| Citation: | ZHAO Xiang-ling, LI Yun-fei. Weight balance problem modeling and benders decomposition algorithm design of preighter[J]. Journal of Traffic and Transportation Engineering, 2023, 23(2): 199-211. doi: 10.19818/j.cnki.1671-1637.2023.02.014
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The relative weight of passenger transport containers is relatively small, which will not cause structural overloading such as local floors and beams, and there are fewer restrictions on load balance; In addition, passenger transportation generally assumes that the average quality of passengers remains unchanged, and considers selling seats as much as possible to obtain maximum revenue, which greatly reduces the difficulty of loading. Therefore, there is relatively little research on passenger loading itself, and the main research issues are the automation implementation and revenue control of loading. Zhang Hong[1]A load optimization system was developed using an improved ant colony algorithm for the domestic aircraft C919; Wong et al[2]We studied a multi product model with quality capacity constraints and price dependence, and found that airlines can increase profits by reducing baggage restrictions for aircraft passengers; Kang Shiyue[3]We studied the cabin control and revenue distribution issues of airline alliances, proposed a decentralized dynamic cabin control model for alliance routes based on ticket price allocation, and analyzed the revenue distribution problem using game theory; Qin Ying and others[4]A dynamic model has been established to describe the repeated competitive game of seat allocation, which can reserve seats for high priced tickets.
Air cargo loading is currently a hot topic in academic research due to the complexity of its issues and the diversity of actual transportation needs. According to the characteristics of the problem, it can be divided into early research on simplified load theory models and research on various practical constraints of heavy machinery after 2000.
There is no literature that provides a complete description of the loading issue of passenger to cargo aircraft. This article aims to describe, model, design algorithms, and solve the problem.
Due to the relatively simple problem of passenger plane loading, airline loading personnel can obtain better passenger position allocation plans based on personal experience. Research mainly focuses on the use of heuristic algorithms in the development of loading systems, such as Zhang Hong[1]The ant colony algorithm used. In addition, regarding the issue of cargo loading in the belly compartment of passenger planes, Meng Chao[16]Aiming at maximizing the comprehensive load utilization rate and volume utilization rate of the belly compartment of civil aviation wide body aircraft, heuristic algorithms and hybrid genetic particle swarm optimization algorithms are used for solving; aspen[17]A loading optimization mixed integer programming model with the minimum total cost of container loading and a mixed integer programming model with the maximum profit were established for the belly compartment of passenger aircraft.
Due to the NP hard nature of freight loading, research methods can be divided into heuristic algorithms and linear integer programming methods.
In terms of integer programming methods, Brosh[21]A linear integer programming model for cargo allocation in aircraft cargo holds is proposed to study the problem of finding the optimal layout of cargo in aircraft cargo holds, with the goal of achieving maximum payload capacity under given volume, mass, and center of gravity conditions; Mongeau et al[9]A linear integer programming model was proposed to achieve the maximum total load capacity and minimum center of gravity distance target value of the aircraft, and the linearization of multi cabin and spatial constraints was described in detail. The model was solved using commercial software, but the solution time was relatively long; Kaluzny et al[22]A mixed integer linear programming model was established for the cargo loading balance problem of military aircraft, with the goal of minimizing the center of gravity deviation and maximizing the value of transported cargo. However, in addition to spatial and center of gravity constraints, the model did not consider aircraft structure and safety constraints, such as floor strength and cumulative load constraints, which play a key role in commercial air cargo transportation; Limburg et al[11]A mixed integer linear programming model has been established, which can allocate a set of selected containers with a quantity less than the predefined positions of the aircraft to the predefined positions of the aircraft. The model aims to minimize the torque of the cargo, structural stress of the aircraft, and fuel consumption to improve the stability and save costs of the aircraft. However, the model cannot solve the problem of more containers than the predefined positions. In the experiment, the solution time for the minimum center of gravity obtained was less than 441.9 seconds; Vancronenburg et al[23]A mixed integer linear programming model has been established to solve the problems of aircraft mass and balance. The goal is to minimize the deviation of the aircraft's center of gravity from the target value and maximize the value of the loaded cargo. This model ensures the integrity and stability of the aircraft structure, the safety of cargo and crew, and the safe and efficient loading and unloading of cargo. The solving speed of this model is fast. However, the center of gravity constraint does not use the aircraft's center of gravity envelope, which poses a safety risk to the aircraft in actual operation. In the experiment, the parameter influence of deviation from the target center of gravity was not discussed; Lurkin et al[12]A mixed integer linear programming model has been established to solve the loading problem of picking up and delivering goods at intermediate airports, with the aim of minimizing time and operating costs while considering both the aircraft's center of gravity and cargo constraints; Brandt [24]A mixed linear integer programming model for multi segment cargo loading problem has also been established, with the goal of maximizing the loading capacity, minimizing the deviation of the flight center of gravity from the specified center of gravity, and minimizing unnecessary container loading and unloading operations at intermediate airports. Although the model considers various practical operational constraints, the position quality constraint, cumulative quality constraint, and joint upper and lower cabin quality constraint are generalized into a unified quality constraint, which is far from actual operation; Zhao Xiangling and others[25]A mixed integer multi-objective programming model with multiple segments was established considering the loading and unloading sequence of containers. The effectiveness of the model in three scenarios was verified using the Gurobi solver, which can effectively reduce the number of loading and unloading times and optimize the center of gravity deviation between the two segments; Zhao et al[13]The nonlinear centroid envelope is described as a multi-stage linear constraint model, and a linearization strategy is designed to establish a mixed integer programming model with maximum total payload and minimum centroid deviation, considering constraints such as position, mass, and balance. The model can obtain the optimal solution within seconds using a commercial solver, and maintain the center of gravity within the specified range of the center of gravity envelope in all test instances.
There is still a certain gap between the model and practical application in China. Research on center of gravity balance abroad is still in the simplified mode of center of gravity deviation constraint, and the center of gravity envelope method used in actual load distribution work has not been adopted for balance constraint. And there are few reports on research in areas such as intermodal transportation loading, non-standard cargo transportation loading, and mixed loading of containers and bulk cargo.
Customer to goods conversion is a research direction that has emerged in recent years. Currently, there are few studies on load balancing in the field of customer to goods conversion, with only a few invention patent reports. In the future passenger to cargo market, due to its low modification costs, it can reduce the initial investment of airlines and will be more favored.
'Passenger to cargo' refers to the conversion of passenger aircraft into cargo aircraft. There are three types of passenger to cargo conversion. The first type is to carry passengers in the cabin and only use the belly compartment of the aircraft for cargo transportation; The second method is to place the cargo to be loaded on the passenger seat without removing it, which is called minor modification; The third method is to dismantle passenger service components such as seats and kitchens in the cabin and install necessary cargo systems for storing, securing, and transporting goods, which is called major renovation. This article mainly studies the issue of load balancing between loading containers in the main cargo hold and loading loose cargo in the lower cargo hold of an aircraft, which is a major reform.
The passenger to cargo aircraft has undergone significant changes in load balance compared to the original passenger aircraft. The passenger plane was converted into a cargo plane, seats and other related equipment were removed, the empty weight of the aircraft was reduced, the cargo hold capacity was expanded, the load-carrying capacity was increased, and the utilization rate was improved. The conversion from passenger to cargo has caused significant changes in performance parameters such as the original center of gravity of the aircraft, the weight of the used aircraft, the force arm, the moment index, the available load, the overall moment of the aircraft, and the center of gravity index, which have a significant impact on the loading. The aircraft needs to be re weighed to determine new relevant loading parameters, and the loading itself has changed. When a passenger plane is loaded, it is generally believed that the passenger mass is fixed and relatively light, and will not have a destructive impact on the aircraft structure. Balancing is mainly achieved by assigning different numbers of passengers to different cabin segments, so that the center of gravity of the aircraft is within its reasonable range. The only problem is the longitudinal balance constraint, which belongs to a one-dimensional allocation problem.
Passenger to cargo and pure cargo aircraft are also different. Usually, both the main cargo hold and the lower cargo hold of a cargo aircraft are transported in the form of containers. Different types of predefined positions and assembly schemes for containers are designed inside the cargo hold, and the loading problem can be considered as the assignment problem of assigning containers to different positions. The main cargo hold for passenger to cargo conversion also pre-defined a few or a single type of container positions during the major conversion process, which can be seen as an assignment problem. However, due to the small capacity of the cargo hold, it is usually transported in the form of loose cargo. A large number of loose cargo of different volumes and masses need to be reasonably allocated to several limited capacity cargo holds, similar to the problem of backpacks. Therefore, the loading problem of passenger to cargo conversion is a combination optimization problem with assignment and backpack characteristics.
In addition, during the design phase of the cargo aircraft, its structural strength and balance are based on the cargo, and the issue of loading large objects such as containers has been considered, with a complete loading process in place. The load balancing sheet of the cargo aircraft specifies predefined positions, joint mass limits for upper and lower compartments, cumulative mass limits, center of gravity envelope limits, etc. These key constraints need to be redesigned and formulated in the conversion of passenger to cargo aircraft. Turbofan type large commercial cargo aircraft are generally double aisle aircraft, and also need to consider conditions such as asymmetric loading restrictions on the weight of the left and right containers, and the balance restrictions on the left and right of the aircraft. However, passenger to cargo conversion is usually a single aisle medium-sized aircraft, and these issues are usually not considered. In this regard, customer to cargo conversion is more like a simplified version of a cargo plane. The research motivation of this article is to establish a relatively complete theoretical model for load balancing of passenger to freight conversion aircraft models.
On the one hand, freight loading should fully utilize the aircraft's capacity to achieve maximum transport capacity; On the other hand, it is hoped that the center of gravity of the load will be as close as possible to the ideal center of gravity position, in order to achieve load balance or save fuel. In loading, the main cargo hold loading is a one-to-one matching assignment problem where containers are assigned to predefined positions; Cargo loading in the lower cargo hold refers to the allocation of bulk cargo to different lower cargo holds, which is a backpack issue. In addition, due to the constraints of the overall aircraft's center of gravity balance, cumulative weight limit, and joint upper and lower weight limit, the main cargo hold and lower cargo hold interact with each other. Therefore, the loading of the aircraft cannot be separated into these two sub problems for independent research. In order to express clearly, mathematical modeling should be carried out according to the above three parts.
| xij={1 集装器 i 分配给舱位 j0 其他 | (1) |
The main cargo hold loading model is
| maxz1=NU∑i=1Np∑j=1Wixij | (2) |
| s.t. NP∑j=1xij⩽ | (3) |
| \sum\limits_{i=1}^{N_{\mathrm{U}}} x_{i j} \leqslant 1 | (4) |
| \sum\limits_{i=1}^{N_{\mathrm{U}}} W_i x_{i j} \leqslant W_j | (5) |
| \sum\limits_{i=1}^{N_{\mathrm{U}}} H_i x_{i j} \leqslant H_j | (6) |
| \sum\limits_{j=s}^t \sum\limits_{i=1}^{N_{\mathrm{U}}} W_i x_{i j} \leqslant W_{s t} \quad(s, t) \in C_{\mathrm{P}} | (7) |
The loading of the cargo hold needs to consider the allocation of multiple goods to different cargo holds. usebRepresenting the index of bulk cargo,hIndicate the cargo hold index and set a two-dimensional 0-1 decision variableybhIndicating to be loaded with bulk cargobIs it allocated to the bulk cargo holdhIf it is 1, otherwise it is 0, that is
| y_{b h}= \begin{cases}1 & \text { 散货 } b \text { 装载在散货舱 } h \text { 内 } \\ 0 & \text { 其他 }\end{cases} | (8) |
The loading model for the cargo hold is
| \max z_2=\sum\limits_{b=1}^{N_{\mathrm{B}}} \sum\limits_{h=1}^{N_{\mathrm{H}}} w_b y_{b h} | (9) |
| \text { s.t. } \sum\limits_{h=1}^{N_{\mathrm{H}}} y_{b h} \leqslant 1 | (10) |
| \sum\limits_{b=1}^{N_{\mathrm{B}}} w_b y_{b h} \leqslant W_h | (11) |
| \sum\limits_{b=1}^{N_{\mathrm{B}}} v_b y_{b h} \leqslant V_h | (12) |
| \sum\limits_{h=q}^r \sum\limits_{b=1}^{N_{\mathrm{B}}} w_b y_{b h} \leqslant W_{q r} \quad(q, r) \in C_{\mathrm{H}} | (13) |
In the formula:z2Target function for cargo hold loading capacity;wbTo be loaded with bulk cargobThe quality;NBThe quantity of bulk cargo to be loaded;NHNumber of cargo compartments for aircraft;WhFor the lower cargo holdhMaximum quality limit;vbTo be loaded with bulk cargobThe volume;VhFor the lower cargo holdhThe maximum volume;WqrFor the lower cargo holdq~rAccumulated weight limit in the area;CHAssemble the lower cargo hold group, including B757-200CH={(1,2), (3,4)}.
| \min z_3=\left|P_{\mathrm{CG}}-P_{\mathrm{CGT}}\right| | (14) |
| \text { s. t. } \sum\limits_{i=1}^{N_{\mathrm{U}}} \sum\limits_{j=1}^{N_{\mathrm{P}}} W_i x_{i j}+\sum\limits_{b=1}^{N_{\mathrm{B}}} \sum\limits_{h=1}^{N_{\mathrm{H}}} w_b y_{b h} \leqslant W_{\mathrm{MPL}} | (15) |
| \begin{gathered} I_{\mathrm{F}, w} \leqslant I_w \leqslant I_{\mathrm{A}, w} \quad w \in\left\{W_{\text {TOW }}, \right. \\ \left.W_{\mathrm{LW}}, W_{\mathrm{ZFW}}\right\} \end{gathered} | (16) |
| \begin{aligned} & \sum\limits_{j=1}^f\left(\sum\limits_{i=1}^{N_{\mathrm{U}}} W_i x_{i j}+\lambda_j \sum\limits_{b=1}^{N_{\mathrm{B}}} w_b y_{b 1}+\right. \\ & \,\,\,\,\,\,\,\,\,\,\, \left.\beta_j \sum\limits_{b=1}^{N_{\mathrm{B}}} w_b y_{b 2}\right) \leqslant W_f \quad f \in Z_{\mathrm{F}} \\ & Z_F=\{1, 2, 3, 4, 5, 6, 7\} \end{aligned} | (17) |
| \begin{aligned} & \sum\limits_{j=15}^a\left(\sum\limits_{i=1}^{N_{\mathrm{U}}} W_i x_{i j}+\lambda_j \sum\limits_{b=1}^{N_{\mathrm{B}}} w_b y_{b 4}+\right. \\ & \,\,\,\,\,\,\,\,\,\,\, \left.\beta_j \sum\limits_{b=1}^{N_{\mathrm{B}}} w_b y_{b 3}\right) \leqslant W_a \quad a \in Z_{\mathrm{A}} \\ & Z_{\mathrm{A}}=\{15, 14, 13, 12, 11, 10, 9, 8\} \end{aligned} | (18) |
In the formula:z3Target function for center of gravity deviation;PCGThe center of gravity corresponding to the loading plan;PCGTFor the designated target center of gravity;WMPLMaximum Payload (MPL) for flights;WTOWTakeoff Weight (TOW) for aircraft takeoff;WLWLanding Weight (LW) for aircraft;WZFWZero Fuel Weight (ZFW) for aircraft;IwFor aircraft of different qualitieswThe index of time;IF, wThe quality of the aircraft iswThe forward limit index of the center of gravity at time,IA, wThe quality of the aircraft iswThe posterior limit index of the center of gravity of time;ZFAssemble the front sections 1-7 of the aircraft（Table 1)；fdoZFThe interval in the middlef，f∈ZF；WfFor joint interval 1~fAccumulated weight limit;ZAGather for the rear section 8-15 of the aircraft,adoZAThe interval in the middlea，a∈ZA；WaJoint interval 15~aAccumulated weight limit;λjandβjrespectivelyTable 1Lower cargo holdjThe quality coefficient of the interval.
| 区间 | 实际区间联合载量计算公式 | 区间限重/kg |
| 1 | 舱位1 | 2 716 |
| 2 | 区间1+舱位2+(下货舱1)×10% | 4 360 |
| 3 | 区间2+舱位3+(下货舱1)×80% | 7 871 |
| 4 | 区间3+舱位4+(下货舱1)×10%+ (下货舱2)×40% | 11 557 |
| 5 | 区间4+舱位5+(下货舱2)×40% | 14 043 |
| 6 | 区间5+舱位6+(下货舱2)×20% | 15 893 |
| 7 | 区间6+舱位7 | 17 712 |
| 8 | 区间9+舱位8 | 23 748 |
| 9 | 区间10+舱位9 | 21 909 |
| 10 | 区间11+舱位10+(下货舱3)×20% | 20 102 |
| 11 | 区间12+舱位11+(下货舱3)×50% | 18 257 |
| 12 | 区间13+舱位12+(下货舱3)×30%+ (下货舱4)×10% | 15 733 |
| 13 | 区间14+舱位13+(下货舱4)×40% | 11 969 |
| 14 | 区间15+舱位14+(下货舱4)×30% | 8 239 |
| 15 | 舱位15+(下货舱4)×20% | 3 476 |
DownLoad:
CSV
Objective function equation (14) ensures the takeoff center of gravity of the aircraftPCGCompared to the designated target center of gravityPCGTThe deviation is minimized. Airlines usually specify a target center of gravity position when loadingPCGTOn the one hand, it ensures that the aircraft maintains balance during flight and guarantees flight safety; On the other hand, reducing fuel consumption, lowering operating costs and carbon emissions, therefore, during loading,PCGShould be as close as possiblePCGTTo achieve center of gravity optimization.PCGdo
| P_{\mathrm{CG}}=100 \frac{B_{\mathrm{TOW}}-B_{\mathrm{M}}}{B_{\mathrm{MAC}}} | (19) |
Equation (19) is used to calculate the relative center of gravity, which is the position of the aircraft's center of gravity relative to the MAC, commonly represented as% MACBTOWdo
| B_{\mathrm{TOW}}=\frac{M_{\mathrm{TOW}}}{W_{\mathrm{TOW}}} | (20) |
In the formula:MTOWThe quality of the aircraft isWTOWThe moment of time.
| \begin{aligned} W_{\mathrm{TOW}}= & W_{\mathrm{OEW}}+W_{\mathrm{TOF}}+\sum\limits_{i=1}^{N_{\mathrm{U}}} \sum\limits_{j=1}^{N_{\mathrm{P}}} W_i x_{i j}+ \\ & \sum\limits_{b=1}^{N_{\mathrm{B}}} \sum\limits_{h=1}^{N_{\mathrm{H}}} w_b y_{b h} \end{aligned} | (21) |
| \begin{aligned} M_{\mathrm{TOW}}= & B_{\mathrm{OEW}} W_{\mathrm{OEW}}+W_{\mathrm{TOF}} B_{\mathrm{TOF}}+ \\ & \sum\limits_{i=1}^{N_{\mathrm{U}}} \sum\limits_{j=1}^{N_{\mathrm{P}}} B_j W_i x_{i j}+\sum\limits_{b=1}^{N_{\mathrm{B}}} \sum\limits_{h=1}^{N_{\mathrm{H}}} B_h w_b y_{b h} \end{aligned} | (22) |
In the formula:WOEWOperational Empty Weight (OEW) for the use of aircraft;WTOFTakeoff Fuel Weight (TOF) for aircraft takeoff;BOEWThe torque generated for OEW;BTOFThe torque generated for TOF;BjMain cargo hold spacejThe arm of strength;BhFor the lower cargo holdhThe arm of strength.
Constraint (15) is the maximum payload limit. Constraint (16) is the balance constraint of the aircraft. For narrow body machines for customer to goods conversion, due to the small lateral mass deviation, lateral balance is generally not considered. The main mass distribution of an aircraft is in the longitudinal axis direction, which affects the pitch balance during flight and must be ensured. The company uses a center of gravity envelope to limit the range of the aircraft's center of gravity during operation, ensuring pitch balance. The aircraft's center of gravity envelope describes the aircraft's center of gravity and its range through an index. The index is the torque that decreases proportionally according to regulations, and is a function of mass and force arm. There are various indices based on the fuel content in the fuel tank of the aircraft at different stages, usually including takeoff quality index, landing quality index, and no fuel quality indexIwexpress,w∈{WTOW, WLW, WZFW}That is
| \begin{aligned} I_w= & I_{\mathrm{OEW}}+I_{\mathrm{Fuel}, w}+\sum\limits_{i=1}^{N_{\mathrm{U}}} \sum\limits_{j=1}^{N_{\mathrm{P}}} \frac{W_i x_{i j}\left(B_j-C_{\mathrm{D}}\right)}{C_1}+ \\ & \sum\limits_{b=1}^{N_{\mathrm{B}}} \sum\limits_{h=1}^{N_{\mathrm{H}}} \frac{w_b y_{b h}\left(B_h-C_{\mathrm{D}}\right)}{C_1} \end{aligned} | (23) |
For the load balancing optimization problem of passenger to cargo aircraft, the BD algorithm is used to decompose the problem into a main problem and a sub problem. The main problem is
| \begin{array}{ll} & \max z=\rho\left(z_1+z_2\right)-(1-\rho) z_3 \\ \text { s.t. } & \sum\limits_{j=1}^{N_{\mathrm{P}}} x_{i j} \leqslant 1 \\ & \sum\limits_{i=1}^{N_{\mathrm{U}}} W_i x_{i j} \leqslant W_j \\ & \sum\limits_{i=1}^{N_{\mathrm{U}}} H_i x_{i j} \leqslant H_j \\ & \sum\limits_{h=1}^{N_{\mathrm{H}}} y_{b h} \leqslant 1 \\ & \sum\limits_{b=1}^{N_{\mathrm{B}}} w_b y_{b h} \leqslant W_h \\ & \sum\limits_{b=1}^{N_{\mathrm{B}}} v_b y_{b h} \leqslant V_h \end{array} | (24) |
The sub problem is
| \left\{\begin{array}{l} \sum\limits_{j=s}^t \sum\limits_{i=1}^{N_{\mathrm{U}}} W_i x_{i j} \leqslant W_{st} \quad(s, t) \in C_{\mathrm{P}} \\ \sum\limits_{h=q}^r \sum\limits_{b=1}^{N_{\mathrm{B}}} w_b y_{b h} \leqslant W_{q r} \quad(q, r) \in C_{\mathrm{H}} \\ \sum\limits_{i=1}^{N_{\mathrm{U}}} \sum\limits_{j=1}^{N_{\mathrm{P}}} W_i x_{i j}+\sum\limits_{b=1}^{N_{\mathrm{B}}} \sum\limits_{h=1}^{N_{\mathrm{H}}} w_b y_{b h} \leqslant \mathrm{~W}_{\mathrm{MPL}} \\ \sum\limits_{j=1}^f\left(\sum\limits_{i=1}^{N_{\mathrm{U}}} W_i x_{i j}+\lambda_j \sum\limits_{b=1}^{N_{\mathrm{B}}} w_b y_{b 1}+\right. \\ \, \, \, \, \, \, \, \, \, \, \left.\beta_j \sum\limits_{b=1}^{N_{\mathrm{B}}} w_b y_{b 2}\right) \leqslant W_f \quad f \in Z_{\mathrm{F}} \\ \sum\limits_{j=15}^a\left(\sum\limits_{i=1}^{N_{\mathrm{U}}} W_i x_{i j}+\lambda_j \sum\limits_{b=1}^{N_{\mathrm{B}}} w_b y_{b 4}+\right. \\ \, \, \, \, \, \, \, \, \, \, \left.\beta_j \sum\limits_{b=1}^{N_{\mathrm{B}}} w_b y_{b 3}\right) \leqslant W_a \quad a \in Z_{\mathrm{A}} \\ I_{\mathrm{F}, w} \leqslant I_w \leqslant I_{\mathrm{A}, w} \quad w \in\left\{W_{\text {TOW }}, W_{\mathrm{LW}}, W_{\text {ZFW }}\right\} \end{array}\right. | (25) |
The BD algorithm used in this article and Cote[27]Similarly, combining complex constraints other than the main problem into SP to check if the solution of MP is correct is calledy-Check algorithm.
For those who did not passy-checkSolution for algorithm verificationx*(n）In order to preventn+Obtain the same solution in MP through one iteration. We need to add Benders' Cut constraints in MP to avoid duplicate infeasible solutions. Therefore, adding constraints is necessaryx≠x*(n）Then, then+One iteration, the Benders' Cut constraint that needs to be added to the main problem is
| \begin{aligned} & \sum\limits_{j=1}^{N_{\mathrm{P}}} \sum\limits_{i=1}^{N_{\mathrm{U}}}\left|x_{i j}-x_{i j}^*(n)\right|+ \\& \, \, \, \, \, \, \, \, \, \, \, \, \sum\limits_{h=1}^{N_{\mathrm{H}}} \sum\limits_{b=1}^{N_{\mathrm{B}}}\left|y_{b h}-y_{b h}^*(n)\right|>0 \end{aligned} | (26) |
The main cargo hold assignment problem uses integer encoding in the encoding method, using a one-dimensional array to number the container allocation order according to the cargo hold position.Figure 3It is an individual coding example, indicating that cabin 1 is loaded with container 10, cabin 2 is loaded with container 7, cabin 3 is loaded with container 8, and so on. This encoding method can ensure that the algorithm always satisfies the uniqueness constraint of the pre-defined positions of the container and the main cargo hold, namely constraint equations (3) and (4).
Figure 4As an example of the coding method for the cargo hold, represented by a two-dimensional matrix, the list shows the cargo holdhThe line represents the goodsb.Figure 4In the middle, cargo hold 1 is loaded with loose cargo 1, 3, and 7, cargo hold 2 is loaded with loose cargo 2, 4, and 8, cargo hold 3 is loaded with loose cargo 6 and 9, and cargo hold 4 is loaded with loose cargo 5 and 10. This encoding method can enable the algorithm to satisfy the uniqueness constraint between bulk cargo and cargo hold, i.e. constraint (10).
The variation method for the cargo hold is to randomly assign a new cargo hold to each loose cargo, such asFigure 6As shown. For the first line, which is the first bulk cargo, assuming a random number 4 is generated, the 1 at the original position is exchanged with the 0 at the new position 4, so that the first bulk cargo is adjusted from the first compartment to the fourth compartment, and so on.
During initialization and mutation, constraints (5), (6), (11), and (12) are synchronized with the generation of new solutions until these constraints are met before outputting new solutions.
| 参数 | 数值 | 参数 | 数值 | |
| PCGT/%MAC | 23 | WMPL/kg | 30 708 | |
| WTOF/kg | 10 000 | IFuel, WTOW | 4.5 | |
| 航程燃油/kg | 8 000 | IFuel, WLW | 0.4 | |
| WOEW/kg | 52 752 | IOEW | 31.3 | |
| BM/m | 25.19 | BMAC/m | 5.07 | |
| C1 | 70 000 | CD/m | 26.36 |
DownLoad:
CSV
| 舱位 | 力臂/m | 最大载荷/kg | 最大高度/m | 分区载荷/kg |
| C1 | 9.91 | 2 716 | 2.00 | C1~C5质量和不超过18 000 |
| C2 | 12.17 | 2 948 | 2.00 | |
| C3 | 14.43 | 2 948 | 2.00 | |
| C4 | 16.69 | 2 948 | 2.00 | |
| C5 | 18.95 | 2 948 | 2.00 | |
| C6 | 21.21 | 2 948 | 2.00 | C6~C10质量和不超过24 000 |
| C7 | 23.47 | 2 948 | 2.00 | |
| C8 | 25.73 | 4 264 | 2.00 | |
| C9 | 27.99 | 4 264 | 2.00 | |
| C10 | 30.25 | 2 948 | 2.00 | |
| C11 | 32.51 | 2 948 | 2.00 | C11~C15质量和不超过29 000 |
| C12 | 34.77 | 2 948 | 2.00 | |
| C13 | 37.03 | 2 948 | 2.00 | |
| C14 | 39.29 | 2 948 | 2.00 | |
| C15 | 41.55 | 2 948 | 1.95 |
DownLoad:
CSV
| 下货舱 | 下货舱1 | 下货舱2 | 下货舱3 | 下货舱4 |
| 货舱限重/kg | 2 469 | 4 672 | 3 773 | 5 606 |
| 舱位累积限重/kg | 4 672 | 7 393 | ||
| 力臂/m | 13.58 | 18.64 | 32.73 | 38.29 |
| 最大容积/m3 | 4.9 | 13.9 | 14.2 | 16.7 |
DownLoad:
CSV
The parameter design for the simulated annealing algorithm in MP is as follows: the initial temperature is set to 1.0 × 106Set the final temperature to 7.4 × 10-12The annealing coefficient is set to 0.95, and the chain length of iterations at the initial temperature is set to 1000.
| f(x, y)=\max \left[\rho \frac{z_1+z_2}{W_{\mathrm{MPL}}}-(1-\rho) \frac{z_3}{12.5}\right] | (27) |
In the formula:ρFor the sake of weight, this article considers that economic benefits are more importantρ=0.8.
| 算例 | z1+z2 | |||
| 本文算法 | Gurobi | Lingo | 人工配载 | |
| 1 | 25 583.6 | 27 166.0 | 27 194.0 | 25 543.0 |
| 2 | 28 706.8 | 30 705.0 | 30 708.0 | 27 733.0 |
| 3 | 29 876.7 | 29 711.0 | 29 711.0 | 27 605.0 |
| 4 | 28 209.5 | 28 901.0 | 28 901.0 | 26 503.0 |
| 5 | 28 229.5 | 29 820.0 | 29 760.0 | 27 798.0 |
| 6 | 27 652.0 | 28 486.0 | 28 466.0 | 26 101.0 |
| 7 | 27 753.2 | 29 052.0 | 29 052.0 | 26 639.0 |
| 8 | 28 685.1 | 29 669.0 | 29 669.0 | 27 839.0 |
| 9 | 29 121.7 | 29 676.0 | 29 676.0 | 27 327.0 |
| 10 | 28 760.2 | 29 548.0 | 29 538.0 | 27 543.0 |
| 11 | 28 503.0 | 30 050.0 | 30 035.0 | 28 012.0 |
| 12 | 28 692.6 | 30 368.0 | 30 368.0 | 28 117.0 |
| 13 | 27 791.3 | 28 669.0 | 28 653.0 | 25 786.0 |
| 14 | 27 502.6 | 28 691.0 | 28 691.0 | 26 023.0 |
| 15 | 28 338.2 | 29 348.0 | 29 348.0 | 26 479.0 |
| 16 | 28 441.3 | 30 078.0 | 29 989.0 | 27 721.0 |
| 17 | 29 138.2 | 30 240.0 | 30 240.0 | 28 032.0 |
| 18 | 28 771.7 | 29 860.0 | 29 834.0 | 26 841.0 |
| 19 | 28 707.9 | 29 725.0 | 29 725.0 | 27 054.0 |
| 20 | 29 116.3 | 30 583.0 | 30 583.0 | 27 942.0 |
| 均值 | 28 379.1 | 29 517.3 | 29 507.1 | 27 131.9 |
DownLoad:
CSV
| 算例 | z3 | |||
| 本文算法 | Gurobi | Lingo | 人工配载 | |
| 1 | 0.05 | 0.02 | 3.47 | 5.05 |
| 2 | 0.04 | 0.02 | 2.71 | 6.01 |
| 3 | 0.08 | 0.02 | 2.98 | 5.15 |
| 4 | 0.05 | 0.02 | 3.23 | 5.65 |
| 5 | 0.01 | 0.02 | 2.85 | 4.37 |
| 6 | 0.02 | 0.02 | 3.20 | 5.45 |
| 7 | 0.09 | 0.02 | 3.12 | 5.23 |
| 8 | 0.05 | 0.02 | 2.99 | 5.79 |
| 9 | 0.03 | 0.02 | 2.99 | 5.32 |
| 10 | 0.03 | 0.02 | 3.00 | 5.54 |
| 11 | 0.02 | 0.02 | 3.36 | 5.65 |
| 12 | 0.10 | 0.02 | 2.87 | 5.02 |
| 13 | 0.05 | 0.02 | 2.95 | 4.97 |
| 14 | 0.10 | 0.02 | 3.06 | 4.87 |
| 15 | 0.03 | 0.02 | 2.99 | 5.32 |
| 16 | 0.01 | 0.02 | 2.78 | 5.09 |
| 17 | 0.06 | 0.02 | 3.33 | 4.86 |
| 18 | 0.07 | 0.02 | 3.41 | 5.45 |
| 19 | 0.09 | 0.02 | 2.99 | 5.33 |
| 20 | 0.06 | 0.02 | 3.49 | 5.01 |
| 均值 | 0.05 | 0.02 | 3.09 | 5.26 |
DownLoad:
CSV
| 算例 | 时间/s | |||
| 本文算法 | Gurobi | Lingo | 人工配载 | |
| 1 | 24.89 | 0.21 | 7 358.00 | 546.00 |
| 2 | 20.12 | 0.19 | 7 365.00 | 557.00 |
| 3 | 20.12 | 0.07 | 7 366.00 | 532.00 |
| 4 | 20.18 | 0.09 | 7 523.00 | 603.00 |
| 5 | 20.19 | 0.08 | 7 201.00 | 549.00 |
| 6 | 20.21 | 0.09 | 7 358.00 | 587.00 |
| 7 | 20.15 | 0.13 | 7 289.00 | 592.00 |
| 8 | 20.16 | 0.10 | 7 527.00 | 563.00 |
| 9 | 20.18 | 0.08 | 7 284.00 | 610.00 |
| 10 | 20.12 | 0.17 | 7 252.00 | 546.00 |
| 11 | 19.95 | 0.09 | 7 356.00 | 598.00 |
| 12 | 20.92 | 0.08 | 7 432.00 | 599.00 |
| 13 | 19.94 | 0.08 | 7 267.00 | 567.00 |
| 14 | 19.92 | 0.09 | 7 256.00 | 642.00 |
| 15 | 19.91 | 0.09 | 7 389.00 | 609.00 |
| 16 | 19.97 | 0.27 | 7 426.00 | 597.00 |
| 17 | 19.91 | 0.14 | 7 396.00 | 609.00 |
| 18 | 19.93 | 0.25 | 7 298.00 | 548.00 |
| 19 | 19.96 | 0.10 | 7 589.00 | 579.00 |
| 20 | 19.88 | 0.09 | 7 481.00 | 602.00 |
| 均值 | 20.33 | 0.13 | 7 370.65 | 581.75 |
DownLoad:
CSV
(2) Designed a heuristic based Benders decomposition algorithm. In the main problem, the simulated annealing algorithm was improved to obtain the solution of the main problem; In the sub problem, designedy-Check algorithm, used to check the feasibility of the main question solution.
(3) Experimental verification was conducted using the B757 passenger to cargo aircraft model and operational data as an example. The results show that the average payload of the algorithm in this paper can reach 28379.1 kg, with an average center of gravity deviation of 0.05% MAC, which well meets the requirements of maximizing flight profits and efficient and safe operation, proving the correctness of the model and the effectiveness of the algorithm.
(4) The current model is applicable to single channel narrow body passenger to cargo aircraft models, while wide body passenger to cargo aircraft need to further consider practical constraints and make adjustments; In addition, with the development of the freight industry, some airlines have proposed the demand for multi airport and multi segment joint loading, as well as issues such as mixed loading of multiple types of cargo and loading efficiency, which have put forward requirements for further research.
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Figures(10) / Tables(7)
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ZHAO Xiang-ling, LI Yun-fei. Weight balance problem modeling and benders decomposition algorithm design of preighter[J]. Journal of Traffic and Transportation Engineering, 2023, 23(2): 199-211. doi: 10.19818/j.cnki.1671-1637.2023.02.014
| 区间 | 实际区间联合载量计算公式 | 区间限重/kg |
| 1 | 舱位1 | 2 716 |
| 2 | 区间1+舱位2+(下货舱1)×10% | 4 360 |
| 3 | 区间2+舱位3+(下货舱1)×80% | 7 871 |
| 4 | 区间3+舱位4+(下货舱1)×10%+ (下货舱2)×40% | 11 557 |
| 5 | 区间4+舱位5+(下货舱2)×40% | 14 043 |
| 6 | 区间5+舱位6+(下货舱2)×20% | 15 893 |
| 7 | 区间6+舱位7 | 17 712 |
| 8 | 区间9+舱位8 | 23 748 |
| 9 | 区间10+舱位9 | 21 909 |
| 10 | 区间11+舱位10+(下货舱3)×20% | 20 102 |
| 11 | 区间12+舱位11+(下货舱3)×50% | 18 257 |
| 12 | 区间13+舱位12+(下货舱3)×30%+ (下货舱4)×10% | 15 733 |
| 13 | 区间14+舱位13+(下货舱4)×40% | 11 969 |
| 14 | 区间15+舱位14+(下货舱4)×30% | 8 239 |
| 15 | 舱位15+(下货舱4)×20% | 3 476 |
DownLoad:
CSV
| 参数 | 数值 | 参数 | 数值 | |
| PCGT/%MAC | 23 | WMPL/kg | 30 708 | |
| WTOF/kg | 10 000 | IFuel, WTOW | 4.5 | |
| 航程燃油/kg | 8 000 | IFuel, WLW | 0.4 | |
| WOEW/kg | 52 752 | IOEW | 31.3 | |
| BM/m | 25.19 | BMAC/m | 5.07 | |
| C1 | 70 000 | CD/m | 26.36 |
DownLoad:
CSV
| 舱位 | 力臂/m | 最大载荷/kg | 最大高度/m | 分区载荷/kg |
| C1 | 9.91 | 2 716 | 2.00 | C1~C5质量和不超过18 000 |
| C2 | 12.17 | 2 948 | 2.00 | |
| C3 | 14.43 | 2 948 | 2.00 | |
| C4 | 16.69 | 2 948 | 2.00 | |
| C5 | 18.95 | 2 948 | 2.00 | |
| C6 | 21.21 | 2 948 | 2.00 | C6~C10质量和不超过24 000 |
| C7 | 23.47 | 2 948 | 2.00 | |
| C8 | 25.73 | 4 264 | 2.00 | |
| C9 | 27.99 | 4 264 | 2.00 | |
| C10 | 30.25 | 2 948 | 2.00 | |
| C11 | 32.51 | 2 948 | 2.00 | C11~C15质量和不超过29 000 |
| C12 | 34.77 | 2 948 | 2.00 | |
| C13 | 37.03 | 2 948 | 2.00 | |
| C14 | 39.29 | 2 948 | 2.00 | |
| C15 | 41.55 | 2 948 | 1.95 |
DownLoad:
CSV
| 下货舱 | 下货舱1 | 下货舱2 | 下货舱3 | 下货舱4 |
| 货舱限重/kg | 2 469 | 4 672 | 3 773 | 5 606 |
| 舱位累积限重/kg | 4 672 | 7 393 | ||
| 力臂/m | 13.58 | 18.64 | 32.73 | 38.29 |
| 最大容积/m3 | 4.9 | 13.9 | 14.2 | 16.7 |
DownLoad:
CSV
| 算例 | z1+z2 | |||
| 本文算法 | Gurobi | Lingo | 人工配载 | |
| 1 | 25 583.6 | 27 166.0 | 27 194.0 | 25 543.0 |
| 2 | 28 706.8 | 30 705.0 | 30 708.0 | 27 733.0 |
| 3 | 29 876.7 | 29 711.0 | 29 711.0 | 27 605.0 |
| 4 | 28 209.5 | 28 901.0 | 28 901.0 | 26 503.0 |
| 5 | 28 229.5 | 29 820.0 | 29 760.0 | 27 798.0 |
| 6 | 27 652.0 | 28 486.0 | 28 466.0 | 26 101.0 |
| 7 | 27 753.2 | 29 052.0 | 29 052.0 | 26 639.0 |
| 8 | 28 685.1 | 29 669.0 | 29 669.0 | 27 839.0 |
| 9 | 29 121.7 | 29 676.0 | 29 676.0 | 27 327.0 |
| 10 | 28 760.2 | 29 548.0 | 29 538.0 | 27 543.0 |
| 11 | 28 503.0 | 30 050.0 | 30 035.0 | 28 012.0 |
| 12 | 28 692.6 | 30 368.0 | 30 368.0 | 28 117.0 |
| 13 | 27 791.3 | 28 669.0 | 28 653.0 | 25 786.0 |
| 14 | 27 502.6 | 28 691.0 | 28 691.0 | 26 023.0 |
| 15 | 28 338.2 | 29 348.0 | 29 348.0 | 26 479.0 |
| 16 | 28 441.3 | 30 078.0 | 29 989.0 | 27 721.0 |
| 17 | 29 138.2 | 30 240.0 | 30 240.0 | 28 032.0 |
| 18 | 28 771.7 | 29 860.0 | 29 834.0 | 26 841.0 |
| 19 | 28 707.9 | 29 725.0 | 29 725.0 | 27 054.0 |
| 20 | 29 116.3 | 30 583.0 | 30 583.0 | 27 942.0 |
| 均值 | 28 379.1 | 29 517.3 | 29 507.1 | 27 131.9 |
DownLoad:
CSV
| 算例 | z3 | |||
| 本文算法 | Gurobi | Lingo | 人工配载 | |
| 1 | 0.05 | 0.02 | 3.47 | 5.05 |
| 2 | 0.04 | 0.02 | 2.71 | 6.01 |
| 3 | 0.08 | 0.02 | 2.98 | 5.15 |
| 4 | 0.05 | 0.02 | 3.23 | 5.65 |
| 5 | 0.01 | 0.02 | 2.85 | 4.37 |
| 6 | 0.02 | 0.02 | 3.20 | 5.45 |
| 7 | 0.09 | 0.02 | 3.12 | 5.23 |
| 8 | 0.05 | 0.02 | 2.99 | 5.79 |
| 9 | 0.03 | 0.02 | 2.99 | 5.32 |
| 10 | 0.03 | 0.02 | 3.00 | 5.54 |
| 11 | 0.02 | 0.02 | 3.36 | 5.65 |
| 12 | 0.10 | 0.02 | 2.87 | 5.02 |
| 13 | 0.05 | 0.02 | 2.95 | 4.97 |
| 14 | 0.10 | 0.02 | 3.06 | 4.87 |
| 15 | 0.03 | 0.02 | 2.99 | 5.32 |
| 16 | 0.01 | 0.02 | 2.78 | 5.09 |
| 17 | 0.06 | 0.02 | 3.33 | 4.86 |
| 18 | 0.07 | 0.02 | 3.41 | 5.45 |
| 19 | 0.09 | 0.02 | 2.99 | 5.33 |
| 20 | 0.06 | 0.02 | 3.49 | 5.01 |
| 均值 | 0.05 | 0.02 | 3.09 | 5.26 |
DownLoad:
CSV
| 算例 | 时间/s | |||
| 本文算法 | Gurobi | Lingo | 人工配载 | |
| 1 | 24.89 | 0.21 | 7 358.00 | 546.00 |
| 2 | 20.12 | 0.19 | 7 365.00 | 557.00 |
| 3 | 20.12 | 0.07 | 7 366.00 | 532.00 |
| 4 | 20.18 | 0.09 | 7 523.00 | 603.00 |
| 5 | 20.19 | 0.08 | 7 201.00 | 549.00 |
| 6 | 20.21 | 0.09 | 7 358.00 | 587.00 |
| 7 | 20.15 | 0.13 | 7 289.00 | 592.00 |
| 8 | 20.16 | 0.10 | 7 527.00 | 563.00 |
| 9 | 20.18 | 0.08 | 7 284.00 | 610.00 |
| 10 | 20.12 | 0.17 | 7 252.00 | 546.00 |
| 11 | 19.95 | 0.09 | 7 356.00 | 598.00 |
| 12 | 20.92 | 0.08 | 7 432.00 | 599.00 |
| 13 | 19.94 | 0.08 | 7 267.00 | 567.00 |
| 14 | 19.92 | 0.09 | 7 256.00 | 642.00 |
| 15 | 19.91 | 0.09 | 7 389.00 | 609.00 |
| 16 | 19.97 | 0.27 | 7 426.00 | 597.00 |
| 17 | 19.91 | 0.14 | 7 396.00 | 609.00 |
| 18 | 19.93 | 0.25 | 7 298.00 | 548.00 |
| 19 | 19.96 | 0.10 | 7 589.00 | 579.00 |
| 20 | 19.88 | 0.09 | 7 481.00 | 602.00 |
| 均值 | 20.33 | 0.13 | 7 370.65 | 581.75 |
DownLoad:
CSV
| 区间 | 实际区间联合载量计算公式 | 区间限重/kg |
| 1 | 舱位1 | 2 716 |
| 2 | 区间1+舱位2+(下货舱1)×10% | 4 360 |
| 3 | 区间2+舱位3+(下货舱1)×80% | 7 871 |
| 4 | 区间3+舱位4+(下货舱1)×10%+ (下货舱2)×40% | 11 557 |
| 5 | 区间4+舱位5+(下货舱2)×40% | 14 043 |
| 6 | 区间5+舱位6+(下货舱2)×20% | 15 893 |
| 7 | 区间6+舱位7 | 17 712 |
| 8 | 区间9+舱位8 | 23 748 |
| 9 | 区间10+舱位9 | 21 909 |
| 10 | 区间11+舱位10+(下货舱3)×20% | 20 102 |
| 11 | 区间12+舱位11+(下货舱3)×50% | 18 257 |
| 12 | 区间13+舱位12+(下货舱3)×30%+ (下货舱4)×10% | 15 733 |
| 13 | 区间14+舱位13+(下货舱4)×40% | 11 969 |
| 14 | 区间15+舱位14+(下货舱4)×30% | 8 239 |
| 15 | 舱位15+(下货舱4)×20% | 3 476 |
| 参数 | 数值 | 参数 | 数值 | |
| PCGT/%MAC | 23 | WMPL/kg | 30 708 | |
| WTOF/kg | 10 000 | IFuel, WTOW | 4.5 | |
| 航程燃油/kg | 8 000 | IFuel, WLW | 0.4 | |
| WOEW/kg | 52 752 | IOEW | 31.3 | |
| BM/m | 25.19 | BMAC/m | 5.07 | |
| C1 | 70 000 | CD/m | 26.36 |
| 舱位 | 力臂/m | 最大载荷/kg | 最大高度/m | 分区载荷/kg |
| C1 | 9.91 | 2 716 | 2.00 | C1~C5质量和不超过18 000 |
| C2 | 12.17 | 2 948 | 2.00 | |
| C3 | 14.43 | 2 948 | 2.00 | |
| C4 | 16.69 | 2 948 | 2.00 | |
| C5 | 18.95 | 2 948 | 2.00 | |
| C6 | 21.21 | 2 948 | 2.00 | C6~C10质量和不超过24 000 |
| C7 | 23.47 | 2 948 | 2.00 | |
| C8 | 25.73 | 4 264 | 2.00 | |
| C9 | 27.99 | 4 264 | 2.00 | |
| C10 | 30.25 | 2 948 | 2.00 | |
| C11 | 32.51 | 2 948 | 2.00 | C11~C15质量和不超过29 000 |
| C12 | 34.77 | 2 948 | 2.00 | |
| C13 | 37.03 | 2 948 | 2.00 | |
| C14 | 39.29 | 2 948 | 2.00 | |
| C15 | 41.55 | 2 948 | 1.95 |
| 下货舱 | 下货舱1 | 下货舱2 | 下货舱3 | 下货舱4 |
| 货舱限重/kg | 2 469 | 4 672 | 3 773 | 5 606 |
| 舱位累积限重/kg | 4 672 | 7 393 | ||
| 力臂/m | 13.58 | 18.64 | 32.73 | 38.29 |
| 最大容积/m3 | 4.9 | 13.9 | 14.2 | 16.7 |
| 算例 | z1+z2 | |||
| 本文算法 | Gurobi | Lingo | 人工配载 | |
| 1 | 25 583.6 | 27 166.0 | 27 194.0 | 25 543.0 |
| 2 | 28 706.8 | 30 705.0 | 30 708.0 | 27 733.0 |
| 3 | 29 876.7 | 29 711.0 | 29 711.0 | 27 605.0 |
| 4 | 28 209.5 | 28 901.0 | 28 901.0 | 26 503.0 |
| 5 | 28 229.5 | 29 820.0 | 29 760.0 | 27 798.0 |
| 6 | 27 652.0 | 28 486.0 | 28 466.0 | 26 101.0 |
| 7 | 27 753.2 | 29 052.0 | 29 052.0 | 26 639.0 |
| 8 | 28 685.1 | 29 669.0 | 29 669.0 | 27 839.0 |
| 9 | 29 121.7 | 29 676.0 | 29 676.0 | 27 327.0 |
| 10 | 28 760.2 | 29 548.0 | 29 538.0 | 27 543.0 |
| 11 | 28 503.0 | 30 050.0 | 30 035.0 | 28 012.0 |
| 12 | 28 692.6 | 30 368.0 | 30 368.0 | 28 117.0 |
| 13 | 27 791.3 | 28 669.0 | 28 653.0 | 25 786.0 |
| 14 | 27 502.6 | 28 691.0 | 28 691.0 | 26 023.0 |
| 15 | 28 338.2 | 29 348.0 | 29 348.0 | 26 479.0 |
| 16 | 28 441.3 | 30 078.0 | 29 989.0 | 27 721.0 |
| 17 | 29 138.2 | 30 240.0 | 30 240.0 | 28 032.0 |
| 18 | 28 771.7 | 29 860.0 | 29 834.0 | 26 841.0 |
| 19 | 28 707.9 | 29 725.0 | 29 725.0 | 27 054.0 |
| 20 | 29 116.3 | 30 583.0 | 30 583.0 | 27 942.0 |
| 均值 | 28 379.1 | 29 517.3 | 29 507.1 | 27 131.9 |
| 算例 | z3 | |||
| 本文算法 | Gurobi | Lingo | 人工配载 | |
| 1 | 0.05 | 0.02 | 3.47 | 5.05 |
| 2 | 0.04 | 0.02 | 2.71 | 6.01 |
| 3 | 0.08 | 0.02 | 2.98 | 5.15 |
| 4 | 0.05 | 0.02 | 3.23 | 5.65 |
| 5 | 0.01 | 0.02 | 2.85 | 4.37 |
| 6 | 0.02 | 0.02 | 3.20 | 5.45 |
| 7 | 0.09 | 0.02 | 3.12 | 5.23 |
| 8 | 0.05 | 0.02 | 2.99 | 5.79 |
| 9 | 0.03 | 0.02 | 2.99 | 5.32 |
| 10 | 0.03 | 0.02 | 3.00 | 5.54 |
| 11 | 0.02 | 0.02 | 3.36 | 5.65 |
| 12 | 0.10 | 0.02 | 2.87 | 5.02 |
| 13 | 0.05 | 0.02 | 2.95 | 4.97 |
| 14 | 0.10 | 0.02 | 3.06 | 4.87 |
| 15 | 0.03 | 0.02 | 2.99 | 5.32 |
| 16 | 0.01 | 0.02 | 2.78 | 5.09 |
| 17 | 0.06 | 0.02 | 3.33 | 4.86 |
| 18 | 0.07 | 0.02 | 3.41 | 5.45 |
| 19 | 0.09 | 0.02 | 2.99 | 5.33 |
| 20 | 0.06 | 0.02 | 3.49 | 5.01 |
| 均值 | 0.05 | 0.02 | 3.09 | 5.26 |
| 算例 | 时间/s | |||
| 本文算法 | Gurobi | Lingo | 人工配载 | |
| 1 | 24.89 | 0.21 | 7 358.00 | 546.00 |
| 2 | 20.12 | 0.19 | 7 365.00 | 557.00 |
| 3 | 20.12 | 0.07 | 7 366.00 | 532.00 |
| 4 | 20.18 | 0.09 | 7 523.00 | 603.00 |
| 5 | 20.19 | 0.08 | 7 201.00 | 549.00 |
| 6 | 20.21 | 0.09 | 7 358.00 | 587.00 |
| 7 | 20.15 | 0.13 | 7 289.00 | 592.00 |
| 8 | 20.16 | 0.10 | 7 527.00 | 563.00 |
| 9 | 20.18 | 0.08 | 7 284.00 | 610.00 |
| 10 | 20.12 | 0.17 | 7 252.00 | 546.00 |
| 11 | 19.95 | 0.09 | 7 356.00 | 598.00 |
| 12 | 20.92 | 0.08 | 7 432.00 | 599.00 |
| 13 | 19.94 | 0.08 | 7 267.00 | 567.00 |
| 14 | 19.92 | 0.09 | 7 256.00 | 642.00 |
| 15 | 19.91 | 0.09 | 7 389.00 | 609.00 |
| 16 | 19.97 | 0.27 | 7 426.00 | 597.00 |
| 17 | 19.91 | 0.14 | 7 396.00 | 609.00 |
| 18 | 19.93 | 0.25 | 7 298.00 | 548.00 |
| 19 | 19.96 | 0.10 | 7 589.00 | 579.00 |
| 20 | 19.88 | 0.09 | 7 481.00 | 602.00 |
| 均值 | 20.33 | 0.13 | 7 370.65 | 581.75 |
DownLoad:
DownLoad:
