A Location-Inventory-Routing Problem in Forward and Reverse Logistics Network Design

We study a new problem of location-inventory-routing in forward and reverse logistic (LIRP-FRL) network design, which simultaneously integrates the location decisions of distribution centers (DCs), the inventory policies of openedDCs, and the vehicle routing decision in serving customers, in which new goods are produced and damaged goods are repaired by a manufacturer and then returned to themarket to satisfy customers’ demands as newones.Our objective is tominimize the total costs ofmanufacturing and remanufacturing goods, building DCs, shipping goods (new or recovered) between the manufacturer and opened DCs, and distributing new or recovered goods to customers and ordering and storage costs of goods. A nonlinear integer programmingmodel is proposed to formulate the LIRP-FRL. A new tabu search (NTS) algorithm is developed to achieve near optimal solution of the problem. Numerical experiments on the benchmark instances of a simplified version of the LIRP-FRL, the capacitated location routing problem, and the randomly generated LIRP-FRL instances demonstrate the effectiveness and efficiency of the proposed NTS algorithm in problem resolution.


Introduction
The need for designing distribution network to achieve a variety of the supply chain objects with the overall productivity has received considerable attentions and becomes more and more stronger in recent years.As Javid and Azad [1] said, location allocation problem, inventory control problem, and the vehicle routing problem are the most considered subproblems in designing a distribution network.Liu and Lee [2] showed that the routing and inventory decisions affect the location decision.Also, Viswanathan and Mathur [3] found that transportation costs will increase when the order quantity during each production runs decreases.Similar conclusions have also been found by Zhang et al. [4]; they pointed out that the above three subproblems are strongly correlated and mutually affect each other.Considering the three subproblems at the same time may provide a comprehensive estimation in constructing an efficient productiondistribution network.This gives rise to the researches on the location-inventory-routing problem (LIRP).
Differing in inventory policies used, the types of facilities located, and the vehicle routing strategies adopted, lots of researches on LIRP in forward logistics network design have been reported.Liu and Lee [2] studied a LIRP using order up to level inventory policy.A route-first and location-allocation second heuristic method was proposed by them to solve LIRP.Ambrosino and Scutella [5] studied a LIRP with the consideration of customers service level.Using (, ) inventory policy, Shen and Qi [6] proposed a Lagrangian relaxation and a rank and search algorithm to solve the LIRP.Following the researches of [6], Javid and Azad [1] solved the same problem by using a tabu search and simulated annealing.Guerrero et al. [7] proposed a column generation, Lagrangian relaxation, and local search combined heuristic for a multiperiod LIRP.
As Ravi et al. [8] said, the forces of economic factors, legislation, corporate citizenship, and environmental protection problems are driving the increasing interests and investments of enterprises in reverse logistics.These motivate researchers to develop optimization models for reverse supply network design in the past decades (Vahdani and Naderi-Beni [9]).
Chen et al. [10] concluded that the consideration of reverse logistics in distribution network design can greatly reduce the logistics costs without interrupting the forward flows.Taking both the forward logistics and reverse logistics into account, many researches on facility location, vehicle routing, and inventory management had been presented.
Vahdani et al. [11] focused on a fuzzy facility locationallocation problem in closed loop logistics.Differing from [11], Vahdani et al. [12,13] proposed a fuzzy possibility programming method to solve multiobjective facility problems in forward and reverse logistic.
Under the background of forward and reverse logistic, different inventory models had also been developed in the literature.Richter [14] proposed the first inventory model in a closed logistic using EOQ model.It assumed that customers' demands can be satisfied by newly made products or by repaired products.For a given fixed waste disposal rate, the optimal minimum cost expression was derived.Shi et al. [15] considered an inventory policy under uncertain customer demand.A Lagrangian relaxation based approach was proposed.
Vehicle routing decision is another important research subject in forward and reverse logistic management.Zachariadis and Kiranoudis [16] studied a VRP problem under the background of forward and reverse logistic.A tabu search method was proposed.For a similar VRP problem, Goksal et al. [17] developed a particle swarm optimization (PSO) based approach for its resolution, in which a variable neighborhood descent (VND) algorithm was used as a local search to explore solution space.
Different combination of facility location decision, inventory management decision, and vehicle routing decision gives rise to different research problems in forward and reverse logistics.For example, Liu et al. [18] proposed a hybrid particle swarm optimization to solve a location routing problem using grey systems theory in reverse logistics.Yu and Lin [19] presented a location routing problem with simultaneous pickup and delivery, which was solved via a multistart simulated annealing algorithm with multistart hill climbing.Modeling the problem as a bilevel programming problem, Wang et al. [20] studied realistic location-inventory problem in reverse logistic of a China B2C company.Li et al. [21] solved a location and inventory control problem by a two-stage heuristic algorithm that combined Lagrangian relaxation method with an ant colony algorithm.
Due to the intrinsic complexity of the problem in mathematics, to our best knowledge, only a few researches, except that of Li et al. [22], simultaneously considered the decisions of the facility location, the inventory management, and vehicle routing in forward and reverse logistics in the literature.In their model, they consider retrieving those products that customers dislike to the DCs.As these products are new or unused, they can be sold to customers directly with no need to be returned to manufacturer for repair.
However, as the work of Adler et al. [23] indicated, used (old or damaged) products (e.g., the engines of Caterpillar Company) can also be retrieved, recovered, and returned to the market to satisfy customers' demands as new ones.
This realistic new feature motivates us to focus on goods production, distribution, and recycling.
A location-inventory-routing problem in forward and reverse logistic (LIRP-FRL) network design is studied in this paper.The LIRP-FRL simultaneously integrates the location decisions of distribution centers (DCs), the inventory policies of opened DCs, and the vehicle routing decision in serving customers, in which new goods are produced and damaged goods are repaired by a manufacturer and then transported to opened DCs.Vehicles that start from and end in the same DC distribute new or recovered goods to satisfy the demands of customers and retrieve damaged goods.Our objective is to minimize the total costs of manufacturing and remanufacturing goods, building DCs, shipping goods (new or recovered) between the manufacturer and opened DCs, distributing new or recovered goods to customers, and retrieving damaged goods from customers to DCs and the inventory costs of DCs including ordering new or recovered goods and holding these goods.
Without taking the inventory strategy of DCs into account, our problem is simplified to the classical capacitated location routing problem (CLRP) [24], which is NP-hard in strong sense.Consequently, our problem is also NPhard.Further, the LIRP-FRL is a nonlinear problem as the EOQ inventory policy is adopted.Due to its complexity, we focus on finding near optimal solutions for the LIRP-FRL.A new tabu search (NTS) algorithm is proposed to find approximately optimal solutions.This algorithm probabilistically accepts a second best solution to change search direction to achieve this goal.Numerical experiments of the CLRP benchmarks demonstrate the effectiveness and the efficiency of the proposed NTS method in obtaining high quality solutions in a very reasonable time.Computational and sensitivity analysis results on randomly generated LIRP-FRL instances are also reported.
The rest of this paper is organized as follows.In Section 2, a mathematical model is developed to describe the studied LIRP-FRL.A tabu search that seeks to search near optimal solution for the LIRP-FRL is proposed in Section 3. Algorithm performance evaluations are conducted in Section 4. Finally, we conclude the paper in Section 5.

Mathematical Model
Given a set of potential sites of distribution centers (DCs), the studied location-inventory-routing problem in forward and reverse logistic (LIRP-FRL) network design is to optimally determine the locations of DCs and their order quantities from a manufacturer, the product flows (forward and reverse) between the manufacturer and the opened DCs, and the paths of vehicles in serving customers.We assume that each route starts from and ends in the same DC, by which new or recovered products are delivered to satisfy the demands of customers for each cycle, and damaged products are retrieved.The objective of the LIRP-FRL is to minimize the sum of the opening costs and the inventory costs of DCs, the setup costs of vehicles, and the transportation costs of (new, repaired, and damaged) products.An example of the studied LIRP-FRL is illustrated in Figure 1.

Cost Analysis.
To simplify the presentation, we first abbreviate the costs related to the LIRP-FRL as follows: (1) The total fixed cost of opening distribution centers is (2) The total cost of ordering new and recovered products is (3) The total cost of producing new products and recovering retrieved products is (4) The total cost of disposal of useless retrieved products is (5) The transportation cost between the manufacturer and distribution centers is (6) The transportation cost between distribution centers and customers is Given the locations of DCs and the routes of vehicles, the LIRP-FRL is reduced to the inventory model of Teunter [25].Some of their valuable conclusions can be directly applied to our LIRP-FRL.
Proposition 1 (Teunter [25])."It is reasonable to restrict our attention to those policies with either  = 1 or  = 1." Teunter [25] concluded that the policies with either  = 1 or  = 1 can obtain near optimal solution with less computational time by comparing with that of achieving optimal solution.Since our problem is more complicated than that of Teunter [25], to simplify the resolution of LIRP-FRL, we consider only the two cases in our implementation.The two cases are as follows.
Case 1 (1, M).That means  = 1; in this case the order times of recovered products for each DC are equal to one.
Case 2 (R, 1).That means  = 1; in this case the order times of new products for each DC are equal to one.
For Case 1 (1, M), the holding cost is Consequently, the total cost is denoted as For Case 2 (R, 1), the holding cost is The corresponding total cost is Proof.It can be observed that, no matter case (1, M) or case (R, 1), decision variables    and    appear only in the objective function.For the case (R, 1), the optimal value of    and    can be obtained by checking the extreme conditions: TCM/   = 0 and TCM/   = 0.

New Tabu Search
Using tabu list to avoid visited solutions to be revisited, the tabu search proposed by Glover [26] is one of the most effective approaches for solving mixed integer programming problems, which has been successfully applied to a variety of distribution network design problems.For the detailed introduction and application of the tabu search, we refer readers to Glover [27] and Habet [28].
In this section, a tabu search (NTS) algorithm that probabilistically accepts the second best solution in search process when a local optimum is reached is proposed to effectively solve the LIRP-FRL.The key components of the NTS, including a solution representation technique, an initial solution generation method, neighbourhood structures, and the general framework of the new tabu search, are presented in detail as follows.
An example was given in Figure 2. In this example, the first number is distribution center 21, followed by distribution center 23.Since there are no customers between distribution center 21 and distribution center 23, distribution center 21 is closed.Customers (19, 16, 15, 14, 8, 11, and 6) between distribution center 23 and distribution center 22 are real customers; thus distribution center 23 is opened to service these customers.The first route of distribution center 23 services customers 19, 16, 15, and 14.Because adding customer 8 exceeds the vehicle capacity, the second route services customers 8, 11, and 6.Customer 6 is followed by distribution center 22, so the second route is terminated.For a similar reason as mentioned above, distribution centers 22 and 25 are open with two routes and one route, respectively.This representation has determined which distribution center is open and the customers on each route.With that decided, we can easily calculate the objective function value, because the inventory cost is related to LRP, which means when LRP is solved the inventory cost can be uniquely determined by Proposition 2. Note that the capacity of distribution center is not taken into consideration.So during the decoding process, a per unit penalty cost  with a big value is added to the objective function value when the total demand serviced by a distribution center exceeds its capacity.
To simplify the presentation of the algorithms proposed in following subsections, we say that an element   of the solution is before another element   or element   is after element   if and only if  < .Further, with the smallest value | − |, if DC   is before DC   , we say that DC   is nearest before DC   , or DC   is nearest after DC   .

Constructing Initial Solution.
A greedy nearest neighbor method is proposed to construct an initial feasible solution of the LIRP-FRL, which is used as an input for the tabu search.Details of the greedy algorithm are given as follows.
Step 1. Assign each customer in Φ 0 to its closest DCs in Ω 0 .
Step 2. Open the DC in Ω 0 with the largest number of customers assigned to say DC * .
Step 3. Sort the customers of set Φ 0 in nondecreasing order of the distance between it and the DC * .
Step 4. With the order, let Φ = { 1 ,  2 , . . .,  |Φ 0 | } be the subset of customers of Φ 0 with the largest |Φ| and Step 5.If Φ 0 ̸ = ⌀, then return to Step 1; Otherwise, open the DCs in Ω 1 and assign each customer to a DC according to the customer-DC assignments determined in Step 4.
Step 6.For each opened DC, sort the customers assigned to it into sequence by using a nearest neighbor heuristic (refer to Rosenkrantz et al. [30]).Split the sequence into several feasible routes so that vehicle capacity constraints are satisfied.
In our implementation of Step 2, if more than one DC has the largest number of customers assigned to, we select the DC with the highest capacity.

Insertion Neighbourhood.
The insertion neighborhood is defined as the set of solutions that can be reached by insertion move, which deletes one element of the solution  and then inserts it to another place.For example, delete element   and insert it into the th place; we obtain a new solution: = ( 1 ,  2 , . . . , −1 ,  +1 , . . . , −1 ,   ,   ,  +1 , . . . ,  ) .(23) According to the element selected (customer or depot) and the position it is inserted in, there are four cases: . . , −1 ,   ,  +1 , . . . , −1 ,   ,  +1 , . . . ,  ) .(24) Similar to the insert move, the swap move has also four cases: (e) If   and   are both customers, then we exchange the two DCs they are currently assigned to and obtain two new customer-DC assignments.
(f) If   and   are both DCs, then we exchange all the customers currently assigned to them.
(g) If   is a customer and   is a DC, then the customers after   assigned to the same DC are reassigned to DC   , and customer   is reassigned to the DC nearest before DC   .All customers previously assigned to DC   will be reassigned to the DC nearest before it.
(h) If   is a DC and   is a customer, this case is similar to case g.

Tabu Search Framework.
To increase the probability of obtaining high quality solutions, a diversification strategy is implemented.This strategy probabilistically accepts the second best solution in the search process while no better solution can be found in the neighborhoods of the current solution.By doing so, we can effectively diversify the search direction and thus more solution space will be explored.Consequently, better solution may be found.The computation results in Section 4 prove this strategy.The proposed tabu search can be summarized as follows.
Step Step 4. Output the best solution objective function value Obj * .
In the proposed tabu search algorithm, Iter counts the number of iterations and  0 represents the initial solution obtained by greedy nearest neighbour method (Section 3.2). curr denotes the current solution.Obj * is the objective function value corresponding to the best solution found so far.The function Obj() represents the objective function value of solution .
In Step 2, comparing with the current solution  curr if no better solution can be found in its neighbourhood, we need to probabilistically move  curr to  tabu-best or  tabu-2nd .In implementation, we randomly generate a value  in interval [0, 1].If  ≥ , then  curr is moved to  tabu-best ; and otherwise  curr is moved to  tabu-2nd , where  is a given parameter.
At each iteration of the tabu search procedure, the best admissible nontabu insertion, swap or 2-opt move is performed.The two elements of solution  curr involved in such move are declared as tabu-active, which is forbidden to be removed in the next  iterations. is the tabu-tenure parameter selected randomly from a range [ min ,  max ].
The tabu search iteration is terminated if the maximum number of iterations maxIter is reached or if the best solution found so far has been improved in successive SucIter iteration.

Computational Results
The proposed tabu search algorithm was coded in the C program and ran on a computer with Intel(R) Core(TM) CPU (3.2 GHz) with 4 GB of RAM under the Microsoft Windows 7 operation system.Customers' demands are randomly selected from interval [11,20].
(2) Randomly generated LIRP-FRL instances: based on the data of CLRP benchmarks and using the setting of [25] for the parameters, such as the order cost of new products of DCs   (=200), the order cost of recovered products of DCs   (=100), the cost of manufacturing   (=60), recovering   (=50), disposal   (−10), and the return rate of product, we randomly generate the LIRP-FRL instances as follows: (i) The holding cost for each distribution center is uniformly distributed in  [5,10] Using the diversification strategy of probabilistically accepting the second best solution, the NTS increases the probability of exploring more solution space.As a result, the NTS takes more computational time than the TS.However, from Table 1, we can observe that although the average computational time of NTS is about two times that of TS, it is still in a very acceptable range.The NTS provides more competitive upper bound than the TS.The average gap between the best upper bound found so far by NTS and the best known upper bound in the literature is about 1.00%, compared to the 4.22% of the TS.The NTS outperforms the TS in the quality of solutions obtained with the average deviation of the gap DEV = −3.22%.
Table 2 reports the computational results on test bed of randomly generated LIRP-FRL instances.From the table, we can also find that the NTS performs better than the classic TS with an average gap between the best upper bound of NTS and that of TS, about −2.79%.The average computational time of NTS is slightly bigger than that of TS.

Sensitivity Analysis.
Sensitivity analysis is also conducted to investigate the effects on the total costs when the input parameters such as the return rate() or the rate between the unit producing cost of new products and recovered products (  :   ) changes.
Table 3 reports the computational results when the product return rate is changed for each randomly generated instance.To manifest the trend more clearly, for each return rate value, we calculate the arithmetic average objective function values of all tested instances.The results are shown in Figure 3. From the figure, we can find that the total costs decrease with the increase of return rate .Focusing on those products with high return rate can effectively reduce the system costs.Table 4 shows the numerical results when the rate between the cost of producing a new product and that of recovering a damaged or old product varies.For each rate, we average the best upper bound obtained for all tested instances.The results are depicted in Figure 4. From the figure, we can observe that the total costs decrease with the increase of the rate.That means reducing the unit recovering cost of product can effectively reduce the total costs, which indicates us to retrieve those subnew products or to improve repairing technique so as to reduce recovering cost.

Conclusions and Future Works
We study a location-inventory-routing problem in forward and reverse logistics (LIRP-FRL) in this paper.A nonlinear mixed integer programming model is proposed to formulate the LIRP-FRL.Based on this model, a new tabu search that probabilistically accepts the second best solution in its neighborhoods is proposed to find near optimal solution for the problem.Numerical experiments on CLRP benchmarks        and randomly generated instances of LIRP-FRL with various problem sizes demonstrate the effectiveness and the efficiency of the new tabu search, which can find high quality solution with a reasonable time.Sensitivity analyses are also conducted to investigate the intrinsic management insights of the problems.
For future research, to be more close to the reality, the stochastic demand of customer should also be considered in the LIRP-FRL.

Figure 1 :
Figure 1: Location-inventory-routing problem in forward and reverse logistic.

2. 1 .
Notations.The notations are described as follows: : the index set of customers, : the index set of potential distribution centers, : the index set of vehicles, : the merged set of customers and potential distribution centers ( ∪ ),   : the maximum capacity of the distribution center ,  ∈ , FC  : the fixed opening cost of establishing distribution center ,  ∈ , : the maximum capacity of homogeneous vehicle,   : the transportation cost of a vehicle on arc(, ), ,  ∈ ,  ̸ = ,   : the setup cost of using a vehicle at distribution center ,  ∈ ,

Figure 2 :
Figure 2: An example of solution representation.

4. 1 .( 1 )
Test Beds.Two test beds were used to evaluate the performance of the proposed tabu search algorithms: The benchmarks of CLRP (Prins et al. [31]), which contains 30 instances: two potential distribution center sizes || = 5 or 10 are considered.The number of customers is || = 20, 50, 100, and 200.The customers and DCs are uniformly distributed in a [1, 50]×[1, 50] square.The fixed opening costs of DCs are randomly generated from interval [5000, 125000].The traveling cost between two nodes equals their Euclidean distance multiplied by 100 and rounded up to the next integer.The setup cost of a vehicle is 1000.The vehicle capacity  equals 70 or 150.

Figure 3 :
Figure 3: Results of different return rates.

Figure 4 :
Figure 4: Results of different rate between the cost of recovered products and the cost of new products.
and   are both customers, then customer   is reassigned to the DC that customer   is assigned to.(b)If   and   are both DCs, then reassign the customers previously assigned to the DC nearest before it.Close DC   and assign its customers to DC   .
(c) If   is a customer and   is a DC, then customer   is reassigned to DC   .(d) If   is a DC and   is a customer, reassign customers that are nearest after customer   to DC   .All customers assigned to DC   are reassigned to the DC nearest before it.3.3.2.Swap Neighbourhood.The swap neighbourhood, a set of solutions, contains the solutions that can be reached by performing a swap move, which exchanges the places of two different elements of solution .For example, exchange element   and element   of ; the resulting new solution is 1. Explore the neighborhoods of  curr and denote the best solution in the neighborhoods, the best nontabu solution, and the second best nontabu solution as  best ,  tabu-best , and  tabu-2nd , respectively.
Step 2. If Obj( best ) < Obj * , then set  curr fl  best and add the corresponding move to tabu list; otherwise, if Obj( tabu-best ) < Obj( curr ), then  curr fl  tabu-best and add the corresponding move to tabu list; otherwise, probabilistically move to  tabu-best or  tabu-2nd ,  curr fl Prob{ tabu-best ,  tabu-2nd }, and add the corresponding move to tabu list.Step 3. Iter fl Iter + 1; If no stopping criterion is reached, return to Step 1.

Table 1 :
Computational results on CLRP benchmarks.

Table 1 .
The maximum number of iterations for NTS maxIter is set to 10000 and the search iteration is terminated if the best upper bound has not been improved in successive 800 iteration (SusIter = 800).Parameter  is uniformly distributed in  [0.01, 0.8].The lower and upper bound of tabu-tenure  min and  max are set to 5 and 12, respectively.The column BKS is the best known solution objective function value reported in the literature.Gap is calculated by the following formula: Gap = 100 * (UB − BKS)/UB.DEV is the deviation of the gap of NTS from the gap of TS; that is, DEV = Gap NTS − Gap TS .

Table 3 :
Results of different return rates.

Table 4 :
Results of different rate between the cost of recovered products and the cost of new products.