Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2018 / Article

Research Article | Open Access

Volume 2018 |Article ID 8237925 | 27 pages | https://doi.org/10.1155/2018/8237925

Lagrangian Relaxation for an Inventory Location Problem with Periodic Inventory Control and Stochastic Capacity Constraints

Academic Editor: Konstantina Skouri
Received06 May 2018
Accepted16 Sep 2018
Published08 Oct 2018

Abstract

We studied a joint inventory location problem assuming a periodic review for inventory control. A single plant supplies a set of products to multiple warehouses and they serve a set of customers or retailers. The problem consists in determining which potential warehouses should be opened and which retailers should be served by the selected warehouses as well as their reorder points and order sizes while minimizing the total costs. The problem is a Mixed Integer Nonlinear Programming (MINLP) model, which is nonconvex in terms of stochastic capacity constraints and the objective function. We propose a solution approach based on a Lagrangian relaxation and the subgradient method. The decomposition approach considers the relaxation of different sets of constraints, including customer assignment, warehouse demand, and variance constraints. In addition, we develop a Lagrangian heuristic to determine a feasible solution at each iteration of the subgradient method. The proposed Lagrangian relaxation algorithm provides low duality gaps and near-optimal solutions with competitive computational times. It also shows significant impacts of the selected inventory control policy into total system costs and network configuration, when it is compared with different review period values.

1. Introduction

Aggressive competition and strong economic turbulence in today’s global markets drive companies to improve the performance of their supply chains in order to achieve a sustainable competitive advantage. The performance of a supply chain depends strongly on its design. Hence the managers’ focus is there. In this context, supply chain network design (SCND) is a widely studied problem, which currently plays an important role in supply chain management and logistics [1, 2]. SCND consists of locating plants, warehouses, and distribution centers, allocating customers to open facilities while minimizing system-wide costs and satisfying service level requirements. Historically, the SCND problem has been tackled through a sequential approach that omits related tactical and operational decisions (e.g., inventory control, fleet design, and warehouse design). In this way, the omitted decisions are addressed after SCND has been solved. This means that strategic decisions, like the facility location, are made without regard to tactical decisions such as inventory control policy. This implies obtaining suboptimal SCND configurations because tactical decisions are subordinates to this network design [3].

This paper is focused on a three-level supply chain, where a single plant serves a set of warehouses, as Figure 1 shows. This set of warehouses serves a set of end retailers in a single commodity scenario. Unlike major previous inventory location models that assume a continuous review policy for warehouse inventory control, we use a periodic review policy for each warehouse, where is the period review, is the reorder point, and is the inventory objective level.

Thus, we study an inventory location model, in which stochastic inventory capacity constraints, expected inventory, and ordering costs are defined using a periodic review strategy. We formulate this inventory location model with periodic review control using an analysis of the expected safety stock, cyclic inventory and order quantities, and peak inventory levels for each potential warehouse. This MINLP model is NP-hard because it is an extension of the Capacitated Facility Location Problem (CFLP), which is already NP-hard.

Considering the high complexity of the analyzed problem, we propose an approximate solution approach based on Lagrangian relaxation and the subgradient method. The decomposition approach considers the relaxation of a different combination of problem constraints, including customer assignment, warehouse demand, and variance constraints. Then, we decompose the relaxed problem in a subproblem for each warehouse, which in turn is disaggregated in an inventory and location subproblem. In addition, a Lagrangian heuristic is developed to achieve a feasible solution at each iteration of the subgradient method. This Lagrangian heuristic is made up of warehouse selection and retailers greedy assignment, followed by local search improvements. We solve instances up to 20 potential warehouses and 40 retailers. The Lagrangian relaxation algorithm proposed in this paper provides low duality gaps and near-optimal solutions with competitive computational times. These results imply that this solution approach may be used in larger problem instances and more complex inventory location problems (ILP) as multicommodity and multiperiod formulations. In addition, the inclusion of periodic review policy in this model is relevant for those companies in which a continuous review policy is not feasible or there is a need to reduce costs for the inventory control system, especially for items in high demand. Considering all these attributes, ILP models could represent more accurately the complexity faced by distribution companies today.

This paper is organized as follows. In Section 2, we review the literature related to inventory location models. In Section 3, we discuss inventory control and capacity constraint issues. In Section 4, we present the formulation of the inventory location model with periodic review and stochastic capacity constraints. Section 5 presents the proposed solution approach based on Lagrangian relaxation. Section 6 presents and analyzes the numerical results. Finally, Section 7 presents conclusions, managerial insights, and suggestions for future research.

2. Literature Review

Over the last twenty years, several authors have studied how the inventory control decisions impact the Facility Location Problem (FLP) through the different integrated inventory location models. Barahona and Jensen [4] present an integer programming (IP) model for the location of a plant with cycle inventory costs, that is, the inventory required to satisfy the demands between two consecutive orders. These inventory costs are incorporated into the objective function as parameters, constituting a third term that is added to the fixed facility costs and transportation costs of Uncapacitated Fixed Charge Location Problem (UFLP). The linear relaxation of the model is solved through Dantzig-Wolfe decomposition. Nozick and Turnquist [5] develop a linear approach to the safety stock of a set of products based on the number of distribution centers through a simple linear regression. This allows safety stock costs to be directly included in the fixed cost coefficient of the UFLP. The resolution of the model is carried out through a hybrid heuristic established by Daskin [6]. Using the same previous framework, Nozick and Turnquist [7] expand their analysis by now considering a two-tier system (plant or central warehouse and DCs), where decisions are made considering whether products should have safety stock on the DCs or at the plant. Nozick and Turnquist [8] modify the previous formulations [5, 7] and now present a maximum covering location model, which ensures finding a proportion of the demand that meets a specific “coverage” distance of a DC. Later, using the approach proposed by Nozick and Turnquist [5], Lin et al. [9] solve a strategic design model of a multilevel and multiproduct distribution system, incorporating economies of scale in transportation and safety stock levels of the various products that are kept on the DCs through a greedy heuristic. All the previous models presented incorporate the operation stock and safety stock costs indirectly in the objective function and therefore, a linear term is added to it, so these models are classified as mixed integer programming models (MIP).

Erlebacher and Meller [10] are the first researchers to formulate a MINLP to address the ILP, in which the locations of the clients are continuously represented. Later Daskin et al. [11] present a location model of DCs that incorporate working and safety inventory costs, extending the UFLP model. In addition, the model includes transport costs from suppliers to DCs that explicitly combine economies of scale into a fixed cost term. The model is formulated as MINLP, where the average demand and total variance served by the DCs are calculated as the sum of the average demands and variances of the clients assigned to them, respectively. These average demands of the DCs are incorporated directly into the objective function through the economic order quantity (EOQ) expression, which in turn structures the working inventory costs. The variances of demand give the expression of the safety stock costs. It should be noted that they consider the ratio between the average demand and the variance of all customers constant, which simplifies the resolution of the problem. The authors propose a Lagrangian relaxation solution algorithm, in which they relax the restrictions of allocation customers to DCs. Shen et al. [12] restructure the model of Daskin et al. [11] as an IP model of Set-Covering; then they solve though branch-and-price approach, a variant of branch-and-bound in which nodes are processed by solving linear relaxations through column generation. Shu et al. [13] modify the model of Shen et al. [12], incorporating a generalization of the assumption that the demands and variances of the clients are proportional, making it more realistic. Similar to Shen et al. [12], they first restructure the model as a Set-Covering problem and solve it with the branch and price method, but making it more efficient. Snyder et al. [14] present a stochastic programming version of the Daskin et al. [11] model, where allocation decisions are made under random parameters such as the average daily demand and variance of the average demand of each retailer, which are described by discrete scenarios. The model minimizes the total expected cost (including location, transportation, and inventory costs) of the system in all scenarios. The location model explicitly handles the effects of economies of scale and risk pooling that result from the consolidation of inventory sites. They present an algorithm based on Lagrangian relaxation, which, as Daskin et al. [11] and Shen et al. [12], relaxes allocation constraints.

Miranda and Garrido [15] solve SCND through a simultaneous approach and incorporate inventory control decisions (EOQ and safety stock) within a CFLP, considering a stochastic demand distributed in a normal form, also modeling the phenomenon of risk pooling. This MINLP model is called a distribution network design model with risk pooling (DNDRP). The DNDRP includes, as constraints, the calculation of the total demands and variances served by each DC. This contrasts with the formulation of Daskin et al. [11], Shen et al. [12], Snyder et al. [14], and Ozsen et al. [16, 17], which incorporate them directly into the objective function through operation inventory costs and safety stock costs, respectively. Another difference between the models mentioned above is that Miranda and Garrido [15] do not explicitly consider economies of scale in transport costs. The deterministic capacity constraint of the DCs is formulated as described by Daskin [6]. The authors do not consider any assumption that may restrict the relationship between customer demands and variances.

The traditional deterministic capacitated location models do not consider inventory decision, and therefore capacity is typically calculated in an exogenous manner. As a result, to count enough inventory capacity, additional DCs must be installed. However, by ordering more frequently, we could have a lower average stock level and therefore lower costs. The papers that most resemble our work are the CFLP with stochastic inventory capacity and risk pooling proposed by Miranda and Garrido [18, 19] and Ozsen et al. [16, 17]; however, we consider a periodic review inventory control policy. Miranda and Garrido [18] use the same framework introduced in Miranda and Garrido [15] replacing the deterministic inventory capacity constraint in DCs by a stochastic constraint based on chance constrained programming. This constraint ensures that the inventory capacity for each DC is at least with respect to one 1- probability. Additionally, they incorporate an order quantity restriction for each DC. One of the relevant conclusions of the modeling approach that they propose is that a decrease in the inventory capacity does not certainly imply an increase in the number of opened warehouses. In fact, decreasing the order size allows the optimal allocation of customers (those with more significant variances) in different warehouses, reducing the total cost of the system. Miranda and Garrido [19] use the same formulation of Miranda and Garrido [18]; nevertheless, the authors explain in detail the exact method of resolution to find solutions to the subproblems of each warehouse. This procedure is based on the incorporation of a constraint that represents a set of inequalities valid for and , where Ω is the domain of all the possible values of each combination of clients. The authors present a heuristic approach based on Lagrangian relaxation and the subgradient method. They relax the demand and variance constraints of DCs and allocation constraints. Lagos et al. [20] consider the Miranda and Garrido [18] model and solve it using a hybrid algorithm combining Ant Colony Optimization (ACO) and Lagrangian relaxation. They use ACO to assign clients to a subset of stores that is previously generated by Lagrangian relaxation. The results show that the hybrid approach is quite competitive, obtaining almost optimal solutions within a reasonable time.

The study by Ozsen et al. [16] is based on the model of Daskin et al. [11] to formulate a capacitated location model with risk pooling (CLMRP). The model captures the interdependence between capacity and inventory management in DCs. They assume that there is no correlation between daily retailer demands and that it follows a Poisson process [11, 12, 14]. This implies that the variance of the daily demand is equal to the daily demand average for each retailer. The model simultaneously determines warehouse locations, order sizes from the plant to warehouses, working and safety stock levels at warehouses, and the allocation of retailers to the warehouses. Similar to Miranda and Garrido [18, 19], the inventory capacity constraint is stochastically modeled by chance constrained programming. The authors propose a Lagrangian relaxation solution algorithm, in which they relax the allocation constraints, offering low gaps with moderate computational requirements for large-scale instances. Ozsen et al. [17] slightly modify the formulation developed by Oszen et al. [16], allowing retailers to be supplied by more than one DC on a probabilistic basis.

Jin et al. [21] propose a simultaneous localization and inventory model with multiple products. The model is formulated as the Capacitated P-Median Problem (CPMP). They assume that the stochastic demands of retailers are normally distributed. The model is formulated as a MINLP and solved through a combined simulation annealing algorithm (CSA). Chen et al. [22] discuss a reliable ILP, where facilities are subject to disruption risks. When a facility fails, customers can be reassigned to a different facility that exists to avoid high costs associated with loss of services. They propose a MINLP that minimizes the sum of installation costs, expected inventory costs, and costs expected under normal and breakdown states. They develop a Lagrangian relaxation solution framework, including an exact algorithm for relaxed nonlinear subproblems.

Several recent studies, including Atamtürk et al. [23], Shahabi et al. [24], and Schuster and Tancrez [25], have reformulated ILP with uncertain demand as Conic Quadratic Mixed-Integer Program (CQMIP). Atamtürk et al. [23] propose a joint inventory location model with stochastic demand considering various cases with uncapacitated and capacitated facilities, correlated retailer demand, stochastic lead times, and multiple products. Later, Shahabi et al. [24] study a location problem with a three-level inventory, where the demand for retailers is assumed to be correlated. Besides, they propose a solution approach, based on an external approximation algorithm, which shows the advantage of using this methodology. Finally, the authors show that the omission of the effect of correlation can lead to substantially suboptimal solutions. Schuster and Tancrez [25] provide a nonlinear continuous formulation that integrates location, order, inventory, and assignment decisions and includes transport, cycle, and safety stock costs. Then, considering that the model becomes linear when specific variables are fixed, they propose a heuristic algorithm that solves the resulting linear program. Finally, they use the solution to improve the estimates of variables for the next iteration. In order to show the efficiency of the algorithm, they compare their results with those of Atamtürk et al., 2012 [23]. They conclude that safety stock and risk pooling in retailers affect the design of a supply chain.

Petridis [26] addresses the optimal design of a multiproduct and multistep supply network under demand uncertainty. The system consists of multiproduct production sites, warehouses, and distribution centers and decisions are made regarding the selection of facilities and their capacity. Also, decision variables are based on the flow of products transferred and safety stock in each distribution center. The delivery time of an order to a customer is calculated, using the probabilities of excess and deficit of inventory. All these decisions are incorporated in a single period, configuring a MINLP. The author explores linearization techniques for the highly nonlinear terms selected from the models, reducing the computational effort for the solution of the model. Qu et al. [27] propose an ILP with stochastic demand through the application of two replacement policies, joint replenishment (JR) and independent replacement (IR). They solve the problem through three algorithms: Genetic Algorithm (GA), Evolutionary Differential Hybrid Algorithm (HDE), and Hybrid Self-adapting Evolutionary Differential Hybrid Algorithm (HSDE). Their computational results show the effectiveness of these algorithms. The results of the ILP suggest that the policy of JR can obtain better solutions regarding costs than the IR policy, due to the fixed ordering costs being shared in the same order.

All the previous papers and their associated analyzed models tended to focus on the ILP with inventory continuous review policy (s, S), rather than inventory periodic review policy. Yao et al. [28] discuss the latter of the two. They study a problem of location and inventory that incorporates multiple sources of warehouses, similar to that of Ozsen et al. [17]. In this problem, the multiple products are produced in several plants. The problem is formulated as a MINLP model. Berman et al. [29] incorporate a (R, S) periodic review inventory policy in the formulation of a coordinated inventory location model, where the choice of revision intervals in the DCs achieves coordination of the system. They present two types of coordination: total coordination, where all DCs have the same interval of review, and partial coordination, where each DC can choose its own review interval. While total coordination increases location costs and inventory costs, it is likely to reduce overall system operating costs, i.e., if operational costs such as scheduling delivery are taken into account. The problem is determining the location of the DCs, the allocation of retailers to the DCs, and the parameters of the inventory policy of the DCs, so that the total cost of the whole system is minimized. The model is formulated as a nonlinear integer programming problem and they solve it through an efficient Lagrangian relaxation algorithm. The results of their computational experiments and case study suggest that the increased costs due to full coordination, compared to partial coordination, are not significant. Therefore, total coordination, while making the model more practical, is economically justifiable. Cabrera et al. [30] formulate a novel joint localization and inventory model including a stochastic capacity constraint based on an Inventory Location Model Periodic Review (ILM-PR) inventory control policy. One of the modifications that they make regarding the continuous review policy is the incorporation of the undershoot concept that has not been considered in the previous ILP models. Based on this, they design a distribution network for a two-tier supply chain, quantifying the impact of the inventory control period review on the configuration costs of network and system. They do this considering both warehouse location and customer allocation decisions. To solve the problem they apply two heuristics, Tabu Search and Particle Swarm Optimization (PSO). According to the authors, this methodology shows an effective convergence rate. This confirms that inventory control policy decisions have an effect on the design of the distribution network. Vahdani et al. [31] consider an ILP in a three-tier supply chain, where it is assumed that retailer demand is correlated and inventory shortage is allowed. The inventory periodic review control policy is utilized. In order to solve the joint ILP, they propose an optimization model based on MINLP, where the objective function is the minimization of total costs of the supply chain. To solve this MINLP model, they present a GA and a simulated annealing (SA) algorithm. Since the performance of the metaheuristic algorithms depends on the configuration of the parameters, the Taguchi method is used to establish the parameters of the indicated algorithms. Finally, the algorithms proposed by the authors are used in several numerical instances that indicate a better GA performance compared to the SA.

3. Inventory Control Policy and Total System Cost

In this section, we discuss inventory control and capacity constraint issues involved in a periodic review policy within the facility location modeling structure with stochastic demand. We will use the methodology proposed by Miranda and Cabrera [32] and Cabrera et al. [30]. When a periodic review is taken into account in an (si, Si, Ri) inventory control policy, capacity constraints cannot be stated at any moment. In an (si, Si, Ri) inventory control policy, inventory levels are reviewed after Ri periods for each warehouse i. Note that this parameter could be optimized; however, in the present research, it is fixed. In addition, if the inventory level is lower than the level si, then an order is placed to reach the objective level Si. Consequently, order size for each warehouse i must consider the well-known undershoot magnitude (USi), which is the number of items required to be ordered in addition to Si-si, in order to reach Si units of inventory, as shown in Figure 2. In other words, the USi is the difference between the reorder point si and the inventory level directly prior to ordering.

For a given review period , demand mean, and variance of a warehouse ( and ), the average undershoot magnitude is computed as follows [33]:Peak inventory levels are not controlled at any moment, solely in specific moments for each review period. This peak inventory level is reached only when orders arrive at the warehouse, time units after the previous order, and only if an order was submitted to the central warehouse or plant. Accordingly, each time an order arrives at a warehouse the inventory level is When an order is submitted to the plant, it is required that total inventory position reaches the level , and later; inventory level is reduced by lead time demand . Similar to Miranda and Garrido [18, 19], we propose that this inventory capacity constraint must be reviewed for each peak inventory instant (i.e., for each order period) with a fixed and known probability, but now assuming a periodic review, as follows:This constraint is reformulated as a deterministic nonlinear constraint, which guarantees that the probabilistic constraint is fulfilled: We specify the minimum order size as :In consequence, constraint (4) can be written asFinally, the reorder point is set in order to ensure that an order is not submitted at each moment in time (i.e., inventory level is larger than ). The inventory level must be enough to fill demand until the next order has arrived time units, with a probability or service level :Similar to (3), this constraint is reformulated as a deterministic nonlinear constraint:Finally, replacing (8) in (6), the inventory capacity constraint for each warehouse can be written asBased on a periodic inventory control policy, the safety stock to be included in the objective function is the average inventory level just before an order arrives at the warehouse:In addition, expected inventory and ordering costs related to order quantity or cycle inventory are evaluated in terms of the minimum order quantity and the average undershoot , as in EOQ model:

4. Model Formulation

In this section, according to the previous inventory control assumptions, the Inventory Location Model with Stochastic Constraints of Inventory Capacity under Periodic Review (ILM-SCC-PR) is presented as a Stochastic Non-Linear Non-Convex Mixed Integer Programming (SNL-MIP) model. In this model, we tackle the problem of storage and delivery of a single product from a single plant or central warehouse to a collection of retailers through a set of candidate warehouses while minimizing the total system cost.

The parameters of the model are as follows:: number of available sites to install warehouses: number of customers to be served: transportation unit cost between the plant and the warehouse ($/unit): fixed transportation cost between the warehouse and the customer : operating fixed cost for each warehouse ($/day): holding cost per time unit at site ($/day): fixed ordering cost per time unit at site ($/day): deterministic lead time when ordering from warehouse : mean of the daily demand for each customer : variance of the daily demand for each customer : variance of the daily demand for each customer : value of the standard normal distribution, which accumulates a probability of : value of the standard normal distribution, which accumulates a probability of : order capacity of the warehouse : inventory capacity of the warehouse

The variables considered in the mathematical formulation are as follows:

: it takes the value 1, if a warehouse is located on site , and 0 otherwise

: it takes the value 1, if warehouse serves customer , and 0 otherwise

: order size at the warehouse (units)

: served demand by each warehouse (units)

: variance of the served demand by each warehouse

Consequently, the SNL-MIP model to solve the problem isThe objective function (12) minimizes the total system cost. The first term is the fixed and operating costs when opening warehouses. The second term is the transportation cost between each warehouse and its allocated customers, plus the transportation and ordering costs between the plant and warehouses. The third term contains fixed and inventory costs related to warehouse order size. The fourth term represents the storage cost associated with safety stock at each warehouse. Constraints (13) ensure that each customer is served exactly by one warehouse. Constraints (14) state that customers can only be assigned to open warehouses (). Constraints (15) ensure that inventory capacity for each warehouse is fulfilled at least with a probability . Constraints (16) ensure that the order size is below the capacity order size allowed to warehouse . Equations (17) and (18) determine the mean and variance of the served demand by each warehouse. Equations (19) calculate average undershoot magnitude for each warehouse. Finally, (20) indicates the domain of decision variables.

The objective function and the two stochastic constraints are nonlinear, resulting in a model that is very hard to solve for large-scale instances. The complexity of the problem motivated us to propose a heuristic approach to solve it. An explanation of the algorithm is described in the next section.

5. Solution Approach

Most of the conventional location models have been solved successfully by Lagrangian relaxation-based heuristics. Fisher [34, 35] provides a detailed analysis of Lagrangian relaxation. Likewise, Daskin [6] applies the same solution approach to solve the UFLP and the CFLP obtaining reasonably good results. Because ILM-SCC-PR is an extension of the UFLP, we implement a Lagrangian relaxation algorithm and subgradient method to solve it. We develop two relaxations to solve the ILM-SCC-PR. First, we relax constraints (17) and (18), decoupling binary network design variables (X and Y) from inventory control decisions (Q) and mean and variance for demand (D and V) in each warehouse. In addition, we relax customer assignment constraints (13), similar to several Lagrangian relaxation applications for standard FLP and ILP. Second, we relax only constraints (17) and (18).

5.1. First Lagrangian Relaxation Algorithm

Associating the dual variables vectors and with the constraints (17) and (18), respectively, and with constraint (13), we obtain the following relaxed problem:For fixed values of the Lagrangian multipliers, , , and , we minimize (21) over location variables, , and the assignment variables . For the given , , and vectors, the problem decouples to the following subproblem for each warehouse :We include a set of valid inequalities to solve previous subproblems and to reduce duality gaps by increasing upper bounds. Valid inequalities are defined as a set of constraints, which bound all feasible solutions of dependent variables and [19].

Each subproblem (22) may be decoupled for the fixed values of the Lagrangian multipliers for each iteration , , as follows:   denotes the benefit of facility and represents the contribution of opening facility to the objective function (12). This decomposition consists of solving to compute and then solving to calculate , based on the computed , as explained in Section 5.3.

5.2. Second Lagrangian Relaxation Algorithm

Associating the dual variables vectors and with constraints (17) and (18), respectively, we obtain the following relaxed problem:For fixed values of the Lagrangian multipliers, and , we want to minimize (25) over location variables, , and the assignment variables . For the given and vectors, the problem decouples to the following subproblems:Each subproblem (26) may be decoupled for the fixed values of the Lagrangian multipliers for each iteration , , in the following subproblems for each warehouse :This decomposition consists of solving to compute and then solving to calculate , based on the computed , as explained in Section 5.3.

5.3. Subproblem Solving
5.3.1. First Lagrangian Relaxation

For fixed values of the Lagrangian multipliers , which are associated with relaxing constraints (17), (18), and (13), respectively, we obtain an infeasible solution of the primal problem in each iteration of the algorithm. This solution generates a lower bound on the optimal value of the primal problem.

First, we solve to calculate the value of of subproblems (23), for which an exact procedure is found in Miranda [36]. Once is obtained, is solved based on the value of according to Algorithm 1 (see Appendix A).

for  
Compute
If    then
, and
, and retain the values computed from the resolution of subproblem (23)
If    then
,
5.3.2. Second Lagrangian Relaxation

For fixed values of the Lagrangian multipliers and , which are associated with relaxing constraints (17) and (18), respectively, we obtain an infeasible solution of the primal problem in each iteration of the algorithm. As in the first Lagrangian relaxation, this solution corresponds to a lower bound on the optimal value of the primal problem. First, we solve to calculate the value of of subproblems (26), which is identical to subproblem (23). Once is obtained, is solved based on the value of through the solver CPLEX.

5.4. Lagrangian Heuristic and Subgradient Optimization

At each iteration k of the Lagrangian algorithm, we use the current lower bound solution to obtain a feasible solution, which is an upper bound to the optimal value of the primal problem. The Lagrangian heuristic considers three main procedures: warehouse selection, greedy assignment of customers, and K-OPT improvements. These three procedures are run for different numbers of warehouses, from 1 to N, based on the results and dual information of the subproblems . Namely, the complete heuristic is executed N times, and the best solution is selected. Notice the high complexity of the heuristic, especially, K-OPT improvement procedure, in contrast to the standard, simple Lagrangian heuristic observed in the literature. In order to avoid a potential high time consumption, only the K-OPT procedure is executed every 30 iterations of the algorithm. The three main procedures are described as follows.

5.4.1. Warehouse Selection

This procedure assumes that the optimal solution of the subproblems and the Lagrange multipliers () are known. For the warehouse selection, the optimal costs of subproblems are taken as initial values. Then, the best warehouses are chosen (see Algorithm 2 in Appendix A).

for   to
return   sites in ascending order of
5.4.2. Greedy Assignment of Customers

Once the warehouses are chosen, the customers are greedy assigned to the chosen warehouses; i.e., each client is assigned to the nearest warehouse, based on the transportation cost , respecting the constraint of ordering capacity and maximum inventory. In order to satisfy these constraints, we calculate three types of order quantities at each warehouse i. First is , which is economic order quantity in absence of capacity constraints. Second is , which is the available inventory capacity once inventory associated with variances are discounted, based on the inventory capacity constraint. Third is , which is the available order quantity once undershoot is subtracted, based on the order capacity constraint. Then, the optimal order quantity is the minimum of the three previous different values for , as long as has a nonnegative value. Otherwise, delete i from the potential site’s pool. The heuristic is described according to Algorithm 3 (see Appendix A).

for   to N
for   to M
for   to
for   to
for    to
for   to
for   to
if      then
break  
else
next   
5.4.3. K-OPT Improvements

Once a feasible solution is obtained through the last two steps (i.e., ), two K-OPT improvements are run, 1-OPT and 2-OPT. The former evaluates the reassignment of each customer to the other installed warehouses, if capacity constraints allow it; then, the best feasible interchange is chosen. If the total cost decreases then the reassignment is permanent. The latter takes pairs of clients in different warehouses and swaps them if capacity constraints allow it. If the total cost decreases the swap becomes permanent. In this algorithm, the optimal value of the dual problem is obtained based on dual maximization, which represents a lower bound to the optimal value of the problem . Thus, the difference between this lower bound and the cost of the best solution obtained through the heuristic previously described is an upper bound to errors of the heuristic solutions.

The update of dual variables in each iteration k is based on the subgradient method [37, 38]. This method employs the slackness/violation vector associated with relaxed constraints. Furthermore, this method utilizes an upper bound UB on the optimal value of the primal problem, which is obtained by solving the Lagrangian heuristic procedure described previously in this section. The procedure is repeated until a standard convergence criterion is met.

6. Numerical Results and Discussion

In this section, we study the quality of the solutions by the proposed heuristic procedure. Furthermore, we validate the model ILM-SCC-PR and its heuristic solutions. We used the instances of Miranda and Garrido [18, 19] as a benchmark for an ILP under continuous review with the assumptions required for a periodic review problem.

We used an Intel Core i3 processor at 2.4 GHz with 6 GB of RAM and Windows 7 to run the heuristic procedure. The program was developed in Microsoft Visual Studio 2010 C++ and the subproblems of Lagrangian relaxation were solved in IBM CPLEX 12.5. The numerical experiments have 20 warehouses and 40 clients (840 binary variables). The main aim of presenting these experiments is to show the quality of the heuristic solutions in terms of their differences with the dual optimal values. This provides lower bounds for the optimal solution for the original problem. In addition, we test the performance with two different Lagrangian relaxations, as we explain previously in Sections 5.1 and 5.2, respectively. The average execution time for the test examples of the first and second Lagrangian relaxation was 42 and 102 seconds, respectively.

The model and the heuristic approach were validated through a sensitivity analysis of the following key parameters: ordering capacity, demand variability, and fixed costs. We considered two levels of order capacity: QCap = 600 and 900. Demand variances and warehouse fixed location costs ranged over seven values from the base case: each. Two values of the review period were considered: R=1, 3. A total of 2×7×7×2 = 196 instances were solved for each one of two Lagrangian relaxations, which sum to 196×2 = 392 instances finally.

The cost parameters are expressed in a generic cost unit, CU. Fixed costs F, ordering costs, OC, and lead times, LT, for each warehouse are reported in Table 2. For holding costs, HC, and transportation costs, RC, a value of 100 CU was assumed. Also, ICap is equal to 1200. and were set to be 1.64 (95% of service level).

The parameters for Lagrangian relaxation used for all experiments are given in Table 1. We determined the Lagrangian procedure based on the maximum number of iterations allowed, or the optimality gap, or the minimum value of (the scale used in calculating the different step sizes for updating each Lagrange multiplier), whichever happened first. The optimality gap is defined as ((UB-LB)/LB) ×100.


ParameterValue

Maximum number of iterations5000
Number of iterations before halving 30
Initial value of 2
Minimum value of 0.0000001
Minimum LB-UB gap0.001%
Initial value for Lagrangian multipliers0.0


W12345678910

F 103,062 81,691 104,051 103,724 89,875 124,375 101,713 87,989 106,199 98,629
OC 61,800 47,150 41,940 88,650 62,100 55,220 41,470 62,650 68,440 69,080
LT3224222233

W11121314151617181920

F 103,648 93,505 76,507 93,668 83,391 100,396 104,592 114,521 123,498 91,817
OC 64,070 45,320 69,690 45,680 77,260 41,000 74,780 53,030 32,930 76,990
LT3232323213

The customer’s mean and variance are shown in Table 3. Both the clients and potential warehouse sites were randomly distributed over a square with 2000 km sides. Transportation costs TC were assumed as 56 CU/km. For more details of TC complete data, see Tables 14 and 15 in Appendix C.


Customer12345678910

Mean73.8168.8670.2464.0769.5269.9676.0161.7463.9274.26
Variance1,249.06979.751,112.21955.501,132.311,152.861,380.28837.35946.121,192.01

Customer11121314151617181920

Mean73.5067.5869.0270.6263.2675.9566.7066.5368.3072.43
Variance1,304.161,129.901,188.831,166.77900.021,378.81958.431,026.321,029.221,153.60

Customer21222324252627282930

Mean57.6582.9257.9965.3261.9977.9663.0375.0660.7964.73
Variance737.091,565.53776.181,035.71908.621,427.99922.581,402.98931.67999.49

Customer31323334353637383940

Mean69.2872.9971.0172.0181.3272.5573.165.2452.7469.88
Variance1,053.061,104.931,146.791,170.891,439.621,334.441,314.541,022.56783.501,215.62

The upper bounds of errors were between 0.5% and 2.5%, and 0.5% and 3.0% for first and second relaxation, respectively, considering R=1, showing the quality of the found solutions. The histogram for the upper bounds of errors is shown in Figure 3. The average error obtained was 1.1% and 1.3% for first and second relaxation, correspondingly.

The upper bounds of errors were between 4.0% and 9.0%, and 5.0% and 9.0% for first and second relaxation, respectively, considering R=3, showing a worst quality of the found solutions comparatively with R=1 solutions. The histogram for the upper bounds of errors is shown in Figure 4. The average error obtained was 6.4% and 6.5% for first and second relaxation, correspondingly.

Table 4 shows the solutions obtained considering both values of ordering capacity (600 and 900), for variances at baseline and R=1 for first and second relaxations. It presents the installed warehouses (W), the served demands, and variance of the served demand by each warehouse (D and V, respectively). It also displays the optimal order quantity in absence of capacity constraints () and the available inventory capacity once the inventory associated with variances is discounted based on the inventory capacity constraint (). It also shows the available order quantity once undershoot is subtracted based on the order capacity constraint () and the order quantity given by the heuristic, . It can be noted that the order quantity given by the heuristic never violates the constraints and in all cases is the same as ,which means that the inventory capacity constraints are active. Correspondingly, Table 5 presents the same outcomes but now considering a period of R=3. In this case, the order quantity additionally takes the same value of ; it can also be equal to , which means that neither inventory nor order capacity constraints are active.


First relaxation: QCap = 600, 900Second relaxation: QCap = 600, 900

WDVWDV

2661.910,433.7451.211.1261.211.12661.910,433.7451.211.1561.211.1
3618.29,990.7402.966.0282.866.03614.09,862.1402.673.6585.073.6
5565.79,332.3547.1135.9308.9135.95496.78,143.5528.9237.7643.5237.7
8486.97,865.7529.6255.5348.5255.58562.89,244.5550.2141.1610.4141.1
14421.26,845.0401.6351.9381.3351.910418.56,783.6543.1277.4682.6277.4


First relaxation: QCap = 600, 900Second relaxation: QCap = 600, 900

WDVWDV

2257.04,004.599.050.2206.750.22254.73,910.3100.461.5210.261.5
3263.63,995.667.230.7197.030.73263.63,995.667.230.7197.030.7
5215.53,510.3186.0198.7268.6186.05216.23,610.1185.5191.7267.3185.5
8268.34,116.9169.711.1190.011.18268.34,116.9169.711.1190.011.1
10217.53,683.5213.5131.2265.2131.210215.83,495.2214.2147.2268.2147.2
11204.63,163.0197.4200.6285.4197.411204.63,163.0197.4200.6285.4197.4
13210.63,615.8217.3156.0275.6156.012223.53,936.9106.0153.9255.9106.0
14263.94,026.687.528.4196.528.413204.83,265.8219.1193.8284.9193.8
15200.53,211.1247.9209.8291.2209.814198.03,169.2120.3269.1295.0120.3
16240.24,384.074.483.1230.674.416231.94,049.379.5123.5243.579.5
19203.23,179.853.2313.0287.453.219263.64,178.613.491.2196.713.4
20209.03,576.4245.2163.0278.0163.020209.03,576.4245.2163.0278.0163.0

The details of the solutions of first and second relaxation are presented in Tables 6 and 7, respectively, in which the columns are as follows: Prob no.: problem number, FC: factor of fixed cost sensitivity (i.e., 0.7 corresponds to a variation -30%), FV: factor of variance sensitivity, DCs opened: the additional DCs that are located compared to the baseline instance, DCs closed: the additional DCs that are closed compared to the baseline instance. No. of open DCs: total number of DCs that are open, Upper Bound: objective value of the best feasible solution, Lower Bound: the best lower bound found for optimal objective function, % Gap: percentage gap between upper bound and lower bound solution, Lag iter: total number of Lagrangian relaxation iterations, and CPU time (s): the number of CPU seconds elapsed when the algorithm terminates.


Prob no.RQCapFCFVDCs opened ()DCs closed ()No. of open DCsUpper BoundLower Bound% GapLag iterCPU time (s)

116000.70.710, 118, 145 2,023,248 2,003,836 0.97178539
416000.71.0 10, 20146 2,070,318 2,056,800 0.66155534
2216001.00.710145 2,161,246 2,131,154 1.41101122
2516001.01.02, 3, 5, 8, 14N/A5 2,221,538 2,202,020 0.89170537
2816001.01.3 10, 20146 2,282,509 2,263,675 0.83134330
4616001.31.0NoneNone5 2,358,720 2,330,384 1.22145331
4916001.31.3NoneNone5 2,422,577 2,406,503 0.67103723
5019000.70.710145 2,022,576 2,002,329 1.01190640
5319000.71.0 10, 20146 2,070,318 2,057,794 0.61143931
7119001.00.710145 2,161,246 2,130,454 1.45116424
7419001.01.02, 3, 5, 8, 14N/A5 2,221,538 2,200,616 0.95183239
7719001.01.3 10, 20146 2,282,509 2,263,675 0.83134330
9519001.31.0NoneNone5 2,358,720 2,330,567 1.2199721
9819001.31.3NoneNone5 2,422,577 2,406,503 0.67103723
993600/9000.70.71215,1611 2,746,539 2,626,621 4.57179151
1023600/9000.71.0NoneNone12 2,884,336 2,736,967 5.38200259
1203600/9001.00.77,1215,16,1911 3,063,005 2,910,209 5.25270776
1233600/9001.01.02,3,5,8,10,11,13, 14,15,16,19,20N/A12 3,224,884 3,040,498 6.06 187954
1263600/9001.01.31,121513 3,380,411 3,144,904 7.49282995
1443600/9001.31.0NoneNone12 3,565,432 3,340,563 6.73237269
1473600/9001.31.312None13 3,765,373 3,481,191 8.16189457

(): with respect to base case, Prob no 25, 74, and 123, respectively; N/A, not applicable.

Prob no.RQCapFCFVDCs opened ()DCs closed ()No. of open DCsUpper BoundLower Bound% GapLag iterCPU time (s)

14816000.70.711852,023,2482,004,4050.941330103
15116000.71.020None62,070,3182,058,6530.571372206
16916001.00.7NoneNone52,161,2462,126,8961.62107183
17216001.01.02, 3, 5, 8, 10N/A52,222,2542,197,1561.1481865
17516001.01.3141052,285,3952,264,9420.9086797
19316001.31.0NoneNone52,360,9242,310,3422.1977349
19616001.31.3141052,422,5772,400,7070.9185554
19719000.70.711852,023,2482,002,2021.051378110
20019000.71.014,201062,072,3842,058,1870.691329188
21819001.00.7NoneNone52,161,2462,126,9001.611318102
22119001.01.02, 3, 5, 8, 10N/A52,222,2542,197,1561.1481868
22419001.01.3141052,285,3952,264,9420.90867102
24219001.31.0NoneNone52,360,9242,310,3422.1977350
24519001.31.3141052,422,5772,400,7070.9185555
2463600/9000.70.7None12112,762,6322,624,2455.2781351
2493600/9000.71.0112122,902,3142,738,0896.001653102
2673600/9001.00.7712,16113,071,1152,908,4365.59146394
2703600/9001.01.02,3,5,8,10,11,12 13,14,16,19,20N/A123,241,3253,040,8136.59157498
2733600/9001.01.315None133,396,7743,160,3917.48142883
2913600/9001.31.01516123,592,9143,343,5197.46153991
2943600/9001.31.31,1519133,761,7913,481,3058.061666101

(): with respect to base case, Prob no 172, 221, and 270, respectively; N/A, not applicable.

Note that upper bound values tend to increase with respect to the increment of the fixed cost (FC). A similar behavior is observed for variation in demand variance (FV). Both tendencies denote a reasonable response of the Lagrangian heuristic since it is expected that system costs increase with respect to both sets of parameters. On the other hand, if we compare results in Table 6 for the first relaxation and results in Table 7 for the second relaxation increasing order capacity constraints a system cost reduction is produced. Finally, when we compare results in Table 6 for first relaxation and results in Table 7 for second relaxation, an increment in the duration of the review period (R = 1 and R = 3) produces worst solutions in terms of system cost and % Gap. These results show the reasonability of the Lagrangian heuristic, based on the tendencies of the objective function when different input parameters are modified (see Tables 813 in Appendix B for more details).


Prob no.FCFVDCs opened ()DCs closed ()No. of open DCsUpper BoundLower Bound% GapLag iterCPU time (s)

10.70.710, 118, 145 2,023,248 2,003,836 0.97178539
20.70.8 10, 20146 2,041,897 2,022,987 0.93136329
30.70.9 10, 20146 2,056,136 2,041,066 0.74161135
40.71.0 10, 20146 2,070,318 2,056,800 0.66155534
50.71.1 10, 20146 2,085,017 2,071,114 0.67149933
60.71.220None6 2,099,931 2,089,759 0.49139731
70.71.3 10, 20146 2,116,294 2,104,572 0.56163336
80.80.710145 2,068,799 2,051,432 0.85155434
90.80.8NoneNone5 2,092,841 2,071,195 1.05182738
100.80.910145 2,109,062 2,091,839 0.82213946
110.81.0 10, 20146 2,125,723 2,109,966 0.75177539
120.81.1 10, 20146 2,140,422 2,115,810 1.16141532
130.81.220None6 2,154,840 2,144,656 0.47201544
140.81.3 10, 20146 2,171,699 2,161,893 0.45183041
150.90.710145 2,115,023 2,091,648 1.12155433
160.90.810145 2,144,384 2,114,680 1.40103622
170.90.910145 2,155,285 2,136,610 0.87178838
180.91.0NoneNone5 2,175,811 2,153,626 1.03203844
190.91.1 10, 20146 2,195,827 2,177,076 0.86168637
200.91.220None6 2,209,749 2,183,209 1.22147032
210.91.3 10, 20146 2,227,104 2,212,965 0.64168837
221.00.710145 2,161,246 2,131,154 1.41101122
231.00.8NoneNone5 2,184,295 2,155,778 1.32191040
241.00.9NoneNone5 2,202,343 2,177,796 1.13271858
251.01.02, 3, 5, 8, 14N/A5 2,221,538 2,202,020 0.89170537
261.01.120None6 2,250,748 2,218,640 1.45147432
271.01.220None6 2,264,658 2,242,602 0.98127428
281.01.3 10, 20146 2,282,509 2,263,675 0.83134330
291.10.710145 2,207,470 2,170,288 1.7192920
301.10.8NoneNone5 2,230,023 2,194,687 1.61113124
311.10.910145 2,247,732 2,219,852 1.26202243
321.11.0NoneNone5 2,267,266 2,244,557 1.01183440
331.11.11235 2,299,007 2,265,427 1.48138931
341.11.21235 2,319,104 2,289,564 1.29145232
351.11.3NoneNone5 2,331,122 2,307,567 1.02180440
361.20.710145 2,253,693 2,208,455 2.05129628
371.20.8NoneNone5 2,275,750 2,235,681 1.7993720
381.20.9NoneNone5 2,293,798 2,264,077 1.31104322
391.21.0NoneNone5 2,312,993 2,288,491 1.0795621
401.21.110145 2,346,628 2,311,584 1.52153134
411.21.2NoneNone5 2,364,918 2,336,833 1.20154834
421.21.3NoneNone5 2,376,850 2,357,551 0.82128828
431.30.7NoneNone5 2,299,884 2,247,278 2.34101122
441.30.8NoneNone5 2,321,478 2,276,995 1.95103222
451.30.9NoneNone5 2,339,525 2,300,269 1.71106422
461.31.0NoneNone5 2,358,720 2,330,384 1.22145331
471.31.110145 2,392,852 2,357,628 1.49182340
481.31.21235 2,408,450 2,382,329 1.10133729
491.31.3NoneNone5 2,422,577 2,406,503 0.67103723

(): with respect to base case, Prob no 25; N/A, not applicable.

Prob no.FCFVDCs opened ()DCs closed ()No. of open DCsUpper BoundLower Bound% GapLag iterCPU time (s)

500.70.710145 2,022,576 2,002,329 1.01190640
510.70.8 10, 20146 2,041,655 2,014,576 1.34363176
520.70.9 10, 20146 2,056,095 2,034,976 1.0495520
530.71.0 10, 20146 2,070,318 2,057,794 0.61143931
540.71.1 10, 20146 2,085,017 2,070,541 0.70142631
550.71.220None6 2,099,808 2,085,217 0.70145432
560.71.3 10, 20146 2,116,294 2,104,572 0.56163336
570.80.710145 2,068,799 2,050,586 0.89235550
580.80.8NoneNone5 2,092,809 2,071,120 1.05191340
590.80.910145 2,109,062 2,090,298 0.90299363
600.81.0 10, 20146 2,125,723 2,109,420 0.77181839
610.81.1 10, 20146 2,140,422 2,127,241 0.62115725
620.81.220None6 2,154,717 2,144,883 0.46184740
630.81.3 10, 20146 2,171,699 2,161,893 0.45183040
640.90.7NoneNone5 2,116,940 2,091,450 1.22198041
650.90.8NoneNone5 2,138,537 2,115,098 1.11117025
660.90.910145 2,155,285 2,136,615 0.87109423
670.91.0NoneNone5 2,175,811 2,155,136 0.96127727
680.91.120None6 2,195,646 2,176,603 0.87142831
690.91.220None6 2,209,626 2,196,774 0.59147832
700.91.3 10, 20146 2,227,104 2,212,965 0.64168837
711.00.710145 2,161,246 2,130,454 1.45116424
721.00.8NoneNone5 2,184,264 2,153,972 1.41204343
731.00.910145 2,201,509 2,179,809 1.00125626
741.01.02, 3, 5, 8, 14N/A5 2,221,538 2,200,616 0.95183239
751.01.1 10, 20146 2,253,266 2,224,064 1.31122727
761.01.220None6 2,264,535 2,244,660 0.89115425
771.01.3 10, 20146 2,282,509 2,263,675 0.83134330
781.10.710145 2,207,470 2,169,785 1.7496521
791.10.8NoneNone5 2,229,992 2,195,949 1.5594820
801.10.910145 2,247,732 2,218,807 1.3099221
811.11.0NoneNone5 2,267,266 2,245,445 0.97192442
821.11.11035 2,297,032 2,264,559 1.4393121
831.11.21235 2,319,104 2,287,954 1.36130328
841.11.3NoneNone5 2,331,122 2,307,567 1.02180439
851.20.710145 2,253,693 2,207,597 2.0997921
861.20.8NoneNone5 2,275,719 2,236,741 1.7498921
871.20.9NoneNone5 2,293,798 2,260,875 1.46145231
881.21.0NoneNone5 2,312,993 2,288,393 1.0797621
891.21.110145 2,346,628 2,311,784 1.51151833
901.21.21235 2,363,777 2,336,760 1.16151433
911.21.3NoneNone5 2,376,850 2,357,551 0.82128828
921.30.7NoneNone5 2,299,850 2,246,840 2.3694620
931.30.8NoneNone5 2,321,446 2,275,764 2.0187018
941.30.9NoneNone5 2,339,525 2,303,075 1.58104722
951.31.0NoneNone5 2,358,720 2,330,567 1.2199721
961.31.110145 2,392,852 2,351,971 1.74239652
971.31.21035 2,410,607 2,382,015 1.20119526
981.31.3NoneNone5 2,422,577 2,406,503 0.67103723

(): with respect to base case, Prob no 74; N/A, not applicable.

Prob no.FCFVDCs opened ()DCs closed ()No. of open DCsUpper BoundLower Bound% GapLag iterCPU time (s)

990.70.71215,1611 2,746,539 2,626,621 4.57179151
1000.70.81,1215,1612 2,821,579 2,668,238 5.75151643
1010.70.9121612 2,851,136 2,698,879 5.64207159
1020.71.0NoneNone12 2,884,336 2,736,967 5.38200259
1030.71.11,1211,1512 2,924,387 2,767,487 5.67165249
1040.71.212None13 2,988,536 2,804,060 6.58176453
1050.71.31,121513 3,005,910 2,839,190 5.87190459
1060.80.71,1215,16,1911 2,855,967 2,722,479 4.90186053
1070.80.8None1511 2,927,783 2,764,501 5.91223264
1080.80.91,1215,1912 2,965,225 2,796,807 6.02193155
1090.81.0NoneNone12 2,997,852 2,838,973 5.60162548
1100.81.11,1211,1512 3,033,260 2,867,744 5.77225467
1110.81.212None13 3,111,402 2,910,038 6.92181454
1120.81.31,121513 3,130,744 2,946,207 6.26178354
1130.90.7715,1611 2,965,807 2,817,936 5.25164546
1140.90.8111,1511 3,027,515 2,862,289 5.77179851
1150.90.91,1215,1912 3,077,708 2,893,453 6.37190554
1160.91.0NoneNone12 3,111,368 2,939,921 5.83178952
1170.91.11,1211,1512 3,147,729 2,972,010 5.91199159
1180.91.212None13 3,234,269 3,017,179 7.20179353
1190.91.31,121513 3,255,577 3,048,281 6.80217065
1201.00.77,1215,16,1911 3,063,005 2,910,209 5.25270776
1211.00.8None1511 3,138,137 2,958,396 6.08183852
1221.00.91,1215,1912 3,190,192 2,992,886 6.59196455
1231.01.02,3,5,8,10,11,13,N/A123,224,8843,040,4986.06187954
14,15,16,19,20
1241.01.11,1211,1512 3,267,793 3,074,058 6.30179753
1251.01.212None13 3,357,135 3,122,780 7.50188256
1261.01.31,121513 3,380,411 3,144,904 7.49282995
1271.10.7715,1611 3,176,424 3,007,463 5.62263374
1281.10.81215,1911 3,226,169 3,055,590 5.58258374
1291.10.91,1215,1912 3,302,676 3,092,201 6.81164747
1301.11.0NoneNone12 3,338,400 3,141,701 6.26203561
1311.11.17,1211,1612 3,380,893 3,173,092 6.55246681
1321.11.21,121913 3,479,214 3,228,727 7.76170659
1331.11.312None13 3,519,640 3,263,281 7.86250780
1341.20.71,1215,16,1911 3,265,744 3,098,273 5.41248471
1351.20.81,7,1211,15,16,1911 3,347,472 3,151,445 6.22213061
1361.20.91,1215,1912 3,415,160 3,190,416 7.04202558
1371.21.0NoneNone12 3,451,916 3,242,612 6.45198158
1381.21.17,1211,1612 3,493,527 3,275,587 6.65242071
1391.21.21,121913 3,600,037 3,331,035 8.08253476
1401.21.31,121513 3,630,078 3,374,142 7.59202261
1411.30.7111,1511 3,386,827 3,191,709 6.11321591
1421.30.81,7,1211,15,16,1911 3,449,723 3,243,352 6.36272877
1431.30.91,1215,1912 3,527,644 3,288,221 7.28169849
1441.31.0NoneNone12 3,565,432 3,340,563 6.73237269
1451.31.17,1211,1612 3,606,160 3,376,022 6.82257676
1461.31.21,121913 3,720,860 3,423,069 8.70208973
1471.31.312None13 3,765,373 3,481,191 8.16189457

(): with respect to base case, Prob no 123; N/A, not applicable.

Prob no.FCFVDCs opened ()DCs closed ()No. of open DCsUpper BoundLower Bound% GapLag iterCPU time (s)

1480.70.71185 2,023,248 2,004,405 0.941330103
1490.70.820None6 2,041,897 2,024,218 0.871344138
1500.70.920None6 2,056,136 2,041,853 0.70999106
1510.71.020None6 2,070,318 2,058,653 0.571372206
1520.71.114,20106 2,093,276 2,070,168 1.123156375
1530.71.212,14,2037 2,113,500 2,086,609 1.291680198
1540.71.320None6 2,120,860 2,104,350 0.781025124
1550.80.7NoneNone5 2,068,799 2,051,220 0.86101779
1560.80.813,14,203,56 2,096,264 2,072,203 1.161186110
1570.80.9NoneNone5 2,109,062 2,091,920 0.8285992
1580.81.020None6 2,126,971 2,108,184 0.891283145
1590.81.114,20106 2,140,929 2,125,481 0.73895105
1600.81.214,20106 2,154,840 2,144,731 0.47907112
1610.81.320None6 2,171,699 2,162,440 0.431071123
1620.90.7NoneNone5 2,115,023 2,087,493 1.321693133
1630.90.814105 2,138,568 2,110,547 1.33110586
1640.90.9NoneNone5 2,155,285 2,136,130 0.9088169
1650.91.014105 2,175,811 2,155,991 0.9289097
1660.91.114,20106 2,195,839 2,178,961 0.771759194
1670.91.214,20106 2,209,749 2,194,826 0.6886593
1680.91.320None6 2,231,671 2,214,517 0.77948100
1691.00.7NoneNone5 2,161,246 2,126,896 1.62107183
1701.00.814105 2,184,295 2,151,423 1.53109283
1711.00.9NoneNone5 2,201,509 2,175,341 1.2098476
1721.01.02, 3, 5, 8, 10N/A5 2,222,254 2,197,156 1.1481865
1731.01.114105 2,257,526 2,219,496 1.7181886
1741.01.214,20106 2,264,658 2,241,920 1.0182391
1751.01.314105 2,285,395 2,264,942 0.9086797
1761.10.7NoneNone5 2,207,470 2,165,608 1.93122598
1771.10.814105 2,230,023 2,191,819 1.7483166
1781.10.9NoneNone5 2,247,732 2,214,688 1.4980854
1791.11.014105 2,267,266 2,244,320 1.02106085
1801.11.114105 2,303,254 2,264,255 1.7280683
1811.11.214105 2,319,191 2,292,148 1.18877106
1821.11.314105 2,331,122 2,308,181 0.99925108
1831.20.7NoneNone5 2,253,693 2,203,834 2.26131488
1841.20.8NoneNone5 2,283,055 2,231,213 2.321341102
1851.20.9NoneNone5 2,293,956 2,252,404 1.8482754
1861.21.0NoneNone5 2,314,701 2,283,059 1.3981564
1871.21.1NoneNone5 2,346,628 2,307,932 1.6880970
1881.21.212, 143,105 2,363,777 2,336,492 1.1784590
1891.21.314105 2,376,850 2,354,471 0.95833113
1901.30.714105 2,299,884 2,242,034 2.58127784
1911.30.8NoneNone5 2,329,278 2,270,847 2.5778150
1921.30.9NoneNone5 2,340,179 2,298,306 1.82121176
1931.31.0NoneNone5 2,360,924 2,310,342 2.1977349
1941.31.1NoneNone5 2,392,852 2,346,581 1.9779750
1951.31.214105 2,410,646 2,371,311 1.6683953
1961.31.314105 2,422,577 2,400,707 0.9185554

(): with respect to base case, Prob no 172; N/A, not applicable.

Prob no.FCFVDCs opened ()DCs closed ()No. of open DCsUpper BoundLower Bound% GapLag iterCPU time (s)

1970.70.71185 2,023,248 2,002,202 1.051378110
1980.70.820None6 2,041,655 2,024,155 0.861588175
1990.70.920None6 2,056,095 2,041,989 0.691078122
2000.71.014,20106 2,072,384 2,058,187 0.691329188
2010.71.120None6 2,087,050 2,069,874 0.833781474
2020.71.214,20106 2,099,808 2,088,349 0.55897106
2030.71.320None6 2,120,860 2,104,350 0.781025115
2040.80.7NoneNone5 2,068,799 2,050,990 0.87101779
2050.80.813,14,203,56 2,095,860 2,072,179 1.141712173
2060.80.914105 2,110,888 2,092,099 0.901377150
2070.81.020None6 2,126,763 2,110,466 0.771465170
2080.81.114,20106 2,140,737 2,126,893 0.65920109
2090.81.214,20106 2,154,717 2,143,841 0.51913123
2100.81.320None6 2,171,699 2,162,509 0.421021121
2110.90.7NoneNone5 2,115,023 2,087,493 1.321693135
2120.90.814105 2,138,537 2,110,820 1.3188671
2130.90.9NoneNone5 2,155,285 2,136,130 0.9088171
2140.91.014105 2,175,811 2,154,123 1.0180690
2150.91.114,20106 2,195,646 2,179,059 0.76972117
2160.91.214,20106 2,209,626 2,196,715 0.59943110
2170.91.320None6 2,231,671 2,214,517 0.77948106
2181.00.7NoneNone5 2,161,246 2,126,900 1.611318102
2191.00.8NoneNone5 2,190,608 2,153,284 1.7387967
2201.00.9NoneNone5 2,201,509 2,175,200 1.21113989
2211.01.02, 3, 5, 8, 10N/A5 2,222,254 2,197,156 1.1481868
2221.01.114,20106 2,250,555 2,222,358 1.2783093
2231.01.214,20106 2,264,535 2,243,913 0.9288296
2241.01.314105 2,285,395 2,264,942 0.90867102
2251.10.7NoneNone5 2,207,470 2,165,608 1.93122596
2261.10.81435 2,236,425 2,191,676 2.04129398
2271.10.9NoneNone5 2,247,732 2,214,688 1.4980854
2281.11.014105 2,267,266 2,241,419 1.1590470
2291.11.1NoneNone5 2,300,405 2,259,632 1.8083063
2301.11.214105 2,319,191 2,283,489 1.5688478
2311.11.314105 2,331,122 2,308,181 0.99925108
2321.20.7NoneNone5 2,253,693 2,203,840 2.26100467
2331.20.8NoneNone5 2,283,055 2,231,213 2.321341104
2341.20.9NoneNone5 2,293,956 2,252,404 1.8482753
2351.21.0NoneNone5 2,314,701 2,283,059 1.3981563
2361.21.1NoneNone5 2,346,628 2,306,865 1.7281165
2371.21.214105 2,364,918 2,333,821 1.33935100
2381.21.314105 2,376,850 2,354,471 0.95833115
2391.30.714105 2,299,850 2,242,041 2.58124985
2401.30.8NoneNone5 2,329,278 2,270,854 2.5778151
2411.30.9NoneNone5 2,340,179 2,298,306 1.82121178
2421.31.0NoneNone5 2,360,924 2,310,342 2.1977350
2431.31.1NoneNone5 2,392,852 2,346,581 1.9779750
2441.31.212105 2,422,495 2,365,429 2.4178650
2451.31.314105 2,422,577 2,400,707 0.9185555

(): with respect to base case, Prob no 221; N/A, not applicable.

Prob no.FCFVDCs opened ()DCs closed ()