## Vehicle Routing Problem for Multiple Product Types, Compartments, and Trips with Soft Time Windows

Read this article. The paper presents a mathematical model to solve vehicle routing challenges. In your own words, explain what vehicle routing is and why it's important to a supply chain.

### Introduction

The vehicle routing problem (VRP) is a well-known problem in the operation research and combinatorial optimization presented by Dantzig and Ramser. The VRP is also an important problem in the fields of transportation, distribution, and logistics. The context is planning the route to deliver goods from a central depot to customers who have placed orders for such goods. The objective of the VRP is to minimize the total route cost. According to the importance of the VRP, there are so many literatures which refer to the models and the solution of the VRP. Many papers present the different views of the problem, the different features of the system, and the different approaches to solve the problem.

The vehicle routing problem with time windows (VRPTW) is a variant of the VRP (Figure 1). A solution of the VRPTW is a set of routes consisting of sequences of customers. Each route is assigned to arrive within their time windows. In addition, if a vehicle arrives before (or after) the lower (or upper) bound of the customer's time window, the vehicle cannot deliver goods to the customer. An exception of VRPTW problem is that the time windows can be violated if the penalty is paid. In this case, it is called the vehicle routing problem with soft time windows (VRPSTW). In VRPSTW, each customer $i$ has a desirable time window with the highest satisfaction $[a_i, b_i]$ as and a hard time window $[LB_i, UB_i]$. A soft time window is shown in Figure 2. In the intervals  $[LB_i, a_i]$ and $[b_i, UB_i]$ the service is allowed for some penalty fee.

Figure 1 Service time interval in VRPTW.

Figure 2 Service time interval in VRPSTW.

The vehicle routing problem with multiple trips (VRPMT) is another variant of the classical VRP. Each vehicle can be scheduled for more than one trip, as long as it corresponds to the maximum distance allowed in the workday.

The vehicle routing problem with multiple compartments (VRPMC) is also a special case of the VRP. Each compartment of the vehicle has a limit and is dedicated to a single type of products.

Many papers present the different views of VRP; there has not been any VRP mathematical model for multiple compartments, trips, and time windows. Several papers presented VRP for single product type. However, in "A mixed-binary linear model for fleet routing and multiple product-type distribution," the author presents a model on distributing multiple items for customer.

In this paper, we address the mathematical model for the VRP for multiple product types, compartments, and trips with soft time windows (VRPMPCMTSTW). The proposed VRP can be regarded as the extension to three problems which are

(i) the vehicle routing problem with multiple compartments,

(ii) the vehicle routing problem with multiple trips,

(iii) the vehicle routing problem with soft time windows.

The summary of the three mentioned problems is presented in Table 1. It is clear that our mathematical model solves the classical VRP since it is a special case for the VRP with multiple compartments, trips, and time windows. The time window cases are studied because they are more practical and the soft time window cases are further investigated due to the feasibility of the problem and, in some cases, due to the fact that it reduces the total cost.

Table 1 Comparison of the problems.

 VRP VRPMC VRPMT VRPSTW This study Vehicle Heter/homo Single Heter/homo Heter/homo Heter/homo Compartment Single Multiple Single Single Multiple Trip Single Single Multiple Single Multiple Time window Not considered Not considered Not considered Considered Considered

In this paper, we present our mathematical models in three cases to see the development process of our study. In the first case, we consider the VRP for multiple product types, compartments, and trips where there is not a time window (model 1). Then we consider the case with time window (model 2) and, finally, the case with soft time window (model 3).

From the above content, model 3 generalizes all previous models mentioned in this paper. It is worth pointing out that another goal of the formulation is to try to formulate the mathematical models which are feasible and solvable by available solvers in a reasonable time. In this research, all models are solved by using AIMMS 3.13 on a personal computer. The remainder of the paper is organized as follows. In Section 2, the problem is defined. Also the assumptions, notations, and mathematical models are presented. Section 3 contains the computational results of our formulations. We use the same set of data from "A mixed-binary linear model for fleet routing and multiple product-type distribution," to examine the feasibility and solutions of the model. Finally, in Section 4, we summarize our conclusion and provide pointers for further research.