Outbound supply chain network design with mode selection, lead times and capacitated vehicle distribution centers

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1 European Journal of Operational Research 165 (2005) Production, Manufacturing and Logistics Outbound supply chain network design with mode selection, lead times and capacitated vehicle distribution centers Erdem Eskigun a, Reha Uzsoy b, *, Paul V. Preckel c, George Beaujon a, Subramanian Krishnan a, Jeffrey D. Tew a a General Motors Corporation, Warren, MI 48090, USA b Laboratory for Extended Enterprises at Purdue, School of Industrial Engineering, Purdue University, 1287 Grissom Hall, West Lafayette, IN 47907, USA c Laboratory for Extended Enterprises at Purdue, Department of Agricultural Economics, Purdue University, West Lafayette, IN 47907, USA Received 8 July 2002; accepted 24 November 2003 Available online 16 March 2004 Abstract Most distribution network design models considered to date have focused on minimizing fixed costs of facility location and transportation costs. Measures of customer satisfaction driven by the operational dynamics such as lead times have seldom been considered. We consider the design of an outbound supply chain network considering lead times, location of distribution facilities and choice of transportation mode. We present a Lagrangian heuristic that gives excellent solution quality in reasonable computational time. Scenario analyses are conducted on industrial data using this algorithm to observe how the supply chain behaves under different parameter values. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Supply chain; Network design; Lagrangian heuristic; Capacitated distribution centers 1. Introduction The outbound supply chain network of an automotive company, shown in Fig. 1, consists of the activities involved in transporting finished vehicles from the assembly plants to dealers. Vehicle distribution centers (VDCs) are used to consolidate and distribute vehicles from different plants to dealers. However, for long-term planning purposes, individual dealerships are aggregated into demand areas that are county-level geographic areas that cover essentially the entire North American auto market. Each assembly plant produces a different vehicle type with some plants producing more than one type within the same facility. * Corresponding author. Tel.: ; fax: address: uzsoy@ecn.purdue.edu (R. Uzsoy) /$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi: /j.ejor

2 E. Eskigun et al. / European Journal of Operational Research 165 (2005) HWY PLANTS PLANTS HWY HWY RAIL RAIL HWY Demand Areas RAIL RAIL RAIL RAIL VDC VDC HWY HWY HWY HWY Demand Areas Demand Areas Fig. 1. Outbound supply chain network of a large-scale automotive company. All vehicle types from the same plant are delivered to a destination using the same transportation mode to take advantage of economies of scale and to simplify the delivery process (e.g., loading, unloading, tracking, etc.) of the vehicles. Therefore, different vehicle types produced in a plant are aggregated into one type. Vehicles produced in the plants are delivered to demand areas via one of two basic modes (see Fig. 1). Vehicles can be loaded on to trains at the plants and delivered to vehicle distribution centers (VDCs). They are then loaded on to trucks at the VDCs and sent to dealers. Vehicles can also be delivered directly from plants to dealers by truck, if the demand area is relatively close to the plant. An important development in the automotive industry in recent years has been an increased interest in reducing the lead-time required to deliver vehicles from the assembly plants to the customer. The potential benefits of lead-time reduction in supply chain management have been widely documented, and include responsiveness to market changes, reduced pipeline inventory and improved customer satisfaction. The lead-time required to move vehicles through the outbound supply chain is the sum of the lead times at the nodes of the network (plants and vehicle distribution centers) and the transportation time. The lead-time at the nodes, in turn, is affected by the volume of flow through each node. Hence we study a network design model that includes lead-time related costs as well as the more traditional fixed costs of locating facilities and transportation costs. In the network design model of an outbound supply chain with capacitated VDCs, the decisions to be taken can be outlined as follows: (1) Where should the vehicle distribution centers (VDCs) be located? (2) What should the size (capacity) of the VDCs be? (3) How should the vehicles be delivered to demand areas, by trucks or through a VDC? (4) What should the volume at each distribution location be? The objective function is to minimize total cost, given by the sum of transportation cost, lead-time cost and fixed costs.

3 184 E. Eskigun et al. / European Journal of Operational Research 165 (2005) This research extends our previous work (Eskigun et al., 2001) by considering capacity restrictions on VDCs in the form of a limit on the number of vehicles delivered through a single VDC over a planning period. The imposition of a capacity limit at each VDC is important to prevent an excessive number of vehicles being delivered through a single VDC, causing congestion in the facility and increasing the total delivery lead-time for the network. The size of a VDC does not affect the fixed cost of establishing a new VDC, which is independent of the volume of vehicles delivered through the VDC. In the remainder of this paper we first review previous related work and then present the network design model with capacitated VDCs. A solution algorithm based on Lagrangian relaxation is given in Section 4, and a number of scenario analyses are discussed in Section 5. We conclude with future research directions in Section Previous related work In the United States, non-military logistics costs are estimated to be over 11% of the Gross National Product (GNP) and constitute about 30% of the cost of the products sold (Thomas and Griffin, 1996). Hence, an extensive literature addresses the coordination of logistics operations and the design of effective production and distribution systems. This body of work includes Bluemenfeld et al. (1987), Brown et al. (1987), Cohen and Lee (1988), Van Roy (1989), Benjamin (1990), Chandra and Fisher (1994), and Arntzen et al. (1995). A critical review of strategic production and distribution models with emphasis on global supply chain models can be found in Vidal and Goetschalckx (1997). Geoffrion and Powers (1995) review the evolution of the strategic distribution system design since Beamon (1998) reviews strategic, stochastic, simulation, and economic models in the supply chain literature and presents research opportunities. In a more recent review, Erenguc et al. (1999) note that the network design models in the literature do not consider operational issues such as lead times in making location/allocation decisions. A closely related area is that of capacitated facility location problems (CFLPs). The literature is rich in solution algorithms for CFLPs under a variety of different assumptions. An important paper in this area is by Geoffrion and Graves (1974) who proposes a Bender s decomposition approach to solve a capacitated single source, multi-commodity network flow problem. Van Roy (1986) develops a cross-decomposition algorithm to solve a CFLP that combines Bender s decomposition and Lagrangian into a single framework. Cournuejols et al. (1991) compare different heuristics and relaxations for the CFLP. Beasley (1993) discusses the performance of Lagrangian heuristics on four different location problems. Both Cournuejols et al. (1991) and Beasley (1993) conclude that Lagrangian heuristics are quite robust compared to other greedy heuristics. Sridharan (1995) presents a more extensive review of different versions of CFLP and their solution techniques. Pirkul and Jayaraman (1998), and Mazzola and Neebe (1999) develop Lagrangian heuristics for multi-commodity CFLPs under different assumptions. A more recent contribution related to our model is presented by Melkote and Daskin (2001) who develop a capacitated facility location/network design model with both fixed facility costs and arc costs among the network nodes. They propose a branch and bound algorithm using bounds obtained by Lagrangian relaxation. A special case of the CFLP is the single sourced capacitated facility location problems (SSCFLPs) in which each demand point can be served by only one network node. SSCFLPs are mostly purely binary models and are considered to be more difficult than CFLPs. For example, when the demand satisfaction constraints are relaxed in a Lagrangian relaxation scheme, CFLPs have continuous knapsack subproblems while more difficult binary knapsack subproblems are obtained in SSCFLPs. Some of the work on SSCFLP includes Klincewicz and Luss (1986), Pirkul (1987), Sridharan (1993), Klose (1999), and Holmberg et al. (1999). The network design model presented in this paper extends the model of Eskigun et al. (2001) by considering capacity restrictions on the VDCs. It is a large-scale integer linear programming (ILP) model

4 E. Eskigun et al. / European Journal of Operational Research 165 (2005) with single source constraints, rendering all integer variables binary. Both location and allocation decisions are taken in the same model. In addition to fixed costs of the VDCs, fixed arc costs between each network locations are included in the objective function. In this respect, it resembles the MIP model of Melkote and Daskin (2001). Our model also considers the waiting time of vehicles at plants and VDCs as well as the total delivery lead-time from plants to demand areas. Consideration of the waiting time in the network nodes distinguishes this work from most other network design models. One of the few papers addressing waiting time is that of Kara and Tansel (2002) addressing the latest arrival hub location problem in cargo delivery systems. They develop a hub location model considering flight times as well as the transient times at hubs for unloading, loading and sorting operations. The departure time of a flight from a hub to a destination location is determined by the arrival time of the latest unit to the hub. Hence routing more units to the same hub might delay the departure times of the flights. However, the total volume from a hub to the destination has little or no effect on the load make-up time. On the other hand, in our model, routing more vehicles from an origin to a destination reduces the load-make up time of the vehicles at the origin locations, i.e., the time spent by the vehicles waiting for a truck or a rail car to be filled to capacity before departing. 3. Capacitated network design model (NDMC) Our capacitated network design model (NDMC) is a static model in the sense that it considers a fixed set of demands that have to be met by the outbound supply chain throughout the planning period. While it is clearly desirable to incorporate time-varying stochastic demands, the sheer size of the problem faced in the industrial context renders it computationally difficult to address these issues directly. Hence we choose to begin by developing a static deterministic model, which will permit us to address the stochastic aspects of the problem through scenario analyses if a fast enough solution procedure can be obtained. Each possible VDC location has a capacity restriction on the number of vehicles it can process within a planning period. The delivery lead-time of vehicles from plants to demand areas consists of the travel time of the carriers and the waiting time (i.e., the dwell time) of vehicles at plants and VDCs. Before presenting the network design model with its parameters, variables, and functions, we first discuss the dwell time and its components Dwell time and its components Trucks and rail cars can carry up to 8 10 and 8 15 vehicles, respectively, and are not sent to their destinations unless they are fully loaded. This causes vehicles going to the same destination to wait until the truck or rail car is fully loaded. This waiting time is called load make-up time. Vehicles may also wait in queues due to congestion at the docks at plants or VDCs. In addition, there may be some waiting time due to administrative issues or inefficiencies in the system that is unrelated to the volume of vehicles through these locations. The sum of the load make-up time, the time losses due to congestion and the other factors constitute the total waiting time or dwell time (DT) of vehicles at plants or VDCs before they are sent to their next locations. Dwell time can thus be closely approximated as: b DT ¼ a þ Volume by destination ; where a and b are constants. The volume by destination in the second component represents the total volume of vehicles sent to a specific destination (VDC or demand area) over the planning period. The specific model of dwell time used in this paper was developed with our industrial partners, who deemed this model to be of sufficient validity to proceed with developing the mathematical model of network design based on this assumption.

5 186 E. Eskigun et al. / European Journal of Operational Research 165 (2005) Model parameters We shall use the following notation throughout this paper: d ik total yearly demand for vehicles of plant i at demand area k, cpv ij unit cost of transporting a vehicle by rail from plant i to VDC j (including the operating cost of the VDCs per unit vehicle), cvd jk unit cost of transporting a vehicle by truck from VDC j to demand area k, cpd ik unit cost of transporting a vehicle from plant i to demand area k directly by truck, tpv ij transit time from plant i to VDC j, tvd jk transit time from VDC j to demand area k, tpd ik transit time from plant i to demand area k, fv j fixed cost of opening VDC j (this cost does not depend the volume or the capacity of the VDC but it is the fixed cost of establishing a new VDC), kv j maximum number of vehicles that can be shipped through VDC j within a year, h the monetary value of the lead-time ($/day) assessed by the company Decision variables ijk ¼ Z ijk ¼ V j ¼ 1 if vehicles are delivered from plant i; to demand area k through VDC j; 0 otherwise; 1 if vehicles are delivered from plant i; to demand area k directly by truck; 0 otherwise; 1 if VDC j is opened; 0 otherwise; LTPVD ijk ¼ lead-time of transporting vehicles from plant i to VDC j and then to demand area k. LTPVD ijk ¼ DTPV ij þ DTVD jk þ tpv ij þ tvd jk ; where DTPV ij, DTVD jk, tpv ij and tvd jk are defined in the dwell time functions and parameters sections, respectively. LTPD ik ¼ lead-time of transporting vehicles directly from plant i to demand area k by truck. LTPD ik ¼ DTPD ik þ tvd ik ; where DTPD ik and tvd ik are defined in dwell time functions and parameters sections, respectively Dwell time functions DTPV ij ¼ Average dwell time of vehicles shipped from plant i to VDC j. ( DTPV ij ¼ c1 ij þ P c2 ij k d ik ijk if P k d ik ijk > 0; 0 otherwise; where c 1 ij and c2 ij are constants.

6 DTPD ik ¼ ( c3 ik þ c4 ik Z ik if Z ijk > 0; 0 otherwise; where c 3 ij and c4 ik are constants. DTVD jk ¼ Dwell time for vehicles at VDC j which are shipped to demand area k. ( DTVD jk ¼ c5 jk þ P c6 jk if P d i d ik ijk > 0; i ik ijk 0 otherwise; where c 5 jk and c6 jk are constants NDMC optimization problem E. Eskigun et al. / European Journal of Operational Research 165 (2005) The NDMC optimization problem can now be defined as follows: Minimize ðcpv ij þ cvd jk Þd ik ijk þ ðcpd ik Þd ik Z ik i;k ð1þ þ hðltpvd ijk Þd ik ijk þ i;k hðltpd ik Þd ik Z ik þ j ðfv j ÞV j ; subject to: ijk þ Z ik ¼ 1 8i; k; ð2þ j d ik ijk 6 ðkv j ÞV j 8j; ð3þ i;k ijk ; Z ik ; V j 2f0; 1g 8i; j; k: ð4þ The first term in the objective function represents the total transportation cost from plants through VDCs to demand areas, the second the transportation costs associated with direct shipment from plants to demand areas. The subsequent two summations represent the lead-time costs from shipment through VDCs and direct shipment, respectively. The final summation represents the fixed costs of opening the VDCs. Constraints (2) guarantee that demand area k is served either through a single VDC or by direct shipment but not both. Constraints (3) ensure that the number of vehicles delivered through a VDC within a planning period does not exceed its capacity limit. As initially formulated, this is a non-linear integer programming model. However, with the introduction of new binary variables and constraints, the problem can be reformulated as an integer linear programming (ILP) model as shown below: Define A ij ¼ 1 if P k ijk > 0; 0 otherwise; B jk ¼ 1 if P i ijk > 0; 0 otherwise; E ik ¼ 1 if Z ik > 0; 0 otherwise:

7 188 E. Eskigun et al. / European Journal of Operational Research 165 (2005) These new variables are arc selection variables for arcs from plants to VDCs (A ij ), from VDCs to demand areas (B jk ), and from plants directly to demand areas (E ik ). Clearly we need additional constraints to ensure that the relationships defined above are maintained. These constraints can be written as follows: ijk 6 A ij 8i; j; k; ð5þ ijk 6 B jk 8i; j; k; ð6þ Z ik 6 E ik 8i; k: ð7þ Together, these constraints guarantee that the variables take the value of one when there is at least one vehicle delivered between two given locations. Otherwise they become zero. When the lead-time and dwell time equations are rewritten in terms of the newly defined variables and substituted into the objective function, the previous formulation can be reduced to an integer linear programming (ILP) model with the help of the new binary variables and the redefined dwell time functions. Proposition 1 hðltpvd ijk Þd ik ijk ¼ hðc 1 ij þ c5 jk þ tpv ij þ tvd jk Þd ik ijk þ i;j hðc 2 ij ÞA ij þ j;k hðc 6 jk ÞB jk: Proposition 2 hðltpd ik Þd ik ik ¼ hðc 3 ik þ tpd ikþd ik Z ik þ i;k hðc 4 ik ÞE ik: Proofs of these propositions are straightforward and given in Appendix A. Having these two propositions, the NDMC can now be stated as an ILP model. Before presenting the model, a new set of parameters is introduced to have an easy reading of the optimization model. d r ijk linear cost of delivering one unit of vehicle from plant i to demand area k through VDC j ¼ cpv ij þ cvd jk þ hðc 1 ij þ c5 jk þ tpv ij þ tvd jk Þ, d t ik linear cost of delivering one unit of vehicle from plant i to demand area k directly by trucks ¼ cpd ik þ hðc 3 ik þ tpd ikþ, c r ij fixed cost of opening an arc between plant i and VDC j ¼ hðc 2 ij Þ, c v jk fixed cost of opening an arc between VDC j and demand area k ¼ hðc 6 jk Þ, c t ik fixed cost of opening an arc between plant i and demand area k ¼ hðc 4 ik Þ Revised NDMC optimization problem We can now state the revised problem formulation as follows: Minimize d r ijk d ik ijk þ d t ik d ikz ik þ c r ij A ij þ c v jk B jk þ c t ik E ik þ i;k i;j j;k i;k j subject to: A ij ; B jk ; E ik 2f0; 1g 8i; j; k; and constraints ð2þ ð7þ: ðfv j ÞV j ; This revised form of the network design problem is an ILP model with all variables binary. When the arc selection constraints (5) (7) are relaxed, NDMC reduces to a single sourced capacitated facility location problem (SSCFLP). When the capacity constraints (3) are not considered, the problem becomes a ð8þ

8 E. Eskigun et al. / European Journal of Operational Research 165 (2005) fixed-charge network flow problem (Nemhauser et al., 1989). When constraints (3) and (5) (7) are relaxed, NDMC reduces to the standard assignment problem. 4. Solution approach NDMC is a large-scale integer linear programming (ILP) model. The size of the problem instance increases as O(IJK) with an increase in the number of plants I, the number of possible VDC locations J, and the number of demand areas K. Hence, it is impractical to obtain exact solutions for NDMC in reasonable computational time. We have thus developed a Lagrangian heuristic to obtain near-optimal solutions in short computation times. In our Lagrangian heuristic some of the complicating constraints are relaxed in NDMC, and the problem is decomposed into relatively easy subproblems. A subgradient algorithm is used to solve the resulting dual problem. The subgradient algorithm uses an initial upper bound during its iterations and the quality of the bound affects the number of iterations conducted for convergence. Hence, a greedy heuristic is developed to quickly obtain a good initial upper bound on the optimal objective function value of the problem. Since the subgradient algorithm usually gives a dual feasible solution that is not primal feasible, a feasibility algorithm is developed to restore primal feasibility. The combination of these three algorithms (i.e., the initial greedy heuristic, the subgradient algorithm and the feasibility algorithm) is referred to as a Lagrangian Heuristic for NDMC (LH-NDMC). The main steps of the heuristic and the details of each algorithm are as follows Initial greedy heuristic When the constraints (5) (7) in NDMC are relaxed, the problem reduces to a SSCFLP. A greedy heuristic for set functions has been used successfully to obtain quick solutions to uncapacitated facility location (UFL) problems (Nemhauser et al., 1989). We use a similar greedy algorithm considering the capacity constraints to obtain an initial upper bound. The details of the algorithm are as follows: Algorithm. IGH-NDMC: Step 0 (Initialization): Initialize set S ¼fiji ¼ 1...Ig where I is the total number of plants. Initialize set V ¼fjjj ¼ 1...Jg where J is the total number of possible VDC locations. Min_Total_Cost ¼ 0; Prev_Total_Cost ¼ 0. Step 1 (Iteration): For each VDC j 2 V S ¼ S [fjg; Set_Cost ¼ 0; Set_Cost ¼ Set_Cost + (minimum total cost of assigning all plant-demand area (ik) pairs to the locations in S without exceeding their capacity limits) Set_Cost ¼ Set_Cost + (Fixed cost of all VDCs in set S and all opened arcs) If (Set_Cost < Min_Total_Cost) Min_Total_Cost ¼ Set_Cost Min_VDC_Index ¼ j S ¼ S nfjg Step 2: If (Min_Total_Cost < Prev_Min_Total_Cost) S ¼ S [fmin VDC Indexg V ¼ V nfmin VDC Indexg

9 190 E. Eskigun et al. / European Journal of Operational Research 165 (2005) Prev_Min_Total_Cost ¼ Min_Total_Cost If V 6¼ Ø, GOTO Step 1 Else STOP Else STOP 4.2. Lagrangian relaxation When constraint set (2) is relaxed, the problem decomposes into independent subproblems which are relatively easy to solve. Constraints (5) and (6) are also relaxed to make solutions to the subproblems easier. The relaxed model and the resulting Lagrangian dual problem are described below. The u, w and v vectors in the objective function of the relaxed problem are the Lagrange multipliers associated with the constraints (2), (5), (6), respectively. Relaxed NDMC optimization problem: Minimize kðu; w; vþ ¼ d r ijk d ik ijk þ d t ik d ikz ik þ c r ij A ij þ c v jk B jk þ c t ik E ik i;k i;j j;k i;k þ ðfv j ÞV j þ u ik 1! ijk Z ik þ w ijk ð A ij þ ijk Þ j i;k j þ v ijk ð B jk þ ijk Þ; subject to: ð3þ; ð4þ; ð7þ; ð8þ: The Lagrangian dual problem can then be described as follows: Lagrangian dual problem: Maximize kðu; w; vþ; subject to: w P 0; v P 0; u free: The relaxed NDMC problem can be decomposed and written as independent subproblems given the values of the multipliers from the Lagrangian dual problem. The NDMC Lagrangian relaxation consists of three types of subproblems, namely VDC (j) subproblems, Plant (i) subproblems, and an Arc-Selection subproblem. VDC (j) subproblem: After decomposing the NDMC problem, a subproblem for each VDC j is obtained as follows: Maximize ½u ik w ijk v ijk d r ijk d ikš ijk ðfv j V j Þ; subject to: i;k d ik ijk 6 ðkv j ÞV j ; i;k ijk ; V j 2f0; 1g 8i; k: The VDCðjÞ subproblem is a 0 1 knapsack problem. There are numerous exact solution techniques for this problem in the literature, which can be classified in three groups. The first are pseudopolynomial dynamic programming algorithms, such as Pisinger s (1997) minknap algorithm. The second class are branch and bound (B&B) algorithms. Among the most successful of these are Horowitz and Sahni (1974), Fayard and

10 E. Eskigun et al. / European Journal of Operational Research 165 (2005) Plateau (1975) and Martello and Toth (1977a, 1997). The third type of solution technique, core algorithms, initially reduce the number of items examined for the knapsack to a core set and dynamically expand the core if necessary in later iterations. Some effective core algorithms especially for large knapsack problems (e.g. with 10,000 items) are due to Plateau and Elhikel (1985), Martello and Toth (1988) and Pisinger (1995). A survey of new trends in knapsack solution algorithms can be found in Martello and Toth (2000). Using an exact solution algorithm for the VDC j subproblem yields a tighter lower bound on the NDMC optimal value but leads to significant computational time when dealing with the millions of variables and constraints in the industrial application of the NDMC problem we consider. Hence, we use Dantzig s (1957) upper bound on the objective function value of a VDC subproblem. The resulting lower bound on the NDMC optimal objective function value is less tight, but saves significant computing time. The Dantzig bound is considered to be very tight for the knapsack problems and its worst-case performance ratio is computed as 0.5 (Dantzig, 1957). We refer the reader to Dantzig (1957) for a description of this wellknown procedure. Plant(i) Subproblem: The PlantðiÞ subproblem can be stated as follows: Maximize ðu ik d t ik d ikþz ik c t ik E ik; i;k k subject to: Z ik 6 E ik 8k; Z ik ; E ik 2f0; 1g 8k: The solution to the plant (i) subproblem can be obtained by observation as follows: if the net benefit of delivering vehicles from plant i to a demand area k (u ik d t ik d ik) exceeds the arc cost c t ik, Z ik is set to 1, otherwise it is set to zero. Arc-selection subproblem: Minimize u ik þ c r ij! w ijk A ij þ c v jk v ijk!b jk ; i;k i;j k j;k i subject to: A ij ; B jk 2f0; 1g 8i; j; k: In this subproblem, if the net cost of arc (ij) is negative, A ij is set to 1. Similarly, if the coefficient value of B jk is negative, B jk is set to 1. Otherwise, they take the value of zero. Given the relaxed NDMC with its subproblems and their solutions, a subgradient algorithm is used to solve the NDMC Lagrangian dual problem. The values of the parameters used in the subgradient algorithm were obtained using preliminary experimentation. The Lagrange multipliers are initialized using the solution from the initial greedy heuristic so that a Lagrangian objective value close to the initial upper bound is obtained. This gives a better starting solution and accelerates convergence. We then solve each plant (i) subproblem and the arc-selection subproblem to optimality, and compute the subgradient values for all Lagrange multipliers. In order to accelerate convergence, we calculate the average value of the previous N Lagrangian objective function values at each N iterations. When the absolute value of the difference between the last two averages is smaller than a threshold value or the total number of iterations exceeds the maximum number of iterations, the upper bound is reduced to avoid convergence problems. In standard subgradient algorithms, convergence problems are observed when the upper bound is loose (Wolsey, 1999). In the literature, there are two ways to handle this problem: In the first of these, the upper bound is updated at intermediate iterations of the subgradient algorithm using a feasibility algorithm (Fumero, 2001). In the second, the upper bound is reduced such that its value does not drop below the Lagrangian objective function value (Wolsey, 1999).

11 192 E. Eskigun et al. / European Journal of Operational Research 165 (2005) Since the feasibility algorithm in LH-NDMC takes too much computational time to use at intermediate iterations, we choose to reduce the upper bound when the Lagrangian objective function value does not improve over the last two N iterations. We then calculate a new step size and a new step direction for each Lagrange multiplier by taking the convex combination of the last three subgradient values. In this way, the typical nervous behavior of a standard subgradient algorithm is reduced (Fumero, 2001; Kokott and Lobel, 1996). Finally, the Lagrange multipliers are updated in the new directions just calculated, and checks for the stopping criteria are carried out. Details of this procedure can be found in Eskigun (2002) Feasibility algorithm The solution obtained from the subgradient algorithm may violate some of the constraints that are relaxed in NDMC. To restore primal feasibility, a feasibility algorithm is developed considering the capacity limits on the VDCs. The presence of the capacity constraints at the VDCs renders this problem significantly more complex than the uncapacitated problem treated in Eskigun et al. (2001). If the VDCs do not have capacity limitations, it is relatively simple to assign flow from unassigned plant-demand area pairs to some VDC in a manner that myopically minimizes costs. However, when capacity constraints are present this is no longer possible, since the allocation of this additional flow may cause violation of the capacity constraints. Since each VDC has a capacity restriction, infeasible plant-demand area (ik) pairs cannot be assigned to VDC locations without considering available capacities. If the total volume of an infeasible (ik) pair exceeds the available capacity of a VDC, either the infeasible pair can be assigned to another VDC or it can be assigned for direct shipment by trucks. However, there is a third possibility to be considered. Some infeasible (ik) pairs can be assigned to a VDC together to share the new arc cost incurred, and some other pairs already assigned to the current VDC can be transferred to another VDC (see Fig. 2). For example, if VDC j does not have enough capacity, some infeasible (ik) pairs can still be assigned to it but some other Left VDCs= {p, s, r} Right VDCs = {m, n} VDC s VDC p Direct Shipment VDC m VDC r Direct Shipment VDC j Direct Shipment VDC n Left Move Right Move Infeasible Subsets (H kn ) Infeasible Set [k] Fig. 2. Illustration of the feasibility algorithm.

12 E. Eskigun et al. / European Journal of Operational Research 165 (2005) (ik) pairs already assigned to VDC j must be moved to another VDC, such as VDC m. If this new VDC does not have enough available capacity for the total volume of transferred pairs either, some other already assigned pairs on VDC m can be transferred to another new VDC. This process continues until we find enough capacity on a VDC for the total volume of demand for the transferred pairs from one VDC to another. However, even the total available capacity on all VDCs might not be enough to assign the infeasible (ik) pairs. In this case, some of the pairs in the current infeasible set can be assigned to plants for direct shipment and the total infeasible volume for VDC assignment can be reduced. It might also be more economical to assign some infeasible pairs to the plants than to a VDC. This possibility is also checked before making any assignment of infeasible pairs or transfer of already assigned pairs to any VDC. Since any assignment or transfer of plant-demand area (ik) pairs might result in opening new arcs between locations (i.e., between plants and VDCs, VDCs and demand areas, and plants and demand areas), it might be more economical to cluster the infeasible (ik) pairs in demand area (k) sets and assign the pairs in a set together to share the incurred new fixed arc cost. In other words, each demand area set (k) contains the infeasible plant indices for the demand area k and these plant indices share the fixed arc cost while being assigned to a VDC. Moreover, in order to have further cost reduction, each demand area set (k) can also be divided into subsets and all elements within the subset only are assigned together. An important issue in implementing the algorithm is that the subgradient algorithm sometimes cannot overcome the fixed cost of VDCs and cannot open enough of them to assign all (ik) pairs. Instead, it adjusts the Lagrange multipliers to obtain a tight bound. This not only increases the number of infeasible pairs the subgradient algorithm generates but also makes the feasibility algorithm perform poorly without opening new VDCs. In this case, new VDCs should be opened considering the total volume of demand of all infeasible pairs. In order to do that, the initial greedy heuristic can be used with the set of candidate VDC locations set to those not opened by the subgradient algorithm and with the initial open VDC set being empty. It is also assumed in this implementation of the greedy heuristic that the new VDCs are used only to assign the infeasible pairs of the subgradient solution. A new VDC location, in this way, is added to the open VDC set as long as the improvement this new VDC brings over the total cost of assigning infeasible pairs is larger than its own fixed cost. The feasibility algorithm, thus, is run for each of these options (i.e., with and without adding new VDCs), and the smaller objective function value is accepted as the best solution value. Given this basic idea of the feasibility algorithm, the details can be found in Eskigun (2002). 5. Scenario analyses Scenario analyses are conducted to evaluate the performance of LH-NDMC and to observe how the supply chain behaves under different conditions. In all scenarios, industrial data representing a substantial portion of the company s actual network is used. The monetary value of the lead-time and the capacity limits on the number of vehicles delivered through a single VDC are varied so that their effects on some supply chain performance indicators are observed. In total, 56 scenarios are generated (see Table 1). Table 1 Design of scenarios Values of parameters Time value ($/day) 0, 5, 10, 25, 50, 75, 100 Capacity value (units/year) 10.0, 20.0, 40.0, 60.0, 80.0, 100.0, 150.0, No. of scenarios 56

13 194 E. Eskigun et al. / European Journal of Operational Research 165 (2005) Results of scenario analyses and discussion Table 2 summarizes the performance of the heuristic over all scenarios performed. The Percent gap column in the table represents the percent error between the best lower bound from the subgradient algorithm and the best upper bound obtained either from the feasibility algorithm or from the initial greedy heuristic, whichever gives a better result. CPU times of all constituent algorithms as well as the total CPU time are given in the other columns of the table. In the computations of the scenario analyses, 375 MHz IBM Power3-II processors are used on the IBM WinterHawk-II SP nodes and AI platforms. Each node has 4 processors and 4GB memory available, but we were not the only user while conducting these scenarios. Each scenario is run on a single processor. As seen from Table 2, the percent gap over all scenarios is 0.24% on average and 0.61% at most. The total CPU time of the heuristic is about 10.1 hours on average and 24.1 hours at maximum. Considering the fact that the NDMC has 23.1 million binary variables and 66.8 million constraints, this solution quality and the computational time even with tight capacity constraints are quite satisfactory. Table 3 shows detailed performance results for each scenario conducted. As seen from the table, when the capacity at the VDCs increases, the percent gap decreases significantly. When the capacity limits become less tight, the problem becomes easier to solve. In addition, the optimality gap between the best lower bound and the optimal solution becomes smaller. Thus, the percent gap between the best upper bound and the best lower bound obtained from the Lagrangian heuristic decreases. Table 3 also shows that for a larger monetary value of the lead-time, the gap between the minimum and maximum percent gap values over all capacity levels becomes smaller. This is due to the fact that as the lead-time gains importance, the direct shipment percentage increases significantly and fewer VDCs are used for vehicle deliveries. In this case, the total volume on VDCs decreases and the capacity limits become less tight. In addition to evaluating the performance of the heuristic, sensitivity analyses are conducted to observe the effects of different monetary values of the lead-time and the capacity levels on some performance measures. The number of open VDCs, the percentage of direct shipment volume, the average lead-time, the average transportation cost, and the average total cost are the performance indicators considered in all scenarios. The percentage of direct shipment volume is the percentage of the total number of vehicles delivered directly by trucks from plants to demand areas. The average transportation cost is the average unit cost without including the fixed cost of open VDCs and the lead-time cost. On the other hand, the average total cost is the average unit cost, including the transportation cost, lead-time cost and the fixed cost of the open VDCs over all vehicles delivered within a year. All sensitivity analyses results are reported based on the values in Scenario 6 with lead-time value of $0/day and a capacity level of In Table 4, the results of the sensitivity analyses with changing monetary values of the lead-time are presented under different capacity levels. As seen from the table, when the value of the lead-time increases, the number of VDCs decreases significantly. As the lead-time gains importance, the percentage of direct shipment by trucks increases due to the fact that trucks are much faster than rail (see Fig. 3). Thus, as Fig. 4 shows, the total rail volume and accordingly the number of VDCs decreases. On the other hand, when the capacity limits become less tight, the gap between the minimum and the maximum values of the number of open VDCs over different lead-time values gets smaller. Above a capacity level of 20.0, the problem starts Table 2 Summary of the performance of the heuristic Percent (%) gap Initial heuristic CPU (seconds) Subgradient algorithm CPU (seconds) Feasibility algorithm CPU (seconds) Total CPU (seconds) Average Maximum

14 E. Eskigun et al. / European Journal of Operational Research 165 (2005) Table 3 Performance of the heuristic Scenario # Time value ($/day) Capacity (units/year) Percent (%) gap Initial heuristic CPU (seconds) Subgradient algorithm CPU (seconds) Feasiblity algorithm CPU (seconds) Total CPU (seconds) (continued on next page)

15 196 E. Eskigun et al. / European Journal of Operational Research 165 (2005) Table 3 (continued) Scenario # Time value ($/day) Capacity (units/year) Percent (%) gap Initial heuristic CPU (seconds) Subgradient algorithm CPU (seconds) Feasiblity algorithm CPU (seconds) Total CPU (seconds) Table 4 The value of lead-time sensitivity analyses Scenario # Time value ($/day) Capacity (units/year) No. of open VDCs (%) Direct shipment volume (%) Average leadtime (days/ unit) (%) Average transportation cost ($/unit) (%) Average total cost ($/unit) (%)

16 E. Eskigun et al. / European Journal of Operational Research 165 (2005) Table 4 (continued) Scenario # Time value ($/day) Capacity (units/year) No. of open VDCs (%) Direct shipment volume (%) Average leadtime (days/ unit) (%) Average transportation cost ($/unit) (%) Average total cost ($/unit) (%) Percentage of Direct Shipment Capacity_Level = 10.0 Capacity_Level = Time Value ($/day) Fig. 3. Results of percentage of direct shipment vs. time value. behaving like an uncapacitated model. The capacity constraints for a few VDCs are still active but do not change the results significantly. Hence, for a fixed value of the lead-time, almost identical results are obtained at capacity levels of 40.0 or more. Table 4 and Fig. 6 also indicate that due to higher truck usage, the average lead-time decreases considerably as the lead-time gains importance. However, it does not drop below 65.34% of our base case (i.e., Scenario 6) due to high constant dwell time at plants and VDCs. Thus, having lower average lead-time values may be possible only by reducing the constant dwell time, mostly due

17 198 E. Eskigun et al. / European Journal of Operational Research 165 (2005) Capacity_Level = 10.0 Capacity_Level = No. Open VDCs Time_Value ($/day) Fig. 4. Results of number of open VDCs vs. time value Capacity_Level = Capacity_Level = Avg. Transp. Cost Time Value ($/day) Fig. 5. Results of average transportation cost vs. time value Capacity_Level =10.0 Capacity_Level=100.0 Avg. Lead Time Time Value ($/day) Fig. 6. Results of average lead-time vs. time value. to inefficiencies and/or congestion in the system. This can be accomplished by reducing the flow through the VDCs where the congestion occurs and by conducting an operational study to improve the efficiency of the system. As seen from the table, at different capacity levels, lead-time does not change significantly for a fixed monetary value of lead-time. The reasons for this behavior will be discussed later. Table 4 and Fig. 5

18 E. Eskigun et al. / European Journal of Operational Research 165 (2005) Capacity_Level = 10.0 Capacity_Level = Avg. Total Cost Time Value ($/day) Fig. 7. Results of average total cost vs. time value. Table 5 Capacity levels sensitivity analyses Scenario # Capacity (units/year) Time value ($/day) No. of open VDCs (%) Direct shipment volume (%) Average leadtime (days/ unit) (%) Average transportation cost ($/unit) (%) Average total cost ($/unit) (%) (continued on next page)

19 200 E. Eskigun et al. / European Journal of Operational Research 165 (2005) Table 5 (continued) Scenario # Capacity (units/year) Time value ($/day) No. of open VDCs (%) Direct shipment volume (%) Average leadtime (days/ unit) (%) Average transportation cost ($/unit) (%) Average total cost ($/unit) (%) also shows that the higher percentage of truck usage when the lead-time gains importance results in an increase in the average transportation cost since the trucks are more costly than the rail. The average total unit cost also shows a similar behavior with changing value of the lead-time and the capacity levels like the average transportation cost (see Fig. 7). Table 5 presents results for the capacity level sensitivity analyses. As seen from the table and Fig. 9, when the capacity levels become less tight, the number of VDCs decreases. However, after the capacity level of 20.0, the change in the number of open VDCs becomes minor due to the model s almost uncapacitated behavior. As the value of the lead-time increases, the gap between the minimum and the maximum number of VDCs becomes even smaller. Table 5 also indicates that as the capacity levels get tighter, the values of other performance indicators (i.e., the percentage of direct shipment volume, the average lead-time, the average transportation cost) show only a slight change (see Figs. 8, 10, 11). For example, when the value of lead-time is $0 per day and the capacity level changes from 20.0 to 10.0, the number of VDCs opened increases significantly. However, other performance measures do not change much. This is due to the fact that as new VDCs are added to the system because of insufficient capacities at the already open VDCs, these new VDCs are opened at demand areas very close to the locations of currently open VDCs. Hence, the values of the performance measures are not affected much by adding these new VDCs. The slight decrease in the transportation cost, despite a slight increase in the direct shipment percentage, on the other hand, can be explained by the fact that when these new VDCs are added to the system, vehicles delivered

20 E. Eskigun et al. / European Journal of Operational Research 165 (2005) Time_Value = $0.0/day Time_Value = $50/day Percentage of Direct Shipment Capacity_Level Fig. 8. Results of percentage of direct shipment vs. capacity level Time_Value = $0.0/day Time_Value = $50.0/day 120 No. Open VDCs Capacity Level Fig. 9. Results of number of open VDCs vs. capacity level. 140 Time_Value = $50/day Time_Value = $50/day Avg. Transp. Cost Capacity Level Fig. 10. Results of average transportation cost vs. capacity level. through VDCs travel less. The gain from the travel cost through VDCs, in this case, outweighs the higher direct shipment cost. However, since the volume going through each VDC falls with tighter capacity limits, the dwell time at VDCs increases and this exceeds the gains in the travel time due to shorter trips and higher truck usage. Thus, the lead-time increases slightly as the capacity limit gets tighter from 20.0 to Although the trade-off between time and cost values (i.e., travel time, dwell-time, travel cost, etc.), explains the slight changes in the values of these three performance indicators, some of them can also be attributed