Discrete Optimization
Stochastic single vehicle routing problem with delivery and pick up and a predefined customer sequence

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Abstract

In this paper we study the routing of a single vehicle that delivers products and picks up items with stochastic demand. The vehicle follows a predefined customer sequence and is allowed to return to the depot for loading/unloading as needed. A suitable dynamic programming algorithm is proposed to determine the minimum expected routing cost. Furthermore, the optimal routing policy to be followed by the vehicle’s driver is derived by proposing an appropriate theorem. The efficiency of the algorithm is studied by solving large problem sets.

Highlights

► We study routing of a vehicle that delivers/picks up products with stochastic demand. ► We develop and analyze an optimal dynamic programming algorithm. ► We identify the optimal routing policy by proposing and proving an appropriate theorem.

Introduction

In this paper we address a special case of vehicle routing. We consider a single vehicle that delivers products to customers and picks up returned items (if any). The vehicle serves customers at a predefined sequence, and is allowed to return to the depot for stock replenishment and unloading of returned items. The customer demands for either pickup or delivery are random and are modelled by independent random variables with known statistics (which may be derived from historical data). The distances, or travel times, among all points in the network (depot and customer sites) are deterministic and known. The delivery product quantities as well as the returned items loaded onto the vehicle share the same storage compartment (the vehicle’s loading compartment).

Upon completion of service at each customer, the driver has to make a decision: (a) Proceed to the next customer, as long as the probability of being able to fully satisfy its demand (for both pickup and delivery) is acceptable; (b) return to the depot in order to refill/unload, and resume the route by visiting the next customer in the predefined sequence. If the vehicle proceeds to the next customer but the actual demand of this customer for either pickup or delivery turns out to be higher than either the stock carried or the empty space left on board, the vehicle will serve the customer as much as feasible, and will then visit the depot to refill/unload, and return to the customer to fully satisfy its demand.

An additional decision should be made concerning the quantity to be loaded to the vehicle each time the vehicle returns to the depot. This is because unnecessarily high stock levels may prevent the collection of returned items, therefore causing additional depot returns and lower customer service. It is noted that the vehicle may have to visit a customer twice (but not more), if it cannot fully meet the demand of this customer during the first visit (for either delivery or pickup).

The objective of the problem is to serve all customers and minimize travel cost in an expected value sense. We refer to this problem as the stochastic vehicle routing with pickup & delivery (SVRPD). Practical applications of the SVRPD may arise in different settings. A prime example is the Ex-van deliveries model, in which during each customer visit the driver simultaneously delivers high-demand commodities to retail outlets, and picks up returned pallets, containers, or nonconforming product. The concept behind Ex-van deliveries is to facilitate regular stock replenishment of certain types of commodities, so that a retail outlet (super market, kiosk, etc.) can maximise its sales. The mission of each Ex-van vehicle is to visit all assigned customer sites, typically in a predefined sequence, replenish the stock of selected products, and collect any return packaging or nonconforming product. The demand of each customer point is not known in advance but it is revealed upon arrival. Therefore, the total demand of a scheduled trip typically exceeds the total capacity of the vehicle, forcing it to return to the depot in order to unload any returned items and replenish its own stock, before resuming its route.

Similar delivery characteristics may also arise in other practical settings. For example, material handling systems in a manufacturing shop often operate along fixed pathways that connect the material warehouse with workcenters located along this pathway in a predefined sequence. Note that in addition to the main pathway connecting the workcenters, there are spurs connecting each workcenter with the material warehouse, allowing the direct return of the material handling vehicle (e.g. AGV). In this case, the demand of each workcenter may also be stochastic, depending on the state of the buffers of the workcenters been served. Return items may be finished parts to be delivered to the warehouse (depot).

In this paper we develop for the SVRPD (a) a method and a related algorithm to determine the minimum expected routing cost, and (b) a policy that allows the vehicle’s driver to make the optimal decisions after serving each customer based on the level of the stock on board and the remaining empty spaces in the vehicle.

The rest of the paper is organized as follows. In the next section we present relevant research to date. In Section 3 we present the method to determine the minimum expected cost and in Section 4 we develop and prove the optimal routing policy. We illustrate the results obtained through an appropriate example in Section 5. In Section 6 we study the efficiency of the proposed algorithm. The conclusions of this work are summarized in Section 7.

Section snippets

Background

The objective of the vehicle routing problem (VRP) is to deliver goods to a set of customers with known demand following multiple minimum-cost vehicle routes originating from and terminating to a depot (see [1], [2], [3], [4], [5]). A survey of significant work in VRP is given by Toth and Vigo (see [6]). The stochastic vehicle routing problem (SVRP) refers to a family of problems that combine the characteristics of stochastic and integer programs (see [7]). A characteristic problem is the VRP

Dynamic programming formulation

We consider a set of nodes V = {0, 1,  , n}, with node 0 denoting the depot and nodes 1,  , n corresponding to customers, and a set of arcs A = {(j, j + 1) : j, j + 1  V}  {(0, j) : j  V, j  0}  {(j, 0) : j  V, j  0} that join the customers along the route 1  2    n, as well as all customers to the depot. The travel cost (distance) of each arc (j, j′) is denoted by cjj > 0. The cjj values satisfy the triangular inequality.

Let ξj be the stochastic product demand (to be delivered to) customer j  {1,  , n}; ξj follows a discrete

The optimal routing policy

The value of the minimum expected cost can be estimated using Eq. (1). However, the decision required to guide the distribution operation in the SVRPD environment is the following: Having served customer j, and given the vehicle remaining load and empty space (z, e) respectively, which is the preferred action; (a) to proceed to customer j + 1 directly, or (b) to visit the depot prior to serving customer j + 1. This decision can be modelled by the following variable:xj(z,e)=0if part(a)part(b)in

Implementation

We developed an appropriate algorithm that uses dynamic programming to derive the optimal solution of the pickup and delivery problem in a reasonable amount of time. The solution algorithm proceeds as follows: For each combination z and e and for each step of the algorithm, both terms of Eq. (1) are calculated and the one with the lowest value is selected. For calculating each term of Eq. (1) all allowable values of θ and s are tested and the appropriate minima are selected.

Fig. 11 illustrates

Algorithm complexity and performance

The complexity of the algorithm can be derived from Eq. (1) by examining the complexity of its terms as follows:

  • The complexity of the first term of part (a) is of order O(Q2), where Q is the vehicle capacity.

  • The complexity of the remaining terms of part (a) is of order O(Q3 log Q), where Q log Q is the complexity of the procedure to derive the minimum of each term.

  • The complexity of part (b) is of order O(Q4(log Q)2). This can be derived by considering the number of operations in part (b), which is

Conclusions

In this paper we presented the pickup and delivery case of the stochastic vehicle routing with depot returns for stock replenishment problem (SVRPD). In this case the vehicle not only delivers products to the customers but it also picks up returned items from each customer (e.g. damaged goods, or empty packaging) to deliver to the depot. The objective is to serve all customers by minimizing travel cost under random customer demand.

The characteristics of the problem were presented, together with

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