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Taking the distribution route optimization of refined oil as background, this paper studies the inventory routing problem of refined oil distribution based on working time equilibrium. In consideration of the constraints of vehicle capacity, time window for unloading oil, service time and demand of each gas station, we take the working time equilibrium of each vehicle as goal and establish an integer programming model for the vehicle routing problem of refined oil distribution, the objective function of the model is to minimize the maximum working time of vehicles. To solve this model, a Lingo program was written and a heuristic algorithm was designed. We further use the random generation method to produce an example with 10 gas stations. The local optimal solution and approximate optimal solution are obtained by using Lingo software and heuristic algorithm respectively. By comparing the approximate optimal solution obtained by heuristic algorithm with the local optimal solution obtained by Lingo software, the feasibility of the model and the effectiveness of the heuristic algorithm are verified. The results of this paper provide a theoretical basis for the scheduling department to formulate the oil distribution plan.

Inventory and transportation are the most important issues of the logistics system, which are the two main links of the logistics to obtain the “time value” and the “space value”, and their consumption accounts for 2/3 of the total logistics cost [

Most of the research on inventory routing optimization problem regarded the total distribution time or the total distribution cost minimum as the objective function, but few researchers considered the vehicle’s working time equilibrium problem. In addition to the total distribution cost, another objective is to equilibrate each vehicle’s working time as far as possible in actual arrangements for distribution planning. In this paper, we study the vehicle routing problem of refined oil distribution with hard time window, and formulate the problem into an integer programming model. The objective function of the model is to minimize the maximum working time of all vehicles. In the actual vehicle scheduling, it is often required to equilibrate the working time of the tankers as far as possible, so the problem of this paper has some practical significance. This model includes the constraints of vehicle capacity, demand of gas station and so on. We will design a heuristic algorithm to solve this model efficiently.

Taking the refined oil distribution logistics system into account which is managed by the gas station, the inventory routing problem of the refined oil distribution based on the working time equilibrium can be described as: an oil depot supplies a certain kind of refined oil to n gas stations, supposed that the oil depot has k tankers; When a tanker was filled with refined oil in the oil depot, it started from the oil depot and distributed refined oil to several gas stations, and then returned to oil depot after the distribution task was finished. Each gas station has a fixed unloading time window; the tanker must unload oil for the gas station at its time window. If the arrival time of tanker is earlier than the earliest unloading time of the gas station, the tanker must wait. If the arrival time of tanker is later than the latest time, the gas station will out of stock. The demand of a gas station can be carried out by multiple tankers; the tanker capacity, the demand of refined oil for each gas station are given the distance between gas stations, and distance between gas stations and oil depot are given; the service time and time window of each gas station are given. The working time of a tanker in the above problem refers to the total duration of the tanker from the depot to the end of the return to the oil depot. When all the working hours of the tankers are exactly same, the working time can reach a complete equilibrium state and the maximum working time of all tankers is the minimum. In practice, it is difficult to find an optimal routing scheme which makes the full equilibrium of the working time of each tanker because of the difference of tanker capacity and gas stations’ service time window. When the operating time of each tanker is not exactly the same, the corresponding maximum working time of all vehicles will increased. According to this thought, a mathematical model for the inventory routing problem of refined oil distribution based on the working time equilibrium can be established.

Defines the following symbols and variables at first:

y: the maximum time that all the tankers return to the oil depot after the tasks is finished;

The mathematical model can be described as follows:

The objective function (1) represents the minimization of the maximum working time of all tankers;

Constraint (2) indicates that each gas station is served by at least one tanker;

Constraints (3)-(4) indicate that the start point and the end point of each tanker’s distribution route must be the oil depot;

Constraint (5) indicates that if a tanker drives into a gas station, it must leave the gas station when the task is completed;

Constraint (6) indicates that the total amount of the refined oil which is loaded in a tanker is no more than the capacity of the tanker;

Constraint (7) indicates the relationship of arrival time between two adjacent gas stations on the same distribution route;

Constraint (8) means that the time for tanker to arrive at a gas station must be in its time window;

Constraint (9) indicates that the amount of the refined oil of all tankers deliver to a certain gas station is equal to its demand;

Constraint (10) ensures that the time that all tankers return to the oil depot is no longer than the maximum working time;

Constraints (11)-(12) are the value constraints of variables.

The mathematical model of inventory routing problem based on the working time equilibrium is an integer programming model. Lingo software can be used to solve small size problems directly, but it needs a long time to get the solution of the large scale problems, so this paper will design a heuristic algorithm to solve this model.

Several definitions are given as follows.

Set of feasible successor gas stations: the set of feasible successor gas stations that can be served when a tanker unload oil at a gas station. Every gas station has a fixed time window to accept the service, and tanker needs a certain time to drive from current gas station to the successor one. If the arrival time to the successor gas station just falls in its service time window, the gas station will be a feasible successor gas station. Since each gas station has a time window, it is impossible that each gas station becomes a successor gas station.

According to the following rules, we can calculate the set of feasible successor gas stations.

Because we consider the calculation of the set of feasible successor gas stations with hard time window, in order to avoid tankers waiting and gas station out of stock, it is necessary to calculate the feasible successor gas stations of oil depot and every gas station in the heuristic algorithm calculation process.

Assumed

If there is an overlap between the time window of gas station i and the revised time window of gas station j corresponding to the gas station i, gas station j can be a successor gas station of gas station i, then gas station j can be added into the set of feasible successor gas stations of gas station i, otherwise, gas station j is not allowed to join in the set of feasible successor gas stations of gas station i.

For every gas station j in the set of the feasible successor gas stations of gas station i, we calculate its transfer probability

where

After finishing the service for gas station i, the tanker will select a gas station from the set of feasible successor gas stations based on the transfer probability. Firstly, the transfer probability is calculated, and the successor gas station is chosen by the roulette wheel method, and added to the tanker’s route.

The heuristic algorithm can be described as follows:

Input: time window, demand and service time of each gas station, distance matrix and time matrix between gas stations and oil depot, the number of tankers, capacity of each tanker;

Step 0: Initialization

The departure time of tanker k from the oil depot is

The set of gas stations that need to be served is

Step 1: Calculate the revised time window of each gas station j in the set of V to the gas station (or oil depot) where the tanker k is, and check whether there is an overlap between the time window of gas station (or oil depot) where tanker k is and the revised time window of gas station j. If there is an overlap, then gas station j will be added into the set

Step 2: Calculate the transfer probability of tanker k to each feasible successor gas station by equation

Suppose the successor gas station is j, update

Step 3: If the demand of gas station j is less than the residual capacity of tanker k, then the replenishment quantity for gas station j is

Step 4: If the set

Step 5: If the set V is empty, terminate the algorithm; otherwise,

Step 6: If all tankers have been used up, terminate the algorithm; otherwise, go to Step 1.

Output: the distribution route and loading capacity of each tanker, the unloading capacity of the refined oil per tanker at each gas station, the working time of each tanker, and the maximum working time to complete the distribution task.

Suppose there is an oil depot supplies refined oil to 10 gas stations.0 indicates oil depot; 1 - 10 indicate gas stations. There are 3 tankers in the oil depot, the driving speed of each tanker is 50 km/h, the capacity of each tanker is shown in

Firstly, we use the Lingo software to solve the integer programming model which is established in this paper. When the solution option set to use global solver, Lingo cannot get the global optimal solution after 130 hours

Tanker | 1 | 2 | 3 |
---|---|---|---|

Capacity | 52 | 48 | 54 |

Gas station | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

q | 14 | 17 | 15 | 13 | 16 | 14 | 20 | 12 | 15 | 17 |

TW | [0.4, 1.2] | [0.6, 1.5] | [0.8, 1.7] | [0.7, 1.6] | [1, 1.8] | [1, 1.6] | [0.4, 1.5] | [0.6, 2] | [1.4, 1.9] | [1.3, 2] |

s | 0.1 | 0.15 | 0.12 | 0.08 | 0.13 | 0.1 | 0.17 | 0.06 | 0.12 | 0.15 |

Gas station | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|---|

0 | - | 20 | 25 | 24 | 28 | 27 | 22 | 23 | 20 | 21 | 26 |

1 | - | 16 | 23 | 18 | 28 | 24 | 19 | 34 | 18 | 11 | |

2 | - | 9 | 16 | 22 | 27 | 30 | 17 | 13 | 24 | ||

3 | - | 30 | 27 | 39 | 31 | 12 | 18 | 24 | |||

4 | - | 18 | 24 | 21 | 33 | 12 | 26 | ||||

5 | - | 26 | 20 | 14 | 30 | 27 | |||||

6 | - | 34 | 26 | 18 | 27 | ||||||

7 | - | 21 | 32 | 25 | |||||||

8 | - | 26 | 35 | ||||||||

9 | - | 25 | |||||||||

10 | - |

Gas station | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|---|

0 | - | 0.4 | 0.5 | 0.48 | 0.56 | 0.54 | 0.44 | 0.46 | 0.40 | 0.42 | 0.52 |

1 | - | 0.32 | 0.46 | 0.36 | 0.56 | 0.48 | 0.38 | 0.68 | 0.36 | 0.22 | |

2 | - | 0.18 | 0.32 | 0.44 | 0.54 | 0.60 | 0.34 | 0.26 | 0.48 | ||

3 | - | 0.60 | 0.54 | 0.78 | 0.62 | 0.24 | 0.36 | 0.48 | |||

4 | - | 0.36 | 0.48 | 0.42 | 0.66 | 0.24 | 0.52 | ||||

5 | - | 0.52 | 0.40 | 0.28 | 0.60 | 0.54 | |||||

6 | - | 0.68 | 0.52 | 0.36 | 0.54 | ||||||

7 | - | 0.42 | 0.64 | 0.50 | |||||||

8 | - | 0.52 | 0.70 | ||||||||

9 | - | 0.50 | |||||||||

10 | - |

operating. Therefore, the solution option set to the local solver. After 70 hours of operation, the local optimal solution of the model is obtained by Lingo, and the results are as follows:

The distribution routes of the 3 tankers are respectively: 0-1-4-6-9-0; 0-8-7-5-0; 0-1-2-3-10-0.

The arrival time of each tanker to the gas stations is shown in

From

The amount of refined oil distributed for each gas station is shown in

Next, according to the heuristic algorithm coded by Matlab software, we obtain an approximate optimal distribution route for 3 tankers: 0-1-3-2-8-0; 0-1-4-6-9-0; 0-7-5-10-0.

The arrival time of each tanker to the gas station is shown in

From

The amount of refined oil per tanker distributed for each gas station is shown in

By comparison, we find that the approximate optimal solution obtained by the heuristic algorithm is approximately the same as the local optimal solution obtained by Lingo. But the local optimal solution of Lingo is better in terms of eliminating the working time difference of each tanker. But it needs a long time to obtain local optimal solution by Lingo, which cannot satisfy the requirement to obtain the optimal solution in short time. The operation time is greatly shorter by using heuristic algorithm coded by Matlab than using Lingo software to solve the integer programming model.

i r k | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 0 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0 | 0.4 | 0 | 0 | 0.86 | 0 | 1.42 | 0 | 0 | 1.88 | 0 | 2.42 |

2 | 0 | 0 | 0 | 0 | 0 | 1.75 | 0 | 1.18 | 0.6 | 0 | 0 | 2.42 |

3 | 0 | 0.4 | 0.82 | 1.15 | 0 | 0 | 0 | 0 | 0 | 0 | 1.75 | 2.42 |

i indicates the gas station; r represents arrival time; k refers to the tanker.

i d k | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 0 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0 | 9 | 0 | 0 | 13 | 0 | 14 | 0 | 0 | 15 | 0 | 0 |

2 | 0 | 0 | 0 | 0 | 0 | 16 | 0 | 20 | 12 | 0 | 0 | 0 |

3 | 0 | 5 | 17 | 15 | 0 | 0 | 0 | 0 | 0 | 0 | 17 | 0 |

i indicates the gas station; d represents distribution quantity; r refers to the tanker.

i r k | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 0 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0 | 0.4 | 1.26 | 0.96 | 0 | 0 | 0 | 0 | 1.75 | 0 | 0 | 2.21 |

2 | 0 | 0.4 | 0 | 0 | 0.86 | 0 | 1.42 | 0 | 0 | 1.88 | 0 | 2.42 |

3 | 0 | 0 | 0 | 0 | 0 | 1.03 | 0 | 0.46 | 0 | 0 | 1.70 | 2.37 |

i indicates the gas station; r represents arrival time; k refers to the tanker.

i d k | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 0 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0 | 8 | 17 | 15 | 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |

2 | 0 | 6 | 0 | 0 | 13 | 0 | 14 | 0 | 0 | 15 | 0 | 0 |

3 | 0 | 0 | 0 | 0 | 0 | 16 | 0 | 20 | 0 | 0 | 17 | 0 |

i indicates the gas station; d represents distribution quantity; k refers to the tanker.

The inventory routing problem is the key problem in making the distribution plan of the refined oil. The working time equilibrium of each tanker often needs to be considered in the actual arrangement of the oil delivery scheme. In this paper, we study the inventory routing problem, which is to make the working time equilibrium of the tanker as far as possible. The mathematical model of the problem is established and Lingo program is written for solving the model. We further design a heuristic algorithm to solve the problem quickly. The model and algorithm in this paper provide a theoretical basis for the formulation of the oil distribution plan.

This paper only considers the inventory routing problem of single refined oil distribution and assumes that the demand of each gas station is determined, and a tanker is allowed to unload at multiple gas stations. In practice, the demand of gas stations is usually a random variable and the tanker for distribution of the refined oil usually has a plurality of compartments of different volumes. In order to facilitate the measurement of gas stations, the oil in one compartment must be unloaded to one gas station, in other words, the oil in one compartment can’t be unloaded to multiple gas stations. We will consider a variety of refined oil distribution problem in the future research and add the vehicle compartment constraints and other conditions in order to get the results that are more suitable to the actual distribution plan.

This work was supported by the National Natural Science Foundation of China (11131009, 71540028, F012408), the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (CIT & TCD20130327), and Major Research Project of Beijing Wuzi University. Funding Project for Technology Key Project of Municipal Education Commission of Beijing (ID: TSJHG 201310037036); Funding Project for Beijing Key Laboratory of Intelligent Logistics System; Funding Project of Construction of Innovative Teams and Teacher Career Development for Universities and Colleges Under Beijing Municipality (ID: IDHT20130517); Funding Project for Beijing Philosophy and Social Science Research Base Specially Commissioned Project Planning (ID: 13JDJGD013).

ZhenpingLi,ZhiguoWu, (2016) Study on the Inventory Routing Problem of Refined Oil Distribution Based on Working Time Equilibrium. American Journal of Operations Research,06,17-24. doi: 10.4236/ajor.2016.61003