Facility location, inventory management, and vehicle routing are three important decisions in supply chain management, and location-inventory-routing problems consider them jointly to improve the performance and efficiency of today’s supply chain networks. In this paper, we study a location-inventory-routing problem to minimize the total cost in a closed-loop supply chain that has forward and reverse logistics flows. First, we formulate this problem as a nonlinear integer programming model to optimize facility location, inventory control, and vehicle routing decisions simultaneously in such a system. Second, we develop a novel heuristic approach that incorporates simulated annealing into adaptive genetic algorithm to solve the model efficiently. Last, numerical analysis is presented to validate our solution approach, and it also provides meaningful managerial insight into how to improve the closed-loop supply chain under study.
Supply chain management is critical for many business organizations to gain advantage in a competitive environment, and its impact has increased steadily in the past decades [
Closed-loop supply chains (CLSCs) [ For a manufacturer, how to decide HDCC locations in a supply chain network when forward and reverse logistics flows are both considered, and how to use those HDCCs to fulfill the demands and collect returns from retailers? What is the optimal stock replenishment policy for those HDCCs? How to optimize vehicle routes in the forward and reverse flows when retailers are served by those HDCCs?
The rest of this paper is organized as follows: Section
The location-inventory-routing problem (LIRP) comprises three subproblems: facility location, inventory control, and vehicle routing. Since they are highly correlated in the real-world business, many research efforts have been conducted to study those problems jointly. The location inventory problem (LIP) is the integrated form of the first two problems, and they are first proposed by Daskin et al. [
LIRPs incorporate all three decisions above, and hence they are a more comprehensive form. In early days, Shen and Qi [
In this paper, a nonlinear integer program model is formulated to study a LIRP in a closed-loop logistics system by considering many real-world business scenarios such as vehicle capacity and the disposal of different types of returned products. To solve this model efficiently, we develop a novel solution approach that extends the power of the adaptive genetic algorithm by incorporating simulated annealing, and numerical study shows that it is more powerful and efficient than other similar heuristics in the literature.
In this paper, we study a closed-loop supply chain network that comprises a manufacturing factory, multiple hybrid distribution-collection centers (HDCCs), and several retailers. This network can be represented by a directed graph in which vertices are the factory, HDCCs, and retailers, and the edges can be directed from the factory to retailers via HDCCs, or vice versa. More specifically, in the forward flow, new products are first shipped from the factory to HDCCs and then from HDCCs to retailers by vehicles on certain routes. In the reverse flow, returned products are sent from retailers to HDCCs for inspection first. A returned product will be disposed immediately at a HDCC if it cannot be refurbished. Otherwise, it will be sent from HDCCs to the factory for repair. In this system, HDCCs operate as warehouses and return collection centers on working days in the forward and reverse flows, respectively. Vehicles are used to deliver new products from HDCCs to retailers as well as to collect returned products from retailers to HDCCs, and a vehicle must return to the same HDCC after it visits all retailers on a route. Figure
A closed-loop supply chain network.
For simplicity, we consider a single type of products and vehicles and assume that a retailer will be assigned to a same HDCC in the forward and reverse flows. Given the locations of the factory and retailers, HDCCs will be built at selected locations, and a HDCC will order new products from the factory and serve at least one retailer in the forward and reverse flows. To minimize the total cost in this system, the following decisions will be optimized:
HDCC location and retailer assignment: selecting locations to build HDCCs and assigning retailers to those HDCCs Inventory replenishment: deciding the optimal order frequency and quantity for each HDCC Vehicle routing: designing circular vehicle routes starting from and ended by each HDCC
In the closed-loop supply chain under study, the total cost is composed by the following: (1) location cost which is the fixed cost of building and operating HDCCs; (2) working inventory cost including order, holding, and shipping costs; (3) routing cost between HDCCs and retailers; (4) return cost. The individual costs per year are calculated as follows:
Location cost: Working inventory cost The working inventory cost comprises three individual terms. The first term is the order cost that is incurred when placing orders to the factory at HDCCs, the second term is the holding cost of new products in inventory, and the third term is the shipping cost of new products from the factory to HDCCs. Similar to [ Order cost: Holding cost of new products: Shipping cost from the factory to HDCCs: Consequently, the total working inventory cost per year is given as follows:
Vehicle routing cost
Forward logistics: Reverse logistics: Therefore, the total annual routing cost is given as follows:
Return cost
Inspection cost: Disposal cost at HDCCs: Cost of refurbish returned products at the factory: Shipment cost from HDCCs to the factory: Holding cost of returned products:
For simplicity, we assume that the holding cost of a returned product is independent of how long it stays in inventory. Therefore, the total return cost per year is given as follows:
According to the individual costs above, the total annual cost in the CLSC is calculated as follows:
Therefore, the location-inventory-routing problem under study can be formulated as follows:
The constraints of this model are explained as follows. Constraint (
It is obvious that the objective function is convex with respect to
By substituting
Facility location and vehicle routing problems are NP-hard in general [
GA is a popular search technique to solve optimization problems based on the principles of natural selection and genetics [
When GA is applied to solve an optimization problem, chromosomes are usually used to represent the candidate solutions to this problem, and they will evolve to better solutions iteratively. In this study, the solutions to the location and routing problems will be first encoded as chromosomes and then solved by AHSAGA. Once the location and routing problems are solved, inventory decisions can be easily optimized by solving (
The length of a chromosome is
Once a population is created, chromosomes or candidate solutions will be evaluated by their fitness to decide whether they will be kept in its offspring population. In this study, the fitness of an individual is measured as follows:
In AHSAGA, roulette-wheel selection is adopted to select and copy solutions with higher fitness values into new populations. Let
In general, a GA process will start with an initial population that is generated randomly, and the fitness of solutions will be improved iteratively by applying selection, crossover, mutation, and replacement operators. In AHSAGA, crossover operator will be applied in an iteration by the following three steps to recombine individuals for a better offspring:
Choose two parents from a population randomly and decide two crossover points arbitrarily. Generate two intermediate chromosomes by moving all the alleles positioned between the crossover lines in a parent to the beginning of the other. In each intermediate chromosome, remove the same alleles which appear in the string moved from the other parent.
An example of this procedure is illustrated in Figure
An example of crossover operation.
Usually, fitness values of chromosomes will be significantly different at the beginning of a GA process, and hence crossover is greatly beneficial to speed up the evolution. In AHSAGA, the probability of crossover is given by (
Crossover operators cannot work effectively if individuals have similar fitness values in a population. For example, in some cases, new chromosomes cannot be generated by crossover if two parents have the same allele at a given gene. To solve this problem, mutation is designed to add diversity to the population and make it possible to explore the entire search space [
If a chromosome starts with a retailer, then the initial allele will be inverted with the first allele that represents a candidate HDCC location.
In AGAs, individuals will be replaced by new ones for evolution. AHSAGA adopts SA as the steady-state technique [
The pseudocode of AHSAGA is shown in Algorithm
Input: Parameters in Section Output: Optimal location-route decisions Begin Choose population size Create an initial population for ( Calculate fitness } Choose while ( while ( Apply select operator to create the mating pool; Choose two chromosomes (parents) from the mating pool randomly; Generate random number Calculate crossover probability if ( Apply crossover operator; } Generate random number Calculate mutation probability if ( Apply mutation operator; } } for ( Calculate fitness } if } else { Generate random number If ( } else { Replace the individual with } } } end
Initialize parameters such as population size
Create an initial population randomly.
Generate an offspring population by applying selection, crossover, and mutation operators.
Calculate the fitness values of the individuals in a new population, and identify those with the maximal and minimal fitness values.
Check whether the maximal fitness value in an offspring population is greater than that in its parent population. If yes, go to Step
Check whether the termination condition is satisfied. If yes, return the chromosome with the maximal fitness value. Otherwise, go to Step
In this study, AHSAGA is implemented by Matlab R2014a and all numerical experiments are conducted on a workstation equipped with an Intel Core i7-4790 CPU at 3.60 GHz and 8.0 GB of RAM under Windows 7.
To validate its performance, AHSAGA has been tested on five data sets that are adapted from LRP files provided by the University of Aveiro [
Gaskell67-21 × 5 (retailer).
Retailer | Coordinates | Demand |
---|---|---|
(151,264) | 55 | |
(159,261) | 35 | |
(130,254) | 40 | |
(128,252) | 70 | |
(163,247) | 105 | |
(146,246) | 20 | |
(161,242) | 40 | |
(142,239) | 5 | |
(163,236) | 25 | |
(148,232) | 30 | |
(128,231) | 60 | |
(156,217) | 65 | |
(129,214) | 65 | |
(146,208) | 15 | |
(164,208) | 45 | |
(141,206) | 105 | |
(147,193) | 50 | |
(164,193) | 45 | |
(129,189) | 125 | |
(155,185) | 90 | |
(139,182) | 35 |
Remark: in this table, the daily demands are adapted by dividing the original quantities by 20 due to the vehicle capacity parameter used in this study.
Gaskell67-21 × 5 (HDCC).
Depot | Coordinates | Fixed cost |
---|---|---|
1 | (136,194) | 50 |
2 | (143,237) | 50 |
3 | (136,216) | 50 |
4 | (137,204) | 50 |
5 | (128,197) | 50 |
LIRP parameters.
Parameter | Description | Value |
---|---|---|
Shipping cost per unit of product between a manufacturing plant and HDCC |
||
Vehicle capacity | 1500 | |
Fixed cost per shipment from a plant to HDCC |
||
Fixed administrative and handling cost of placing an order to a plant at HDCC |
||
Disposal cost per unit of returned product which cannot be refurbished at HDCC |
2 | |
Holding cost per unit of new product per year at HDCC |
2 | |
Holding cost per unit of returned product at HDCC |
1 | |
Fixed cost of repairing and repacking one unit of returned product at a manufacturing plant | 2 | |
Inspection cost per unit of returned product at HDCC |
1 | |
Daily returns from retailer |
||
Shipping cost per unit of product and distance | 5 | |
Probability that a returned product cannot be refurbished | 0.3 | |
Working days per year | 300 |
Since the performance of AHSAGA can be affected significantly by its parameters, a sensitivity analysis is conducted on the parameters shown in Table
AHSAGA parameters.
Parameter | Description | Sensitivity analysis | Experimental setting | |
---|---|---|---|---|
Range | Median | |||
Number of iterations | {200, 400, 600, 800, 1000} | 600 | 600 | |
Population size | {20, 40, 60, 80, 100} | 60 | 60 | |
Initial crossover probability | {0.5, 0.6, 0.7, 0.8, 0.9} | 0.7 | 0.8 | |
Initial mutation probability | {0.05, 0.1, 0.15, 0.2, 0.25} | 0.15 | 0.25 | |
Adaptive coefficient | {0.08, 0.09, 0.1, 0.11, 0.12} | 0.1 | 0.09 | |
Adaptive coefficient | {0.1, 0.125, 0.15, 0.175, 0.2} | 0.15 | 0.175 | |
Adaptive coefficient | {1, 2, 3, 4, 5} | 3 | 3 | |
Adaptive coefficient | {1, 2, 3, 4, 5} | 3 | 2 | |
Initial temperature | {50, 100, 150, 200, 250} | 150 | 200 | |
Cooling rate | {0.95, 0.96, 0.97, 0.98, 0.99} | 0.97 | 0.98 |
Sensitivity analysis on AHSAGA parameters.
From Figure When The optimal value will always decrease when When A higher initial temperature
In this section, Gaskell67-21 × 5 files [
Initial HDCC locations and vehicle routes.
HDCC number | Vehicle number | Route | Number of orders |
---|---|---|---|
2 | 1 | 11-16 | 24 |
4 | 2 | 18-20-6-17 | 62 |
3 | 24-22-19-23 | ||
5 | 4 | 9-10-8-26-7-14 | 65 |
5 | 21-15-12-13-25 |
Trend in objective values.
Trend in adaptive probabilities.
Crossover probability
Mutation probability
To validate its performance, AHSAGA is compared with other two heuristics in the literature, which are adaptive annealing genetic algorithm (IAGA) [
Objective values from AHSAGA, HGSAA, and IAGA (Gaskell67-21 × 5).
Computational results on Gaskell67-21 × 5.
Instance name | HGSAA | IAGA | AHSAGA | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
M1 | SD | CV | M2 | SD | CV | M3 | SD | CV | ||||
Gaskell67-21 × 5 | Computational time | 2.09 | 0.02 | 0.01 | 1.85 | 0.03 | 0.02 | 1.76 | 0.02 | 0.01 | 15.59% | 4.80% |
Convergence generation | 221.22 | 121.62 | 0.55 | 211.02 | 112.90 | 0.54 | 202.18 | 107.53 | 0.53 | 8.61% | 4.19% | |
Total cost | 50860938.99 | 2778602.09 | 0.05 | 50276017.05 | 2991285.98 | 0.06 | 49977998.24 | 2709926.51 | 0.05 | 1.74% | 0.59% |
Remark: in Table
The section presents a comprehensive comparison between AHSAGA, HGSAA, and IAGA on three types of problems by the number of retailers. More specifically, the number of retailers is less than 50 in small-size problems, between 50 and 100 in medium-size problems, and more than 100 in large-size problems. The numerical results on small-size, medium-size, and large-size problems are shown in Tables The mean objective values from AHSAGA are significantly lower than those from IAGA and HGSAA for most problems. This indicates that AHSAGA has a great capability to search global optimums and hence can provide better solutions. AHSAGA takes less computational times and convergence generations to find the optimal solution than IAGA and HGSAA for all problems. This indicates that AHSAGA is the most efficient approach. The variation of the optimal values from AHSAGA, which is measured by the coefficient of variation, is lower than that from IAGA and HGSAA for all problems. This indicates that AHSAGA is more robust and consistent than the other two algorithms.
Computational results for small-size problems.
Instance name | HGSAA | IAGA | AHSAGA | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
M1 | SD | CV | M2 | SD | CV | M3 | SD | CV | ||||
Srivastava86-8 × 2 | Computational time | 1.30 | 0.02 | 0.02 | 1.09 | 0.01 | 0.01 | 1.10 | 0.01 | 0.01 | 15.73% | −0.43% |
Convergence generation | 201.66 | 116.19 | 0.58 | 203.34 | 115.63 | 0.57 | 195.14 | 108.22 | 0.55 | 3.23% | 4.03% | |
Total cost | 58357282.29 | 3318088.97 | 0.06 | 58972563.90 | 3408120.16 | 0.06 | 58080565.56 | 3150230.79 | 0.05 | 0.47% | 1.51% | |
Perl83-12 × 2 | Computational time | 1.39 | 0.02 | 0.01 | 1.20 | 0.02 | 0.01 | 1.19 | 0.01 | 0.01 | 14.24% | 1.01% |
Convergence generation | 197.06 | 109.60 | 0.56 | 199.30 | 133.01 | 0.67 | 187.28 | 116.90 | 0.62 | 4.96% | 6.03% | |
Total cost | 9879584.53 | 514520.39 | 0.05 | 9866462.28 | 494570.55 | 0.05 | 9857315.61 | 490727.85 | 0.05 | 0.23% | 0.09% | |
Gaskell67-22 × 5 | Computational time | 2.01 | 0.04 | 0.02 | 1.71 | 0.03 | 0.02 | 1.71 | 0.03 | 0.01 | 14.99% | 0.11% |
Convergence generation | 206.52 | 115.21 | 0.56 | 217.36 | 123.96 | 0.57 | 198.64 | 109.48 | 0.55 | 3.82% | 8.61% | |
Total cost | 90392692.15 | 4533650.78 | 0.05 | 90140435.61 | 5993545.10 | 0.07 | 89430743.81 | 4280679.66 | 0.05 | 1.06% | 0.79% | |
Min92-27 × 5 | Computational time | 2.37 | 0.01 | 0.01 | 1.99 | 0.02 | 0.01 | 1.99 | 0.02 | 0.01 | 16.38% | 0.00% |
Convergence generation | 175.82 | 106.58 | 0.61 | 188.10 | 103.53 | 0.55 | 175.68 | 94.84 | 0.54 | 0.08% | 6.60% | |
Total cost | 363157783.99 | 25693304.31 | 0.07 | 367201239.25 | 27220166.35 | 0.07 | 356654161.93 | 24405194.09 | 0.07 | 1.79% | 2.87% | |
Gaskell67-29 × 5 | Computational time | 2.51 | 0.03 | 0.01 | 2.08 | 0.03 | 0.01 | 2.07 | 0.02 | 0.01 | 17.74% | 0.87% |
Convergence generation | 212.40 | 117.98 | 0.56 | 208.40 | 134.58 | 0.65 | 200.86 | 101.74 | 0.51 | 5.43% | 3.62% | |
Total cost | 85966292.78 | 4976794.74 | 0.06 | 85879276.51 | 5292913.64 | 0.06 | 85053221.84 | 4861829.37 | 0.06 | 1.06% | 0.96% | |
Gaskell67-32 × 5 | Computational time | 2.62 | 0.04 | 0.02 | 2.28 | 0.03 | 0.01 | 2.24 | 0.03 | 0.01 | 14.48% | 1.55% |
Convergence generation | 202.96 | 119.95 | 0.59 | 201.96 | 119.24 | 0.59 | 197.00 | 113.53 | 0.58 | 2.94% | 2.46% | |
Total cost | 142731259.91 | 7358411.37 | 0.05 | 142691917.33 | 7239503.55 | 0.05 | 140407259.50 | 7051863.18 | 0.05 | 1.63% | 1.60% | |
Gaskell67-36 × 5 | Computational time | 2.86 | 0.04 | 0.01 | 2.45 | 0.04 | 0.02 | 2.45 | 0.03 | 0.01 | 14.33% | 0.03% |
Convergence generation | 192.58 | 113.85 | 0.59 | 205.52 | 122.26 | 0.59 | 189.00 | 111.51 | 0.59 | 1.86% | 8.04% | |
Total cost | 51830112.18 | 3370888.67 | 0.07 | 52045789.31 | 2998029.49 | 0.06 | 51240375.41 | 2951876.85 | 0.06 | 1.14% | 1.55% |
Computational results for medium-size problems.
Instance name | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
M1 | SD | CV | M2 | SD | CV | M3 | SD | CV | ||||
Christofides69-50 × 5 | Computational time | 3.52 | 0.04 | 0.01 | 2.96 | 0.05 | 0.02 | 2.97 | 0.04 | 0.01 | 15.60% | −0.14% |
Convergence generation | 185.52 | 118.83 | 0.64 | 179.58 | 114.49 | 0.64 | 173.90 | 106.46 | 0.61 | 6.26% | 3.16% | |
Total cost | 64622673.96 | 4896193.44 | 0.08 | 65120367.91 | 4880581.60 | 0.07 | 63410536.90 | 4447087.25 | 0.07 | 1.88% | 2.63% | |
Perl83-55 × 15 | Computational time | 4.55 | 0.04 | 0.01 | 4.14 | 0.06 | 0.01 | 4.09 | 0.05 | 0.01 | 10.01% | 1.27% |
Convergence generation | 202.68 | 120.89 | 0.60 | 199.34 | 117.33 | 0.59 | 194.64 | 114.00 | 0.59 | 3.97% | 2.36% | |
Total cost | 75608999.15 | 4218565.18 | 0.06 | 75488435.46 | 4821373.57 | 0.06 | 74318631.82 | 3372606.19 | 0.05 | 1.71% | 1.55% | |
Christofides69-75 × 10 | Computational time | 5.53 | 0.06 | 0.01 | 4.76 | 0.09 | 0.02\ | 4.80 | 0.05 | 0.01 | 13.24% | −0.73% |
Convergence generation | 203.52 | 119.05 | 0.58 | 201.04 | 115.46 | 0.57 | 193.18 | 110.48 | 0.57 | 5.08% | 3.91% | |
Total cost | 140932062.97 | 10696126.94 | 0.08 | 137838194.63 | 13260464.21 | 0.10 | 137205765.38 | 10092485.60 | 0.07 | 2.64% | 0.46% | |
Perl83-85 × 7 | Computational time | 6.03 | 0.04 | 0.01 | 5.21 | 0.11 | 0.02 | 5.21 | 0.10 | 0.02 | 13.48% | −0.11% |
Convergence generation | 211.78 | 119.39 | 0.56 | 186.32 | 113.21 | 0.61 | 182.14 | 102.34 | 0.56 | 14.00% | 2.24% | |
Total cost | 407102768.27 | 10738366.84 | 0.03 | 404838974.84 | 12150698.81 | 0.03 | 404614214.54 | 9236869.46 | 0.02 | 0.61% | 0.06% | |
Daskin95-88 × 8 | Computational time | 6.23 | 0.04 | 0.01 | 5.47 | 0.10 | 0.02 | 5.43 | 0.08 | 0.02 | 12.84% | 0.75% |
Convergence generation | 214.16 | 125.73 | 0.59 | 206.94 | 112.33 | 0.54 | 206.64 | 110.85 | 0.54 | 3.51% | 0.14% | |
Total cost | 95326029.80 | 6322807.23 | 0.07 | 94077056.19 | 5903902.59 | 0.06 | 94002405.72 | 5837160.63 | 0.06 | 1.39% | 0.08% |
Computational results for large-size problems.
Instance name | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
M1 | SD | CV | M2 | SD | CV | M3 | SD | CV | ||||
Christofides69-100 × 10 | Computational time | 7.19 | 0.08 | 0.01 | 6.55 | 0.14 | 0.02 | 6.48 | 0.07 | 0.01 | 9.97% | 1.18% |
Convergence generation | 190.72 | 114.09 | 0.60 | 191.74 | 116.64 | 0.61 | 188.22 | 106.69 | 0.57 | 1.31% | 1.84% | |
Total cost | 229586535.05 | 13462701.82 | 0.06 | 231296925.10 | 15532578.99 | 0.07 | 227749748.70 | 10551026.83 | 0.05 | 0.80% | 1.53% | |
Or76-117 × 14 | Computational time | 8.70 | 0.07 | 0.01 | 7.71 | 0.20 | 0.03 | 7.71 | 0.10 | 0.01 | 11.34% | −0.08% |
Convergence generation | 209.78 | 126.74 | 0.60 | 198.64 | 119.05 | 0.60 | 188.28 | 108.04 | 0.57 | 10.25% | 5.22% | |
Total cost | 5062462547.35 | 255122955.85 | 0.05 | 5079364715.42 | 256027550.46 | 0.05 | 5010781647.96 | 244879865.55 | 0.05 | 1.02% | 1.35% | |
Min92-134 × 8 | Computational time | 9.57 | 0.10 | 0.01 | 8.53 | 0.17 | 0.02 | 8.48 | 0.10 | 0.01 | 11.40% | 0.61% |
Convergence generation | 191.82 | 133.67 | 0.70 | 188.68 | 111.54 | 0.59 | 184.32 | 106.78 | 0.58 | 3.91% | 2.31% | |
Total cost | 3727217265.35 | 200966165.84 | 0.05 | 3750482276.62 | 201259019.17 | 0.05 | 3721752794.82 | 166680491.26 | 0.04 | 0.15% | 0.77% | |
Daskin95-150 × 10 | Computational time | 10.95 | 0.10 | 0.01 | 9.07 | 0.19 | 0.02 | 9.07 | 0.13 | 0.01 | 17.22% | 0.05% |
Convergence generation | 223.24 | 112.92 | 0.51 | 204.74 | 107.05 | 0.52 | 195.22 | 95.69 | 0.49 | 12.55% | 4.65% | |
Total cost | 17531706417.34 | 1394570344.68 | 0.08 | 14896666385.68 | 1140033562.24 | 0.08 | 14688680807.96 | 1050208955.68 | 0.07 | 16.22% | 1.40% | |
Perl83-318 × 4 | Computational time | 9.50 | 0.10 | 0.01 | 8.40 | 0.18 | 0.02 | 8.39 | 0.08 | 0.01 | 11.64% | 0.03% |
Convergence generation | 195.98 | 113.32 | 0.58 | 189.50 | 105.69 | 0.56 | 187.38 | 102.74 | 0.55 | 4.39% | 1.12% | |
Total cost | 3758638540.59 | 174594029.53 | 0.05 | 3764348485.23 | 174118187.75 | 0.05 | 3718507764.20 | 166085079.53 | 0.04 | 1.07% | 1.22% |
Closed-loop supply chains are an emerging and important topic due to the tremendous economic and environmental impact of consumer returns. In this paper, we study a location-inventory-routing problem in a closed-loop supply chain by formulating it as a nonlinear integer programming model. Since the problem is NP-hard, we also design a novel adaptive genetic algorithm by incorporating simulated annealing to solve this model efficiently. To make this study more practical, many real-world business scenarios such as vehicle capacity and the disposal of different types of returned products are also considered and modeled precisely.
This study can be extended in several directions in the future: first, this problem will be more practical and flexible if some assumptions are relaxed. For example, it will be flexible to allow a many-to-many relationship between vehicles and retailers, and it will also be more practical to relax the assumption that a retailer will be visited by a vehicle every working day. Second, since secondary markets have become an important channel to sell used products, it will be greatly beneficial to study LIRPs in a CLSC by considering those markets. Third, our model will be more valuable if it incorporates more business scenarios such as supply risk and multiple sourcing.
set of candidate HDCC locations, where
set of vehicles, where
set of retailers, where
set of locations, which is the union of HDCCs and retailers (i.e.,
fixed cost of building and operating a HDCC at location
shipping cost per unit of product between the factory and HDCC
vehicle capacity
daily demand of retailer
fixed cost per shipment from the factory to HDCC
fixed administrative and handling cost of placing an order to the factory from HDCC
disposal cost per unit of returned product which cannot be refurbished at HDCC
holding cost per unit of new product per year at HDCC
holding cost per unit of returned product at HDCC
fixed cost of repairing and repacking one unit of returned product at the factory
inspection cost per unit of returned product at HDCC
daily returns from retailer
distance from HDCC
distance from retailer
shipping cost per unit of product and distance
probability that a returned product cannot be refurbished
workdays per year (remark: similar to [
number of orders placed at HDCC
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research was supported by the National Natural Science Foundation of China under Grant nos. 71672074 and 71772075.