This paper considers a multiperiod vehicle lease planning problem for urban freight consolidation centers (UFCCs) in the urban freight transport network where short-term-leased and long-term-leased vehicles are hired together. The objective is to allocate the two kinds of leased vehicles optimally for direct transportation services from the associated origin node to the associated UFCC or from the associated UFCC to the associated destinations so as to satisfy a given set of period-to-period freight demands over a given planning horizon at total minimum vehicle allocation cost subject to demand-dependent transportation time restriction. The problem is formulated as an integer programming model and proven to be NP-hard in a strong sense. Thus, a Lagrangian heuristic is proposed to find a good solution efficiently. Numerical experiments show that the proposed algorithm finds good lower and upper bounds within reasonable time.

1. Introduction

Recent worldwide economic growth has accelerated urbanization, and people have been moving to cities for better jobs and more fulfilling lifestyle. In the late 2000s, it was reported that more people lived in cities than in rural areas, and it is expected that more people will move to urban areas in the near future [1, 2]. According to Blanco and Fransoo [3], the number of megacities with at least 10 million people is increasing, and in a decade, their contribution to world economy will steadily grow and constitute more than 20% of world GDP. As people start to live in concentrated and sometimes congested areas, traffic jams and air pollution continue to increase. As a result, many researchers have started to investigate the impacts of congestion due to urban transportation systems [2, 4].

Freight transportation is one of the key contributors to traffic congestion and harmful pollutants in cities. Unlike typical passenger cars, freight vehicles are bigger in size and move more frequently due to the nature of their business. Furthermore, a recent trend toward just-in-time delivery makes the problem even worse [5]. To survive in competition and to meet increased customer expectation about timely delivery, companies have dispatched small packages more frequently to customers inside metropolitan areas. As average vehicle utilization went down and the number of vehicles dispatched increased, traffic congestion and the subsequent problems have been exacerbated to such an extent that they could cause serious social issues. To deal with the problems caused by urban freight transportation, integrated approaches for city logistics systems have been proposed. Crainic et al. [5] proposed an integrated short-term scheduling of operations management of logistics resources. Ehmke [1] and Ehmke et al. [6] proposed efficient routing systems integrating traffic information and logistics. Yang et al. [7] investigated a problem of designing a city logistics network considering green-house gas emissions. Thus, the research goal of urban freight transportation and city logistics should be aligned with congestion and pollution reduction with less degradation of the city center commercial activities. In this regard, multitier urban freight transportation system with consolidation and in-out synchronization, so-called urban freight consolidation, has been proposed as an alternative transportation infrastructure for city logistics.

Urban freight consolidation center (UFCC) is a logistics facility located at the boundary of urban areas to serve freight transportation to city centers including retailers, buildings, and construction sites [8]. Products are transported from their origins such as manufacturers to their destinations in cities through urban consolidation centers. They are first moved to urban consolidation centers, where incoming shipments are unloaded, sorted, and consolidated with other products from different origins. Sorted and consolidated products are then transferred directly to outbound vehicles without storage. In just-in-time delivery environment, it is quite important to send orders to customers when necessary, even though shipment size is not enough to fill a full truck load. By consolidating products in an intermediate point between origins and destinations, urban consolidation centers can contribute to an increase of vehicle utilization, an increase of average size of vehicles involved, and the reduction of delivery frequency. In addition to this, the usage of environmentally friendly vehicles such as electric and clean natural gas power vehicles could often contribute to the decrease of overall harmful emissions from freight transportation vehicles. By improving the overall loading factor of a vehicle destined for congested city centers, urban consolidation centers could effectively reduce the total travel distance and further reduce the impact of freight operations on traffic congestion [9]. In fact, the concept of UFCC has been tested with real business practice in European countries and Toronto, Canada [1012].

Research on urban freight consolidation centers focuses on the economic analysis of the consolidated freight transportation system. Su and Roorda [12] and Triantafyllou et al. [13] showed that urban consolidation centers can be successfully operated in real urban environments. It was reported that the trial systems were able to reduce harmful emissions and traffic congestion in city centers when properly managed and synchronized. Marcucci and Danielis [8] showed that, in their analysis, urban consolidation centers could attract a considerable amount of freight shipments bound to urban areas. Zhou and Wang [14] studied the issues related to development and construction of consolidation centers and showed that proper strategies based on public-private partnership can increase overall economic benefits of the system to participants in city logistics. While the feasibility of urban freight consolidation centers has been studied in depth, operation and network planning issues have been discussed (e.g., [1517]) but still not been fully investigated yet. The detailed analysis of operation and planning of urban freight consolidation system should be done to be deployed in real business practice. In the supply chain context, an urban consolidation center is similar to the transportation system with cross-docking terminals. In cross-docking network, goods are moved through cross-docking terminals where shipments are sorted, consolidated, and transferred to outbound vehicles in a synchronized manner.

Research on operation and management of the cross-docking based consolidation strategy has progressed in two directions. One is concerned with problems that are related to the internal operations at the consolidation center. Gue [18] analyzed the effects of scheduling trailers into doors on a layout of a freight consolidation center. Bartholdi III and Gue [19, 20] designed the layout of a freight consolidation center. Li et al. [21] studied a scheduling problem to minimize storage and order picking in a consolidation center. On the other hand, planning problems on a network level have been considered to locate consolidation centers, to allocate vehicles, and to make vehicle consolidation schedules. Ratliff et al. [22] and Chen et al. [23] considered a problem of making vehicle consolidation schedules for a transportation network. Donaldson et al. [24] considered a problem of allocating vehicles and making vehicle consolidation schedules. Sung and Song [25] and Sung and Yang [26] studied an integrated model of locating consolidation centers and allocating vehicles. All the works on a network level about consolidation based transportation have considered static freight demands, not varying with time. However, freight demands in various industries such as food, apparel, electronic goods, and logistics may be dynamic. Especially in urban freight transportation, demands tend to fluctuate over time. As a cost-efficient way of allocating vehicles to satisfy dynamic freight demands, the issue of vehicle supply on lease has received much research attention, in the situation where the unit-period vehicle lease cost depends on the lease term. The unit-period long-term vehicle lease cost is generally cheaper than the unit-period short-term vehicle lease cost [2729]. In the case of static freight demands, the long-term vehicle lease is obviously better than the short-term vehicle lease, while in the case of dynamic freight demands, either one does not dominate over the other so that the two lease options need to be considered together. Furthermore, in an urban freight consolidation setting, we need to consider consolidation of shipments which makes the lease planning more difficult.

Therefore, this paper considers a multiperiod vehicle lease planning problem in an urban freight consolidation network (MVLPUC). The problem, denoted by , is concerned with optimally allocating the two kinds of leased vehicles for inbound and outbound transportation services so as to satisfy a given set of period-to-period freight demands over a given planning horizon at total minimum vehicle allocation cost subject to demand-dependent transportation time restriction. The planning horizon is divided into discrete time periods such as weeks or months. It is assumed that each freight demand is transported through a single path via one urban consolidation center (where the operations of sorting and consolidating are handled) located between origin and destination nodes, and each outbound vehicle at each urban consolidation center departs as soon as all the associated inbound vehicles arrive and the associated freight demands are sorted appropriately. It is also assumed that an unlimited number of homogeneous capacitated vehicles can be acquired through either long-term lease or short-term lease, with the unit-period short-term vehicle lease cost being greater than or equal to the unit-period long-term vehicle lease cost.

This paper is organized as follows: Section 2 introduces the problem formulation for . In Section 3, we discuss the procedure for solving based on Lagrangian relaxation and problem decompositions. We present the numerical experiments in Section 4 and then conclude our discussion in Section 5.

2. Model

In this section, we present the mathematical formulation for Problem . Before introducing the model formulation, all the parameters and decision variables used in this paper are given as follows.

Sets and Parameters : sets of origin nodes representing manufacturers, destination nodes representing retailers, intermediate nodes representing UFCCs, and all nodes (i.e., ), respectively.: sets of edges representing potential direct services from origin nodes to intermediate nodes and from intermediate nodes to destination nodes, respectively ( and ).: set of all edges (i.e., ).: set of freight demands, defined by an ordered pair of two nodes for all and .: subset of , each demand of which can be transported through the intermediated node within the associated transportation time restriction (to be explained later).: set of time periods.: quantity of freight demand at time period .: unit-period long-term vehicle lease cost for and , respectively.: unit-period short-term vehicle lease cost for and , respectively, at time period ( and ).: vehicle capacity.: transportation time elapsed for each and , respectively.: handling (sorting and consolidating) time at the intermediate node .: transportation time restriction required for freight demand .Decision Variables : 1 if freight demand is transported in period through the intermediate node and 0 otherwise.: numbers of long-term-leased vehicles allocated for and , respectively.: numbers of short-term-leased vehicles allocated for and , respectively, in period .We note that, for notational simplicity, notations (or subscripts) and (or ) are used alternately to refer to freight demand (or ) on the corresponding edge . Furthermore, the two terms “edge” and “direct service” are used interchangeably in this paper. Then, we now present the problem formulation for Problem as follows.

Problem The objective function (1a) represents the cost of allocating any long-term-leased and short-term-leased vehicles for edges. Constraints (1b) imply that all the freight demands have to be serviced for each period. Constraints (1c) and (1d) require that the total amount of demands transported through any edge should not exceed the total capacity of any allocated vehicles for each period. In regard to the set in constraints (1b), (1c), and (1d), we assume that there exists the transportation time limit for each freight demand such that the sum of the transportation times between nodes and the handling time at an intermediate node does not exceed (i.e., ). Thus, for each , the elements of set can be identified as freight demands in that satisfy the aforementioned condition with at a preprocessing stage.

3. Solution Approach

In this section, we propose a heuristic approach based on Lagrangian relaxation. Problem can be proven as NP-hard in a strong sense in the same manner as shown in Sung and Song [25]. Moreover, considers time-varying demands, so it may be too complex to derive an exact algorithm to find optimal solutions even for small-sized problem instances. Therefore, we here propose an efficient heuristic method based on the Lagrangian relaxation.

3.1. Lagrangian Relaxation

Before relaxing constraints in for Lagrangian relaxation heuristic, the following problem is derived by introducing another decision variable (equivalent to variables ) and adding a set of constraints to .

Problem Problem has some obviously redundant variables and the associated constraints (2d) while it yields the interesting problem structure which is good to apply the Lagrangian relaxation method. That is, if constraints (2b) and (2d) in are Lagrangian relaxed, then the resulting problem can be decomposed into single-edge problems. Thus, constraints (2b) and (2d) are Lagrangian relaxed with Lagrange multipliers (unrestricted) and (≥0), respectively. For given , the resulting problem can be derived as follows.

Problem It is evident that is a lower bound on for any given . Let , and let be a lower bound (on ) obtained by solving the linear programming (LP) relaxation problem of . Then, the proposed Lagrangian relaxation provides a good lower bound as indicated in Proposition 1. Since the proposition below is obvious to show, we omit the proof.

Proposition 1. Consider .

Furthermore, as mentioned earlier, Problem can be decomposed into single-edge problems of selecting demand and allocating vehicles (SEPDV), resulting in and for each and , respectively, as follows.

Problem Problem It is straightforward to show that, for given ,and hence, can be evaluated through and . Since and have the same problem structure, we only discuss the procedure for solving in the next section.

3.2. Solving

In this section, we present the solution procedure for so that we can eventually evaluate the value of for each . First, for given , Proposition 2 and Corollary 3 are characterized so as to reduce the solution space of and variables, respectively, in .

Proposition 2. For given , the following statements hold in : (a)If for some and , then there is an optimal solution with .(b)If for some and , then there is an optimal solution with .

Proof. (a) Let us compare the situations of and for some and such that . The value of the objective function (4a) for the former case cannot be smaller than that for the latter case because the coefficient of the variable in (4a) is nonnegative. Also, the former case makes the constraint (4b) tighter than the latter case. Therefore, the former case (i.e., ) cannot give a better solution than the latter case (i.e., ).
(b) This can be shown in the same manner as in the proof of (a), and hence we omit its proof.

Let be a set of freight demands that can be transported through where and . Note that . For , let and be subsets of satisfying Propositions 2(a) and 2(b) in , respectively, and let .

Corollary 3. For given and fixed at the associated values according to Proposition 2, there is an optimal solution in with , where

Proof. The constraint (4b) can be rewritten asfor all . Since all the variables are binary variables, holds for all . Since it is assumed that for all , any feasible solution with , where cannot yield a better solution value than the feasible solution with . Moveover, any feasible solution with , where cannot yield a better solution value than the feasible solution with . Therefore, there is an optimal solution with . Also, with fixed at , it is obvious that there is an optimal solution with .

Then, the restricted problem with fixed at in , denoted by , can be decomposed into single-period bounded knapsack problems (BKP), , as follows.

Problem Note that . We next discuss how to solve . Let denote the optimal value of . We first transform into the associated 0-1 knapsack problem, and the knapsack problem can be solved by the dynamic programming algorithm proposed by Toth [30]. We remark that takes the finite number of integer values in the set . Furthermore, for each value of in , it is straightforward to show that the problem reduces to a 0-1 knapsack problem with decision variables for all . Therefore, we iteratively solve a 0-1 knapsack problem for each value of in , compare the objective function values for each case, and then identify the optimal solution for . Once all the problems of are solved for each value of as mentioned above, the optimal value of the problem , , is computed asThe number of problems and to be solved can be further reduced by Propositions 4(a) and 4(b), respectively. Let be the optimal solution of with explicit dependence on , and furthermore, let us denote as for and . Note that for .

Proposition 4. (a) Given for some and some , if for some , then .
(b) Given for some and all , if for some , then for .

In summary, we now present the solution procedure for , PROC(), in Algorithm 1 based on the whole discussion above. For reference, Figure 1 illustrates the relationship among all the problems discussed in this section.

(1) Let for all and for all be , and -variables of
   the optimal solution of respectively;
(2) Set ;
(3) Fix some at the associated values according to Proposition 2 and determine ,
    and for all by Corollary 3;
(4) Set for all and ;
(5) Set and for all and where and
    are binary variables having 1 only if and are not necessary
   to be solved respectively, and 0 otherwise;
(6) Set ;
(7) while     do
(8)  if     then
(9)     Set ;
(10)     while     do
(11)      if     then
(12)       Solve and set the resulting solution as (;
(13)       Set for all and ;
(14)     Set and for all
      by Proposition 4(a);
(15)     Set ;
(16)     Set for all by Proposition 4(b) where
      such that ;
(17)     if     then
(18)     Set (refer to (10));
(19)     Set and for all ;
(20)   Set ;

3.3. Finding the Lagrange Multipliers

So far, we examined how to address the problem for each when the Lagrangian multiplier is given. We now discuss the procedure for finding the Lagrangian multiplier . For the purpose, we adapt the subgradient optimization procedure which is one of the most popular methods to find a good set of Lagrange multipliers [31]. Specifically, the Lagrange multipliers are generated iteratively as(i) for all , ,(ii) for all , , ,where and are the subgradients for constraints (2b) and (2d), respectively, and is the stepsize at iteration .

When it comes to the subgradients and in the formulas above, they are determined by the solution of given as follows: for all , , and for all ,, . Moreover, the stepsize at iteration is determined bywhere represents the best upper bound of found up to iteration and is a control parameter for . In this paper, is set to 2 at the beginning and then halved if the lower bound is not improved in a predetermined number of consecutive iterations.

3.4. Finding a Feasible Solution

A solution obtained by solving based on the discussion in Section 3.2 may be infeasible to , and hence we propose a two-phase heuristic to deal with infeasibility issues as follows: In the first phase, a construction heuristic (CH) is derived to modify any infeasible solutions to into feasible ones, and then, in the second phase, a tabu search-based heuristic (TSH) is utilized to improve the solution of (CH). The details are presented in the subsequent sections.

3.4.1. Construction Heuristic (CH)

As mentioned above, a construction heuristic (CH) intends to deal with possible infeasibility issues of solutions obtained by solving , and (CH) is performed at each subgradient iteration. Let and be the - and -variable values of the optimal solution of and , respectively. First, the corresponding demand for all and is reassigned to the UFCC with the lexicographical maximum value of the three-tuple relation . Then, from the reassigned demands, demand quantity for all and is computed as or . Finally, to allocate vehicles for each at the minimum vehicle cost, we solve the following vehicle allocation problem (VAP), .

Problem Proposition 5 characterizes the optimal solutions of .

Proposition 5. There is an optimal solution of with , where and is the th largest .

Proof. As in Corollary 3, it is straightforward to show that there is an optimal solution of with . For an integer such that and for ,Let be the value of the objective function (12a) associated with fixed at in . Then, . It implies that is a linear function in over the range . Therefore, there is an optimal solution of with .

By Proposition 5, can be solved by finding a solution of having the minimum cost among number of solutions which are associated with fixed at for all .

3.4.2. Tabu Search-Based Heuristic (TSH)

After the first phase (CH), a tabu search-based heuristic (TSH) is performed to improve the solution obtained from (CH). We note that (TSH) is expected to require much more computational time than (CH), and hence, to save the elapsed time of the overall algorithm, (TSH) is executed only at the subgradient iterations satisfying the following two conditions ((a) and (b)) together: (a) the best lower bound is improved and (b) the solution obtained by (CH) at the current subgradient iteration is different from the solutions obtained by (CH) so far. In this paper, we call the conditions (a) and (b) as “(TSH)-activating conditions.

Tabu search is one of the popular metaheuristics designed to escape the trap of local optimality [32], and it has been used frequently in various network-related problems as in Sung and Song [25], Crevier et al. [33], and Üster and Maheshwari [34]. We now explain the main features of the tabu search used in (TSH), including move operation, tabu restriction and aspiration criterion, move evaluation, and termination condition.

The move operation used in this paper is to modify assignment of some freight demand, say , from its current UFCC, say , to the alternative UFCC, say , for some period, say . It corresponds to modifying the value of from to and the value of from to in Problem . The tabu restriction is employed such that the UFCC assignment for any freight demand should not be changed during tabu_size number of iterations except for the case that satisfies the aspiration criterion (to be explained later). To implement the tabu restriction, as a new solution is selected, the information of the freight demand associated with the move operation is recorded in a list (called tabu list) and kept in the list over the tabu_size number of iterations. In this paper, the tabu_size is chosen randomly from the discrete uniform distribution in the closed interval . The aspiration criterion adapted here is as follows: a solution from the move operation for the freight demands recorded in the tabu list is eligible to be chosen as a new solution only if the move improves the best upper bound. Moreover, when we choose a new solution among neighbor solutions at each iteration, the move evaluation function used in this paper is to maximize where , , and represent the decrease amount of the objective function value, the frequency that the corresponding move is used to produce a new solution so far, and the associated penalty with the frequency, respectively. When it comes to , it is computed simply by subtracting the objective value of the new solution from the objective value of the current solution, and if is positive, is set to and otherwise. We note that this move evaluation function is known to have an advantage in diversifying the search direction efficiently [32]. Lastly, if the total number of performed iterations amounts to max_iter or the best upper bound has not been improved during max_no_imp consecutive iterations, (TSH) is terminated. In this paper, max_iter and max_no_imp are set at and , respectively.

3.5. Overall Solution Algorithm

We now summarize all the discussion and present the algorithm PROC() for solving in Algorithm 2. It can be terminated either after a predetermined number ( in this paper) of iterations or when gets too small (i.e., in this paper). It can also be terminated when is less than , where and represent the best upper and lower bounds searched so far, respectively.

(1) Initialize the necessary parameters and Lagrange multipliers (i.e., , , ,
   , , and );
(2) Initialize where represents the number of consecutive iterations for
   which the best lower bound has not been improved;
(3)    Solve the Lagrangian problem , and let be the resulting objective value;
      if     then
(4)     Set and