The location inventory routing optimization model and algorithm for the islands shipping network

: This paper studies a hub-and-spoke shipping network of islands and integrates a maritime location inventory routing problem (MLIRP) for the islands shipping network. By determining a series of decisions including the location of the hub islands, the number of shipping routes, the schedule of every route, the running mode of every route, the ship size, the wharf scale and the inventory capacity, the objective of the paper is to minimize the total cost of the islands shipping network. For solving the integrated MLIRP model, a hybrid genetic algorithm based on the stepwise configuration (SC-GA) is presented. Finally, instances are put forward to evaluate the performance of the algorithm. The results of sensitivity analysis show that the wharf construction cost affects the total costs the most, so more attention should be paid to it when making decisions. This paper can provide theoretical and practical information to design and optimize an islands shipping network.


Introduction
Generally, islands are far away from the mainland and have extremely poor supply and infrastructure. The residents on islands have to rely on the supplies from the mainland, and thus ships frequently run between the mainland and islands and archipelago inside. In this setting, a closed and hub-and-spoke islands shipping network is developed. In the network, there are strong correlations between the location of the hub islands, the number of shipping routes, the schedule of every route, the running mode of every route, the ship size, the wharf scale and the inventory capacity. Therefore, the paper considers a maritime location-inventory-routing problem (MLIRP) with deterministic demand, multi-size carrier, and multi-running-mode.
With respect to the LIP, Shen et al. (2003) studied the problem by allowing some retailers to serve as distribution centres for other retailers to reduce the cost of inventory and structured a set-covering integer-programming model. For the LRP, Tuzun and Burke (1999) expounded on the interdependent relationship of location and routing. Wu et al. (2002) studied the multi-distribution centre location and routing problem. They divided the original problem into two sub-problems, the location-allocation problem and the general routing problem, and used anneal arithmetic to solve them. For the IRP, Viswanathan and Mathur (1997) noted that the core of IRP is how to optimize the inventory replenishment strategy and routing to minimize the sum of the long-term inventory costs and transportation costs. Song and Furman (2013) introduced a flexible modelling framework for IRP that can accommodate various practical features. What's more, for the LIRP, Guerrero et al. (2015) proposed a partition using a Dantzig-Wolfe formulation on the routing variables and presented a heuristic with hybridization between column generation, Lagrangian relaxation, and local search to solve the LIRP. Liu et al. (2015) studied a three-phase supply chain distribution system consisting of one supplier, a set of retailers, and a single product with a continuous review (Q, r) inventory policy, and a pseudo-parallel genetic algorithm integrating simulated annealing was proposed.
In addition, previous studies about the MLIRP have been relatively few recently (Ronen, 2002;Hsu and Hsieh, 2007). For example, Christiansen et al. (2011) and Uggen et al. (2013) developed heuristic algorithms to solve the problem. The former established the model by fleet transport capacity constraints and then proposed a construction heuristic embedded in a genetic algorithm framework. The latter presented a new way of optimizing MIRP by using a heuristics based on fix-and-relax time decomposition. In addition, Engineer et al. (2012), Hewitt et al. (2013), and Rakke et al. (2015) used the branch and bound method to solve MIRP. Due to ship routing with uncertainty in weather conditions and unpredictable waiting times at ports, Agra et al. (2015) considered stochastic sailing time and port time, and a two-stage stochastic programming 4 model with recourse was presented where the first stage consists of routing, loading, and unloading decisions and the second stage consists of scheduling and inventory decisions.
The islands shipping network design is the generalization of maritime transport network optimization problem (MTNOP). Nowadays, there are a large number of studies of MTNOP (Iakovou et al. 1999;Gelareh et al., 2010;Meng and Wang, 2011;Reinhardt and Pisinger, 2012;Wang et al., 2013;Brouer et al., 2014;Tran and Haasis, 2015). For example, Agarwal and Ergun (2008) presented an integrated model, a mixed-integer linear program, to solve the ship-scheduling and cargo-routing problems simultaneously. Moreover, some scholars focused on the hub-and-spoke marine transport network. Dong et al. (2015) proposed a two-stage stochastic programming model with recourse to address MTNOP, with the first stage determining the optimal service capacity and the second focused on the optimal routing of shipments. Although the two-stage method can solve the problem, it cannot comprehensively consider all cases in the solving process. Jin (2003) and Xu et al. (2007) studied the ship-transport network of China and the surrounding area. The former discussed the strategies of shipping centres and the transport network, and the latter found that geographical constraints plays an important role in the network topology of the ship-transport network of China.
Despite a wide range of studies about MTNOP and related LIRPs, few studies involve an islands transport system with multi-size carrier and multi-running-mode. Therefore, this paper considers a MLIRP for an islands shipping network. The contributions of the paper are as follows: The islands shipping network is a closed system where ships run inside the network only, and it is essentially different from general maritime networks that are open. In the closed islands network, there are strong relations among routing, ships and wharf, but the optimization of transportation costs, construction costs and inventory costs may be contrary. To minimize the total cost under the precondition of uninterrupted supply, this paper considers the location of the hub islands, the number of shipping routes, the schedule of every route, the running mode of every route, the ship size, the wharf scale and the inventory capacity. Finally, for solving the MLIRP, we introduce a genetic algorithm based on the stepwise configuration (SC-GA) and evaluate its performance.
The rest of this paper is organized as follows: In Section 2, the MLIRP with deterministic 5 demand, multi-size carrier and multi-running-mode is formulated. Section 3 presents a genetic algorithm based on the stepwise configuration that can optimize the problem. Section 4 describes computational experiments to evaluate the performance of the SC-GA. Section 5 presents the conclusions and provides insights towards future research topics.

Problem description
The island shipping network usually consists of the mainland, the hub islands and the satellite islands. In the network, the supplies are transported from the mainland to the hub island via the main network and then distributed from the hub island to its satellite islands via the branch network. Since the islands shipping network is a one-way closed transport network, there are some complicated problems in this network. To minimize the total cost under the precondition of uninterrupted supply, we need to optimize the location of hub islands, the routes in the main network and branch network, the running mode of every route, and the ship size and schedule of every route, wharf scale and inventory capacity and cycle supply of every island. For instance, if we use a large ship in a route and try to load it fully, it can prolong the schedule and reduce the shipping cost, but it will lead to increasing wharf and warehouse construction costs, which may increase the total cost. Hence, the overall network needs to be determined as a whole.

Notation
The notations used in the paper are listed as follows, unless otherwise specified (Table 1).
where tr s c is the shipping cost per nautical mile of the s -type ship. Eqs. (1) and (2)  Then, the ship using cost model can be stated as follows: where pro s c and run s c are the purchase cost and monthly maintenance cost of the s -type ship.
Eq. (5) is the sum of the ship purchase costs and monthly maintenance costs during the total operation time of all the BFRGs and CRs in the branch network and the main network.

Wharf construction cost model
Then, the wharf construction cost model can be stated as follows: where port s c is the construction cost of a wharf in which s -type ship can berth. In Eq. (6), the first and second items are the sums of the wharf construction costs of all the satellite islands respectively passed through by BFRGs and CRs in the branch network, the third and the fourth items are the sums of the wharf construction cost of all hub islands respectively passed through by BFRGs and CR in the main network, and the fifth item is the sum of the construction costs of all hub island wharfs corresponding to ship sizes appearing in the branch network of every island.

Inventory cost model
In the islands shipping network, all shipping has to interrupt when the tropical cyclone comes, so that the daily consumption of each island will be supplied by its inventory until the tropical cyclone is over. Thus, the minimum inventory risk Q is proposed to resist the risk of tropical cyclone, and it can be given by where t  is the number of days that the minimum inventory of every island can supply its daily consumption, and q is the average daily consumption.
In this paper, we make the following assumptions. (1) The daily consumption of each island is continuous and deterministic. (2) The inventory time is in the start of each period, and sup Q denotes the supply. Therefore, the inventory state function is as follows: ()  10 where Eq. (9) is the sum of the inventory costs of all islands passed through by the BFRGs and CRs in the branch network and the main network. Eq. (10) is the sum of the warehouse construction costs of all islands passed through by BFRGs and CRs in the branch network and the main network.

Formulation
The integrated MLIRP model can thus be described as the following formulations: Minimize total dr cy mdr mcy ship port stf stb The objective function Eq. (11) is the total logistics cost during the whole operation period.
Constraints (12) and (13) show that, in the branch network, there is only one route from a hub island to its each satellite island. Constraints (14) and (15) show that, in the main network, from the mainland to the hub island there is only one route. Constraint (16) (19) shows that the total cycle supply of the hub islands transported by any CR in the main network should not exceed the deadweight of the ship equipped on the CR. Constraints (20) and (21) respectively show the value range of the schedule of the BFRGs and CRs in the branch network.

Algorithm
The MLIRP with multi-size carrier and multi-running-mode is an NP-hard problem. For

Chromosome representation
To express the information of the location of each hub island, the number of routes of the main network and branch network, which islands are passed through by each route and the sequence of the islands, we design the chromosome respectively for the main and branch networks by the manner of saturated division to fully express the whole solution space (under an extreme situation, all routes are BFRs and transported by respective ships). That is to say, a chromosome consists of the shipping network gene segment of transport from the mainland to hub islands (shorted as hub island gene segment) and the shipping network gene segments of transport from the hub island to its satellite islands (shorted as each island gene segment), and we use n-1 (assuming a gene segment contain n islands) separators to divide the gene segment into n segments, so the gene between two separators signifies the islands in one group. One possible chromosome is shown in Fig. 2.  Fig. 2, the hub island gene segment means that island 2 # , 10 # , and 14 # are respectively the hub islands 1, 2, and 3; there are two route groups in the main network, and they are respectively transport from mainland to island 10 # and 14 # successively (using the same ship) and from mainland to island 2 # . The island 1 # gene segment means that there are three routes in island 1 # of the branch network, and they are respectively transport from island 2 # to island 6 # and 4 # successively (using the same ship), from island 2 # to island 1 # , 7 # , and 3 # successively (using the same ship) and from island 2 # to island 5 # . The island 2 gene segment means that there are two 13 routes in island 2 of the branch network, and they are respectively transport from island 10 # to island 11 # and 9 # successively (using the same ship) and from island 10 # to island 8 # and 12 # successively (using the same ship). The island 3 gene segment means that there are three routes in island 3 of the branch network, and they are respectively transport from island 14 # to island 13 # and 16 # successively (using the same ship), from island 14 # to island 18 # and from island 14 # to island 17 # and 15 # successively (using the same ship). The above information expressed in the chromosomes encoding the hub island location and routing will provide data support for the SC module.

SC module
The cycle supply and inventory capacity of each island transported by same route are determined by the route's shipping schedule t , the ship size of any route in the branch and main network under different running mode is determined by the cycle supply of each island transported by the same route, the wharf scale of each island transported by same route is determined by the ship size of the route; and the range of t can be obtained according to Eqs. (20-23). Then, we regard t as an independent variable and regard the cycle supply, inventory capacity, ship size, wharf scale, total costs of route and inventory as dependent variables of every route in the branch and main network under different running modes. Thus, the steps of SC module are as follows: ① For a route of the branch network, we use the enumeration method, taking t as an independent variable, to calculate the minimum total costs respectively under back-and-forth transport and cycle transport running mode, choose the running mode with the lower minimum total cost and the corresponding shipping schedule, ship size, wharf scale and cycle supply and inventory capacity of every island passed though by this route as the optimal configuration solution of this route.
Then, according to the above method, we successively optimize and configure all routes in the branch network. ② We take the total daily consumption of every island as the daily consumption of every hub island; ③ For a route of the main network, we use the enumeration method, taking t as an independent variable, to calculate the minimum total costs respectively under back-and-forth transport and cycle transport running mode, choose the running mode with the 14 lower minimum total cost and the corresponding shipping schedule, ship size, wharf scale and cycle supply and inventory capacity of every island passed though by this route as the optimal configuration solution of this route. Then, according to above method, we successively optimize and configure all routes in the main network. The procedure is described as follows: Step 1: Decode the information of the hub island location and a route in branch network from the chromosome.
Step 2: Set the value range of schedule t of the route under back-and-forth transport and cycle transport running modes according to Eq. (20) and Eq. (21).
Step 3: Use an exhaustive method, taking t as an independent variable, calculate the minimum total costs of the route under the back-and-forth and cycle transport running modes.
Step 4: Choose the running mode with the lower minimum total cost, and define the corresponding shipping schedule, ship size, wharf scale and cycle supply and inventory capacity of each island as the optimal configuration solution of the route.
Step 5: Repeat step 1 to step 4 , and configure all routes in the branch network.
Step 6: Take the total daily consumption of every island as the daily consumption of every hub island.
Step 7: Decode the information of the hub island location and a route in main network from the chromosome.
Step 8: Set the value range of schedule t of the route under the back-and-forth transport and cycle transport running modes according to Eq. (22) and Eq. (23).
Step 9: Use the exhaustive method, taking t as the independent variable, to calculate the minimum total costs the route under the back-and-forth and cycle transport running modes.
Step 10: Choose the running mode with the lower minimum cost, and define the corresponding shipping schedule, ship size, wharf scale, cycle supply and inventory capacity of each island as the optimal configuration solution of the route.
Step 11: Repeat step 7 to step 10, and configure all routes in the main network.
Step 12: Output the total cost of the chromosome and the optimal configuration solution of every route in the branch and main networks. 15 The specific steps are shown in Fig. 3.
Read the information of centre island location and a route in underlying network from the chromosome Set the value range of schedule t of the route under back-and-forth transport running mode according to equation (20) Choose the running mode with the lower minimum cost and define the corresponding shipping schedule, ship size, wharf scale and cycle supply and storage capacity of each island as the optimal configuration solution of the route

Optimization for upper network
Set the value range of schedule t of the route under cycle transport running mode according to equation (21) Use an exhaustive method to calculate the minimum cost of the route under the back and forth transport running mode Use an exhaustive method to calculate the minimum cost of the route under the cycle transport running mode Set the value range of schedule t of the route under the back-and-forth transport running mode according to equation (22) Set the value range of schedule t of the route under the cycle transport running mode according to equation (23) Use the exhaustive method to calculate the minimum cost of the route under the back-and-forth transport running mode Use the exhaustive method to calculate the minimum cost of the route under the cycle transport running mode Choose the running mode with the lower minimum cost and dedine the corresponding shipping schedule, ship size, wharf scale and cycle supply and storage capacity of each island as the optimal configuration solution of the route Take the total daily consumption of every archipelago as the daily consumption of every centre island 16 Fig. 3. Framework of SC module

Fitness
For each chromosome in SC-GA, the fitness value is equal to the total cost of the chromosome obtained by the SC module. Obviously, the lower the fitness value is, the better the chromosome is.

Crossover
According to the characteristics of chromosome, the delimiters and the island are crossed respectively in the crossover operator of SC-GA. This means that the offspring inherits the position of delimiters in parent 1 and the order of island numbers in parent 2. An example of the crossover operator is shown in Fig. 4.

Mutation
Due to complex chromosomes, the mutation operator of SC-GA has three different methods.
The first operator is the exchange between island numbers in the same gene segment; the second is 17 the exchange between the delimiter and island number in the same gene segment; and the third is the exchange between the island numbers of the hub island gene segment and each island gene segment. An example of the mutation operator is shown in Fig

Basic data
In this subsection, one example is put forward to assess the effectiveness of the method, in which 3 sets of archipelagos are assumed and the number of islands is 10, 7 and 5, respectively.
The coordinates of every island are shown in table 2. The alternative ship sizes are 100, 500, 1000 and 5000 tons. Table 3 shows the daily consumption of each island. Table 4 shows the relevant costs of ship and wharf construction. The total time of system operation is 20 years, and t  of the inventory system is 5 days.   The optimal result was obtained after 2815 generations, and the convergence process is shown in Fig. 6. The result shows that the 3 # , 14 # , and 20 # islands are selected as the hub islands of the three sets of islands; there are 12 ships equipped in the islands shipping network: 2 ships of 5000 tons, 1 ship of 1000 tons, 5 ships of 500 tons, and 4 ships of 100 tons. There are 28 wharfs equipped in the islands shipping network. Four wharfs of 5000 tons, 1000 tons, 500 tons, and 100 tons are equipped on 3 # island, two wharfs of 5000 tons and 100 tons are equipped on 14 # island, two wharfs of 5000 tons and 500 tons are equipped on 20 # island, and every satellite island equips a wharf of 1000 tons, 500 tons, or 100 tons. The inventory capacity of all islands is 27210 tons.
The optimized routings are shown in tables 5~8. The wharf, the inventory capacity, and the cycle supply of every island are shown in table 9. The islands shipping network is shown in Fig. 7.   Fig. 6. The algorithm converge process

Algorithm comparison
In this subsection, we compare the greedy algorithm and saving algorithm with the SC-GA (Table 10). The strategy of greedy algorithm, according to the size order of all costs, is to optimize the location of hub islands in terms of the consumption of every island first, then set wharf, then optimize the routings, ship size, and schedule of each route, and then set the inventory system. The strategy of the saving algorithm is seeking the minimum transport distance and the minimum ship size of each route. Compared with the greedy algorithm, the ship using cost, the wharf construction cost and the warehouse construction cost of the result of SC-GA is lower, but the shipping cost and the inventory cost are higher; compared with the saving algorithm, the shipping cost, the ship using cost, and the wharf construction cost of the result of SC-GA is lower, but the inventory cost and the warehouse construction cost are higher; however, for the total cost during the total time of system operation, the result of SC-GA can save 59605 thousand yuan over the greedy algorithm and 1025964 thousand yuan over the saving algorithm, and the total cost savings, respectively, are 1.6% and 28.0%. Thus, SC-GA can take various factors into account and effectively optimize the islands shipping network. 23

Sensitivity analysis
To analyse the sensitivity of every cost element to the total cost, we establish a sensitivity analysis model.  ,  ,  ,  and  respectively denote the weight coefficients of shipping cost, ship using cost, wharf construction cost, inventory cost, and warehouse construction cost.
Then, the objective function (11) of the cost optimization model can be changed as follows: min stb stf port ship mcy mdr cy dr total The value range of  ,  ,  ,  , and  is from 0 to 2. The results for different weight coefficients are shown in According to table 10, we perform a linear regression analysis, using  ,  ,  ,  , and  as independent variables and using the shipping cost, the ship using cost, the wharf construction cost, the inventory cost, the warehouse construction cost, and the total cost during the total time of system operation as the dependent variables. The corresponding regression coefficients are shown in table 12. The data in the second to sixth columns are the regression coefficients of each cost when the weight coefficients of the cost are regarded as independent variables. We found that the regression coefficient of the total cost during the total time of system operation in the sixth column is higher than that of the other columns, which means that the total cost during the total time of system operation has a higher sensitivity to  , that is to say, the impact of the weight coefficient of the wharf construction cost on the optimal solution is greater than that of the other weight coefficients. In the islands shipping network, the wharf scale is determined according to the ship size, and the ship size of every route is determined according to the routing, the schedule of every route, and location of hub islands, so the impact of the transport system, containing route, ship and wharf on the total cost during the total time of system operation is larger compared with the inventory system. In conclusion, more attention should be paid to the optimization of the transportation system and hub island location compared with that of the inventory system when making decisions.

Conclusions
This paper proposes the MLIRP for the islands shipping network. We establish a logistics cost model that considers the location of each island's hub island, the number of routes of the main network and branch network, the running mode of every route and which islands are passed through by each route and the sequence of the islands, ship size and schedule of each route, wharf scale and inventory capacity and cycle supply. Then, we develop the SC-GA to solve the integrated MLIRP model, and the performance of algorithm is assessed by a set of examples. The sensitivity analysis shows that more attention should be paid to the optimization of the transportation system and hub island location comparing with inventory system when making decisions. This paper has important theoretical significance and application value for establishing and optimizing islands shipping networks.
Notably, this paper only studies the shipping transportation in the islands logistics network.
In addition, air transportation can also be considered in the network. Therefore, in the further study, we could propose the integrated model of shipping and air transportation, thus enhancing the security of supply in the island logistics network.

Compliance with Ethical Standards
Funding: This paper was funded by Key Program of National Natural Science Foundation of