Analysis, modeling and solution of the concrete delivery problem

doi:10.1016/j.ejor.2007.11.011

European Journal of Operational Research

Volume 193, Issue 3, 16 March 2009, Pages 820-835

https://doi.org/10.1016/j.ejor.2007.11.011 Get rights and content

Abstract

This paper describes a specific local search approach to solve a problem arising in logistics which we prove to be NP-hard. The problem is a complex scheduling or vehicle routing problem where we have to schedule the tours of concrete mixer vehicles over a working day from concrete-producing depots to concrete-demanding customers and vice versa. We give a general mixed integer programming model which is too hard to solve for state of the art mixed integer programming optimizers in the case of the usually huge problem instances coming from practice. Therefore we present a certain local search approach to be able to handle huge practical problem instances.

Introduction

Scheduling concrete mixer vehicles to deliver concrete from concrete-producing depots to concrete-demanding customers introduces some new combinatorial challenges from the viewpoint of traditional vehicle routing and scheduling problems. Besides the sheer size of usual practical problem instances of the concrete-producing/delivering industry we have to take into account constraints and properties that are not considered in standard vehicle routing and scheduling problems.

Most of the additional constraints necessary come directly from the material to deliver. Ready-mixed concrete is a perishable product, which has several ramifications for our problem. First, concrete may only reside in the concrete mixer vehicle for a certain amount of time before it looses quality for the customer and eventually hardens and can even destroy the barrel resulting in disproportional maintenance costs. Second, it is not advisable that vehicles transport a non-full load of concrete, because this could result in an increased rate of hardening of the concrete. This property prevents delivering concrete to more than one construction site before refilling the vehicle at a depot (delivering more concrete to a construction site than the demanded amount poses usually no problem or costs). Finally, if more than one delivery of concrete is necessary to satisfy the demand of the corresponding customer then the temporal spacings between the consecutive deliveries may not exceed certain limits (time lags) as the concrete could partially harden at the construction site before the subsequent supplies arrive.

Besides above mentioned complications for the scheduling part directly induced by the perishability of concrete, typical concrete delivering scenarios differ further from standard vehicle routing problems in the following aspects. Usually, we cannot expect total homogeneity of the vehicles. In our considered practical applications the available vehicle fleet contains concrete mixer vehicles capable of only transporting 2 m³ of concrete to vehicles that can deliver 8 m³. The demands of construction sites often exceed the vehicle capacities and thereby require chains of subsequent deliveries. Furthermore, there exists usually more than one depot and there is enough concrete demand during a given working day calling for decisions from which depot to deliver to which customer.

Altogether we have a highly complex scheduling problem at the intersection of several research areas, such as logistics, supply chains and just-in-time production. We will try to investigate the concrete delivery problem in a general form, but still being able to solve problem instances coming from practice. For this aim this paper is organized as follows. In Section 2, we will give a short review on existing papers on the subject. From Section 3 on we will formulate a general model of the concrete delivery problem by first defining a notation for the input parameters of the problem. Section 4 continues with the definition of a network flow model based on the input data and results eventually in a mixed integer programming model (MIP) which describes formally the general concrete delivery problem that we consider and could be used to solve such problem instances. In Section 5 we present a certain general local search approach, which we utilize to solve our problem in Section 6. The paper finishes in Sections 7 Computational results, 8 Conclusions where the computational results are given and conclusions drawn.

Section snippets

Related work

The logistical problems arising in concrete delivery seem to represent a rather young research field. Following are most publications in this area that are known to us.

Durbin researches in his dissertation [6] the concrete delivery problem both practically and theoretically. The result is a decision support system which is used in practice and incorporates different models described in his thesis. The problem he considers is a special case of ours as it assumes homogenous vehicles. He mostly

Problem parameters and notation

Since the concrete delivery problem that we consider is similar to the vehicle routing problem with time windows (VRPTW), we have based our notation especially on [5], which contains the prevalent notation for VRPTW problems (for an overview over VRPTW problems and the plethora of algorithms we refer to [3], [4] besides the aforementioned paper). Thus, we will simply call the concrete mixers just vehicles and the construction sites customers.

We consider a static model where all data are

Flow network

Given the input parameters defined in the last section we can now define a graph or flow network on which the mixed integer programming model will be based on. The basic idea of the graph is that we represent each possible supply/delivery of a customer, each possible reload of a vehicle at a depot and each starting and ending point of a vehicle as a node and then add all edges to the graph that correspond to a part of a route that could be contained in an optimal solution of our problem. We

Generic local search approach

Because of its size for realistic instances from practice the aforementioned MIP model does not permit generation of an optimal solution using CPLEX. Therefore, we have tried to combine this model with a certain local search approach motivated by Kilby et al. and Schrimpf et al. [9], [12].

The basic idea of our local search approach is the following. Consider a generic local search algorithm $A$ . This algorithm will start with an initial solution s – the incumbent, the current best solution – and

Heuristic for solving the subproblems

The heuristic algorithm begins with fixing all depot node times to constants as follows: $w_{D_{i 1}} ≔ a (D_{i}), w_{D_{i 2}} ≔ a (D_{i}) + mintl (D_{i}), \dots, w_{D_{i, n (D_{i})}} = a (D_{i}) + (n (D_{i}) - 1) mintl (D_{i}) ⩽ b (D_{i})$ for all $D_{i} \in D$ . This may already cut optimal solutions from the solution space, but makes subsequent algorithms much simpler and its impact on the solution quality seems to be rather small on bigger instances, where the depots can reload vehicles at a much faster rate than in our example in Fig. 1.

The core of the heuristic is presented

Computational results

To evaluate our algorithms we tested them on some real world data coming from a typical working week from a big concrete distributing firm which operates in several geographical distinct regions with different vehicle fleets. The real world data and constraints are modelled by our aforementioned MIP model and can therefore be solved with the preceding local search algorithm. All solutions with given run times were achieved on a 1700 MHz Intel Pentium M processor notebook with 512 MB RAM on the

Conclusions

The topic of this paper is to model and solve the logistical problem of delivering concrete from concrete-producing depots to concrete-demanding customers.

To the best of our knowledge our model and the problem itself is the most general one considered in the literature so far and describes the practical problem in our opinion intuitively and rather easily in a formal way. Unfortunately, the size of the model does not allow solving real world instances with an MIP solver. Therefore, we have

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Analysis, modeling and solution of the concrete delivery problem

Abstract

Introduction

Section snippets

Related work

Problem parameters and notation

Flow network

Generic local search approach

Heuristic for solving the subproblems

Computational results

Conclusions

European Journal of Operational Research

Automation in Construction

European Journal of Operational Research

Journal of Computational Physics

Vehicle routing problem with time windows, part I: Route construction and local search algorithms

Transportation Science