Ann Oper Res (2010) 181: 185–197 DOI 10.1007/s10479-010-0714-1
Solving multi-objective optimization formulation for fleet planning in a railway industry Hamid Reza Sayarshad · Nikbakhsh Javadian · Reza Tavakkoli-Moghaddam · Nastaran Forghani
Published online: 12 February 2010 © Springer Science+Business Media, LLC 2010
Abstract In this paper, we propose a novel three-objective mathematical model and a solution procedure for optimizing fleet planning for rail-cars in a railway industry. These objectives are to: (1) minimize the sum of the cost related to service quality, (2) maximize profit calculated as the difference between revenues generated by serving customer demand and the combined costs of rail-car ownership and rail-car movement and, (3) minimize the sum of the rail-car fleet sizing, simultaneously. The Pareto optimal set is depicted and used for a trade-off analysis. A number of numerical examples are given to illustrate the presented model and solution methodology. Keywords Multi-objective model · Fleet planning · Rail-car size · Railway
1 Introduction The fundamental problem of optimum distribution or allocation arises in a variety of industries from Industrial Engineering to Civil Engineering, with applications that can range from traffic analysis to delivery systems. Consequently, any theoretical contribution that solves these kinds of problems in general can have far reaching implications and applications. A common area, in which optimum distribution becomes critical, is that of rail transport with fleets of rail-cars circulating in networks. In recent years, investment in rail-based freight transportation has increased, and this has been the catalyst for complex managerial problems, attracting attention from industry and academia point of view. The capacity of a rail transport system is directly related to the number of available rail-cars (i.e., fleet H.R. Sayarshad () · N. Javadian Department of Industrial Engineering, Mazandaran University of Science and Technology, Babol, Iran e-mail:
[email protected] R. Tavakkoli-Moghaddam Department of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran N. Forghani Department of MBA, College of Ness Wadia, University of Pune, Pune, India
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size) and their distribution among rail yards. Rail yards are owned by railroads and used to build outbound trains, break down inbound trains, classify inbound cars for assignment to outbound trains, and provide facilities for refueling, crew change, and maintenance functions. Owners and operators of rail transport systems invest in rail-cars in order to provide the capacity needed to meet demands. Consequently, determining this optimum number of rail-cars (i.e., optimum fleet size) can save money. However, determining this optimal for a particular system requires a trade-off between the cost of owning rail-cars and the potential penalties associated with not meeting demands as a result of not using enough rail-cars. This implies a need for multi-objective optimization. In addition, serving demands results in the relocation of rail-cars; however, this consequent movement between various locations is often imbalanced. This implies the need for an optimal allocation strategy for both loaded and empty rail-cars over networks. This paper is organized as follows. Section 2 provides a brief literature review summarizing the current state of the art and highlighting deficiencies that the proposed work responds to. The mathematical model for the problems is detailed in Sect. 3, in the form of a threeobjective model. This includes a thorough discussion of the design variables, the objective functions, and the constraints. Section 4 summarizes basic concepts concerning threeobjective and then outlines the methods that are used to provide a single solution point as well as depict the complete Pareto optimal set. Section 5 provides an example problem with extensive discussion of the results. Finally, Sect. 6 summarizes the primary contributions, discusses broader implications of the work, and itemizes potential areas for future work.
2 Literature review Fleet planning problems in general represent a critical area of Industrial Engineering and Planning. Consequently, a number of models have been developed for a variety of application areas. Two common applications are the trucking industry (Hall and Racer 1995; Du and Hall 1997; and Ozdamar and Yazgac 1999) and airline express package services (Barnhart and Schneur 1996). Furthermore, fleet sizing problems can be related to service design (Cranic 2000). Fleet planning is also important in material handling systems used for manufacturing operations (Beamon and Deshpande 1998; and Beamon and Chen 1998). Although it is not extensive, some studies have been completed with fleet sizing problems for vehicles. Sherali and Tuncbilek (1997) and Sherali and Maguire (2000) proposed a static and dynamic time-space network representation of fleet sizing models for the automobile and railroad industries, concerned with the problem of shipping automobiles via railroad auto racks. These studies do not consider both the issues of rail-car fleet sizing and rail-car allocation. Much of the work with fleet sizing, which is related to the distribution of loaded and empty freight cars, belongs to the class of practical decision-making problems. Regarding such problems, Dejax and Grainic (1987) provided a thorough survey of the models for management and distribution of empty vehicles as well. A survey of optimization models for trail routing and scheduling can be found in Cordeau et al. (1998). Powell (1988) and Mendiratta and Turnquist (1982) introduced network flow models for the empty vehicle distribution. Their efforts concern empty vehicle allocation, not fleet size decisions. Frantzeskakis and Powell (1990) and Yafeng and Hall (1997) treated a problem of fleet sizing and empty equipment redistribution from the standpoint of the inventory theory and developed decentralized stock control policies for empty equipment; however, their focus is on utilization of a given vehicle fleet, not on fleet sizing decisions. Beaujon and Turnquist (1991) described a model for fleet sizing and vehicle allocation that includes uncertainties in both demands and travel
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times. Their focus is on developing a general model for fleet sizing, not necessarily on fleet sizing for rail-car operations. Turnquist and Jordan (1986), and Powell and Carvallho (1998), developed a taxonomy of vehicle fleet sizing by distinguishing between deterministic and stochastic models and dividing them into the sub problems of fully and partially loaded vehicles, but this work involves building general models in transportation systems, not on fleet sizing in railroad operations. In general, these existing models do not involve fleet-sizing decisions. Under the topic of fleet sizing, much work has been completed with transportation problems associated with hazardous material in general. This work tends to focus on routing and scheduling of individual truck trips (Nozick et al. 1997; Alumur and Kara 2007; Tarantilis and Kiranoudis 2001; and Miller-Hooks and Mahmassani 1998) or issues associated with either disposal facilities (e.g., ReVelle et al. 1991; List and Mirchandani 1991) or emergency response teams (e.g., List 1993; List and Turnquist 1998). Surveys of works related to hazardous materials transportation are available (List et al. 1991; Erkut and Alp 2007 and Erkut and Verter 1995). In general, their studies do not consider fleet sizing. Rather, it focuses on routing and scheduling. In addition to providing basic models for fleet sizing problems, some researchers have begun to consider fleet sizing problems in the context of computational optimization problems. Fu and Ishkhanov (2004) used a heuristic approach to determine the optimum mix of different types of vehicles in order to maximize operating efficiency. Wu et al. (2005) also modeled fleet sizing as a linear problem and optimize the fleet size and mix for the truckrental industry. Bojovic (2002) addressed the problem of determining an optimal number of rail-cars to satisfy demand, while minimizing the total cost. As suggested earlier, fleet sizing problems often involve multiple goals and objectives, and thus require the use of bicriteria to optimize performance. However, work in this area has been limited. Chang et al. (2000) use a multi-objective fuzzy mathematical programming method to minimize operating cost and minimize the passenger’s total travel time loss. This work focuses on routing and scheduling problems for vehicles. Most recently, Sayarshad and Tavakkoli-Moghaddam (2010) presented a single objective model for the fleet size and freight car allocation with stochastic demands. They proposed a two-stage solution procedure based on a SA algorithm for solve the given problem. Sayarshad and Ghoseiri (2009) use simulated annealing (SA) to maximize profit in a multi-period problem. However, in this paper, we consider the following capabilities, namely (1) incorporating three objectives and using the Pareto optimal set for the presented model, (2) introducing both fleet sizing and allocation, and (3) applying this concept to a rail-cars fleet planning problem with the associated rail-yard restrictions. 3 The mathematical model This section details a three-objective mathematical model for fleet planning problems, stemming from the work of Sayarshad and Marler (2009). In this paper, the aim is to determine how to distribute and transport rail-cars over a specified number of rail yards and a specified period of time in order (1) to maximize the naturally conflicting objectives of profit, (2) to minimize the unmet demands (or maximize the service quality), and (3) to minimize the rail-car fleet size. 3.1 Design variables Following, we describe the design variables for the given problem. We assume that the planning horizon (T ) has been divided into discrete decision periods, and we use t to denote
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one such period. A set of rail network locations is denoted by N , which is divided to two subsets of N1 and N2 with respect to the origin and destination points in the network. We also assume that there exist demands for transportation service between points i ∈ N1 , and j ∈ N2 in period t ∈ T , and these demands can be represented as dij (t). Demands induce the loaded rail-car movements that constitute the first design variables and are represented as Lij (t). Because the demand for loaded rail-cars into any point i may not be the same as the demand from i to other points, movements of empty rail-cars balance the flow. In general, there may be several different types of cars available to serve demands, so we can expand the notation to Lij k (t) to represent flows of cars of type k. However in this paper, we concern only one type of car. These empty- rail-car movements also represent design variables and are denoted by Ej i (t). We denote the number of rail-cars present at origin i and destination j at the end of period t respectively by Soi (t) and Sdj (t). We also denote the number of rail-cars present at origin i and destination j at the initial and end of period t by Soi (1) and Sdj (T ), respectively. Fleet sizing is the sum of the Soi (1) for all origin i ∈ N1 : Fleet size =
i
Soi (1) =
Sdj (T )
(1)
j
If insufficient rail-cars are available at origin i in period t to meet all demands, some demands are either backordered or lost from the system. We denote the number of units of demand (i.e., rail-car loads) from i to j that remains unmet at the end of period t by Qij (t). The design variables are summarized as follows: Lij (t): Ej i (t): Soi (t): Sdj (t): Qij (t):
number of loaded rail-cars dispatched from i ∈ N1 to j ∈ N2 in period t ∈ T . number of empty rail-cars dispatched from j ∈ N2 to i ∈ N1 in period t ∈ T . number of rail-cars present at origin i ∈ N1 at the end of period t ∈ T . number of rail-cars present at destination j ∈ N2 at the end of period t ∈ T . unmet demand from i ∈ N1 to j ∈ N2 in period t ∈ T .
The total number of design variables is equal to T (3N1 N2 + N1 + N2 ). 3.2 Objective functions Following, we discuss the components of the three objective functions; namely (1) minimizing the sum of the cost related to service quality, (2) maximizing profit calculated as the difference between revenues generated by the serving customer demand and the combined costs of rail-car ownership and rail-car movement, and (3) minimizing the sum of the rail-car fleet sizing. The treatment of travel time plays an important role in the formulation of fleet management models. In many systems, travel times are unknown due to equipment failures and/or external interference. Rather than introduce travel time directly, we choose instead to formulate the problem in terms of the rail-car arrivals. In particular, given that Lij (τ ) loaded rail-cars are dispatched from point i to point j in period τ . Thus, we define αij (τ, t) as the proportion of loaded rail-cars that are dispatched in period τ from i to j , and that arrive in period t . We define βj i (τ, t) as the proportion of empty rail-cars that are dispatched in period τ from j to i, and that arrive in period t . We assume a constant revenue per loaded rail-car sent from i to j , denoted as rij .We also assume that the unit costs of moving rail-cars from point i to j are constant and denoted by λij for loaded rail-cars and θj i for empty rail-cars. q is constant number representing the daily cost of the ownership and rent of the rail-car. We define hi as the unit cost of
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holding a rail-car for one period at origin i, where hi ≥ q. We also define wj as the unit cost of holding a rail-car for one period at destination j , where wj ≥ q. The penalty cost for the unmet demand (i.e., backordered rail-cars) is also important to the presented model. We denote by ρij the unit penalty cost per period for rail-cars loaded waiting at i to be transported to j . The revenues and costs associated with operating the system are summarized as follows: rij : revenues per loaded rail-car sent from i ∈ N1 to j ∈ N2 . λij : cost of moving a loaded rail-car from i ∈ N1 to j ∈ N2 . θj i : cost of moving a empty rail-car from j ∈ N2 to i ∈ N1 . q: cost per period to own or lease a rail-car. hi : cost of holding a rail-car for one period at origin i ∈ N1 . wj : cost of holding a rail-cars for one period at destination j ∈ N2 . ρij : penalty cost per period for one unit of unmet demand from i ∈ N1 to j ∈ N2 . SCi (t): yard capacity at origin yard i ∈ N1 at the end of period t ∈ T . SCj (t): yard capacity at destination yard j ∈ N2 at the end of period t ∈ T . αij (τ, t): proportion of loaded rail-cars dispatched from i ∈ N1 to j ∈ N2 in period τ ∈ T which arrive in period t ∈ T , such that: αij (τ, t) = 1 ∀i, j, t (2) τ
