Discrete Dynamics in Nature and Society

Volume 2015 (2015), Article ID 365367, 10 pages

http://dx.doi.org/10.1155/2015/365367

## The Optimization of Transportation Costs in Logistics Enterprises with Time-Window Constraints

School of Economics and Management, North China Electric Power University, Beijing 102206, China

Received 25 November 2014; Revised 26 March 2015; Accepted 27 March 2015

Academic Editor: Delfim F. M. Torres

Copyright © 2015 Qingyou Yan and Qian Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents a model for solving a multiobjective vehicle routing problem with soft time-window constraints that specify the earliest and latest arrival times of customers. If a customer is serviced before the earliest specified arrival time, extra inventory costs are incurred. If the customer is serviced after the latest arrival time, penalty costs must be paid. Both the total transportation cost and the required fleet size are minimized in this model, which also accounts for the given capacity limitations of each vehicle. The total transportation cost consists of direct transportation costs, extra inventory costs, and penalty costs. This multiobjective optimization is solved by using a modified genetic algorithm approach. The output of the algorithm is a set of optimal solutions that represent the trade-off between total transportation cost and the fleet size required to service customers. The influential impact of these two factors is analyzed through the use of a case study.

#### 1. Introduction

In a competitive environment, obtaining the maximum profit plays a key role in the success of an enterprise. Logistics costs make up a large portion of the total costs of enterprises but can be reduced through supply chain optimization. Analysis of the logistics costs of enterprises reveals that transportation costs are an important part of the costs of logistics enterprises. Therefore, it is very important to study how transportation costs can be optimized in logistics enterprises.

The transportation costs of logistics enterprises are influenced by the fixed costs and variable costs involved in the transportation process. However, transportation costs are more closely related to time-window constraints, which are governed by customers’ arrival times. Logistics enterprises must pay penalties when time-window constraints are violated, and this causes increases in transportation costs.

Many past studies have been dedicated to determining how to achieve the lowest possible transportation cost. For example, McCann [1] addressed two interrelated questions: the optimum size of a vehicle or vessel and the structure of transportation costs with respect to haulage distance. C. Pilot and S. Pilot [2] focused on minimizing the total costs involved in a transportation problem. Jha et al. [3] considered a joint-location inventory problem and minimized the transportation cost involved in a joint inventory location model by using a modified adaptive different evolution algorithm. Chanas and Kuchta [4] proposed what they see as an optimal solution to the transportation problem, which makes use of fuzzy cost coefficients and an algorithm determining the nature of the solution.

As exploration of transportation problems has developed, multiobjective transportation cost problems have emerged. For instance, Prakash et al. [5] drew attention to a cost-time trade-off bulk transportation problem, which they solve by using Pareto optimal solutions. Ojha et al. [6] formulated a multiobjective transportation solution, with fuzzy relations under fuzzy logic. The objectives of their model are the minimization of the total transportation cost and total time for transportation required for the system.

The conditions that force each vehicle to start with each customer at a period specified by that customer are called time-window constraints. Existing literature on transportation problems with time-window constraints has mainly concentrated on vehicle routing problems. Vehicle routing problems, with different variations and generalizations, have been studied for several decades, since the pioneering work of Dantzig and Ramser [7] on a truck dispatching problem.

Alvarenga et al. [8] proposed a robust heuristic approach to vehicle routing problems with time windows (VRPTW), using travel distance as the main objective through an efficient genetic algorithm and a set partitioning formulation.

Ghoseiri and Ghannadpour [9] presented a new model and solution for multiobjective VRPTW using goal programming and genetic algorithm, in which decision makers specify optimistic aspiration levels to objectives and deviations from those aspirations are minimized. They used a direct interpretation of VRPTW as a multiobjective problem, in which both total required fleet size and total traveling distance were minimized, while capacity and time-window constraints were secured.

Al-Khayyal and Hwang [10] formulated a model for finding the minimum-cost route in a network for a heterogeneous fleet of ships engaged in the pickup and delivery of several liquid bulk products. They showed that the model can be reformulated as an equivalent mixed-integer linear program with a special structure.

Yu et al. [11] proposed a hybrid approach, which consists of ant colony optimization (ACO) and Tabu search, to solve VRPTW.

Chiang and Hsu [12] proposed their own approach to solve a multiobjective vehicle routing problem with time windows. The objectives were to simultaneously minimize the number of vehicles and the total distance. Their approach was based on an evolutionary algorithm and it aims to find a set of Pareto optimal solutions.

Because of the many applications of different vehicle routing problems, a wide variety of researchers have focused on developing solutions to them. Useful techniques for solving general vehicle routing problems can be found in [13–15].

Our analysis of the works described above has shown that existing literature based on accurate algorithms and heuristic algorithms aims to achieve the lowest transportation cost possible. To our knowledge, no study has considered the time-window constraints in the transportation cost model. Time-window constraints increase transportation costs in logistics enterprises, and so it is necessary for logistics enterprises to take the time-window constraints into consideration when making decisions.

This paper presents a biobjective transportation cost model with time-window constraints, which is modeled through modified genetic algorithm. In our study, the simultaneous minimization of fleet size and total transportation cost are considered objective functions.

The model is formulated under the following assumptions:(1)Time-window constraints are soft, and the time windows specified by customers are elastic.(2)The service time for a vehicle at its destination is equal to zero.(3)A route is defined as starting from a depot, going through a number of customers, and ending at the depot. Every customer on the route must be visited only once by one of the vehicles.

#### 2. Model Formulation

This paper assumes that a logistics enterprise is the single supplier in a transportation process. The logistics enterprise distributes goods to customers according to the number of orders. The customer number for a route is uncertain, and the vehicle route must be determined in order to optimize transportation cost.

The system of logistics enterprise transportation can be regarded as a simple network. In this network, the start node and end node are both the central depot of the logistics enterprise. Each arc of the network represents the transportation relationship between customers. The number of the arc represents the travel time between the two customers. Let us assume there are customers, , and for simplicity denote the depot as customer 0. Figure 1 presents a network representation of the transportation process.