A method for determining optimal paths of a combined rail and road network under uncertain transit time

Abstract

The application discloses a method for determining an optimal path of an iron-water combined transportation network under uncertain transfer time conditions, comprising the following steps: 1, collecting road network data under railway and waterway transportation modes; 2, establishing an optimal path selection model of the iron-water combined transportation network under uncertain transfer time conditions; 3, transforming uncertain time constraints in the optimal path selection model; 4, loading the iron-water combined transportation network based on the transformed model, and determining road section states and node calculation sequences based on topological sorting; 5, based on the node calculation sequences, establishing a transfer time set of all nodes of the iron-water combined transportation network, an iron-water combined transportation scheme road section time cost set and an iron-water combined transportation scheme road section transportation cost set; 6, inputting model time and cost sets, and solving the optimal path of the iron-water combined transportation network by using topological sorting; and 7, determining and outputting an optimal path set of the iron-water combined transportation network with the minimum transportation cost under the condition of meeting time cost constraints.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of intermodal transportation, more particularly relates to a method for determining an optimal path of intermodal transportation network under uncertain transfer time. BACKGROUND

[0002] Intermodal transportation refers to a transportation process in which railway and waterway transportation tools are connected and transferred to achieve the transportation together. Railway transportation and waterway transportation are important components of the comprehensive transportation system and are the main carriers of modern logistics. Intermodal transportation gives full play to the comparative advantages and combination effects of railway and waterway transportation, which is beneficial to improving transportation efficiency and reducing logistics cost, and has become one of the main research directions of the development of China's logistics industry. With the rapid development of the logistics industry, the requirements of intermodal transportation carriers for transportation cost and transportation time are becoming more stringent, and they tend to choose the scheme with the minimum total transportation cost and meeting the time requirements of the transportation scheme. Therefore, when providing an optimal path of the network for different intermodal transportation users, the system needs to balance the time cost, node transfer time, cost and speed of the road sections between railway and waterway transportation modes, and provide an overall optimal path scheme, rather than only involving a single variable factor.

[0003] At present, there are few studies on intermodal transportation composite network in the field of network optimal path solving, and most of them are for urban traffic road networks. The existing optimal path solving method for urban road networks cannot effectively reflect the change of path type in the process of railway to waterway intermodal transportation, and it is difficult to avoid the waste of transportation time and the decline of transportation safety caused by too high transfer times. In addition, it cannot meet the demand of intermodal transportation users to find a path scheme that meets the target constraint conditions of cost and time cost to a certain extent under the condition of variable transfer time. SUMMARY

[0004] To solve the above technical problems, the present application adopts the following technical solution: the present application provides a method for determining an optimal path of intermodal transportation network under uncertain transfer time, which comprises the following steps:

[0005] Step 1: Collecting road network data under railway and waterway transportation modes, and abstracting an intermodal transportation network through the connection information of each road section and node;

[0006] Step 2: Under the condition of uncertain transfer time, establishing an optimal path selection model of intermodal transportation network with the minimum transportation cost as the objective function;

[0007] Step 3: Converting the uncertain time constraint in the optimal path selection model, defuzzifying the uncertain transfer time triangular fuzzy number, and obtaining the expected value of the transfer time interval;

[0008] Step 4, for the transformed model, load the iron-water combined transport network, determine the link state and node calculation order based on topological sorting;

[0009] Step 5, based on the node calculation order, establish the transfer time set of all nodes of the iron-water combined transport network, the time cost set of the iron-water combined transport scheme link, and the transportation cost set of the iron-water combined transport scheme link;

[0010] Step 6, input the model time and cost set, and solve the optimal path of the iron-water combined transport network by topological sorting;

[0011] Step 7, determine and output the optimal path set of the iron-water combined transport that meets the time cost constraint and has the minimum transportation cost.

[0012] Further, the specific method of step 1 is as follows: an iron-water combined transport network G(V, E) is constructed, where V is the set of iron-water combined transport nodes, E is the set of iron-water combined transport links, O represents the set of starting nodes, D represents the set of ending nodes, let W represent the set of all OD pairs in the iron-water combined transport network, t ij represents the link time between node i and node j, Trn max is the maximum allowed number of intermodal transfers.

[0013] Further, the specific method of step 2 is as follows: let K represent the set of transportation modes between node i and node j, d ij is the distance between node i and node j, C k represents the cost of transportation mode k per unit of transportation distance, D jc represents the demand for product c at destination j, represents the single transfer time fuzzy number of node j from transportation mode k to k', represents the speed of transportation from i to j by transportation mode k, T D represents the contract transportation time that the iron-water combined transport scheme needs to meet, represents the 0-1 judgment variable of whether to select transportation mode k and whether transfer occurs, 1 if yes, 0 if no;

[0014] Based on the above definitions, let the transportation cost of the iron-water combined transport be Z, which includes the comprehensive transportation cost borne by the railway and waterway in the iron-water combined transport process:

[0015] The total transportation time of the transportation scheme is which includes the total transportation time of the railway and waterway and the total transfer time of the transfer nodes

[0016] According to the above symbol definitions, the optimal path selection model of the iron-water combined transport network is represented by formulas (1)-(4):

[0017]

[0018]

[0019]

[0020]

[0021] Equation (1) is the objective function of the minimum transportation cost, equation (2) represents that the total time of the transportation scheme satisfies the contract time requirement constraint, equation (3) represents the constraint that only one transportation mode can be selected for each road section, and equation (4) represents the constraint that transfer can only occur once at each transfer node, that is, the transfer task cannot be disassembled.

[0022] Further, the specific method of step 3 is as follows:

[0023] Step 3.1, the transfer time triangular fuzzy number is expressed as α,β≥0, where δ, represent the boundary parameters of the triangular fuzzy number, and α, β represent the left and right floating distances. The membership function is expressed as

[0024] Equation (5):

[0025]

[0026] Step 3.2, based on the membership function of , the defuzzification of using the expected value method can obtain the expected interval value of .

[0027]

[0028] Step 3.3, let and wherein represents the boundary parameter of the transfer time triangular fuzzy number, and . Substitute equation (6) to defuzzify and obtain the interval expected value of .

[0029]

[0030] Further, the specific method of step 4 is as follows:

[0031] Step 4.1, for the loaded intermodal transportation network, delete the entry arc of the starting point O and the exit arc of the ending point D, and mark the state variables ξ of these road sections ij= -1, create an empty set M and an empty queue Q, and put O into M;

[0032] Step 4.2: Determine whether M is an empty set. If it is an empty set, it indicates that the queue Q is a topological sequence, and go to step 5. Otherwise, go to step 4.3.

[0033] Step 4.3. Search for all nodes i in M ​​whose incoming arc state is 1 or -1, delete them from M and put them into Q, and mark all nodes whose outgoing arc state is 0 as ξ. ij =1, add the end point of the road segment to the set M and go to step 4.2.

[0034] Furthermore, the specific method of step 5 is as follows: establish a transit time set Tr of all nodes in the water-jumping transport network, where tr(j) represents the transit time of node j, and tr(j)∈Tr; the transit time node set is determined based on the defuzzified node transit time, where the transit time of nodes that do not undertake transit is set to 0; establish a section time cost set of the iron-water intermodal transport plan, including the total transportation time of railways and waterways and the defuzzified total transit time of the transit nodes; establish a section transportation cost set of the iron-water intermodal transport plan, including the comprehensive transportation costs borne by the two modes of railways and waterways during the iron-water intermodal transport process.

[0035] Furthermore, the specific method of step 6 is as follows:

[0036] Step 6.1: Based on the equivalent iron-water transport network and topological sequence obtained by topological sorting, define node v j degree Node v j The cost weight equation is j=2,3,...,n, where u i 、w ij is the weight equation parameter, and the node v j Marked as (1,d j ,u j ), j≠1 and v j ∈V, where j=2,3,..,n,permanent label S={v1},temporary label R={v2,v3,...,v n};

[0037] Step 6.2, let u1 = 0, and use the cost weight equation to calculate the cost weight of the road section. End, u j That is the shortest (v1,v j ) road right; otherwise, go to step 6.3;

[0038] Step 6.3, check the search section node, if dj > 0, any node in R has an arc in D, i.e. G contains a loop, end; otherwise, search another node v j = 0 in R i , let S = S U {v i}, R = R \ {v i}, go to step 6.4;

[0039] Step 6.4, initialize the number of times of transfer Trn = 0, judge whether the node v j occurs molten iron transfer, if the node v j occurs transfer process, let T = T t + t ij + tr(j), Trn++, if the node v j does not occur transfer process, let T = T t + t ij , until all adjacent nodes are checked, go to step 6.5;

[0040] Step 6.5, judge whether the number of times of transfer Trn satisfies the maximum allowed number of times of molten iron transfer Trn max constraint, if Trn <= Trn max , go to step 6.6, otherwise, go back to step 6.2;

[0041] Step 6.6, judge whether the total time of the transportation scheme T satisfies the contract time T D constraint, if T <= T D , go to step 6.7, otherwise, go back to step 6.2;

[0042] Step 6.7, judge whether the search of the adjacent node set of all nodes v j has been completed in turn, make node marking operation u' j = min{u j , u i + w ij}, d' j = d j - b ij , if the marking value d j of the node v j decreases, the original marking of the node v j is updated as (j, d' j , u' j ), save the search result of the node, go to step 7; otherwise, only update the value of d j , go to step 6.2.

[0043] Beneficial effects: compared with the prior art, the technical scheme of the present application has the following beneficial technical effects:

[0044] (1) the method provided by the present application firstly defuzzifies the uncertain transfer time triangular fuzzy number in the combined transport of iron and water through the expectation method, can complete the mathematical expression and deterministic transformation of the fuzzy transfer time, takes the minimum transportation cost as the optimization goal, can balance the transfer time and transportation cost between the two different transportation modes of railway and waterway, can effectively avoid the influence of the uncertain transfer time of iron and water on the total transportation time, meet the time constraint requirements of the combined transport contract of iron and water, enhance the reliability of the transportation time, and find the overall optimal combined transport path set of iron and water;

[0045] (2) the method for determining the optimal path of the combined transport of iron and water provided by the present application is based on the topological sorting processing method, can exclude unreasonable road sections in the case that the network exists a loop, and can arrange the cost shortest paths from the nodes to the end points in descending order, so that the redundant road sections can be greatly reduced, and the impedance of the excluded paths is ensured, the method provided by the present application can effectively reduce unnecessary repeated calculation in the solving process, prevent the omission of the searched node paths, and greatly save the solving time;

[0046] (3) the present application adds the maximum allowed transfer times in the determination of the optimal path of the combined transport of iron and water according to the characteristics of the combined transport network of iron and water, increases the limitation of the transfer times in the solving process, deletes the combined transport paths of iron and water whose transfer times exceed the maximum constraint through setting the maximum allowed transfer times, and can effectively avoid the problems of time waste and increase of transportation uncertainty caused by too many transfer times;

[0047] (4) the method for determining the optimal path of the combined transport of iron and water under the condition of uncertain transfer time can scientifically solve the optimal path of the combined transport network of iron and water, save the calculation time, and is a kind of feasible and effective method for solving the optimal path of the combined transport network of iron and water; the present application is the exploration and innovation of the theory and practice of the combined transport of iron and water, and has strong theoretical research significance and practical guiding value. The present application can effectively ensure that the optimal path set of the combined transport of iron and water with the minimum transportation cost, the transportation time meeting the constraint and the transfer times meeting the requirements is selected without repetition and omission under the condition of uncertain transfer time. BRIEF DESCRIPTION OF DRAWINGS

[0048] In order to more clearly illustrate the technical scheme of the embodiments of the present application, the drawings of the embodiments will be briefly introduced as follows. Obviously, the drawings described in the following description only relate to some embodiments of the present application, but are not limited to the present application.

[0049] Figure 1 is the flow chart of the method of the present application;

[0050] Figure 2 is the simple road network of the combined transport of iron and water for verifying the method of the present application. DETAILED DESCRIPTION

[0051] In order to make the purposes, technical solutions and advantages of the present application clearer, the present application will be described in detail below with reference to the drawings and specific embodiments.

[0052] The present application provides a method for determining the optimal path of the multimodal transport network under uncertain transfer time, as shown in the figure, comprising the following steps: Figure 1

[0053] Step 1, collect the road network data under the railway and waterway transportation mode, and abstract the multimodal transport network through the connection information of each road section and node;

[0054] Step 2, under the condition of uncertain transfer time, establish a multimodal transport network optimal path selection model with the minimum transportation cost as the objective function;

[0055] Step 3, transform the uncertain time constraint in the optimal path selection model, defuzzify the uncertain transfer time triangular fuzzy number, and obtain the expected value of the transfer time interval;

[0056] Step 4, for the transformed model, load the multimodal transport network, and determine the road section state and node calculation order based on topological sorting;

[0057] Step 5, based on the node calculation order, establish the transfer time set of all nodes of the multimodal transport network, the time cost set of the multimodal transport scheme road section, and the transportation cost set of the multimodal transport scheme road section;

[0058] Step 6, input the model time and cost set, and solve the optimal path of the multimodal transport network by using topological sorting;

[0059] Step 7, determine and output the optimal multimodal transport path set with the minimum transportation cost under the time cost constraint.

[0060] Further, the specific method of step 1 is as follows: a multimodal transport network G(V, E) is constructed, wherein V is a multimodal transport node set, E is a multimodal transport road section set, O represents a starting node set, D represents a terminal node set, W represents a set of all OD pairs in the multimodal transport network, t ij represents the road section time between node i and node j, Trn max is the maximum allowed number of multimodal transport.

[0061] Further, the specific method of step 2 is as follows: assuming K represents the set of transportation modes between node i and node j, d ij is the distance between node i and node j, C k represents the cost of transportation mode k per unit transportation distance, D jc represents the demand of product c at destination j, represents the single transfer time fuzzy number of node j from transportation mode k to k',​ denotes the speed of transportation from i to j by transportation mode k, T D denotes the contract transportation time that the intermodal transportation scheme needs to meet, denotes the 0-1 judgment variable of whether to select transportation mode k and whether to occur transfer, 1 if yes, and 0 if no;

[0062] Based on the above definitions, let the transportation cost of intermodal transportation be Z, which includes the comprehensive transportation cost borne by railway and waterway in the intermodal transportation process:

[0063] The total time of the transportation scheme is The total transportation time of railway and waterway is And the total transfer time of the transfer node is

[0064] According to the above symbol definitions, the optimal path selection model of the intermodal transportation network is represented as formulas (1)-(4):

[0065]

[0066]

[0067]

[0068]

[0069] Formula (1) is the objective function of the minimum transportation cost, formula (2) represents the constraint that the total time of the transportation scheme meets the contract time requirement, formula (3) represents the constraint that only one transportation mode can be selected for each route, and formula (4) represents the constraint that transfer can only occur once at each transfer node, i.e., the transfer task cannot be disassembled.

[0070] Further, the specific method of step 3 is as follows:

[0071] Step 3.1, the transfer time triangular fuzzy number is represented as α,β≥0, where δ, represent the boundary parameters of the triangular fuzzy number, and α, β represent the left and right floating distances. The membership function is represented as formula (5):

[0072]

[0073] Step 3.2, based on the membership function of , the expected value method is used to defuzzify to obtain the expected interval value of

[0074] the expected interval value​

[0075]

[0076] Step 3.3, set wherein, denote the transport time triangular fuzzy number boundary parameters, then Substitute formula (6) to defuzzify can obtain interval expectation value of

[0077]

[0078] Further, the specific method of step 4 is as follows:

[0079] Step 4.1, for the loaded iron-water intermodal network, delete the entry arc of the starting point O and the exit arc of the ending point D, mark the state variables ξ of these road sections ij =-1, establish empty set M and empty queue Q, and put O into M;

[0080] Step 4.2, judge whether M is empty set, if it is empty set, it indicates that the queue Q is the topological sequence, turn to step 5, otherwise turn to step 4.3;

[0081] Step 4.3, search all nodes i with entry arc state of 1 or -1 in M, delete it from M and put it into Q, mark all nodes with exit arc state of 0 as ij =1, add the end point of the road section to set M, and turn to step 4.2.

[0082] Further, the specific method of step 5 is as follows: establish the transport time set Tr of all nodes in the jumping intermodal network, wherein tr(j) represents the transport time of node j, tr(j)∈Tr; the transport time node set is determined according to the defuzzified node transport time, wherein the transport time of the node not bearing transport is set to 0; establish the road section time cost set of the iron-water intermodal scheme, which contains the total transport time of railway and waterway and the total transport time of the defuzzified transport node; establish the road section transport cost set of the iron-water intermodal scheme, which contains the comprehensive transport cost of railway and waterway in the process of iron-water intermodal transportation.

[0083] Further, the specific method of step 6 is as follows:

[0084] Step 6.1, based on the equivalent iron-water intermodal network and the topological sequence obtained by the topological sorting processing, define the node v j in-degree The cost weight equation of the node v j is wherein u i , w ijFor the weight equation parameter, the node v j is marked as (1, d j , u j ), j≠1 and v j ∈V, wherein, j=2, 3,..,n, permanent label S={v1}, temporary label R={v2, v3,...,v n};

[0085] Step 6.2, let u1=0, and calculate the cost weight of the link by using the cost weight equation, if end, u j is the weight of the shortest (v1, v j ) path in the G in the iron-water combined transport network; otherwise, go to step 6.3;

[0086] Step 6.3, check the search link node, if d j >0, any node in R has an arc in D, that is, G contains a loop, end; otherwise, search another node v j in R that satisfies d i =0, let S=S∪{v i}, R=R\{v i}, and go to step 6.4;

[0087] Step 6.4, initialize the transport number Trn=0, and judge whether the node v j has the iron-water transport, if the node v j has the transport process, let T=T t +t ij +tr(j), Trn++, if the node v j has no transport process, let T=T t +t ij , until all adjacent nodes are checked, and go to step 6.5;

[0088] Step 6.5, judge whether the transport number Trn satisfies the maximum allowed transport number Trn max constraint, if Trn≤Trn max , go to step 6.6, otherwise, return to step 6.2;

[0089] Step 6.6, judge whether the total transport time T satisfies the contract time T D constraint, if T≤T D , go to step 6.7, otherwise, return to step 6.2;

[0090] Step 6.7, judge whether the search of the adjacent node set of all nodes v j has been completed, and make the node marking operation u'j =min{u j ,u i +w ij}, d' j =d j -b ij , if node v j Tag value d j Decreases, then node v j The original mark is updated to (j,d' j ,u' j ), save the node search results and go to step 7; otherwise, only update d j , go to step 6.2.

[0091] The following is a simple rail-water transport network to test the performance of the optimal path determination method under the condition of uncertain transit time, thereby verifying the effectiveness of the present invention. The test network selected a 6-node network containing two types of sections: railway and waterway. The dotted line represents the railway section and the solid line represents the waterway section. The transportation cost, time cost, and transit time set information of the specific network section are as follows: Figure 2 As shown in the figure, the defuzzified transit time using the expected value method is [0.37, 0.7, 2.1]. The optimal route for the iron-water transport is defined to have no more than 2 transit times, and the agreed transport time cost constraint for the iron-water transport contract is 15.

[0092] Solve the optimal path of the iron-water transport network according to the method described in step 6. First, calculate the cost weight of the section, then check the search section nodes, and then determine whether the number of transfers exceeds the maximum allowed number of transfers and whether the total time of the transportation plan meets the contract time constraints. Finally, determine whether the adjacent node set search of all nodes has been completed and establish the optimal path set for iron-water transport.

[0093] Therefore, it can be calculated that according to Figure 1 The method for solving the optimal path for iron-water intermodal transport shown in the figure obtains the optimal path that can achieve the minimum transportation cost goal and meet the time cost constraint: OAED, whose transportation cost is 11, the transportation time cost is 10.07 and meets the time cost constraint of 15.

[0094] The above description is only a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any changes or substitutions that can be easily conceived by a technician familiar with this technical field within the technical scope disclosed by the present invention should be covered by the scope of protection of the present invention.

Claims

1. A method for determining the optimal path of a molten iron intermodal network under uncertain conditions of transit time, characterized in that, The method comprises the following steps: Step 1, construct a water-iron combined transport network G(V, E), wherein V is a set of water-iron combined transport nodes, E is a set of water-iron combined transport links, O represents a set of starting nodes, D represents a set of ending nodes, let W represent a set of all OD pairs in the water-iron combined transport network, t ij represents the link time between node i and node j, Trn max is the maximum number of intermodal transfers allowed; Step 2, under the condition of uncertain transfer time, an optimal path selection model of the combined transport network of molten iron is established with the minimum transport cost as an objective function; Step 3, the uncertain time constraint in the optimal path selection model is converted, the triangular fuzzy number of the uncertain transfer time is defuzzified, and an expected value of the transfer time interval is obtained; Step 4, for the converted model, the combined transport network of molten iron is loaded, and the road section state and node calculation order are determined based on topological sorting; Step 5, based on the node calculation order, the transfer time set of all nodes in the combined transport network of molten iron, the time cost set of the road section of the combined transport scheme of molten iron and the transport cost set of the road section of the combined transport scheme of molten iron are established; Step 6, the model time and cost set are input, and the optimal path of the combined transport network of molten iron is solved by using topological sorting; Step 7, the optimal path set of the combined transport of molten iron with the minimum transport cost under the time cost constraint is determined and output; The specific method of step 2 is as follows: assuming K represents the set of transportation modes between node i and node j, d ij The distance between node i and node j, C k The cost of unit transportation distance of transportation mode k, D jc The demand of destination j for product c, The single transfer time fuzzy number of node j from transportation mode k to k', T The speed of i to j by transportation mode k, T D The contract transportation time that the pig iron intermodal scheme needs to meet, The 0-1 judgment variable of whether to select transportation mode k and whether transfer occurs, 1 if yes, and 0 if no; Based on the above definition, the transportation cost of the combined railway-waterway transportation is Z, which includes the comprehensive transportation cost of the railway and waterway during the combined railway-waterway transportation: Total transport time for the transport solution is Total transport time including rail, water and transshipment nodes According to the above symbol definition, the optimal path selection model of the combined transport network of molten iron is represented by formula (1)-(4): Formula (1) is the objective function of the minimum transport cost, formula (2) represents the constraint that the total transport time of the transport scheme meets the contract time requirement, formula (3) represents the constraint that only one transport mode can be selected for each road section, and formula (4) represents the constraint that transfer can only occur once at each transfer node, i.e. the transfer task cannot be disassembled; The specific method of step 3 is as follows: Step 3.1, Triangular Fuzzy Number for Transit Time is expressed as where δ represents the triangular fuzzy number boundary parameter, and α, β represent the left and right floating distances. The membership function is expressed as equation (5): Step 3.2, based on the membership function using the expectation method defuzzification yields the expected interval value Step 3.3, Set and where, denote the transport time triangular fuzzy number boundary parameters, then Substitute formula (6) to defuzzify can be obtained interval expected value of 2. The method according to claim 1, wherein, The specific method of step 4 is as follows: Step 4.1, for the loaded rail-water network, delete the in-arc of origin O and the out-arc of destination D, mark the state variable ξ of these links ij = -1, establish empty set M and empty queue Q, put O into M; Step 4.2, judge whether M is an empty set, if it is an empty set, it means that the queue Q is the topological sequence, and step 5 is converted, otherwise step 4.3 is converted; Step 4.

3. Search all nodes i in M with in-arc state 1 or -1, remove them from M and put them into Q, and mark all out-arc state 0 node state variables with ξ ij = 1, add the end of the link to the set M, and go to step 4.

2.

3. The method according to claim 2, wherein, The specific method of step 5 is as follows: the transfer time set of all nodes in the combined transport network of molten iron is established, wherein tr(j) represents the transfer time of node j, and tr(j) e Tr; the transfer time node set is determined according to the defuzzified node transfer time, wherein the transfer time of the node not bearing the transfer is set to 0; the time cost set of the road section of the combined transport scheme of molten iron is established, including the total transport time of railway and waterway and the total transfer time of the defuzzified transfer node; the transport cost set of the road section of the combined transport scheme of molten iron is established, including the comprehensive transport cost of railway and waterway in the combined transport process of molten iron.

4. The method according to claim 3, wherein, The specific method of step 6 is as follows: Step 6.1, based on the equivalent water transportation network and the topological sequence obtained by the topological sorting processing, define the node v j in-degree node v j The cost weight equation is where u i , w ij are weight equation parameters, mark the node v j as (1, d j , u j ), j≠1 and v j ∈V, where, permanent label S={v1}, temporary label R={v2, v3,..., v n} Step 6.2, let u1=0, calculate the cost weight of the link by the cost weight equation, if End, u j is the weight of the shortest path from v1to v j in G; otherwise, go to step 6.3; Step 6.

3. Check the search path node, if d j > 0, any node in R has an arc in D, i.e. G contains a loop, end; otherwise, search another node v j in R that satisfies d i = 0, let S = S U {v i}, R = R \ {v i}, go to Step 6.

4. Step 6.4, initialize the number of transits Trn = 0, judge node v j whether the molten iron transits, if node v j has transited, then let T = T t + t ij + tr(j), Trn++, if node v j has not transited, then let T = T t + t ij , until all adjacent nodes are checked, go to step 6.5; Step 6.

5. Determine whether the number of transshipments Trn satisfies the maximum number of transshipments Trn allowed for the combined transport of the iron ore and the molten iron max Constraint if Trn ≤ Trn max Step 6.6, otherwise return to Step 6.2; Step 6.

6. Determine if the total time T of the transport solution satisfies the contract time T D Constraint if T < T D Go to Step 6.7, otherwise go back to Step 6.2; Step 6.7, judge whether all nodes v have been finished in turn j adjacent node set search, make node mark operation u' j = min{u j , u i + w ij}, d' j = d j - b ij , if the node v j mark value d j is reduced, the original mark of the node v j is updated to (j, d' j , u' j ), save the node search result, and turn to step 7; otherwise, only update the value of d j , and turn to step 6.2.

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