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Energy supply is an important system that affects the overall efficiency of urban transportation. To improve the system operational efficiency and reduce costs, we formulate and solve a collaborative multidepot petrol station replenishment problem with multicompartments and time window assignment by establishing a mixed-integer linear programming model. The hybrid heuristic algorithm composed of genetic algorithm and particle swarm optimization is used as a solution, and then the Shapley value method is applied to analyze the profit allocation of each petrol depot under different coalitions. The optimal membership sequence of the cooperation is determined according to the strict monotone path. To analyze and verify the effectiveness of the proposed method, a regional petrol supply network in Chongqing city in China is investigated. Through cooperation between petrol depots in the supply network, the utilization of customer clustering, time window coordination, and distribution truck sharing can significantly reduce the total operation costs and improve the efficiency of urban transportation energy supply. This approach can provide theoretical support for relevant government departments and enterprises to make optimal decisions. The implementation of the joint distribution of energy can promote the sustainable development of urban transportation.

The collaborative multidepot petrol station (PS) replenishment problem (PSRP) with multicompartments and time window assignment (CMPSRPMT) is a variant of the multidepot PSRP with time windows (MPSRPTW) [

Based on operational requirements, each petrol depot (PD) in the current distribution network is only responsible for specific PSs in a certain region. PDs are cut off from others, and sharing of distribution resources in a network is extremely difficult. The result is a temporal and spatial imbalance between the capacity of distribution trucks and the demand for customer service delivery, such as the idling and roundabout transport of distribution trucks. The inefficient distribution capacity and service quality of several PDs limit the efficiency of the entire distribution network. Moreover, customer demands for heterogeneous service time windows (TWS) require depots to increase their number of delivery trucks, which exert pressure on transportation facilities and depot budgets. Therefore, multidepot cooperative distribution must be based on TWS and distribution TS to effectively assign customer service. Through comprehensive resource sharing, multidepot cooperative distribution reduces costs and improves the service quality of the entire distribution network. Indeed, in China, large oil companies are already collaborating on the distribution of refined products [

In this study, multicompartment truck sharing, TWS, and a cooperation mechanism are integrated into the traditional MPSRPTW as CMPSRPMT. To optimize the CMPSRPMT and improve calculation accuracy, an optimal mathematical model is established to minimize the total operating cost. A hybrid heuristic algorithm that combines genetic algorithm (GA) with particle swarm optimization (PSO) algorithm is designed to achieve near-optimal solution. The Shapley value method is applied to allocate the benefits from cooperation. The membership sequence is analyzed according to the strict monotone path (SMP) principle. Therefore, it is conducive that the solution of the CMPSRPMT can improve the efficiency and flexibility of petrol distribution networks. For the CMPSRPMT, first of all, refined products have special distribution nature, that is, product diversification, and cannot be mixed; Secondly, multidepot cooperation, multicompartment vehicle use, and different time-window coordination mechanisms are comprehensively considered; it can propel the sustainable development of distribution theory and the entire intelligent urban transportation system.

The remaining parts of this study are organized as follows. In

Past decades have scant academic research on the replenishment of PSs. Previous studies focus on optimizing the distribution of a single PD to multiple PSs and several variations of PSRP. For example, the multiperiod, time window, trip packing, and multidepots with TWS are separately considered for the PSRP [

The aforementioned works are some of the few that are directly related to PSRP modeling and research. However, the present study is closely related to the multicompartment truck transportation and TS to be considered in modeling. Derigs et al. [

One part of the CMPSRPMT literature focuses on the problem of cooperation in the distribution networks. Cooperation among PDs largely influences the efficiency of their distribution networks. Qi et al. [

Another part of the CMPSRPMT literature focuses on the TWA in distribution. Neves-Moreira et al. [

Generally speaking, the mathematical model of vehicle assignment and routing problem is NP hard. In order to solve this kind of problem, it is very important to select and apply appropriate algorithms. Kuo et al. [

The aforementioned studies tackle numerous PSRP aspects but suffer from the following issues. (1) The replenishment network design procedure rarely considers the cooperation among PDs by regional partitioning. (2) Minimal attention is paid to distribution TS, multicompartment truck application, and transship transportation among the participants of a collaborative multidepot optimization network. (3) A single intelligent algorithm and heuristic approach are difficult to apply directly to a specific scale of CMPSRPMT with numerous PSs.

Combining observations in Table

Comparison between the existing literature and this study.

Study | Number of depots | Stations per trip | Time windows | Multicompartments | Fleet sharing | Regional partitioning |
---|---|---|---|---|---|---|

Cornillier et al. [ | One | Several | No | Yes | No | No |

Cornillier et al. [ | One | Several | Yes | Yes | No | No |

Boctor et al. [ | One | Several | No | No | No | No |

Cornillier et al. [ | Several | Several | Yes | Yes | No | No |

Popović et al. [ | One | Several | No | Yes | No | No |

Vidović et al. [ | One | Several | No | Yes | No | No |

Wang et al. [ | Several | Several | Yes | No | Yes | No |

Carotenuto et al. [ | One | Several | No | Yes | No | No |

Wang et al. [ | One | Several | Yes | Yes | No | No |

This study | Several | One | Yes | Yes | Yes | Yes |

CMPSRPMT integrates the problems of cooperative distribution, TS, and TWAs. Figure

Noncooperative petrol distribution network.

Figure

Cooperative petrol distribution network.

On the basis of the regional partitioning method, the transport time between petrol depots and stations is used to cluster corresponding PSs. Different customer groups are assigned to different PD services, and the demand of an individual PS is combined with the required time window. Different types of petrol tankers are assigned to distribution on the basis of the appropriate time window for each PD.

Five assumptions underline the corresponding mathematical model. (1) In a relatively short time period, each PS only generates one distribution order for refined products’ demand. The demand for each kind of refined products does not exceed the maximum loading capacity of a single compartment of the distribution truck. (2) Each customer (PS) can only be served once in a time period, but its demand can be fulfilled by multiple distribution trucks. Specifically, one PD can organize a distribution task, but multiple trucks can complete the distribution task. (3) The transportation speed of petrol transfer trucks and petrol distribution trucks is constant. (4) During a time period, distribution trucks may be dispatched repeatedly if time permits, regardless of the transfer time across PDs. (5) The loading and unloading service times of PDs and stations are related not to the refined products’ type but to the truck type and operational quantity.

The proposed model is mathematically formulated as an optimization problem to minimize the total cost when each PD is assigned to serve a group of PSs with different trucks and TWS [

Notations and definitions in the CMPSRPMT.

Symbol | Description |
---|---|

Set of PDs | |

Set of PSs | |

Set of petrol transfer trucks | |

Set of petrol trucks, | |

Set of subdivisions of | |

Represents the maximum capacity of the | |

Set of petrol varieties | |

Represents a working period | |

Represents one time period of a working period, | |

The cost of time delay as the penalty coefficient when | |

The average annual maintenance cost of the petrol transfer truck (dollar/ | |

The average annual maintenance cost of different kinds of petrol distribution truck (dollar/ | |

The total demand of | |

The distribution operating quantity of | |

The total distribution operating quantity of all kinds of petrol for PD | |

The demand quantity of all kinds of petrol for all the PSs in a time period (gallon) | |

The demand quantity of | |

Decision matrix, | |

The fixed cost of PD | |

The variable cost factor of the PD; it is related to the operating quantity of the PD | |

The working periodicity | |

The relationship among PD | |

If the petrol | |

The service cost (including labor cost and operation cost) allocated on behalf of the PD participating in the petrol regional joint distribution network, and it is related to the distribution operating volume, | |

The operating volume coefficient of service cost | |

The cost of time delay as the penalty coefficient when | |

The average annual maintenance cost of the petrol transfer truck (dollar/ | |

The average annual maintenance cost of different kinds of petrol distribution truck (dollar/ | |

The total demand of | |

The distribution operating quantity of | |

The total distribution operating quantity of all kinds of petrol for PD | |

The demand quantity of all kinds of petrol for all the PSs in a time period (gallon) | |

The demand quantity of type | |

Decision matrix, | |

The fixed cost of PD | |

The variable cost factor of PD; it is related to the operating quantity of the PD | |

The working periodicity | |

The relationship among PD | |

If the petrol | |

The service cost (including labor cost and operation cost) allocated on behalf of the PD participating in the petrol regional joint distribution network, and it is related to the distribution operating volume, | |

The operating volume coefficient of service cost |

CMPSRPMT is formulated as a mixed-integer linear programming model to minimize the total cost. The cost function contains three components, namely,

Equation (

Equation (

Equation (

The optimization model of the CMPSRPMT is defined as follows:

Equation (

A GA-PSO hybrid algorithm is designed to address the petroleum distribution of the multidepot PS optimization model. GA is an evolutionary computing approach used to mimic the natural selection procedure and study combinatorial optimization problems [

As a hybrid algorithm, GA-PSO is applied to meet the requirements of the multidepot PSRP optimization for more complex algorithms. Existing traditional heuristics are already proven efficient. However, the complexity of petrol distribution networks limits their capacity in finding near-optimal solutions. As such, properly integrating different methods into a hybrid solution approach such as GA-PSO can effectively improve the optimization results. Other hybrid heuristics exist but may display weaknesses compared with GA-PSO in terms of performance. For example, Chen et al. [

Similar to existing research [

Step 1: algorithm initialization: in the hybrid algorithm, the PD performs the refined products’ delivery service. An integer within

Step 2: given the improved chromosome fitness function value, roulette wheel selection is then carried out. Crossover and mutation operations are executed based on the crossover probability _{c} and the mutation probability _{m}, respectively. The optimal solutions and new chromosomes are updated.

Step 3:

Step 4: repeat the loop steps indicated in Figure

Step 5: if the number of iterations reaches its maximum, the loop process ends. The existing best known particle (chromosome) and fitness function value are selected as the optimal solution; otherwise, the above steps are repeated, starting from Step 2.

Step 6: the optimal solution is calculated and selected from all available chromosomes and serves as the final result for the CMPSRPMT.

The flowchart of the hybrid algorithm.

In the optimized CMPSRPMT, each PS arranges for delivery of refined oil from the nearest PD. Refined products’ dispatching and distribution TS are implemented among PDs to reduce costs and increase profits. Alliance members agree to join the cooperation depending on the fairness of profit allocation. Participants are encouraged to join a coalition where the benefits are proportional to their contributions. Therefore, the leader or coordinator needs to implement an effective revenue delivery mechanism to ensure the group stability. The Shapley value model provides an efficient cost and profit allocation according to the participants’ marginal contributions in cooperative game theory [

The Shapley value is a solution concept that provides a unique solution to the cost allocation problem. The computation formula in the following expresses the cost to be allocated to participant

The Shapley value model is based on four fairness properties, namely, efficiency, symmetry, dummy property, and additivity [

The strictly monotonic path (MPS) strategy [

In formula (

A practical study in Chongqing city in China is considered to evaluate the effectiveness of the proposed CMPSRPMT optimization. As the central city of China, Chongqing is an ideal research subject for this study. Figure

Petrol depots’ and petrol stations’ distribution diagram.

Initial petrol stations’ assignment.

Petrol depot | Petrol station |
---|---|

PD1 | PS1 PS2 PS3 PS4 PS5 PS6 PS7 PS19 PS20 PS29 PS30 PS57 PS58 PS59 |

PD2 | PS8 PS9 PS14 PS15 PS16 PS17 PS18 PS31 PS32 PS33 |

PD3 | PS10 PS11 PS21 PS22 PS23 PS24 PS25 PS26 PS27 PS28 PS45 PS46 PS60 PS61 PS62 |

PD4 | PS34 PS35 PS36 PS39 PS40 PS41 PS42 PS43 PS44 PS63 PS64 PS65 |

PD5 | PS12 PS13 PS37 PS38 PS47 PS48 PS49 PS50 PS51 PS52 PS53 PS54 PS55 PS56 |

Transportation times between facilities in the petrol replenishment network (hour).

PD1 | PD2 | PD3 | PD4 | PD5 | |
---|---|---|---|---|---|

PD1 | — | 2.5 | 1.8 | 2.3 | 1.6 |

PD2 | 2.5 | — | 2.1 | 2.7 | 2.8 |

PD3 | 1.8 | 2.1 | — | 1.7 | 2.0 |

PD4 | 2.3 | 2.7 | 1.7 | — | 2.2 |

PD5 | 1.6 | 2.8 | 2.0 | 2.2 | — |

PS1 | 0.7 | 1.8 | 1.2 | 1.8 | 1.3 |

PS2 | 0.8 | 1.7 | 1.3 | 1.9 | 1.4 |

PS3 | 0.8 | 1.9 | 1.2 | 1.8 | 1.2 |

PS4 | 0.4 | 1.6 | 1.0 | 1.6 | 1.2 |

PS5 | 0.9 | 2.0 | 1.1 | 1.7 | 1.0 |

PS6 | 1.2 | 1.6 | 1.3 | 1.9 | 1.5 |

PS7 | 1.0 | 1.7 | 1.1 | 1.7 | 1.4 |

PS8 | 1.3 | 1.6 | 1.5 | 1.8 | 1.7 |

PS9 | 1.4 | 1.6 | 1.7 | 2.0 | 1.8 |

PS10 | 0.6 | 1.8 | 0.8 | 1.5 | 1.2 |

PS11 | 0.6 | 1.8 | 0.9 | 1.3 | 1.0 |

PS12 | 0.3 | 1.7 | 1.1 | 1.5 | 1.2 |

PS13 | 0.4 | 2.1 | 1.2 | 1.8 | 1.0 |

PS14 | 2.2 | 0.7 | 1.6 | 1.8 | 2.4 |

PS15 | 2.3 | 0.8 | 1.6 | 1.7 | 2.5 |

PS16 | 2.2 | 0.9 | 1.5 | 1.6 | 2.3 |

PS17 | 2.0 | 0.8 | 1.8 | 2.0 | 2.6 |

PS18 | 1.7 | 0.5 | 1.5 | 1.6 | 2.4 |

PS19 | 1.8 | 0.8 | 1.5 | 1.5 | 2.0 |

PS20 | 2.0 | 0.6 | 1.2 | 1.3 | 2.2 |

PS21 | 1.2 | 1.1 | 0.3 | 0.6 | 1.3 |

PS22 | 1.5 | 1.2 | 0.6 | 0.8 | 1.6 |

PS23 | 1.1 | 0.8 | 0.6 | 0.9 | 1.3 |

PS24 | 1.0 | 1.1 | 0.3 | 0.7 | 1.1 |

PS25 | 0.8 | 1.3 | 0.2 | 1.0 | 1.2 |

PS26 | 1.1 | 1.6 | 0.4 | 0.6 | 1.2 |

PS27 | 0.9 | 1.5 | 0.4 | 0.7 | 1.1 |

PS28 | 1.8 | 1.1 | 0.7 | 0.8 | 1.7 |

PS29 | 1.2 | 1.1 | 0.6 | 0.9 | 1.0 |

PS30 | 1.3 | 1.5 | 0.5 | 1.2 | 1.6 |

PS31 | 2.2 | 1.6 | 0.7 | 1.5 | 2.3 |

PS32 | 1.6 | 1.3 | 0.6 | 1.6 | 2.0 |

PS33 | 1.5 | 1.4 | 0.6 | 1.8 | 1.9 |

PS34 | 1.3 | 1.3 | 0.4 | 0.7 | 1.1 |

PS35 | 1.4 | 1.2 | 0.5 | 0.8 | 1.2 |

PS36 | 1.5 | 1.2 | 0.6 | 0.8 | 1.3 |

PS37 | 0.6 | 1.4 | 0.5 | 1.2 | 1.3 |

PS38 | 0.7 | 1.4 | 0.4 | 1.0 | 1.2 |

PS39 | 1.5 | 1.6 | 0.7 | 0.5 | 1.3 |

PS40 | 1.4 | 1.6 | 0.6 | 0.2 | 1.2 |

PS41 | 1.5 | 1.7 | 0.9 | 0.5 | 1.1 |

PS42 | 1.4 | 1.6 | 0.8 | 0.1 | 1.0 |

PS43 | 1.4 | 1.7 | 0.9 | 0.6 | 1.0 |

PS44 | 1.2 | 1.6 | 0.7 | 0.4 | 0.7 |

PS45 | 1.4 | 1.5 | 0.6 | 0.5 | 1.1 |

PS46 | 1.3 | 1.6 | 0.6 | 0.3 | 1.0 |

PS47 | 1.0 | 1.8 | 1.0 | 1.1 | 0.5 |

PS48 | 0.9 | 1.7 | 1.1 | 1.2 | 0.4 |

PS49 | 0.9 | 1.8 | 1.2 | 1.3 | 0.3 |

PS50 | 1.0 | 1.9 | 1.3 | 1.4 | 0.4 |

PS51 | 0.9 | 1.8 | 1.4 | 1.5 | 0.5 |

PS52 | 1.0 | 2.0 | 1.3 | 1.3 | 0.2 |

PS53 | 1.1 | 2.1 | 1.1 | 1.1 | 0.3 |

PS54 | 1.1 | 1.9 | 1.0 | 1.0 | 0.4 |

PS55 | 1.0 | 1.8 | 0.9 | 1.0 | 0.4 |

PS56 | 0.9 | 1.6 | 0.8 | 0.8 | 0.6 |

PS57 | 0.7 | 1.5 | 0.8 | 0.9 | 0.6 |

PS58 | 0.7 | 1.5 | 0.9 | 1.1 | 0.6 |

PS59 | 0.7 | 1.6 | 1.1 | 1.3 | 0.4 |

PS60 | 1.1 | 1.9 | 1.2 | 0.8 | 0.6 |

PS61 | 0.9 | 1.7 | 1.0 | 1.1 | 0.7 |

PS62 | 0.7 | 1.5 | 1.2 | 1.5 | 0.5 |

PS63 | 1.1 | 2.0 | 1.3 | 1.2 | 0.6 |

PS64 | 0.9 | 1.8 | 1.2 | 1.0 | 0.5 |

PS65 | 0.9 | 1.7 | 1.1 | 1.0 | 0.6 |

In this study, three kinds of refined products are considered, namely, nos. 92, 95, and 98. Table

The demand quantity and service time window of each petrol station in a time period (gallon).

Petrol station | Demand | Time window | ||
---|---|---|---|---|

No. 92 | No. 95 | No. 98 | ||

PS1 | 3536 | 2929 | 1401 | (600, 800) |

PS2 | 2027 | 2877 | 56 | (800, 1000) |

PS3 | 2264 | 1191 | 1217 | (800, 1000) |

PS4 | 2097 | 1412 | 1284 | (1000, 1200) |

PS5 | 2514 | 1304 | 2000 | (1000, 1200) |

PS6 | 3288 | 1415 | 939 | (1200, 1400) |

PS7 | 3928 | 2856 | 972 | (1400, 1600) |

PS8 | 2617 | 1625 | 969 | (1400, 1600) |

PS9 | 2476 | 2612 | 1596 | (1600, 1800) |

PS10 | 2042 | 1570 | 1010 | (1600, 1800) |

PS11 | 2418 | 1641 | 699 | (1800, 2000) |

PS12 | 3593 | 2532 | 1045 | (2000, 2200) |

PS13 | 2174 | 1115 | 155 | (2200, 2400) |

PS14 | 3098 | 2484 | 1491 | (1400, 1600) |

PS15 | 3946 | 1430 | 260 | (1000, 1200) |

PS16 | 3047 | 1084 | 362 | (1200, 1400) |

PS17 | 2594 | 1453 | 1010 | (800, 1000) |

PS18 | 3177 | 2199 | 112 | (1400, 1600) |

PS19 | 3334 | 2740 | 471 | (1000, 1200) |

PS20 | 3013 | 1642 | 60 | (2000, 2200) |

PS21 | 2575 | 2634 | 956 | (1000, 1200) |

PS22 | 3075 | 2975 | 1237 | (1200, 1400) |

PS23 | 3976 | 1745 | 152 | (800, 1000) |

PS24 | 2737 | 2027 | 570 | (1400, 1600) |

PS25 | 2118 | 2043 | 704 | (1200, 1400) |

PS26 | 2052 | 1911 | 604 | (1800, 2000) |

PS27 | 3591 | 2252 | 1430 | (1000, 1200) |

PS28 | 2738 | 2312 | 308 | (2000, 2200) |

PS29 | 2132 | 2125 | 1437 | (1400,1600) |

PS30 | 3556 | 1463 | 376 | (800, 1000) |

PS31 | 2691 | 1957 | 247 | (600, 800) |

PS32 | 2389 | 1552 | 1612 | (1600, 1800) |

PS33 | 3417 | 2172 | 215 | (1000, 1200) |

PS34 | 3558 | 1077 | 1718 | (1200, 1400) |

PS35 | 3177 | 1122 | 1845 | (2200, 2400) |

PS36 | 2783 | 1349 | 1325 | (800, 1000) |

PS37 | 3361 | 1676 | 1455 | (2000, 2200) |

PS38 | 3703 | 1757 | 1925 | (1600, 1800) |

PS39 | 2010 | 2857 | 409 | (1000, 1200) |

PS40 | 3869 | 1448 | 625 | (2000, 2200) |

PS41 | 3296 | 1147 | 1516 | (1400, 1600) |

PS42 | 2921 | 1954 | 659 | (1800, 2000) |

PS43 | 3392 | 2825 | 223 | (1000, 1200) |

PS44 | 3032 | 1318 | 1807 | (1400, 1600) |

PS45 | 2912 | 2733 | 192 | (800, 1000) |

PS46 | 3480 | 1299 | 455 | (2000, 2200) |

PS47 | 2545 | 1499 | 158 | (1000, 1200) |

PS48 | 3265 | 1655 | 540 | (1400, 1600) |

PS49 | 3285 | 1553 | 1954 | (800, 1000) |

PS50 | 3064 | 2706 | 1589 | (2000, 2200) |

PS51 | 3744 | 2850 | 87 | (2200, 2400) |

PS52 | 3237 | 2556 | 157 | (1200, 1400) |

PS53 | 2367 | 2895 | 60 | (1600, 1800) |

PS54 | 3547 | 2843 | 263 | (1400, 1600) |

PS55 | 3851 | 1504 | 772 | (1000, 1200) |

PS56 | 3164 | 1089 | 593 | (1800, 2000) |

PS57 | 2770 | 2146 | 1751 | (2000, 2200) |

PS58 | 3495 | 2652 | 271 | (1600, 1800) |

PS59 | 3916 | 1207 | 116 | (1400, 1600) |

PS60 | 3496 | 1489 | 138 | (1800, 2000) |

PS61 | 3984 | 1329 | 204 | (2000, 2200) |

PS62 | 2815 | 2650 | 292 | (1200, 1400) |

PS63 | 3032 | 1798 | 717 | (1000, 1200) |

PS64 | 2994 | 1017 | 1565 | (600, 800) |

PS65 | 2811 | 2148 | 885 | (800, 1000) |

To ensure normal operations, PDs need to bear the fixed and variable costs related to their distribution tasks. Several parameter settings in the hybrid algorithm and model are determined according to the associated previous studies [

Objective function parameters: three kinds of petrol distribution trucks are available. The set of petrol distribution trucks and their compartments are

Algorithm parameters: according to Gao et al. [

In this study, a working period can be divided into several time periods. GA-PSO algorithm is used to assign PSs to corresponding PDs and compute the total cost in one time period. In particular, relevant government departments or core enterprises in the industry are encouraged to offer an initial subsidy, set at 7% of the cost, to those who wish to join the cooperation. Table

Comparison between the initial and optimized network over one time period (unit: USD).

Initial cost | Optimized cost | ||
---|---|---|---|

{PD1} | 1382.87 | 1086.07 | 96.80 |

{PD2} | 1086.90 | 1010.82 | 76.08 |

{PD3} | 1231.99 | 1145.75 | 86.24 |

{PD4} | 970.79 | 902.83 | 67.96 |

{PD5} | 1157.70 | 1076.66 | 81.04 |

{PD1 PD2} | 2469.77 | 2074.34 | 395.43 |

{PD1 PD3} | 2614.86 | 2143.21 | 471.65 |

{PD1 PD4} | 2353.66 | 2053.49 | 300.17 |

{PD1 PD5} | 2540.57 | 2102.84 | 437.73 |

{PD2 PD3} | 2318.89 | 1875.41 | 433.48 |

{PD2 PD4} | 2057.69 | 1808.63 | 249.06 |

{PD2 PD5} | 2244.60 | 1949.18 | 295.42 |

{PD3 PD4} | 2202.78 | 1843.52 | 359.26 |

{PD3 PD5} | 2389.69 | 1892.67 | 497.02 |

{PD4 PD5} | 2128.49 | 1780.36 | 348.13 |

{PD1 PD2 PD3} | 3701.76 | 2807.26 | 894.50 |

{PD1 PD2 PD4} | 3440.56 | 2886.14 | 554.42 |

{PD1 PD2 PD5} | 3627.47 | 2895.92 | 731.55 |

{PD1 PD3 PD4} | 3585.65 | 2909.23 | 676.42 |

{PD1 PD3 PD5} | 3772.56 | 2904.58 | 867.98 |

{PD1 PD4 PD5} | 3511.36 | 2804.67 | 706.69 |

{PD2 PD3 PD4} | 3289.68 | 2651.37 | 638.31 |

{PD2 PD3 PD5} | 3476.59 | 2697.79 | 778.80 |

{PD2 PD4 PD5} | 3215.39 | 2700.79 | 514.60 |

{PD3 PD4 PD5} | 3360.48 | 2607.36 | 753.12 |

{PD1 PD2 PD3 PD4} | 4672.55 | 3665.71 | 1006.84 |

{PD1 PD2 PD3 PD5} | 4859.46 | 3651.67 | 1207.79 |

{PD1 PD2 PD4 PD5} | 4598.26 | 3657.80 | 940.46 |

{PD1 PD3 PD4 PD5} | 4743.35 | 3600.45 | 1142.90 |

{PD2 PD3 PD4 PD5} | 4447.38 | 3408.81 | 1038.57 |

{PD1 PD2 PD3 PD4 PD5} | 5830.25 | 4354.57 | 1475.68 |

Table

Petrol stations’ assignment in the grand coalition.

Petrol depot | Petrol station |
---|---|

PD1 | PS1 PS2 PS3 PS4 PS5 PS6 PS7 PS8 PS9 PS10 PS11 PS12 PS13 |

PD2 | PS14 PS15 PS16 PS17 PS18 PS19 PS20 |

PD3 | PS21 PS22 PS23 PS24 PS25 PS26 PS27 PS28 PS29 PS30 PS31 PS32 |

PD4 | PS39 PS40 PS41 PS42 PS43 PS44 PS45 PS46 |

PD5 | PS47 PS48 PS49 PS50 PS51 PS52 PS53 PS54 PS55 PS56 PS57 PS58 PS59 PS60 |

For comparison purposes, we implement and test the proposed GA-PSO hybrid algorithm, PSO, HGA (hybrid genetic algorithm), and TS-ACO (tabu search and ant colony optimization hybrid algorithm) with same data. PSO simulates the social behavior of bird flocking and fish schooling [

Comparison of algorithms’ performance.

Sequence | Total cost(USD) | Time | ||||||
---|---|---|---|---|---|---|---|---|

GA-PSO | PSO | HGA | TS-ACO | GA-PSO | PSO | HGA | TS-ACO | |

1 | 4355 | 4604 | 4463 | 4444 | 184 | 100 | 269 | 186 |

2 | 4338 | 4523 | 4544 | 4452 | 167 | 122 | 219 | 216 |

3 | 4344 | 4562 | 4462 | 4486 | 214 | 103 | 212 | 194 |

4 | 4310 | 4562 | 4540 | 4463 | 177 | 92 | 269 | 180 |

5 | 4388 | 4546 | 4541 | 4419 | 218 | 91 | 275 | 314 |

6 | 4307 | 4550 | 4510 | 4493 | 163 | 114 | 253 | 214 |

7 | 4359 | 4595 | 4502 | 4467 | 243 | 105 | 197 | 258 |

8 | 4374 | 4608 | 4491 | 4436 | 162 | 125 | 172 | 216 |

9 | 4395 | 4565 | 4556 | 4449 | 167 | 95 | 250 | 199 |

10 | 4313 | 4598 | 4552 | 4422 | 215 | 102 | 183 | 253 |

11 | 4389 | 4601 | 4488 | 4484 | 155 | 154 | 249 | 308 |

12 | 4316 | 4540 | 4478 | 4478 | 230 | 107 | 228 | 253 |

13 | 4380 | 4524 | 4484 | 4417 | 181 | 148 | 240 | 279 |

14 | 4311 | 4530 | 4469 | 4465 | 198 | 131 | 225 | 238 |

15 | 4354 | 4597 | 4483 | 4494 | 238 | 123 | 259 | 281 |

16 | 4347 | 4549 | 4515 | 4501 | 163 | 105 | 197 | 189 |

17 | 4374 | 4552 | 4495 | 4437 | 237 | 144 | 255 | 311 |

18 | 4389 | 4563 | 4474 | 4452 | 197 | 103 | 242 | 198 |

19 | 4309 | 4549 | 4491 | 4433 | 151 | 128 | 179 | 261 |

20 | 4377 | 4518 | 4485 | 4453 | 167 | 108 | 230 | 184 |

Average | 4351.45 | 4561.80 | 4501.15 | 4457.25 | 191.35 | 115.00 | 230.15 | 236.60 |

Compared with other three algorithms, the average optimization cost from our proposed algorithm is of better quality. In terms of computing time, in addition to PSO, the average computational time of our proposed algorithm also has significant advantages. Thus, we can conclude that GA-PSO has the following merits: (1) exchanging best-fit chromosomes and worst-fit particles between the GA and the PSO algorithm enhances the GA-PSO’s capability to obtain better solutions. For example, the GA-PSO algorithm has a higher probability to obtain the best solution. (2) The proposed algorithm combines GA and PSO algorithm’s local and global search capabilities. As illustrated, the average cost from the proposed hybrid algorithm is lower than that of PSO, HGA, and TS-ACO algorithms by 4.8%, 3.4%, and 2.4%, respectively. (3) In the 20 runs, PSO is computed faster than the other three algorithms. Although GA-PSO requires slightly more computation time, the low average cost still makes this algorithm desirable in a practical setting where a slightly longer computation time can be tolerated (e.g., in operation management). Therefore, we conclude that the proposed GA-PSO hybrid algorithm is superior to the three existing algorithms when optimizing multidepot refined products’ distribution networks.

Comparison between the cost changes and the use of distribution trucks before and after the cooperation enables the intuitive analysis of the benefits of the collaborative PS replenishment network. Figures

Comparison between initial and optimized networks’ cost solutions.

Cost reduction variations of different coalitions.

Comparison of the truck usage between initial and optimized networks.

S | Initial truck usage | Truck usage after optimization | The saving of truck usage |
---|---|---|---|

{PD1} | 4a1, 3a2, 4a3 | – | – |

{PD2} | 3a1, 3a2, 2a3 | – | – |

{PD3} | 5a1, 3a2, 4a3 | – | – |

{PD4} | 4a1, 4a2, 2a3 | – | – |

{PD5} | 6a1, 3a2, 2a3 | – | – |

{PD1 PD2} | 7a1, 6a2, 6a3 | 3a1, 2a2, 3a3 | 4a1, 4a2, 3a3 |

{PD1 PD3} | 9a1, 6a2, 8a3 | 3a1, 2a2, 2a3 | 6a1, 4a2, 6a3 |

{PD1 PD4} | 8a1, 7a2, 6a3 | 3a1, 2a2, 2a3 | 5a1, 5a2, 4a3 |

{PD1 PD5} | 10a1, 6a2, 6a3 | 3a1, 2a2, 2a3 | 7a1, 4a2, 4a3 |

{PD2 PD3} | 8a1, 6a2, 6a3 | 2a1, 2a2, 2a3 | 6a1, 4a2, 4a3 |

{PD2 PD4} | 7a1, 7a2, 4a3 | 3a1, 2a2, 2a3 | 4a1, 5a2, 2a3 |

{PD2 PD5} | 9a1, 6a2, 4a3 | 3a1, 2a2, 2a3 | 6a1, 4a2, 2a3 |

{PD3 PD4} | 9a1, 7a2, 6a3 | 3a1, 2a2, 2a3 | 6a1, 5a2, 4a3 |

{PD3 PD5} | 11a1, 6a2, 6a3 | 3a1, 2a2, 2a3 | 8a1, 4a2, 4a3 |

{PD4 PD5} | 10a1, 7a2, 4a3 | 3a1, 2a2, 2a3 | 7a1, 5a2, 2a3 |

{PD1 PD2 PD3} | 12a1, 9a2, 10a3 | 4a1, 2a2, 2a3 | 8a1, 7a2, 8a1 |

{PD1 PD2 PD4} | 11a1, 10a2, 8a3 | 4a1, 2a2, 2a3 | 7a1, 8a2, 6a3 |

{PD1 PD2 PD5} | 13a1, 9a2, 8a3 | 4a1, 2a2, 3a3 | 9a1, 7a2, 5a3 |

{PD1 PD3 PD4} | 13a1, 10a2, 10a3 | 4a1, 2a2, 2a3 | 9a1, 8a2, 8a3 |

{PD1 PD3 PD5} | 15a1, 9a2, 10a3 | 4a1, 2a2, 2a3 | 11a1, 7a2, 8a3 |

{PD1 PD4 PD5} | 14a1, 10a2, 8a3 | 4a1, 2a2, 2a3 | 10a1, 8a2, 6a3 |

{PD2 PD3 PD4} | 12a1, 10a2, 8a3 | 4a1, 2a2, 2a3 | 8a1, 8a2, 6a3 |

{PD2 PD3 PD5} | 14a1, 9a2, 8a3 | 3a1, 2a2, 2a3 | 11a1, 7a2, 6a3 |

{PD2 PD4 PD5} | 13a1, 10a2, 6a3 | 4a1, 2a2, 1a3 | 9a1, 8a2, 5a3 |

{PD3 PD4 PD5} | 15a1, 10a2, 8a3 | 3a1, 2a2, 2a3 | 12a1, 8a2, 6a3 |

{PD1 PD2 PD3 PD4} | 16a1, 13a2, 12a3 | 5a1, 3a2, 2a3 | 11a1, 10a2, 10a3 |

{PD1 PD2 PD3 PD5} | 18a1, 12a2, 12a3 | 5a1, 2a2, 3a3 | 13a1, 10a2, 9a3 |

{PD1 PD2 PD4 PD5} | 17a1, 13a2, 10a3 | 5a1, 3a2, 2a3 | 12a1, 10a2, 8a3 |

{PD1 PD3 PD4 PD5} | 19a1, 13a2, 12a3 | 5a1, 3a2, 3a3 | 14a1, 10a2, 9a3 |

{PD2 PD3 PD4 PD5} | 18a1, 13a2, 10a3 | 4a1, 2a2, 2a3 | 14a1, 11a2, 8a3 |

{PD1 PD2 PD3 PD4 PD5} | 22a1, 16a2, 14a3 | 6a1, 3a2, 3a3 | 16a1, 13a2, 11a3 |

In the CMPSRPMT, TS is an important mechanism. A PD joining the cooperation can coordinate the sharing of petrol distribution trucks. Before cooperation, the PSs’ demands and the required service TWS for each PD must be considered. Then, the largest numbers of delivery trucks are needed to certify customer satisfaction. After cooperation, a chronological and overall time window plan can be made for all PDs. Trucks can be shared and reasonably arranged, which greatly decreases the required number of total petrol distribution trucks in the network. For example, Table

Based on the above parameter settings, the cost and the number of distribution trucks of integrated MILP are calculated. Then, we compared them with the model calculation results adopted in this study. The comparison is shown in Table

Model comparison.

Transportation cost ($) | Penalty cost ($) | Cooperation cost ($) | Total cost ($) | The number of distribution trucks | |
---|---|---|---|---|---|

Integrated MILP | 4560.62 | 315.62 | – | 4876.24 | 18a1, 16a2, 10a3 |

This study | 3900.19 | 204.38 | 250 | 4354.57 | 16a1, 13a2, 11a3 |

Many game theory-based approaches to allocating profits are available. In this study, minimum cost-remaining savings (MCRS), GQP, the improved Shapley value model, and the equal profit method (EPM) are adopted to conduct a comparative analysis that tests the stability of alliance [

According to the snowball theory, the proximity of a profit distribution scheme to the core center reflects the stability of the alliance with the given profit distribution scheme [

Distance between schemes and the core center location.

Profit allocation model | Profit allocation schemes | Core center | Distance |
---|---|---|---|

MCRS | (338, 297, 348, 226, 289) | (333, 281, 368, 206, 298) | 70 |

GQP | (336, 290, 351, 214, 273) | – | 62 |

Shapley | (327, 271, 377, 200, 301) | – | 34 |

EPM | (347, 301, 382, 233, 285) | – | 88 |

Table

Upon the CMPSRPMT optimization, the benefits and cost savings should be reasonably allocated among each depot to ensure the long-term group stability. Then, the Shapley value model and the SMP method introduced above are applied to determine the benefit allocation and order analysis of membership.

According to formula (

Cost allocation in the CMPSRPMT optimization with one alliance (unit: USD).

{PD1} | 96.80 | (96.80, 0, 0, 0, 0) |

{PD2} | 76.08 | (0, 76.08, 0, 0, 0) |

{PD3} | 86.24 | (0, 0, 86.24, 0, 0) |

{PD4} | 67.96 | (0, 0, 0, 67.96, 0) |

{PD5} | 81.04 | (0, 0, 0, 0, 81.04) |

{PD1 PD2} | 395.43 | (208.08, 187.35, 0, 0, 0) |

{PD1 PD3} | 471.65 | (241.11, 0, 230.54, 0, 0) |

{PD1 PD4} | 300.17 | (164.51, 0, 0, 135.66, 0) |

{PD1 PD5} | 437.73 | (226.75, 0, 0, 0, 210.98) |

{PD2 PD3} | 433.48 | (0, 211.66, 221.82, 0, 0) |

{PD2 PD4} | 249.06 | (0, 128.59, 0, 120.50, 0) |

{PD2 PD5} | 295.42 | (0, 145.23, 0, 0, 150.19) |

{PD3 PD4} | 359.26 | (0, 0, 188.77, 170.49, 0) |

{PD3 PD5} | 497.02 | (0, 0, 251.11, 0, 245.91) |

{PD4 PD5} | 348.13 | (0, 0, 0, 167.53, 180.60) |

{PD1 PD2 PD3} | 894.50 | (303.40, 273.96, 317.14, 0, 0) |

{PD1 PD2 PD4} | 554.42 | (225.98, 190.07, 0, 138.37, 0) |

{PD1 PD2 PD5} | 731.55 | (290.32, 208.80, 0, 0, 232.43) |

{PD1 PD3 PD4} | 676.42 | (240.92, 0, 265.19, 170.31, 0) |

{PD1 PD3 PD5} | 867.98 | (279.60, 0, 303.97, 0, 284.41) |

{PD1 PD4 PD5} | 706.69 | (249.94, 0, 0, 190.72, 266.03) |

{PD2 PD3 PD4} | 638.31 | (0, 206.43, 266.61, 165.27, 0) |

{PD2 PD3 PD5} | 778.80 | (0, 212.89, 318.77, 0, 247.14) |

{PD2 PD4 PD5} | 514.60 | (0, 146.76, 0, 169.06, 198.78) |

{PD3 PD4 PD5} | 753.12 | (0, 0, 281.62, 198.04, 273.46) |

{PD1 PD2 PD3 PD4} | 1006.84 | (284.71, 250.22, 325.34, 146.57, 0) |

{PD1 PD2 PD3 PD5} | 1207.79 | (325.58, 258.86, 354.03, 0, 269.32) |

{PD1 PD2 PD4 PD5} | 940.46 | (298.02, 194.85, 0, 176.77, 270.82) |

{PD1 PD3 PD4 PD5} | 1142.90 | (290.06, 0, 321.75, 208.50, 322.59) |

{PD2 PD3 PD4 PD5} | 1038.57 | (0, 212.88, 347.74, 198.03, 279.92) |

{PD1 PD2 PD3 PD4 PD5} | 1475.68 | (327.10, 270.79, 376.82, 199.55, 301.42) |

Table

The analysis of cooperation sequences is extremely important for the benefit allocation strategy and the participants’ willingness to join. In other words, the order in which the participants join the cooperation affects the allocation of benefits and the satisfaction of SMP principles. Figure

Cost reduction percentages and grand alliance formation process.

Formula (

Petrol distribution coalition sequences based on the SMP.

PD1 | PD3 | PD2 | PD5 | PD4 | PD2 | PD3 | PD1 | PD5 | PD4 | PD3 | PD5 | PD4 | PD1 | PD2 | PD4 | PD5 | PD3 | PD1 | PD2 | PD5 | PD3 | PD4 | PD1 | PD2 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

7.00% | — | — | — | — | 7.00% | — | — | — | — | 7.00% | — | — | — | — | 7.00% | — | — | — | — | 7.00% | |||||

17.44% | 18.71% | — | — | — | 19.47% | 18.01% | — | — | — | 20.38% | 21.24% | — | — | — | 17.26% | 15.60% | — | — | — | 21.24% | 20.38% | ||||

21.94% | 25.74% | 23.67% | — | — | 23.67% | 25.74% | 21.94% | — | — | 22.86% | 23.62% | 20.40% | — | — | 20.40% | 23.62% | 22.86% | — | — | 23.62% | 22.86% | 20.40% | |||

23.54% | 28.74% | 23.82% | 23.26% | — | 23.82% | 28.74% | 23.54% | 23.26% | — | 26.12% | 27.86% | 21.48% | 20.98% | — | 21.48% | 27.86% | 26.12% | 20.98% | — | 27.86% | 26.12% | 21.48% | 20.98% | ||

23.65% | 30.59% | 24.91% | 26.04% | 20.56% | 24.91% | 30.59% | 23.65% | 26.04% | 20.56% | 30.59% | 27.94% | 21.56% | 23.65% | 24.91% | 21.56% | 27.94% | 30.59% | 23.65% | 24.91% | 27.94% | 30.59% | 21.56% | 23.65% | 24.91% |

In the five sequences, the cost reduction percentage for each PD reflects the results of taking different PDs as first participants. The optimal cooperation sequence of coalition is

Optimal membership sequence based on the SMP principle.

PD3 | PD5 | PD4 | PD1 | PD2 | |
---|---|---|---|---|---|

7.00% | — | — | — | — | |

20.38% | 21.24% | — | — | — | |

22.86% | 23.62% | 20.40% | — | — | |

26.12% | 27.86% | 21.48% | 20.98% | — | |

30.59% | 27.94% | 21.56% | 23.65% | 24.91% |

The variation of the cost reduction percentage of each petrol depot in the optimal sequence.

The design of collaborative PS replenishment networks on the basis of regional partitioning and rational resource sharing propels the sustainable development of refined products’ distribution and of the entire energy supply system. The CMPSRPMT optimization presents a reasonable division of refined products’ distribution service zones and reduces long-distance and cross-transportation phenomena. Thus, costs are minimized, and additional benefits are offered to each PD. Effective TWA coordination and shared strategies among trucks and PSs are important characteristics of the proposed network. These improvements significantly contribute to energy and social resources and cost savings for PD operators and transportation administrations.

Cooperation among logistics facilities recently plays an important role in the optimization of distribution processes. Incorporating further transportation resource sharing can allow for more cost savings. In addition, traffic administration policies that encourage joint distribution are also a sign of political will to achieve the sustainable development of administrated areas. As one of the main development factors, refined products’ distribution activities can be further organized with the coordination of TWAs and reduce the number of petrol distribution trucks. Therefore, encouraging the formation of a grand coalition is a relevant approach that can benefit not only PDs but the entire society.

This study presents an effective method to solve the optimization of the CMPSRPMT, which improves the cooperation level of PDs and the operational optimization of refined products’ distribution. The distribution and resource sharing are optimized through the regional petrol joint distribution formed by the cooperation among PDs. In the optimization process, we consider the mechanisms of PS clustering, time window coordination, and truck coutilization. The comparison of data before and after the cooperation shows that the total number of distribution trucks and the total operating costs are significantly reduced.

The optimization model considers customer clustering, TWA, and multicompartment distribution TS. As a result, the overall transportation distance and the number of trucks used are reduced. A regional petrol distribution network in Chongqing city in China is taken as an example to evaluate the effect of the proposed model and method in practice. We propose a hybrid heuristic approach based on GA and PSO and use the Shapley value method to allocate the benefits of PD in the cooperation. The optimal sequence of the PD joining the cooperation is found according to the SMP principle.

In summary, the study of petrol cooperative optimization is consistent with the reality. The proposed optimization method is better than that of existing research in this field. Based on the analysis, a summary of the conclusions is provided as follows. (1) Regional petrol distribution network through customer clustering, coordination of TWA, and sharing of distribution trucks can considerably shorten the distribution distance, reduce the required number of distribution trucks, and reduce the total operational costs of the network petrol distribution. (2) Optimizing the PSs served by each depot and using each other’s delivery trucks when the time window allows can highly relieve the traffic stress of refined products’ supply in urban areas, reduce the negative effect of the transport energy supply system, and contribute to the sustainable development of urban transportation. (3) The benefit allocation of each PD and the determination of the optimal entry sequence ensure the formation of a more cooperative distribution network and strengthen the group stability.

The results of this study point toward interesting research directions in the future. The following points can be considered. (1) This study only considers the cooperation between the PD and the PS in the process of distribution operation. Thus, the cooperation can be extended to a wider range of the transport energy supply chain. (2) This study assumes that the transport speed of petrol distribution trucks is constant, which is consistent with most of the existing joint distribution literature. Future research can consider real-time urban traffic speed analysis to achieve more realistic results. (3) In the future, the dynamic CMPSRPMT can be built by considering the product inventory of PDs and PSs.

The transport time, demand quantity, and service time window data used to support the findings of this study are available from the corresponding companies and administrative departments.

The authors declare that they have no conflicts of interest.

This research was supported by the National Nature Science Foundation of China (Grant nos. 71471024 and 71871035), the Humanity and Social Science Foundation of Ministry of Education of China (Grant nos. 18YJC630189 and 17YJA630079), and Social Science Planning Project of Chongqing of China (Grant no. 2019YBGL049).