problems, the Tabu Search is adapted here to solve the UPDPFV problem.

A real pickup and delivery problem of a logistics company is introduced to illustrate the proposed method. The application of UPDPFV shows that the proposed algorithm with consideration of time-dependent fuzzy velocity increases the feasibility and robustness of the dispatch result.

Section snippets

Membership function simulation of fuzzy velocity

For urban pickup and delivery problem, the time-window of each customer is demanded by the customer itself. So the arriving time to each customer must be calculated to decide whether the time-window is satisfied or not. In urban transport, the velocity of vehicles on the road is usually fuzzy because of the complex traffic situation. How to calculate the velocity on the road becomes the key factor that influences the results of UPDPFV.

Calculation of fuzzy arriving time to customers

In a UPDP problem, there are usually a dispatch center and many customers. Several vehicles are dispatched to pass through these customers. Fig. 3 illustrates one of the routes from the dispatch center to Customers A and B. There are four locations that the vehicle must pass through, which are Cross1, Customer 1, Cross 2, and Customer 2 before it finally returns to dispatch center 0. We use p, q, r, s, and t to describe the road to these positions. The vehicle first moves to Cross 1 through

Optimization Model of UPDPFV

For urban pickup and delivery problem, the total transport costs compose of fuel costs, driver costs, depreciation costs and carriage arranging costs, in additional to the costs not relative with vehicle driving distance such as insurance. The optimization problem of total transportation costs C(X) is $\min C (X) = C_{g} (X) + C_{d} (X) + C_{v} (X) + C_{a} (X)$ where $C_{g} (X) = K_{g} \cdot \sum_{k} \sum_{i} \sum_{j} (d_{ij} x_{ijk} β_{ijk})$ $β_{ijk} = (β_{0} + (1 - β_{0}) \frac{{\overset{⇀}{w}}_{ijk} + w_{ijk}}{W_{k}})$ $C_{d} (X) = {\bar{C}}_{d} (X) + {\hat{C}}_{d} (X)$ ${\bar{C}}_{d} (X) = {\bar{K}}_{d} \sum_{k} \sum_{j} x_{0 jk}$ ${\hat{C}}_{d} (X) = K_{d} \sum_{k} \sum_{i} \sum_{j} (d_{ij} x_{ijk})$ $C_{v} (X) = K_{v} \sum_{k} \sum_{i} \sum_{j} (d_{ij} x_{ijk})$ $C_{a} (X) = \sum_{k} \sum_{i} (K_{a} w_{i} \sum_{j} z_{ijk})$ $y_{ki} = \{\begin{matrix} 1 \end{matrix}$

Solution flow chart of Tabu Search

To solve the UPDPFV expressed in Eq. (10) we need to calculate the fuzzy arriving time to each customer and decide whether time-window constraint for each customer is satisfied or not. Tabu Search based on fuzzy possibility is adapted to solve this problem.

First, an initial solution for the UPDPFV is established by using Clarke–Wright Saving Algorithm. Then the solution is modified with Tabu Search, so as to obtain the final optimized results.

In the Clarke–Wright Saving Algorithm two points

Problem and results

The application is based on real data in a large-scale logistic corporation, which include pickup and delivery tasks for 28 customers in a city, and their locations are shown in Fig. 7. Every circle dot is a location of a customer, and the rectangle place means the pickup and delivery center. The selected parameters are: vehicle maximum capacity V = 15 m³, full load W = 3000 kg, fuel cost coefficient K_g = 0.7 Chinese Yuan/km, weight cost coefficient K_a = 0.5 Chinese Yuan/kg, driver cost per distance

Conclusions

Very few articles, especially, relating to road nets, have considering the influence of fuzzy velocity to the dispatch result in urban cities. Accounting for this factor and integrating it into the urban pickup and delivery model makes the problem complex but more realistic. The time-dependent fuzzy velocity of roads is simulated from the collected traffic data in big cities, and the results shows that the membership function of fuzzy velocity is “pseudo normal distribution”. The fuzzy arriving

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View full text

Urban pickup and delivery problem considering time-dependent fuzzy velocity

Abstract

Introduction

Section snippets

Membership function simulation of fuzzy velocity

Calculation of fuzzy arriving time to customers

Optimization Model of UPDPFV

Solution flow chart of Tabu Search

Problem and results

Conclusions

Computer & Industrial Engineering

Computers and Operations Research

Computers and Operations Research

Fuzzy Set and Systems

Vehicle routing problem with fuzzy due-time using genetic algorithms

Japanese Journal of Fuzzy Theory and Systems

Scheduling of vehicles from a central depot to a number of delivery points

Operations Research

The truck dispatching problem

Management Science