A joint scheduling method for berths and quay cranes in a multi-berth tidal port under uncertain environment

Abstract

A joint scheduling method for berths and quay cranes in a multi-berth tidal port under uncertain environment, comprising: establishing a mathematical model for joint scheduling of berths and quay cranes in multiple berths under uncertain conditions; then generating an initial population by combining two strategies of stochastic generation and greedy construction; using a gene repair algorithm to convert infeasible solutions into feasible solutions; calculating fitness values and selection probabilities; generating offspring populations by combining elitist retention strategy and roulette wheel strategy, and using a fusion crossover operator and a uniform mutation operator based on fitness values to perform a preliminary search of the solution space; finally, further optimizing by combining a local search algorithm based on simulated annealing mechanism; verifying the effectiveness of the joint scheduling model of berths and quay cranes under uncertain conditions and the algorithm efficiency by comparing the SA-AGA algorithm, the algorithm without combining the greedy construction strategy, the algorithm without combining the simulated annealing mechanism, and the traditional adaptive genetic algorithm without using both of them; the present invention improves the operation efficiency of container terminals and reasonably allocates berths and quay cranes.

Description

Technical Field

[0001] The present invention relates to a method for allocating berths and quay cranes in a multi-terminal tidal port. Technical Background

[0002] The development of economic globalization has led to a rapid growth in trade among countries around the world. The fact that 71% of the world is covered by seawater determines that maritime transportation plays a crucial role in the global trade system. According to statistics, from 2019 to 2021, more than 80% of the goods in global trade were completed by sea transportation. As the second-largest economy in the world, since China joined the WTO in 2001, the volume of import and export trade has been growing rapidly, with average annual growth rates of more than 4% and 10% respectively. And from 2009 to 2020, China contributed approximately 65% of the global increment in seaborne trade. Containers, because of their uniform size standards, can significantly reduce the transportation costs of goods, reduce the loss of goods, and improve the loading and unloading efficiency of goods. Therefore, in the process of sea transportation, container transportation has become the main transportation mode, and container ports have become important nodes in the global supply chain. The huge trade demand has brought great opportunities to container ports while also posing new challenges. As a large-scale complex logistics system with many resources including berths, quay cranes, yards, and container trucks, container ports must integrate all resources within the port to enhance their competitiveness.

[0003] In addition to multi-terminals, uncertain ship arrival times, and quay crane operation efficiency, the existence of tides is also an important reason affecting the scheduling results of berths and quay cranes in the port. The existence of tides causes the water depth of the quay line to change periodically over time. At certain moments, the water depth conditions of the quay itself may not be sufficient for large ships to berth, but the water depth affected by tides can meet the draft requirements of the ships. This poses new requirements for determining the berthing positions of ships. In addition, the quay crane itself has a certain width, and there is a safety distance between quay cranes, which means that quay cranes can only reach a part of the quay line, rather than the whole.

[0004] The above factors have an inestimable impact on the production operations of container ports. Only by fully considering these constraints and integrating and utilizing all resources within the port can the operation efficiency and core competitiveness of the port be truly improved.

[0005] Imai et al. [1] First, taking the JCT and SAGT container terminals in the Port of Colombo as examples, they studied the berth scheduling problem when berths of other terminals can be shared unidirectionally. Although their research considered two terminals, it still studied with one terminal as the main body and did not involve the specific scheduling methods of resources such as berths and quay cranes in the other terminal. Frojan et al. [2]For the first time, the single-terminal berth allocation problem was extended to the multi-terminal scenario, and a heuristic algorithm based on genetic algorithm and local search was designed to solve it. However, their research only assumed the most basic constraints and did not consider important factors such as tides, uncertain environments, and other operation links. Since then, the research on multi-terminal resource scheduling has gradually increased. Budipriyanto et al. [3] et al. studied the multi-terminal berth allocation problem under uncertain environments using simulation software. Gutierrez et al. [4] et al. studied the static and dynamic berth allocation problems under uncertain vessel arrival times with a port having two shorelines as the object. By representing the vessel arrival time as a triangular fuzzy number, a fuzzy mixed integer linear programming model and a fully fuzzy linear programming model were established. Krimi et al. [5] et al. first studied the joint scheduling problem of berths and quay cranes under the multi-terminal cooperation mechanism and designed a heuristic algorithm based on a rolling strategy to solve it. However, since their research object was a bulk cargo terminal, their model and algorithm could not be directly applied to the scheduling operations of container ports. Grubisic et al. [6] et al. studied the berth and quay crane allocation problem for medium-sized terminals with multiple shoreline layouts and established a mixed integer programming model with the minimum vessel stay time in port as the objective. Bouzekri et al. [7] studied the joint allocation problem of vessel loading periods, berths, and quay cranes and established an integer programming model after considering the service range of quay cranes and the influence of tides. Lujan et al. [8] Based on the multi-terminal berth and quay crane joint allocation problem, the uncertainty of vessel arrival times was considered, and the vessel arrival time was described using triangular fuzzy numbers.

[0006] Through the above analysis, it can be found that in the field of single-terminal operation scheduling, the independent scheduling of berths and quay cranes has been studied more carefully and deeply because it started earliest. And due to the strong correlation between berths and quay cranes, the research on their joint scheduling has also gradually increased. However, the research on this problem in a deterministic environment has become increasingly mature. Therefore, emerging concepts such as uncertain environments and green environmental protection have received the attention of many scholars, and the research based on these backgrounds has developed rapidly. However, the research on multi-terminal operation scheduling is still very scarce. Most of the research has only studied the multi-terminal berth-quay crane joint scheduling in a deterministic environment, and a few studies have only considered the randomness of vessel arrival times while ignoring the impact of the fluctuation of quay crane operation efficiency.

[0007] This paper systematically studies the berth - quay crane joint scheduling problem of multi - terminal tidal ports in an environment where the ship arrival time and quay crane operation efficiency are uncertain, and fully considers key constraints such as the maximum service range of quay cranes and the interference between quay cranes. This makes up for the deficiencies in the current research on the joint scheduling of berths and quay cranes in multi - terminal ports. In specific operations, a non - linear mixed - integer programming model is established with the goal of minimizing the expectation and variance of the total cost of the scheduling plan under actual samples, so as to obtain a ship berthing plan and a quay crane operation plan with good cost and anti - interference performance. This provides a new reference for the scheduling optimization of container ports and further improves the theory of joint scheduling of berths and quay cranes in multi - terminals. Summary of the Invention

[0008] To overcome the above - mentioned shortcomings of the prior art, in the case of complex and changeable operation tasks in existing container terminals, a method for allocating berths and quay cranes in multi - terminals is proposed, which can reasonably plan the berths and quay crane operations of container terminals, rationally allocate terminal resources, and reduce the terminal operation cost.

[0009] The present invention is implemented by providing the following technical solutions:

[0010] A method for joint scheduling of berths and quay cranes in a multi - terminal tidal port under an uncertain environment, including:

[0011] Step 1: Construct a joint scheduling model of berths and quay cranes in a multi - terminal tidal port under uncertain conditions;

[0012] 1.1. Determine the objective function;

[0013] Since the present invention considers the uncertainty of ship arrival time and quay crane operation efficiency, through an active strategy, buffer time and buffer efficiency are respectively added to the expected ship arrival time and the average quay crane operation efficiency to reduce the impact of the fluctuations of these two on the scheduling plan; and in order to minimize this impact, a certain number of samples are randomly generated to simulate the possible ship arrival times and quay crane operation efficiencies, and the objective function is to minimize the sum of the expectation and variance of the total cost of the scheduling plan under each sample, as shown in Equation (1);

[0014] min E({f s})+σ({f s}) (1)

[0015] In the formula, f s is the total cost of the scheduling plan under sample s, E({f s}) represents the average value of the total cost of the scheduling plan under each sample, σ({f s}) is the variance, and E({fs The smaller σ({f s}), the lower the cost required when the scheduling plan is actually applied. And the smaller σ({f

[0016] 1.2 Set constraint conditions;

[0017] The operation cost of a ship refers to the operation cost required for quay cranes to perform container loading and unloading services for it during the period from the ship's berthing to its departure. It is related to the ship's berthing time, departure time, and the number of quay cranes allocated, as shown in Equation (2); where c y is the unit quay crane unit time operation cost when the ship is operating in the port; C i is the number of quay cranes operating for ship i, i ∈ V; y i is the berthing time of the ship i to be scheduled, i ∈ V; is the actual arrival time of ship i under sample s;

[0018]

[0019] When the arrival time of a ship exceeds its planned berthing time, it will affect the docking and operation of other ships waiting to berth, and the containers to be loaded and unloaded by it need to be detained additionally. At this time, a port arrival delay cost will be generated; its size is related to the quantity of containers to be loaded and unloaded by the ship and the port arrival delay time, as shown in Equation (3); where p after is the unit time penalty cost when the ship arrives after its expected arrival time; is the quantity of export containers of the ship i to be scheduled, i ∈ V; is the quantity of import containers of the ship i to be scheduled, i ∈ V;

[0020]

[0021] Whether the ship can depart normally is very important for both the ship and the terminal. If it cannot depart normally, for the ship, it may not be able to arrive at the next port on time, and for the terminal, it will affect the berthing and operation of subsequent ships. At this time, a departure delay penalty cost is generated, which is related to the quantity of containers of the ship and the time of delayed departure, as shown in Equation (4); where p i is the unit time penalty cost when the actual departure time of ship i exceeds its expected departure time; d i is the actual departure time of ship i; ED i is the expected departure time of the ship i to be scheduled, i ∈ V;

[0022]

[0023] If the arrival time of a ship is earlier than its planned berthing time, the ship can only wait in the anchorage. This situation will reduce the ship's satisfaction and thus affect the competitiveness of the port. Therefore, to reduce this impact, the ship waiting cost is added, which is related to the quantity of containers to be unloaded on the ship and the time of arriving at the port in advance, as shown in Equation (5); where p before is the unit time waiting cost when the ship arrives and waits before its expected arrival time; y i is the berthing time of the ship i to be scheduled, i ∈ V;

[0024]

[0025] Considering the storage location of the containers to be loaded on the ship in the yard, each ship will have its own ideal berthing position. Therefore, when the ship berths at its pre-assigned terminal, if there is a deviation between its berthing position and the ideal berthing position, it will cause an increase in the horizontal transportation distance between the yard and the ship berthing position, thus increasing the transportation cost. Therefore, the deviation cost needs to be added, as shown in Equation (6), which is related to the quantity of containers to be loaded and unloaded on the ship and the deviation distance; where c x is the unit deviation cost when the berthing position of the ship at its pre-assigned terminal deviates from the ideal berthing position; x i is the berthing position of the ship i to be scheduled, i ∈ V; bp i is the best berthing position of the ship i to be scheduled at its pre-assigned terminal, i ∈ V; z im is when the pre-assigned terminal of the ship i is m, z im = 1, otherwise z im = 0, i ∈ V, m ∈ T; u im is when the berthing terminal of the ship i is m, u im = 1, otherwise u im = 0, i ∈ V, m ∈ T;

[0026]

[0027] When the ship does not berth at its pre-assigned terminal, this deviation cost does not need to be considered. However, since the containers to be loaded on the ship need to be transferred from the pre-assigned terminal to the actual berthing terminal, this incurs additional transfer costs, as shown in Equation (7), which is related to the quantity of containers to be loaded on the ship; where c km is the unit quantity transfer cost from terminal k to terminal m, k, m ∈ T;

[0028]

[0029] Therefore, under any sample s, the cost required for the scheduling plan is shown in Equation (8);

[0030] f s = Cost oper + Cost arrd + Cost depd + Cost wait + Cost devi + Cost tran (8)

[0031] The existence of the objective function can determine what kind of scheduling plan is the most ideal, but to obtain a feasible ship berthing plan and quay crane allocation plan, many constraints need to be satisfied;

[0032]

[0033]

[0034] When determining the berthing terminal of a ship, it is first necessary to ensure that all ships to be scheduled have selected a terminal to berth at, and each ship can only berth at one terminal, as shown in Constraint (9); at the same time, as shown in Constraint (10), throughout the ship's stay in the port, the water depth of the terminal it berths at cannot be lower than its draft requirement; where df i is the draft of ship i to be scheduled, i ∈ V; M is a sufficiently large positive number; is the water depth of terminal m at time t, m ∈ T, t ∈ H;

[0035]

[0036]

[0037]

[0038]

[0039] Due to the limitation of the quay length, Constraint (11) is introduced to ensure that when a ship berths, the entire length of the ship from the bow to the stern is within the quay line range of its berthing terminal; in addition, to prevent spatio-temporal conflicts when ships berth, that is, occupying the same quay line at the same time, Constraints (12) to (14) must also be satisfied; where Constraint (12) is the spatio-temporal constraint between ships to be scheduled, indicating that when two ships berth on the same quay line at the same time, the position of the stern of the ship on the left must be to the left of the position of the bow of the ship on the right; similarly, Constraints (13) and (14) respectively represent the spatio-temporal constraints when the ship to be scheduled is on the right and left of the berthed ship; where, x i is the berthing position of ship i to be scheduled, i ∈ V; is the length of the ship to be scheduled \(i\), including the horizontal safety reserve length, \(i\in V\); is the length of the berthed ship \(i\), including the horizontal safety reserve length, \(i\in V\) 0 ; indicates that if ship \(j\) berths on the right side of ship \(i\), otherwise \(i,j\in V\), \(i\neq j\); indicates that if ship \(j\) berths on the right side of ship \(i\), otherwise \(i\in V\) 0 , \(j\in V\), \(i\neq j\); indicates that if ship \(j\) berths on the left side of ship \(i\), otherwise \(i\in V\) 0 , \(j\in V\), \(i\neq j\);

[0040]

[0041]

[0042]

[0043]

[0044] In addition to the constraints that the ship berthing at the terminal and the berthing position need to meet, there are also some constraints on the berthing time that must be considered; first of all, the berthing time of the ship must be later than its arrival time. At the same time, due to the uncertainty of the ship's arrival time, it is stipulated that when scheduling, the ship must pass through a buffer time after its expected arrival time before it can berth, as shown in constraint (15); the departure time of the ship is jointly determined by its berthing time, the quantity of containers to be loaded and unloaded, the quay crane operation efficiency, and the number of quay cranes allocated. Constraint (16) defines the relationship between them; similarly, the quay crane operation efficiency considered during scheduling is the sum of its average operation efficiency and buffer efficiency, thereby weakening the impact brought by the change of the right quay crane operation efficiency; Constraint (17) defines the relationship between the berthing time and departure time of any two ships to be scheduled when they need to occupy the same section of the shoreline, that is, the departure time of the ship that first occupies this section of the shoreline must be earlier than the berthing time of the later occupying ship; similarly, Constraint (18) defines the relationship between the berthing time of the ship to be scheduled and the departure time of the berthed ship; among them, \(y\) i is the berthing moment of the ship to be scheduled \(i\), \(i\in V\); is the expected arrival moment of the ship to be scheduled \(i\), \(i\in V\); is the slack of the arrival time of ship \(i\), \(i\in V\); \(g\) is the interference coefficient between quay cranes; indicates that if ship \(j\) berths after ship \(i\) departs, otherwise i, j ∈ V, i ≠ j; Indicates that if vessel j berths after vessel i departs, otherwise j ∈ V, i ∈ V 0 ;

[0045]

[0046]

[0047] When any two vessels to be scheduled berth at the same terminal, there must be a relationship between them in either the spatial dimension or the temporal dimension; through interaction with constraint (12), constraint (19) defines their connection in the spatial dimension, and interaction with constraint (17) defines the relationship in the temporal dimension; similarly, through interaction with constraints (13), (14), and (18), constraint (20) realizes the definition of the relationship between the vessels to be scheduled and the vessels that have already berthed in both space and time; the existence of these constraints can prevent the situation where two vessels occupy the same section of the shoreline simultaneously; among them, is if the berthing terminal of the berthed vessel i is m, otherwise i ∈ V 0 , m ∈ T;

[0048]

[0049]

[0050]

[0051]

[0052]

[0053]

[0054]

[0055]

[0056]

[0057]

[0058] If the relationship between the ship operation time and the quay crane operation efficiency in constraint (16) is not considered, that is, the ship operation time is regarded as a known constant, then equations (1) to constraint (20) together constitute an optimization model for the berth allocation problem under uncertain environment; however, in the actual production operation process, the ship operation is jointly completed by quay cranes, so the required time is also closely related to quay cranes; and there are also many constraints that need to be satisfied simultaneously when allocating quay cranes, as shown in constraints (21) to (30); specifically, constraint (21) defines the range of the number of quay cranes that can be allocated to each ship, that is, it must be between the minimum number of quay cranes that can be allocated to the ship and the maximum number of quay cranes that can be allocated; constraint (22) defines the relationship between the number of allocated quay cranes and the quay crane numbers; constraints (23) and (24) avoid the situation where one quay crane serves two ships at the same time; constraint (25) makes the quay cranes serving the same ship continuous, that is, when two quay cranes a and c serve the same ship at the same time, all quay cranes in the middle of them must also serve it; as shown in constraints (26) to (28), when there are more than two ships on a quay line at the same time, quay cranes cannot cross each other to provide services, that is, when quay crane a serves a ship, the quay crane on its right cannot serve other ships on the left, and the quay crane on its left cannot serve other ships on the right; the existence of constraints (29) and (30) ensures that all quay cranes allocated to the ship can provide services, that is, the service range of the allocated quay cranes can cover the berthing quay line interval of the ship; among them, is the minimum number of quay cranes allocated to the ship i to be scheduled, i ∈ V; is the maximum number of quay cranes allocated to the ship i to be scheduled, i ∈ V; indicates that if the quay crane q at the terminal m serves the berthed ship i, otherwise i ∈ V 0 , q ∈ Q m , m ∈ T; θ iqm indicates that if the quay crane q at the terminal m serves the ship i, θ iqm = 1, otherwise θ iqm = 0, i ∈ V, q ∈ Q m , m ∈ T; is the maximum service position of the quay crane q at the terminal m, m ∈ T, q ∈ Q m ; is the minimum service position of the quay crane q at the terminal m, m ∈ T, q ∈ Q m ;

[0059] Step 2: Design a multi-terminal berth quay crane joint scheduling algorithm;

[0060] 2.1 Initializing the population: Half of the chromosomes in the initial population are generated by the greedy construction strategy. Let the berthing quay of the ship be its pre-assigned quay, and let the berthing position of the ship be its optimal berthing position bp i , and let the berthing time of the ship be Let the number of quay cranes assigned to the ship be a random integer within, let the starting quay crane number be 1, and let the slack of the ship's arrival time and the slack of the quay crane operation efficiency be and The other half of the chromosomes are generated by the random generation strategy. Let the berthing quay of the ship be a random integer within [1, N T , let the berthing position of the ship be a random integer within, let the berthing time of the ship be Let the number of quay cranes assigned to the ship be randomly generated as an integer within ; let the starting quay crane number be a random integer within, and let the slack of the ship's arrival time and the slack of the quay crane operation efficiency be and

[0061] 2.2 Calculating the fitness function: Step 1.1: For all samples s ∈ S, take the actual berthing time of the ship as the greater of the ship's actual arrival time and the planned berthing time y i , and according to and the actual quay crane operation efficiency of the ship obtain the actual departure time of the ship. Step 1.2: According to the actual time in port of the ship and the number of quay cranes C i assigned to it, calculate the in-port operation cost Cost oper of the ship; Step 1.3: If the actual berthing time of the ship is its actual arrival time , it means that the ship arrives late, go to Step 1.4; otherwise, go to Step 1.5; Step 1.4: According to the late arrival time of the ship, calculate the late arrival cost Cost arrd of the ship, and go to Step 1.6; Step1.5: According to the waiting time of the ship, calculate the waiting cost Cost wait of the ship; Step 1.6: If the actual berthing quay of the ship is the same as its pre-assigned quay, go to Step 7, otherwise go to Step 8; Step 1.7: According to the actual berthing position x i of the ship and its ideal berthing position bpi , calculate the ship's position deviation cost Cost devi ; Step 1.8: Calculate the transshipment cost of the container to be loaded on the ship tran ;

[0062] 2.3 Population crossover operation: For any chromosome P, its length is 7×N V , the mutated chromosome is recorded as C. For any ship i, take a random number r∈[0,1], if r≤p m , then let C[i], C[i+N V ], ..., C[i+6×N V ] is a random number in its feasible domain, otherwise, it remains unchanged. m is the mutation probability. Since the solution obtained by the algorithm gradually converges with the increase of the number of iterations, the difference in individual fitness in the population will decrease in the later stage of the algorithm, and it may fall into a local optimal solution. Therefore, this paper adopts an adaptive strategy to make the mutation probability unchanged with the optimal solution. uc , thus exploring more new areas in the solution space and increasing the possibility of jumping out of the local optimum;

[0063] 2.4 Gene repair operation: Step 2.1: For all terminals m∈T, the set of ships that have berthed at the beginning of the planning period V 0 Get the set of scheduled ships at terminal m The set of ships to be dispatched at terminal m is obtained by sorting the set of ships to be dispatched V according to the berthing time from small to large. Step 2.2: The first element in the current dispatched ship i is the berthing time y i Collection with Pier Water Depth Get the reliable parking time period set H useful ; Step 2.3: If H useful is an empty set, let i dock at other terminals and go to Step 2.1, otherwise go to Step 2.4; Step 2.4: If there is no h∈H useful , so that the berthing time y i Not less than the starting point of the time period And the departure time is d i Not greater than the end of the time period Then go to Step 2.5, otherwise, go to Step 2.6; Step 2.5: If there is no h∈H useful , making the reliable parking time length If it is not less than the minimum operation time of the ship, let i berth at other ports and go to Step 1, otherwise, go to Step 6; Step 2.6: Assume that the number of ships in port during scheduling is For a ship If its departure time d j is greater than the berthing time y i , then add j to the set V now ; Step 2.7: According to the berthing positions x now of all ships k ∈ V that are already in the port during scheduling k , the ship's hull length , the number of quay cranes C k assigned, and the starting quay crane number obtain the set Q of idle shoreline intervals during scheduling idle ; Step 2.8: Let the set of available shoreline intervals during scheduling If there exists r ∈ Q idle such that the shoreline length l r is greater than or equal to and the number of available quay cranes NC r is greater than or equal to then add r to the set Q useful ; Step 2.9: If Q useful is an empty set, let y i be the minimum departure time of all ships in V now , and go to Step 2.1; Step 2.10: If there exists s ∈ Q useful such that and where is the starting position of the available shoreline interval s, is the ending position of s, then go to Step 11, otherwise, go to Step 2.14; Step 2.11: For all available quay cranes within the berthing shoreline interval s If its service range does not overlap with the ship's berthing shoreline interval , then let NC s = NC s - 1; Step 2.12: If the number of quay cranes within the interval then remove s from Q useful and go to Step 2.9; Step 2.13: If the starting quay crane number assigned to i and the last quay crane number assigned to i go to Step 16, otherwise go to Step 2.15; Step 2.14: Arbitrarily take s ∈ Q useful , and let x i be a random integer within ; Step 2.15: Let the number of quay cranes C i assigned be a random integer within , and let be a random integer within; Step 2.16: Remove i from and add it to ; Step 2.17: If all terminals are empty sets, then gene repair is completed, otherwise go to Step 2.6;

[0064] 2.5 Local search based on simulated annealing mechanism: Step 3.1: Initialize the current loop count h = 1, system temperature T = T 0 ; Step 3.2: If the number of generations g uc with the optimal solution unchanged is greater than or equal to 10, and h is less than the maximum loop count H max , then go to the next step; otherwise, exit the simulated annealing process; Step 3.3: Search in the neighborhood Ω max of the current population's optimal solution P P to obtain a new solution P new ; Step 3.4: Perform gene repair operation on P new to make it a feasible solution, and calculate its fitness value f new ; Step 3.5: Determine whether the current fitness value f new is greater than the optimal fitness value f max , if so, then go to the next step, otherwise go to Step 3.7; Step 3.6: Let P max = P new , f max = f new , g uc = 1, and exit the simulated annealing process; Step 7: Calculate the acceptance probability and generate a random number r ∈ [0, 1], determine whether r < p accept , if so, then go to the next step, otherwise go to Step 3.9; Step 3.8: Let the worst solution P min of the current population = P new , f min = f new ; Step 3.9: Let T = ηT, the current loop count h = h + 1, and go to Step 3.2.

[0065] The present invention combines a greedy construction strategy, a genetic algorithm, and a simulated annealing algorithm into a new hybrid algorithm, which has the advantages of the three algorithms; at the same time, the impacts of multi-terminal coordinated scheduling, tidal factor influence, quay crane service range, and other aspects on berth and quay crane scheduling are discussed.

[0066] In summary, the beneficial effects of the present invention are as follows: 1. The joint scheduling operation process of berths and quay cranes in the context of multiple terminals is considered, and the impacts of multiple aspects such as multi-terminal coordinated scheduling, tidal factor influence, and quay crane service scope on berth and quay crane scheduling are discussed; 2. Through an active strategy, buffer variables are added to the ship arrival time and quay crane operation efficiency, and an objective function is designed based on the simulation method to minimize the expected value and variance sum of the sum of ship operation cost, arrival delay penalty cost, ship waiting cost, ship departure delay cost, deviation cost, and transfer cost under randomly generated samples, thereby constructing a non-linear mixed integer programming model; the present invention improves the operation efficiency of container terminals, reasonably allocates berths and quay cranes, and reduces the terminal operation cost. Description of the Drawings

[0067] Figure 1 is the flowchart of the multi-terminal berth and quay crane joint scheduling algorithm of the present invention;

[0068] Figures 2(a) to 2(c) is the scheduling scheme diagram obtained under the S+U mode of the present invention, where Fig. 2(a) is the final scheduling scheme of the No. 1 terminal of this container port obtained by the algorithm under the S+U scheduling mode of this example, Fig. 2(b) is the final scheduling scheme of the No. 2 terminal of this container port obtained by the algorithm under the S+U scheduling mode of this example, and Fig. 2(c) is the final scheduling scheme of the No. 3 terminal of this container port obtained by the algorithm under the S+U scheduling mode of this example;

[0069] Figures 3(a) to 3(c) is the scheduling scheme diagram obtained under the M+C mode of the present invention, where Fig. 3(a) is the final scheduling scheme of the No. 1 terminal of this container port obtained by the algorithm under the M+C scheduling mode of this example, Fig. 3(b) is the final scheduling scheme of the No. 2 terminal of this container port obtained by the algorithm under the M+C scheduling mode of this example, and Fig. 3(c) is the final scheduling scheme of the No. 3 terminal of this container port obtained by the algorithm under the M+C scheduling mode of this example;

[0070] Figures 4(a) to 4(c) is the scheduling scheme diagram obtained under the M+U mode of the present invention, where Fig. 4(a) is the final scheduling scheme of the No. 1 terminal obtained by the algorithm under the M+U scheduling mode of this example, Fig. 4(b) is the final scheduling scheme of the No. 2 terminal obtained by the algorithm under the M+U scheduling mode of this example, and Fig. 4(c) is the final scheduling scheme of the No. 3 terminal obtained by the algorithm under the M+U scheduling mode of this example;

[0071] Figures 5(a) to 5(c)This is a comparison graph of the convergence curves of the SA-AGA algorithm of the present invention and other algorithms. Among them, Fig. 5(a) is the curve of the objective function value varying with the number of iterations obtained by the SA-AGA algorithm under randomly generated cases when the number of ships to be scheduled is 20, 30, and 40. Fig. 5(b) is the curve of the objective function value varying with the number of iterations obtained by the NG-AGA algorithm under randomly generated cases when the number of ships to be scheduled is 20, 30, and 40. Fig. 5(c) is the curve of the objective function value varying with the number of iterations obtained by the NSA-AGA algorithm and the AGA algorithm under randomly generated cases when the number of ships to be scheduled is 20, 30, and 40. Specific implementation method

[0072] The technical solution of the present invention will be further described below in conjunction with the accompanying drawings of the specification and specific embodiments;

[0073] A berth quay crane joint scheduling method for multi-terminal tidal ports under uncertain environments, the method comprising:

[0074] Step 1: Construct a berth quay crane joint scheduling model under uncertain conditions;

[0075] 1.1. Determine the objective function:

[0076] Since the present invention takes into account the uncertainty of the ship arrival time and the quay crane operation efficiency, through an active strategy, buffer time and buffer efficiency are respectively added to the expected arrival time of the ship and the average quay crane operation efficiency to reduce the influence of the fluctuations of these two on the scheduling scheme; and in order to minimize this influence, a certain number of samples are randomly generated to simulate the possible ship arrival time and quay crane operation efficiency, and the objective function is to minimize the sum of the expectation and variance of the total cost of the scheduling scheme under each sample as shown in Equation (1);

[0077] min E({f s})+σ({f s}) (1)

[0078] In the formula, f s is the total cost of the scheduling scheme under sample s, E({f s}) represents the average value of the total cost of the scheduling scheme under each sample, σ({f s}) is the variance, and the smaller E({f s}) is, the lower the cost required when the scheduling scheme is actually applied. And σ({f sThe smaller it is, the stronger the anti-interference ability of the scheduling plan in response to changes in the ship's arrival time and quay crane operation efficiency. The total cost of the model in the present invention mainly includes six aspects: ship operation cost, ship arrival delay penalty cost, ship departure delay penalty cost, ship waiting cost, berthing position deviation cost, and transfer cost.

[0079] 1.2 Set constraint conditions;

[0080] The operation cost of a ship refers to the operation cost required for the quay crane to perform container loading and unloading services for it during the period from the ship's berthing to its departure. It is related to the ship's berthing time, departure time, and the number of quay cranes assigned to it, as shown in Equation (2); where c y is the unit quay crane unit time operation cost when the ship is operating at the port; C i is the number of quay cranes operating for ship i, i ∈ V; y i is the berthing time of the ship i to be scheduled, i ∈ V; is the actual arrival time of ship i under sample s;

[0081]

[0082] When the ship's arrival time exceeds its planned berthing time, it will affect the docking and operation of other ships waiting to berth, and the containers to be loaded and unloaded by it need to be detained additionally. At this time, arrival delay costs will be generated. Its size is related to the quantity of containers to be loaded and unloaded by the ship and the arrival delay time, as shown in Equation (3); where p after is the unit time penalty cost when the ship arrives after its expected arrival time; is the quantity of export containers of ship i to be scheduled, i ∈ V; is the quantity of import containers of ship i to be scheduled, i ∈ V;

[0083]

[0084] Whether the ship can depart normally is very important for both the ship and the terminal. If it cannot depart normally, for the ship, it may not be able to reach the next port on time, and for the terminal, it will affect the berthing and operation of subsequent ships. At this time, departure delay penalty costs are generated, which are related to the quantity of containers of the ship and the time of delayed departure, as shown in Equation (4); where p i is the unit time penalty cost when the actual departure time of ship i exceeds its expected departure time; d i is the actual departure time of ship i; ED i is the expected departure time of ship i to be scheduled, i ∈ V;

[0085]

[0086] If the arrival time of a ship is earlier than its planned berthing time, the ship can only wait in the anchorage. This situation will reduce the satisfaction of the ship and thus affect the competitiveness of the port. Therefore, to reduce this impact, the ship waiting cost is added, which is related to the quantity of containers to be unloaded on the ship and the time of arriving earlier than scheduled, as shown in Equation (5); where p before is the unit time waiting cost when the ship arrives and waits before its expected arrival time; y i is the berthing time of the ship i to be scheduled, i ∈ V;

[0087]

[0088] Considering the storage location of the containers to be loaded on the ship in the yard, each ship will have its own ideal berthing position. Therefore, when the ship berths at its pre-allocated terminal, if there is a deviation between its berthing position and the ideal berthing position, it will cause an increase in the horizontal transportation distance between the yard and the ship berthing position, thus increasing the transportation cost. Therefore, the deviation cost needs to be added, as shown in Equation (6), which is related to the quantity of containers to be loaded and unloaded on the ship and the deviation distance; where c x is the unit deviation cost when the berthing position of the ship at its pre-allocated terminal deviates from the ideal berthing position; x i is the berthing position of the ship i to be scheduled, i ∈ V; bp i is the best berthing position of the ship i to be scheduled at its pre-allocated terminal, i ∈ V; z im is when the pre-allocated terminal of the ship i is m, z im = 1, otherwise z im = 0, i ∈ V, m ∈ T; u im is when the berthing terminal of the ship i is m, u im = 1, otherwise u im = 0, i ∈ V, m ∈ T;

[0089]

[0090] When the ship does not berth at its pre-allocated terminal, this deviation cost does not need to be considered. However, since the containers to be loaded on the ship need to be transferred from the pre-allocated terminal to the actual berthing terminal, this generates additional transfer costs, as shown in Equation (7), which is related to the quantity of containers to be loaded; where c km is the unit quantity transfer cost from terminal k to terminal m, k, m ∈ T;

[0091]

[0092] Therefore, under any sample s, the cost required for the scheduling plan is as shown in Equation (8);

[0093] f s = Costoper +Cost arrd +Cost depd +Cost wait +Cost devi +Cost tran (8)

[0094] The existence of the objective function can determine what kind of scheduling scheme is the most ideal. However, to obtain a feasible ship berthing plan and quay crane allocation plan, many constraints need to be satisfied;

[0095]

[0096]

[0097] When determining the berthing terminal of a ship, it is first necessary to ensure that all ships to be scheduled have selected a terminal for berthing, and each ship can only berth at one terminal, as shown in Constraint (9); at the same time, as shown in Constraint (10), during the entire in-port time of the ship, the water depth of the terminal where it berths cannot be lower than its draft requirement; where df i is the draft of the ship i to be scheduled, i ∈ V; M is a sufficiently large positive number; is the water depth of terminal m at time t, m ∈ T, t ∈ H;

[0098]

[0099]

[0100]

[0101]

[0102] Due to the limitation of the quay line length, Constraint (11) is introduced to ensure that when the ship berths, the section of the quay line from the bow to the stern of the ship is within the quay line range of its berthing terminal; in addition, to prevent spatio-temporal conflicts when the ship berths, that is, occupying the same section of the quay line at the same time, Constraints (12) to (14) must also be satisfied; where Constraint (12) is the spatio-temporal constraint between ships to be scheduled, indicating that when two ships berth on the same quay line at the same time, the position of the stern of the left ship must be to the left of the position of the bow of the right ship; similarly, Constraints (13) and (14) respectively represent the spatio-temporal constraints when the ship to be scheduled is on the right and left sides of the berthed ship; where, x i is the berthing position of the ship i to be scheduled, i ∈ V; is the length of the ship i to be scheduled, including the horizontal safety reserve length, i ∈ V; is the length of the berthed ship i, including the horizontal safety reserve length, i ∈ V0 ; indicates that if ship j berths on the right side of ship i, otherwise i, j ∈ V, i ≠ j; indicates that if ship j berths on the right side of ship i, otherwise i ∈ V 0 , j ∈ V, i ≠ j; indicates that if ship j berths on the left side of ship i, otherwise i ∈ V 0 , j ∈ V, i ≠ j;

[0103]

[0104]

[0105]

[0106]

[0107] In addition to the constraints that the berthing of ships at the terminal and their berthing positions need to satisfy, there are also some constraints on the berthing time that must be considered; first of all, the berthing time of a ship must be later than its arrival time at the port. At the same time, due to the uncertainty of the ship's arrival time, it is stipulated that when scheduling, the ship must pass through a buffer time after its expected arrival time before it can berth, as shown in constraint (15); the departure time of the ship is jointly determined by its berthing time, the quantity of containers to be loaded and unloaded, the operation efficiency of the quay crane, and the number of quay cranes allocated. Constraint (16) defines the relationship between them; similarly, the operation efficiency of the quay crane considered during scheduling is the sum of its average operation efficiency and buffer efficiency, thereby reducing the impact brought by the change in the operation efficiency of the right quay crane; Constraint (17) defines the relationship between the berthing time and departure time of any two ships to be scheduled when they need to occupy the same section of the shoreline, that is, the departure time of the ship that first occupies this section of the shoreline must be earlier than the berthing time of the ship that occupies it later; similarly, Constraint (18) defines the relationship between the berthing time of the ship to be scheduled and the departure time of the ship that has already berthed; among them, y i is the berthing moment of ship i to be scheduled, i ∈ V; is the expected arrival moment of ship i to be scheduled, i ∈ V; is the slack of the arrival time of ship i, i ∈ V; g is the interference coefficient between quay cranes; indicates that if ship j berths after ship i departs, otherwise i, j ∈ V, i ≠ j; indicates that if ship j berths after ship i departs, otherwise j ∈ V, i ∈ V 0 ;

[0108]

[0109]

[0110] When any two ships to be scheduled are berthed at the same terminal, there must be a relationship between them in either the spatial dimension or the temporal dimension; through interaction with Constraint (12), Constraint (19) defines their connection in the spatial dimension, and interaction with Constraint (17) defines the relationship in the temporal dimension; similarly, through interaction with Constraints (13), (14), and (18), Constraint (20) realizes the definition of the relationship between the ships to be scheduled and the berthed ships in both space and time; the existence of these constraints can prevent the situation where two ships occupy the same section of the shoreline simultaneously; among them, if the berthing terminal of the berthed ship i is m, otherwise i ∈ V 0 , m ∈ T;

[0111]

[0112]

[0113]

[0114]

[0115]

[0116]

[0117]

[0118]

[0119]

[0120]

[0121] If the relationship between the ship operation time and the quay crane operation efficiency in constraint (16) is not considered, that is, the ship operation time is regarded as a known constant, then equations (1) to constraint (20) together form an optimization model for the berth allocation problem under uncertain environment; however, in the actual production operation process, the ship operation is jointly completed by quay cranes, so its required time is also closely related to quay cranes; and there are also many constraints that need to be satisfied simultaneously when allocating quay cranes, as shown in constraints (21) to (30); specifically, constraint (21) defines the range of the number of quay cranes that can be allocated to each ship, that is, it must be between the minimum number of quay cranes that can be allocated to the ship and the maximum number of quay cranes that can be allocated to the ship; constraint (22) defines the relationship between the number of allocated quay cranes and the quay crane numbers; constraints (23) and (24) avoid the situation where one quay crane serves two ships at the same time; constraint (25) makes the quay cranes serving the same ship continuous, that is, when two quay cranes a and c serve the same ship at the same time, all quay cranes in the middle of them must also serve it; as shown in constraints (26) to (28), when there are more than two ships on a quay line at the same time, quay cranes cannot cross each other to provide services, that is, when quay crane a serves a ship, the quay crane on its right cannot serve other ships on the left, and the quay crane on its left cannot serve other ships on the right; the existence of constraints (29) and (30) ensures that all quay cranes allocated to the ship can provide services, that is, the service range of the allocated quay cranes can cover the berthing quay line interval of the ship; among them, is the minimum number of quay cranes allocated to the ship i to be scheduled, i ∈ V; is the maximum number of quay cranes allocated to the ship i to be scheduled, i ∈ V; indicates that if the quay crane q at the terminal m serves the berthed ship i, otherwise i ∈ V 0 , q ∈ Q m , m ∈ T; θ iqm indicates that if the quay crane q at the terminal m serves the ship i, θ iqm = 1, otherwise θ iqm = 0, i ∈ V, q ∈ Q m , m ∈ T; is the maximum service position of the quay crane q at the terminal m, m ∈ T, q ∈ Q m ; is the minimum service position of the quay crane q at the terminal m, m ∈ T, q ∈ Q m ;

[0122] Step 2: Design the terminal data;

[0123] Suppose the operators of each terminal reach an agreement to share resources such as berths and quay cranes with each other. Among them, the quay length of Terminal 1 is 1000m, and it has 11 quay cranes with an average working efficiency of 15 TEU / h; the quay length of Terminal 2 is 1140m, and it has 12 quay cranes with an average working efficiency of 15 TEU / h; the quay length of Terminal 3 is 1200m, and it has 13 quay cranes with an average working efficiency of 15 TEU / h. Other data such as the cost coefficient and quay crane interference coefficient of the terminals are shown in Table 1.

[0124] Table 1 Data of Three Terminals in a Container Port

[0125]

[0126]

[0127] Table 2 Water Depths of Each Terminal in the Container Port after Being Affected by Tides

[0128]

[0129] Table 2 shows the water depths of each terminal in the port at each moment after being affected by tides during the planned period. Considering the possibility of ships delaying departure, the time of the water depth of the terminal is longer than the length of the planned period. When formulating a berthing plan for a ship, it is necessary to ensure that the water depth of the terminal where the ship berths during the entire period from entering the port to leaving the port is not less than the draft requirement of the ship. Otherwise, the ship may not be able to enter or leave the port according to the berthing plan, and it will also affect the berthing operations of other ships, thus invalidating the entire berthing plan.

[0130] Table 3 Service Scopes of Quay Cranes at Each Terminal

[0131]

[0132] Since all quay cranes at the same terminal move on the same track and the quay cranes need to occupy a certain quay space, any quay crane has a quay line area that it cannot serve. Table 3 shows the service scopes of quay cranes within the quay lines of each terminal in the port. The service scope of each quay crane is related to the number of quay cranes on its left and right sides. Taking Quay Crane No. 4 at Terminal 1 as an example, there are 3 quay cranes on its left, so the starting point of its service scope is 150 (50×3) m, and there are 7 quay cranes on its right, and the quay length of the terminal is 1100m, so the ending point of the service scope of this quay crane is 750 (1100 - 50×7) m. The same applies to other quay cranes.

[0133] Step 3: Design ship data;

[0134] The situation where all terminals are idle at the start of scheduling is rather special. More commonly, scheduling is carried out when some shorelines and quay cranes at each terminal are already occupied by ships. Therefore, this paper assumes that at the start of the planning period, there are already 2 or 3 ships carrying out loading and unloading operations at each terminal shoreline. So the ship data is divided into berthed ship data and ships to be scheduled data, and the specific values are shown in Table 4 and Table 5.

[0135] Table 4 Berthed Ship Data

[0136]

[0137]

[0138] Table 5 Ships to be Scheduled Data

[0139]

[0140] In the table, U represents following a uniform distribution, and Z represents an integer. The ship length is randomly generated within 120 - 360m, the expected arrival time of the ship is randomly generated within 0 - 48h, the terminal where the ship berths is randomly generated within 1 - 3, the berthing position of the ship is any position within its berthing terminal (not exceeding the boundary), the time in port of the ship is 1.1 - 1.5 times the longest time required for operation, the quantity of containers to be loaded on the ship is randomly generated within 100 - 300, and the quantity of containers to be unloaded is randomly generated between 0.7 - 1.3 times the quantity of containers to be loaded. The draft of the ship is related to the ship length, as shown in Equation (31).

[0141]

[0142] Step 4: Design a multi - terminal berth and quay crane joint scheduling algorithm;

[0143] 4.1 Initialize the population: Half of the chromosomes in the initial population are generated by the greedy construction strategy. Let the terminal where the ship berths be its pre - assigned terminal, let the berthing position of the ship be its optimal berthing position bp i , let the berthing time of the ship be Let the number of quay cranes assigned to the ship be a random integer within, let the starting quay crane number be 1, and let the slack of the ship's arrival time and the slack of the quay crane operation efficiency be and The other half of the chromosomes are generated by the random generation strategy. Let the terminal where the ship berths be a random integer within [1, N T , let the berthing position of the ship be a random integer within, let the berthing time of the ship be Let the number of quay cranes assigned to the ship be randomly generated as an integer within ; let the starting quay crane number be Random integers within, and let the slack of the ship arrival time and the slack of the quay crane operation efficiency be and

[0144] 4.2 Calculate the fitness function: Step 1.1: For all samples s ∈ S, take the actual berthing time of the ship as the actual arrival time of the ship and the larger value of the planned berthing time y i and, according to and the actual quay crane operation efficiency of the ship obtain the actual departure time of the ship Step 1.2: According to the actual time in port of the ship and the number of quay cranes C allocated i , calculate the in-port operation cost Cost of the ship oper ; Step 1.3: If the actual berthing time of the ship is its actual arrival time it means the ship arrives late, go to Step 1.4; otherwise, go to Step 1.5; Step 1.4: According to the late arrival time of the ship calculate the late arrival cost Cost of the ship arrd , and go to Step 1.6; Step1.5: According to the waiting time of the ship calculate the waiting cost Cost of the ship wait ; Step 1.6: If the actual berthing quay of the ship is the same as its pre-allocated quay, go to Step 7, otherwise go to Step 8; Step 1.7: According to the actual berthing position x of the ship i and its ideal berthing position bp i , calculate the position deviation cost Cost of the ship devi ; Step 1.8: Calculate the transfer cost Cost of the containers to be loaded on the ship tran ;

[0145] 4.3 Population crossover operation: For any chromosome P, its length is 7×N V , and the mutated chromosome is denoted as C. For any ship i, take a random number r ∈ [0, 1]. If r ≤ p m , then let C[i], C[i + N V , …, C[i + 6×N V be random numbers within its feasible region, otherwise, remain unchanged. Where p mis the mutation probability. Since the solution obtained by the algorithm gradually converges as the number of iterations increases, in the later stage of the algorithm, the difference in the fitness of individuals in the population will decrease, and it may fall into a local optimal solution. Therefore, this paper adopts an adaptive strategy to make the mutation probability gradually increase with the number of generations g when the optimal solution remains unchanged uc to explore more new regions in the solution space and increase the possibility of jumping out of the local optimum;

[0146] 4.4 Gene repair operation: Step 2.1: For all docks m ∈ T, from the set of ships that have berthed at the beginning of the planning period V 0 obtain the set of ships that have been scheduled at dock m From the set of ships to be scheduled V, obtain the set of ships to be scheduled at dock m in ascending order of berthing time Step 2.2: For the first element in, that is, the current scheduled ship i, from the berthing time y i and the dock water depth set obtain the reliable berthing time period set H useful ; Step 2.3: If H useful is an empty set, let i berth at other docks, go to Step 2.1, otherwise go to Step 2.4; Step 2.4: If there does not exist h ∈ H useful such that the berthing time y i is not less than the starting point of this time period and the departure time d i is not greater than the end point of this time period then go to Step 2.5, otherwise, go to Step2.6; Step 2.5: If there does not exist h ∈ H useful such that the reliable berthing time length is not less than the minimum operation time of the ship, then let i berth at other docks and go to Step 1, otherwise, go to Step 6; Step 2.6: Let the set of ships in the port during scheduling For the ship If its departure time d j is greater than the berthing time y i , then add j to the set V now ; Step2.7: According to the berthing positions x now of all ships k ∈ V that are already in the port during scheduling k , the ship body length , the number of quay cranes C k allocated and the starting quay crane number obtain the set of idle shoreline intervals Q during scheduling idle ; Step 2.8: Let the set of available shoreline intervals during scheduling If there exists r ∈ Q idle, make the shoreline length l r greater than or equal to and the number of available quay cranes NC r greater than or equal to then add r to the set Q useful ; Step 2.9: If Q useful is an empty set, let y i be the minimum departure time of all ships in V now and go to Step 2.1; Step 2.10: If there exists s ∈ Q useful such that and where is the starting position of the available shoreline interval s, is the ending position of s, then go to Step 11, otherwise, go to Step 2.14; Step 2.11: For all available quay cranes within the berthing shoreline interval s If its service range does not overlap with the ship's berthing shoreline interval then let NC s = NC s - 1; Step 2.12: If the number of quay cranes within the interval then remove s from Q useful and go to Step 2.9; Step 2.13: If the starting quay crane number assigned to i and the last quay crane number assigned to i go to Step 16, otherwise go to Step 2.15; Step 2.14: Arbitrarily take s ∈ Q useful , let x i be a random integer within; Step 2.15: Let the number of quay cranes assigned C i be a random integer within, let be a random integer within; Step 2.16: Remove i from and add it to ; Step 2.17: If for all docks are all empty sets, then gene repair is completed, otherwise go to Step 2.6;

[0147] 4.5 Local search based on simulated annealing mechanism: Step 3.1: Initialize the current loop count h = 1, system temperature T = T 0 ; Step 3.2: If the number of generations g uc when the optimal solution remains unchanged is greater than or equal to 10, and h is less than the maximum loop count H max, then proceed to the next step; otherwise, exit the simulated annealing process; Step 3.3: Search in the neighborhood Ω max of the current population's optimal solution P P to obtain a new solution P new ; Step 3.4: Perform gene repair operation on P new to make it a feasible solution, and calculate its fitness value f new ; Step 3.5: Determine whether the current fitness value f new is greater than the optimal fitness value f max . If so, proceed to the next step; otherwise, go to Step 3.7; Step 3.6: Let P max = P new , f max = f new , g uc = 1, and exit the simulated annealing process; Step 7: Calculate the acceptance probability and generate a random number r ∈ [0, 1]. Determine whether r < p accept . If so, proceed to the next step; otherwise, go to Step 3.9; Step 3.8: Let the worst solution P min of the current population be P new , f min = f new ; Step 3.9: Let T = ηT, the current loop generation h = h + 1, and go back to Step 3.2;

[0148] Step Five: Generate test cases;

[0149] Randomly generate a test case with 40 ships to be scheduled. The data of the arriving ships is shown in Table 5. At the same time, according to the characteristics of the problem and the test results in the initial stage of data experiments, set the following parameters: population size popsize = 100, maximum genetic generation G max = 500, initial mutation probability In the simulated annealing mechanism, the initial temperature T 0 = 100, temperature cooling rate η = 0.8, the maximum number of loops H max of the simulated annealing mechanism = 100.

[0150] Taking the randomly generated example as the input of the algorithm, the scheduling plans under the three scheduling modes of S+U, M+C, and M+U can be obtained as shown in Figures 2 to 4 respectively. In the figures, the abscissa represents the quay positions of each terminal, and the ordinate represents the time after the start of the planning period. Each rectangle represents a ship, the number in the middle of the rectangle is the ship number, the left number represents the number of the first quay crane providing services for the ship, and the right number represents the number of the last quay crane providing services for the ship. The rectangles without numbers in the middle represent the ships that were already in port when the scheduling started.

[0151] Figures 2(a), 2(b), and 2(c) are respectively the final scheduling plans of the 1st to 3rd terminals of this container port obtained by the algorithm under the S+U scheduling mode. Taking Terminal 1 as an example, at the start of the scheduling, there were already two ships docked on the quay, and quay cranes No. 2 to 4 and No. 6 to 8 were respectively providing services for them; then Ship 18 berthed at the 464m position of this quay at t = 8, and quay cranes No. 6 to 10 were assigned to this ship and left port at t = 13.6; Ship 9 berthed at the 204m position of this quay at t = 15, and quay cranes No. 4 to 6 were assigned to this ship and left port at t = 19.4; and so on. The berthing time, berthing position, assigned quay crane number, and departure time and other berthing information of all ships can be obtained. It can be found that the water depth conditions of the quay during the time when all ships are in port meet their draft requirements, and the service ranges of the assigned quay cranes can cover the berthing intervals of the ships. In addition, the final berthing terminals of all ships are also their pre-assigned terminals. Therefore, this scheduling plan is feasible, and the solution obtained by this algorithm is a feasible solution.

[0152] Figures 3(a), 3(b), and 3(c) are respectively the final scheduling plans of the 1st to 3rd terminals of this container port obtained by the algorithm under the M+C scheduling mode. Still taking Terminal 1 as an example, under this scheduling mode, Ship 31 berthed first, at the 926m position of this quay at t = 4, and quay cranes No. 9 to 11 were assigned to this ship and left port at t = 12.9; Ship 18 berthed at the 498m position of this quay at t = 8, and quay cranes No. 5 to 8 were assigned to this ship and left port at t = 13.6; and so on. Similarly, the berthing time, berthing position, assigned quay crane number, and departure time and other berthing information of all ships can be obtained. The water depth conditions of the quay during the time when all ships are in port meet their draft requirements, and at the same time, the service ranges of the assigned quay cranes can cover the berthing intervals of the ships. Therefore, the algorithm can obtain a feasible solution under this scheduling mode.

[0153] Figures 4(a), 4(b), and 4(c) are respectively the final scheduling plans of the 1st to 3rd berths obtained by the algorithm under the M+U scheduling mode for this example. According to this scheduling plan, the berthing information of all ships, such as berthing time, berthing location, assigned quay crane number, and departure time, can also be obtained, and all of them meet the constraints. Therefore, the solution obtained by the algorithm under the M+U scheduling mode is also a feasible solution. In addition, as can be seen from Figure 3 and Figure 4, under the M+U and M+C scheduling modes, not all ships berth at their pre-assigned berths in the scheduling plan, but can freely choose their berthing berths on the premise of meeting the constraints.

[0154] Step 6: Verify the effectiveness of the algorithm;

[0155] Compared with the traditional genetic algorithm, the SA-AGA algorithm proposed in this paper combines the greedy construction strategy into the generation process of the initial population and combines the simulated annealing mechanism into the genetic algorithm. Therefore, to study the performance of the SA-AGA algorithm, an instance is randomly generated under the scales of the number of arriving ships being 20, 30, and 40 respectively, and it is used as the input and substituted into the SA-AGA algorithm, the algorithm without combining the greedy construction strategy (NG-AGA), the algorithm without combining the simulated annealing mechanism (NSA-AGA), and the traditional adaptive genetic algorithm (AGA) without adopting both of them for experiments, and the results are compared and analyzed.

[0156] Figures 5(a), 5(b), and 5(c) are respectively the curves of the objective function values varying with the number of iterations obtained by the SA-AGA algorithm, the NG-AGA algorithm, the NSA-AGA algorithm, and the AGA algorithm for the randomly generated cases when the number of ships to be scheduled is 20, 30, and 40. From the changes of the three curves of SA-AGA, NSA-AGA, and AGA, it can be seen that by combining the simulated annealing mechanism on the solution framework of the genetic algorithm, the algorithm can be effectively prevented from falling into local optimum, thereby reducing the objective function value of the final scheduling plan and improving the quality of the solution. And according to the changing trends of the two curves of SA-AGA and NG-AGA, it can be found that when initializing the population, compared with only using the random generation method, the method of combining the random generation strategy and the greedy construction strategy, although it cannot achieve obvious effects when the number of ships is 20, can make the objective function value of the final plan decrease faster and the obtained objective function value is also lower as the number of ships to be scheduled increases.

Claims

1. A joint scheduling method for berths and quay cranes in a multi-berth tidal port under uncertain environment, Characterized in that, It includes the following steps: Step 1: Construct a joint scheduling model for berths and quay cranes in a multi-berth tidal port under uncertain conditions; 1.1 Determine the objective function: Since the present invention takes into account the uncertainty of the ship's arrival time and the efficiency of the quay crane operation, the expected arrival time of the ship is Average operating efficiency of quay cranes Add buffer time and buffer efficiency To reduce the impact of the fluctuations of the two on the scheduling plan; and in order to minimize this impact, a certain number of samples are randomly generated to simulate the possible arrival time of ships and the efficiency of quay crane operations, and the objective function is to minimize the sum of the expectation and variance of the total cost of the scheduling plan under each sample, as shown in formula (1); min E({f s})+σ({f s}) (1) where f s is the total cost of the scheduling plan under the s sample, E({f s}) represents the average value of the total cost of the scheduling plan under each sample, and σ({f s}) is the variance. The smaller E({f s}) is, the lower the cost required when the scheduling plan is actually applied. The smaller σ({f s}) is, the stronger the anti-interference ability of the scheduling plan to changes in ship arrival time and quay crane operation efficiency. The total cost of the model of the present invention mainly includes six aspects: ship operation cost, ship arrival delay penalty cost, ship departure delay penalty cost, ship waiting cost, berthing position deviation cost, and transfer cost; 1.2 Set the constraint conditions; The operating cost of a ship refers to the operating cost required for the quay crane to provide container loading and unloading services for it during the period from the ship's berthing to its departure. It is related to the ship's berthing time, departure time, and the number of quay cranes allocated, as shown in Equation (2); where, c y is the operating cost per quay crane per unit time when the ship is operating in the port; C i is the number of quay cranes operating for ship i, i ∈ V; y i is the berthing time of the ship i to be scheduled, i ∈ V; is the actual arrival time of ship i under sample s; When the arrival time of a ship exceeds its planned berthing time, it will affect the berthing and operation of other ships waiting to berth, and the containers to be loaded and unloaded by it need to be detained additionally. At this time, the arrival delay cost will be generated; its magnitude is related to the quantity of containers to be loaded and unloaded by the ship and the arrival delay time, as shown in Equation (3); where, p after is the unit time penalty cost when the ship arrives after its expected arrival time; is the quantity of export containers of the ship to be scheduled, i ∈ V; is the quantity of import containers of the ship to be scheduled, i ∈ V; Whether the ship can depart normally is very important for both the ship and the terminal. If it cannot depart normally, for the ship, it may not be able to reach the next port on time, and for the terminal, it will affect the berthing and operation of subsequent ships. At this time, the departure delay penalty cost is generated, which is related to the container volume of the ship and the time of delayed departure, as shown in Equation (4); where, p i is the unit time penalty cost when the actual departure time of ship i exceeds its expected departure time; d i is the actual departure time of ship i; ED i is the expected departure time of the ship i to be scheduled, i ∈ V; If the arrival time of a ship is earlier than its planned berthing time, the ship can only wait in the anchorage. The occurrence of this situation will reduce the satisfaction of the ship and thus affect the competitiveness of the port. Therefore, to reduce this impact, the ship waiting cost is added, which is related to the quantity of containers to be unloaded on the ship and the time of arriving at the port in advance, as shown in Equation (5). Among them, p before is the unit time waiting cost when the ship arrives and waits before its expected arrival time; y i is the berthing time of the ship i to be scheduled, i ∈ V; Considering the storage location of the containers to be loaded onto the ship in the yard, each ship will have its own ideal berthing position; therefore, when the ship berths at its pre-assigned terminal, if there is a deviation between its berthing position and the ideal berthing position, it will cause an increase in the horizontal transportation distance between the yard and the ship's berthing position, thus increasing the transportation cost. Therefore, a deviation cost needs to be added, as shown in Equation (6), which is related to the quantity of containers to be loaded and unloaded on the ship and the deviation distance; where, c x is the unit deviation cost when the berthing position of the ship at its pre-assigned terminal deviates from the ideal berthing position; x i is the berthing position of the ship i to be scheduled, i ∈ V; bp i is the optimal berthing position of the ship i to be scheduled at its pre-assigned terminal, i ∈ V; z im is when the pre-assigned terminal of the ship i is m, z im = 1, otherwise z im = 0, i ∈ V, m ∈ T; u im is when the berthing terminal of the ship i is m, u im = 1, otherwise u im = 0, i ∈ V, m ∈ T; When the ship is not berthed at its pre-assigned terminal, this deviation cost does not need to be considered. However, since the containers to be loaded onto the ship need to be transferred from the pre-assigned terminal to the actual berthing terminal, this incurs additional transfer costs, as shown in Equation (7), which is related to the quantity of containers to be loaded onto the ship. Among them, c km is the transfer cost per unit quantity of containers from terminal k to terminal m, where k, m ∈ T; Therefore, under any sample s, the cost required for the scheduling plan is shown in Equation (8); f s = Cost oper + Cost arrd + Cost depd + Cost wait + Cost devi + Cost tran (8) The existence of the objective function can determine what kind of scheduling plan is the most ideal, but to obtain a feasible ship berthing plan and quay crane allocation plan, many constraints need to be satisfied; When determining the berthing terminal of a ship, it is first necessary to ensure that all ships to be scheduled have selected a terminal to berth at, and each ship can only berth at one terminal, as shown in Constraint (9); meanwhile, as shown in Constraint (10), throughout the ship's stay in the port, the water depth of the terminal it berths at cannot be lower than its draft requirement; where df i is the draft of the ship to be scheduled i, i ∈ V; M is a positive number; is the water depth of terminal m at time t, m ∈ T, t ∈ H; Due to the limitation of the quay length, constraint (11) is introduced to ensure that when a ship is berthed, the quay line from the bow to the stern of the ship is within the quay line range of the berthing quay; in addition, to prevent spatio-temporal conflicts when ships are berthed, that is, occupying the same quay line at the same time, constraints (12) to (14) must also be satisfied; among them, constraint (12) is the spatio-temporal constraint between ships to be scheduled, indicating that when two ships are berthed on the same quay line, the position of the stern of the ship on the left must be to the left of the position of the bow of the ship on the right; similarly, constraints (13) and (14) respectively represent the spatio-temporal constraints when the ship to be scheduled is on the right and left sides of the berthed ship; where, x i is the berthing position of ship i to be scheduled, i ∈ V; is the length of ship i to be scheduled, including the horizontal safety reserve length, i ∈ V; is the length of the berthed ship i, including the horizontal safety reserve length, i ∈ V 0 ; indicates that if ship j is berthed on the right side of ship i, otherwise indicates that if ship j is berthed on the right side of ship i, otherwise i ≠ j; indicates that if ship j is berthed on the left side of ship i, otherwise In addition to the constraints that the berthing of the ship at the terminal and the berthing position need to satisfy, there are also some constraints on the berthing time that must be considered; first of all, the berthing time of the ship must be later than its arrival time at the port. At the same time, due to the uncertainty of the ship's arrival time at the port, it is stipulated that when scheduling, the ship must pass through a buffer time after its expected arrival time before it can berth, as shown in constraint (15); the departure time of the ship is jointly determined by its berthing time, the quantity of containers to be loaded and unloaded, the operation efficiency of the quay crane, and the number of quay cranes allocated. Constraint (16) defines the relationship between them; similarly, the operation efficiency of the quay crane considered during scheduling is the sum of its average operation efficiency and buffer efficiency, thereby reducing the impact brought by the change in the operation efficiency of the right quay crane; Constraint (17) defines the relationship between the berthing time and departure time of any two ships to be scheduled when they need to occupy the same section of the shoreline, that is, the departure time of the ship that first occupies this section of the shoreline must be earlier than the berthing time of the ship that occupies it later; similarly, Constraint (18) defines the relationship between the berthing time of the ship to be scheduled and the departure time of the ship that has already berthed; among them, y i is the berthing time of the ship to be scheduled i, i ∈ V; is the expected arrival time of the ship to be scheduled i at the port, i ∈ V; is the slack of the arrival time of ship i, i ∈ V; g is the interference coefficient between quay cranes; means that if ship j berths after ship i departs, otherwise means that if ship j berths after ship i departs, otherwise When any two vessels to be scheduled are berthed at the same terminal, there must be a relationship between them in either the spatial dimension or the temporal dimension; through interaction with constraint (12), constraint (19) defines their connection in the spatial dimension, and interaction with constraint (17) defines the relationship in the temporal dimension; similarly, through interaction with constraints (13), (14) and (18), constraint (20) realizes the definition of the relationship in space and time between the vessels to be scheduled and the berthed vessels; the existence of these constraints can prevent the situation where two vessels occupy the same stretch of shoreline simultaneously; among them, If the berthing terminal of the berthed vessel i is m, Otherwise If the relationship between the ship operation time and the quay crane operation efficiency in constraint (16) is not considered, that is, the ship operation time is regarded as a known constant, then equations (1) to constraint (20) together constitute an optimization model for the berth allocation problem under uncertain environment; however, in the actual production operation process, the ship operation is jointly completed by quay cranes, so its required time is also closely related to quay cranes; and there are also many constraints that need to be satisfied simultaneously when allocating quay cranes, as shown in constraints (21) to (30); specifically, constraint (21) defines the range of the number of quay cranes that can be allocated to each ship, that is, it must be between the minimum number of quay cranes that can be allocated to the ship and the maximum number of quay cranes that can be allocated; constraint (22) defines the relationship between the number of allocated quay cranes and the quay crane numbers; constraints (23) and (24) avoid the situation where one quay crane serves two ships simultaneously; constraint (25) makes the quay cranes serving the same ship continuous, that is, when two quay cranes a and c serve the same ship simultaneously, all quay cranes in the middle of them must also serve it; as shown in constraints (26) to (28), when there are more than two ships on a quay line at the same time, quay cranes cannot cross each other to provide services, that is, when quay crane a serves a ship, the quay crane on its right cannot serve other ships on the left, and the quay crane on its left cannot serve other ships on the right; the existence of constraints (29) and (30) ensures that all quay cranes allocated to the ship can provide services, that is, the service range of the allocated quay cranes can cover the berthing quay line interval of the ship; among them, is the minimum number of quay cranes allocated to the ship i to be scheduled, i ∈ V; is the maximum number of quay cranes allocated to the ship i to be scheduled, i ∈ V; indicates that if quay crane q at terminal m serves the berthed ship i, otherwise i ∈ V 0 , q ∈ Q m , m ∈ T; θ iqm indicates that if quay crane q at terminal m serves ship i, θ iqm = 1, otherwise θ iqm = 0, i ∈ V, q ∈ Q m , m ∈ T; is the maximum service position of quay crane q at terminal m, m ∈ T, q ∈ Q m ; is the minimum service position of quay crane q at terminal m, m ∈ T, q ∈ Q m ; Step 2: Design a joint scheduling algorithm for multi-berth quay cranes 2.1 Initialize the population: Half of the chromosomes in the initial population are generated by the greedy construction strategy. Let the berthing quay of the ship be its pre-assigned quay, and let the berthing position of the ship be its optimal berthing position bp i , and let the ship's berthing time be Let the number of quay cranes assigned to the ship be a random integer within, let the starting quay crane number be 1, and let the slack of the ship's arrival time and the slack of the quay crane operation efficiency be and The other half of the chromosomes are generated by the random generation strategy. Let the berthing quay of the ship be a random integer within [1, N T , let the berthing position of the ship be a random integer within, let the berthing time of the ship be Let the number of quay cranes assigned to the ship be randomly generated as an integer within ; let the starting quay crane number be a random integer within, and let the slack of the ship's arrival time and the slack of the quay crane operation efficiency be and 2.2 Calculate the fitness function: Step 1.1: For all samples s ∈ S, take the actual berthing time of the ship as the actual arrival time of the ship and the larger value of the planned berthing time y i . Then, according to and the actual quay crane operation efficiency of the ship , obtain the actual departure time of the ship Step 1.2: According to the actual time in port of the ship and the number of quay cranes C allocated i , calculate the in-port operation cost Cost of the ship oper ; Step 1.3: If the actual berthing time of the ship is its actual arrival time , it means that the ship arrives at the port with a delay. Go to Step 1.4; otherwise, go to Step 1.5; Step 1.4: According to the delayed arrival time of the ship , calculate the delayed arrival cost Cost of the ship arrd , and go to Step 1.6; Step 1.5: According to the waiting time of the ship , calculate the waiting cost Cost of the ship wait ; Step 1.6: If the actual berthing dock T of the ship i V is the same as its pre-allocated dock, go to Step 1.7; otherwise, go to Step 1.8; Step 1.7: According to the actual berthing position x of the ship i and its ideal berthing position bp i , calculate the position deviation cost Cost of the ship devi ; Step 1.8: Calculate the transfer cost Cost of the containers to be loaded onto the ship tran ; 2.3 Population crossover operation: For any chromosome P with a length of 7×N V , the mutated chromosome is denoted as C; for any ship i, a random number r ∈ [0, 1] is taken. If r ≤ p m , then let C[i], C[i + N V , ···, C[i + 6×N V be random numbers within its feasible region, otherwise, keep them unchanged; where p m is the mutation probability. Since the solution obtained by the algorithm gradually converges as the number of iterations increases, in the later stage of the algorithm, the difference in individual fitness in the population will decrease, and it may fall into a certain local optimal solution. Therefore, this paper adopts an adaptive strategy to make the mutation probability gradually increase with the increase of the number of generations g uc without change in the optimal solution, so as to explore more new regions in the solution space and increase the possibility of jumping out of the local optimum; 2.4 Gene repair operation: Step 2.1: For all docks m ∈ T, from the set of vessels V that have berthed at the beginning of the planning period 0 Obtain the set of vessels that have been scheduled at dock m From the set of vessels to be scheduled V, obtain the set of vessels to be scheduled at dock m in ascending order of berthing time Step 2.2: For the first element within, that is, the currently scheduled vessel i, from the berthing time y i and the set of dock water depths Obtain the set of reliable berthing time periods H useful ; Step 2.3: If H useful is an empty set, let i berth at other docks and go to Step 2.1, otherwise go to Step 2.4; Step 2.4: If there does not exist h ∈ H useful such that the berthing time y i is not less than the start point of this time period and the departure time d i is not greater than the end point of this time period then go to Step 2.5, otherwise, go to Step 2.6; Step2.5: If there does not exist h ∈ H useful such that the reliable berthing time length is not less than the minimum operation time of the vessel, then let i berth at other docks and go to Step 2.1, otherwise, go to Step 2.6; Step 2.6: Let the set of vessels in the port during scheduling be For vessel If its departure time d j is greater than the berthing time y i , then add j to the set V now ; Step2.7: According to the berthing positions x now of all vessels k ∈ V that are already in the port during scheduling k , the ship length , the number of quay cranes C k assigned, and the starting quay crane number Obtain the set of idle shoreline intervals Q during scheduling idle ; Step 2.8: Let the set of available shoreline intervals during scheduling be If there exists r ∈ Q idle such that the shoreline length l r is greater than or equal to and the number of available quay cranes NC r is greater than or equal to then add r to the set Q useful ; Step 2.9: If Q useful is an empty set, let y i be V now The minimum departure time of all ships in, go to Step 2.1; Step 2.10: If there exists s ∈ Q useful , such that and where is the starting position of the available shoreline interval s, is the ending position of s, then go to Step 2.11, otherwise, go to Step 2.14; Step 2.11: For all available quay cranes within the berthing shoreline interval s If its service range and the ship's berthing shoreline interval [x i , x i +L i V do not overlap, then let NC s = NC s -1; Step 2.12: If the number of quay cranes within the interval then remove s from Q useful and go to Step 2.9; Step 2.13: If the starting quay crane number assigned to i and the last quay crane number assigned to i go to Step 2.16, otherwise go to Step 2.15; Step 2.14: Arbitrarily take s ∈ Q useful , let x i be a random integer within; Step 2.15: Let the number of quay cranes assigned C i be a random integer within, let be a random integer within; Step 2.16: Remove i from and add it to ; Step 2.17: If all docks are empty sets, then gene repair is completed, otherwise go to Step 2.6; 2.5 Local Search Based on Simulated Annealing Mechanism: Step 3.1: Initialize the current loop count h = 1 and the system temperature T = T 0 ; Step 3.2: If the number of generations g uc with unchanged optimal solution is greater than or equal to 10 and h is less than the maximum loop count H max , then proceed to the next step; otherwise, exit the simulated annealing process; Step 3.3: Search within the neighborhood Ω max of the current population's optimal solution P P to obtain a new solution P new ; Step 3.4: Perform gene repair operations on P new to make it a feasible solution and calculate its fitness value f new ; Step 3.5: Determine whether the current fitness value f new is greater than the optimal fitness value f max . If so, proceed to the next step; otherwise, go to Step 3.7; Step 3.6: Let P max = P new , f max = f new , g uc = 1, and exit the simulated annealing process; Step 3.7: Calculate the acceptance probability and generate a random number r ∈ [0, 1]. Determine whether r < p accept . If so, proceed to the next step; otherwise, go to Step 3.9; Step 3.8: Let the worst solution P min of the current population be P new , f min = f new ; Step 3.9: Let T = ηT, the current loop count h = h + 1, and go back to Step 3.2.

Images