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et al. 1999;Spengler et al. 1997; Marìn and Pelegrìn 1998; Jayaraman et al. 1999; Krikke et al. 1999,2001; Fleischmann et al. 2000). In most of the models, transportation and processing costs were minimized while the environmental costs associated with the designed network were often neglected. 2.2 Dynamic lot-sizing problem
The dynamic lot sizing problem in its simplest form considers a facility, possibly a warehouse or a retailer, which faces dynamic demand for a single item over a finite horizon (see Wagner and Whitin 1958). The facility places orders for the item from a supply agency, e.g.,a manufacturer or a supplier, which is assumed to have an unlimited quantity of the product.The model assumes a fixed ordering (setup) cost, a linear procurement cost for each unit purchased, and a linear holding cost for each unit held in inventory per unit time. Given the time varying demand and cost parameters, the problem is to decide when and how much to order at the facility in each period so that all demand is satisfied at minimum cost.
The dynamic lot-sizing problem has been well studied in the past since it was first introduced more than four decades ago. The exact solution technique, known as the Wagner- Whitin algorithm, based on Dynamic Programming is well known in production planning and inventory control. For more information about this model, see the books by Bramel and Simchi-Levi (1997), Johnson and Montgomery (1974) and Silver et al. (1996). A variety of heuristic methods have also been proposed, for example the Silver-Meal heuristic described in Silver and Meal (1973).
In Teunter et al. (2006) a variant of the basic lot sizing model is considered where the serviceable stock may also be made using a remanufacturing operation that utilizes returns and produces serviceable stock that is indistinguishable from the newly manufactured stock. Examples of remanufacturing include single-use cameras and copiers. An inventory system with remanufacturing can be described in Fig . 1. The model studied makes the following assumptions:
– no disposal option for returns;
– holding cost for serviceables is greater than holding cost for returns; – variable manufacturing and remanufacturing costs are not included.
The objective is again to minimize the sum of the set-up costs and holding costs. Two variants are considered. In the first it is assumed that there is a joint set-up cost for manufacturing and remanufacturing which is appropriate when the same production line is used for both processes. The second variant assumes separate set-up costs for manufacturing and remanufacturing. We review these models in the next two sections.
3 Waste management
The widely acknowledged increase in solid waste production, together with the increased concern about environmental issues, have led local governments and agencies to devote resources to solid waste collection policy planning. Waste management is a key process to protect the environment and conserve resources. In recent years, policies of governments towards waste management have focused on waste avoidance, reuse and recycling. As a result there has been significant progress in these management areas, particularly for the more developed nations. The environmental aspects of waste management means that activities concerning the transport of waste materials are clearly part of the Green Logistics agenda.
4 Vehicle routing and scheduling
The Vehicle Routing and Scheduling Problem (VRSP) concerns the determination of routes and schedules for a fleet of vehicles to satisfy the demands of a set of customers. The basic Capacitated Vehicle Routing Problem (CVRP) can be described in the following way.We are given a set of homogeneous vehicles each of capacity Q, located at a central depot and a set of customers with known locations and demands to be satisfied by deliveries from the central depot. Each vehicle route must start and end at the central depot and the total customer demand satisfied by deliveries on each route must not exceed the vehicle capacity, Q. The objective is to determine a set of routes for the vehicles that will minimize the total cost. The total cost is usually proportional to the total distance traveled if the number of vehicles is fixed and may also include an additional term proportional to the number of vehicles used if the number of routes may vary.
The CVRP and many of its variants have been well studied in the literature since its
introduction by Dantzig and Ramser (1959). Its exact solution is difficult to determine for large-scale problems as it is a member of the class of NP-hard problems. Specialised algorithms are able to consistently find optimal solutions for cases with up to about 50 customers; larger problems have been solved to optimality in some cases, but often at the expense of considerable computing time.In practice, other variations and additional constraints that must be taken into consideration usually make the vehicle routing problem even more difficult to solve to optimality.So many solution procedures are based on heuristic algorithms that are designed to provide good feasible solutions within an acceptable computing time, but without a guarantee of optimality.
There are several books and survey articles that summarize different approaches and provide references to the large number of journal articles that have been written on this topic (e.g., Golden and Assad 1988; Toth and Vigo 2001). There are many other research works about the classical CVRP. Some exact methods have been tailored for this problem (e.g., Laporte and Nobert 1987; Agarwal et al. 1989; Lysgaard et al. 2004; Fukasawa et al.2006). Others have proposed approximate methods and heuristics due to the complexity of the problem and the need to solve it in a reasonable computing time (see Gendreau et al.2002; Laporte and Semet 2002; Cordeau and Laporte 2004; Cordeau et al. 2005). Most of these approaches are based on local search techniques.
Most papers assume that the costs and times of traveling between the depot and the customers and between customers are known and fixed. They are either given or calculated using a shortest path algorithm on the graph or network representing the locations. In practice,the times and shortest paths may vary, particularly by time of day.
5 Conclusions
This paper has described the field covered by Green Logistics and described some of the new problems that arise when the objectives considered are not simply economic, but involve wider environmental and social considerations too. There are many different types of operational research models that have key roles to play in dealing with Green Logistics issues, but in this paper we have concentrated on describing areas where combinatorial optimization is central to the design of acceptable solutions. It is expected
that as environmental factors assume increasing importance, the effective use of combinatorial optimization theories and techniques will be needed to meet the challenges of new problems.There is a research consortium in the UK working on many different aspects of Green Logistics models and more information can be found on the website of the Green Logistics project. The Green Logistics project includes several work modules that relate to topics covered in this review such as reverse logistics and the effect of vehicle routing and scheduling policies on the Green Logistics agenda.