Fuel Replenishment Problem of Heterogeneous Fleet in Initiative Distribution Mode

Abstract

Petrol, a vital energy source for residents’ consumption and economically sustainable operation, generates substantial distribution demand. To reduce distribution costs, we propose a fuel replenishment problem using a heterogeneous fleet based on the initiative distribution mode. In this mode, the distribution center determines both the delivery orders of customers and the distribution plan. We develop a mathematical model with minimal operational costs, including transport, employment, and penalty costs. A Two-stage heuristic algorithm K-IBKA based on time-space clustering is proposed, which also combines the advantages of the butterfly optimization algorithm in quick convergence and hierarchical mutation strategy in population diversity. The results demonstrate that: (1) Heterogeneous truck distribution exhibits better cost advantages compared to homogeneous distribution, reducing total costs by 13.07%; (2) Compared to passive distribution mode, the total cost of the initiative distribution mode is reduced by 11.03% and 41.80%, respectively, through small and large-scale instances. (3) Compared with the unimproved BKA, ALNS, and GA, the total cost calculated by K-IBKA is reduced by 37.68%, 35.30%, and 27.26%, respectively, thus demonstrating the contribution of this work to reducing the cost of petrol distribution and achieving sustainable development of distribution.

1. Introduction

With the development of society, the demand for petrol products in automobiles, machinery, agriculture, and other industries is growing. To replenish petrol stations daily, the distribution company must simultaneously determine detailed routes and product assignments for each truck [1,2]. Generally, a tanker truck is separated into several compartments storing several incompatible oil products. At the beginning of the delivery task, the trucks load the product at the oil depot and then travel to one or more petrol stations to deliver the petrol products. Once empty, the trucks return to the depot to resupply, after which they continue serving other stations. At the end of the day, the trucks return to the depot [3]. This problem is called the fuel replenishment problem (FRP), a variant of the multi-trip and multi-compartment vehicle routing problem (MTMCVRP).

The distribution mode of petrol products is mainly classified into two types: passive distribution mode (PDM) and initiative distribution mode (IDM). Most distribution companies adopt the PDM, in which each petrol station orders products determined by the station’s average daily sales and the difference between the capacity of the station’s oil tanks and an estimation of the remaining stock, then the company transports the products [4]. However, PDM has the following problems: (1) The specified replenishment time of different petrol stations may be highly similar, which makes it very difficult to dispatch tanker trucks. (2) It is difficult to arrange the delivery time of multiple products at a petrol station only by experience or inventory control strategy, resulting in increased operating costs. Contrarily, in the IDM, the distribution center fully masters the sales data of the petrol stations and makes replenishment plans for each station by combining the sales forecast with the remaining inventory, thereby improving the efficiency of the distribution system. It is worth noting that transport costs account for a large portion of the total cost of the existing fuel replenishment system. Therefore, improving this distribution mode is crucial.

This paper deals with an FRP with a fleet of heterogeneous vehicles that occurs in the context of IDM. We denote this problem as the fuel replenishment problem in initiative distribution mode (FRP-IDM). It involves the features of MTMCVRP and addresses assigning routes for a heterogeneous fleet of multi-compartment tanker trucks. The objective of FRP-IDM is to generate an optimal distribution routing scheme that minimizes distribution costs while effectively coordinating the time windows for multiple oil tanks. Two decisions need to be made: (1) The arrangement of the distribution schedule of trucks. Every truck is allowed to execute multiple trips. (2) The compartment assignment of each type of petrol product. The contributions of this research work are summarized as follows:

(1)

IDM is introduced to FRP, which coordinates the fuel replenishment system entirely and improves efficiency in the dispatch of trucks.

(2)

By introducing temporal and spatial distance, oil tanks with similar time windows and positions can be clustered and distributed centrally, which significantly reduces the repeated routes and inaccurate distribution.

(3)

A Two-stage heuristic algorithm, K-IBKA, which clusters first and then optimizes with an improved Black-winged kite (I-BKA) algorithm, is developed. This algorithm performs better than previous algorithms.

The remainder of this paper is organized as follows: A literature review is provided in Section 2. The problem description and details on the mathematical formulation are presented in Section 3. The solution of the proposed model and two-stage heuristic algorithm is introduced in Section 4. We demonstrate the model and algorithm’s practicability in Section 5. Finally, the conclusions and ideas for future research are provided in Section 6.

2. Literature Review

The FRP is essentially a kind of vehicle routing problem. The MTMCVRP is closely related to the research in this paper. Therefore, this section introduces MTMCVRP and focuses on the comparative analysis of relevant studies on the optimization problem of FRP-IDM and algorithm.

El-Fallahi first defined the MCVRP and proposed a memetic algorithm with a post-optimization phase based on path re-linking and a tabu search method. In this problem, the distribution center uses tanker trucks of the same or different types for distribution. The tanker truck has limited capacity, and the product in each compartment can only be delivered to one customer [5]. MCMTVRP has gained a lot of attention, mainly in FRP with multiple petrol products but also in several other real-world applications, such as grocery distribution, blood transportation, livestock feed distribution, milk collection, etc. [6].

2.1. PDM

As one of the variants of MCMTVRP, FRP is often studied by scholars primarily based on PDM, which means the demands are known. In the instance given by Avella, petrol stations keep in touch with the distribution company and place orders in advance, and then the company arranges the distribution route the next day [7]. Cornillier divides the FRP into two sub-problems, including a truck loading problem and a routing problem. That is, assigning the products to the corresponding compartments and routes. In later studies, Cornillier added the time window consideration to [8], then described two heuristics based on arc and route preselection to solve it [9]. In former studies, the trucks only picked products from a single oil depot, and then Cornillier added multi-depot to previous studies [10]. Because the enormous customer demands often necessitate multiple deliveries, Wang raised the FRP considering split delivery and a heterogeneous fleet of vehicles. An Adaptive Large Neighborhood Search (ALNS) heuristic is applied to solve a large instance with 60 customers and 20 vehicles [3]. Considering that petrol product is a dangerous commodity, Wu and Li also weighed the risks of storing and transporting petrol while arranging the dispatch [11]. Assuming no time limit and the station could be visited more than once, Onut used a fleet of heterogeneous trucks to distribute petrol products and solved the problem, including the daily demands of 13 stations [12].

2.2. IDM

As for the research on IDM, Wei proposed a replenishment problem of highway petrol stations in this mode. According to predictive time-varying sales at petrol stations, real-time road congestion, and a series of operational constraints, a detailed task assignment is designed [4]. Considering some petrol stations’ small capacity and high sales rate, Liu and Zuo split the demand within a day. They proposed mixed-integer linear programming (MILP) to minimize the total cost, which was solved by a variable neighborhood algorithm of fusion-simulated annealing [13]. To serve multiple tanks in a station on the same trip. Xu proposed a joint replenishment strategy and developed an evolution algorithm with a generalized opposition-based learning (GOBL-DE) algorithm [14].

2.3. Algorithms for FRP

There are three main types of algorithms for solving VRP: exact algorithm, classical heuristic algorithm, and meta-heuristic algorithm. In terms of solving FRP, which is a variant of VRP, most studies have chosen heuristic algorithms. This is because the solution quality of the exact algorithm is higher than others, but the calculated speed and problem scale are limited [4,7]. In contrast, heuristic algorithms are able to solve problems faster and handle larger-scale FRPs. Metaheuristic algorithms include genetic algorithms (GAs) [15], adaptive large neighborhood search (ALNS) heuristic algorithms [11], and ant colony optimization (ACO) [13], which have been shown to have advantages in solving FRP problems.

Meanwhile, many scholars combined the advantages of multiple algorithms to improve the algorithms for higher optimization efficiency for FRP. Ada examined a new variant of petrol station replenishment problem (PSRP) by considering a multi-depot vehicle routing problem with open inter-depot routes (MDVRPOI) and proposed a tabu-based adaptive large neighborhood search (T-ALNS) algorithm [16]. Xu proposed a hybrid heuristic approach based on GA and PSO and used the Shapley value method to allocate the benefits of petrol depots in the cooperative distribution [17].

Table 1. Literature comparison.

	Heterog.	TW	MC	MT	MP	IDM	Customers	Solution Method
Avella et al. (2004) [7]	√		√				60	Branch-and-price
Cornillier et al. (2008) [8]	√		√	√			30	Exact algorithm
Cornillier et al. (2009) [9]	√	√	√	√			30	Two-phrase heuristic
Cornillier et al. (2012) [10]	√	√	√	√			50	Two-phrase heuristic
Onut et al. (2014) [12]	√						13	Exact algorithm
L. Wang et al. (2020) [3]	√		√	√	√		60	Adaptive Large Neighborhood Search (ALNS)
Xu et al. (2020) [17]	√	√	√	√	√		65	Heuristic algorithm composed of genetic algorithm (GA) and particle swarm optimization (PSO)
Wei et al. (2021) [4]		√		√		√	8	Exact algorithm
Xu et al. (2023) [14]		√	√		√	√	40	Differential evolution algorithm with generalized opposition-based learning (GOBL-DE) algorithm
Liu & Zuo. (2024) [13]		√	√	√	√	√	72	A hybrid ant colony optimization with variable neighborhood search (ACO-VNS)
Ada Che et al. (2023) [16]	√		√	√	√		72	Tabu-based adaptive large neighborhood search (T-ALNS)
This study	√	√	√	√	√	√	60	Two-phrase heuristic

TW: Time Window; MC: Multi-Compartment; MP: Multi-Product; MT: Multi-Trip; IDM: Initiative distribution mode; “√” means that this paper takes this factor into consideration.

compares the main references to FRP. In summary, the objective function of FRP is mainly to reduce transportation costs. The solution algorithm is mainly based on heuristic algorithms, such as GA, ALNS, VNS, or a combination of multiple algorithms. The comparison of the customer scale studied shows that the scale of problems solved by the exact algorithm is generally small. The heuristic algorithm has the advantage of a large scale of solutions. In addition, most of the previous related studies belong to the PDM. In the existing studies of this model, although scholars have taken into account the constraints of various actual situations, the petrol stations are the ones who report the demand. In the modeling process, the demand and replenishment time window of the petrol stations are considered to be known, and the distribution center only distributes oil products passively. However, since each petrol station in the PDM is an independent individual, each station only considers its own orders when reporting the demand. The lack of systematic arrangements leads to unreasonable scheduling plans, and the model applicability of this mode begins to decline. Therefore, the replenishment model of petrol stations is gradually changing to IDM based on petrol station sales and inventory. It is more realistic to study the FRP in this mode. In the research on the IDM model shown in Table 1, compared with the PDM, their solution algorithm is more complicated. This is because more constraints are considered in the IDM, and the research scale is gradually increasing.

However, in existing studies on IDM, no study considers heterogeneous fleet, multi-compartment, multi-product, multi-trip, and IDM simultaneously. Therefore, this paper will construct an FRP-IDM model that considers the above factors. In addition, based on the existing algorithms, this paper proposes integrating K-means based on Time-space clustering with the improved BKA algorithm (I-BKA) to improve the quality of problem-solving.

3. Construction of Heterogeneous Fleet of Petrol Distribution Route Planning Model Based on IDM

In this section, we introduce a real-world case of a petrol distribution company. It distributes all types of petrol products from the central oil depot to petrol stations in an area. Figure 1 shows the complete petrol supply system. Each station orders several types of products during the working day and then stores the product in an underground oil tank with a known volume. In this section, petrol products are labeled #92, #95, and #98 according to the ratio of isooctane to n-heptane.

The distribution company adopts IDM to master the sales, capacity, and inventory data of each station to calculate the demand for each station’s product. The objective of the distribution plan is to minimize the transportation cost, the employment cost, and the penalty cost.

Define the entire distribution network as $G\left(O, E\right)$, $E=\left\{\right(i,j)|i,j\in O\}$ is the set of edge. $O=\{0, 1, 2, \dots n\}$ is point set, including a central oil depot $M=\left\{0\right\}$ and oil tanks. $I=\{1, 2, \dots n\}$ is the set of oil tanks. The first trip of a tanker truck is defined as: the truck starts from depot M at the beginning of the distribution cycle and then serves the designated oil tanks before returning to depot M. After a certain time of loading petrol products, it continues to the next trip. A truck charged an employment cost once at the beginning of the first trip. Additional employment costs should be paid if the standard working hour is exceeded. All the trips of a truck are called a scheduling plan. In addition, all tanker trucks stop in the garage of the oil depot at the beginning and end of the distribution cycle.

3.1. Assumptions

Prior to modeling, the following assumptions are proposed:

(1)

The compartments should be completely full or emptied once begin to load or the delivery is started.

(2)

The management and planning, such as inventory holding costs, are not considered, and attention should be focused only on daily delivery schedules.

(3)

Each station sells several kinds of petrol products, which are stored separately in underground tanks. The capacity, initial inventory, and sales speed of each tank are known.

(4)

The delivery volume shall not exceed the capacity of the tank or compartment.

(5)

If the tanker truck arrives but the oil tank cannot accommodate the demand, it needs to wait inside the station. The oil tank continues to sell until it can accommodate the demand. Then, the truck starts unloading.

(6)

Each compartment can only hold one product and can only be delivered to a single oil tank. When returning to the oil depot for the next trip, the compartment can be loaded with any product.

3.2. Mathematical Model

The model symbols are shown in

Table 2. Sets, parameters, and decision variables.

Symbol	Definition
Sets
$I$	$\mathrm{Set} \mathrm{of} \mathrm{oil} \mathrm{tanks}, I=\{1,2 ,\dots i_m$}
O	Set of all node, O = M ∪ I, M is the oil depot
$K$	$\mathrm{Set} \mathrm{of} \mathrm{truck} \mathrm{types}, K=\{1,2 ,\dots ,k_m$}
$P$	$\mathrm{Set} \mathrm{of} \mathrm{the} \mathrm{different} \mathrm{types} \mathrm{of} \mathrm{petrol} \mathrm{products}, P=\{1,2 ,\dots ,p_m$}
$R$	$\mathrm{Set} \mathrm{of} \mathrm{trips} \mathrm{of} \mathrm{the} \mathrm{truck}, R=\{1,2 ,\dots ,r_m$}
$U_k$	$\mathrm{Set} \mathrm{of} \mathrm{compartments} \mathrm{of} \mathrm{the} \mathrm{truck} \mathrm{of} \mathrm{type} k, U_k=\{1,2,\dots ,u_m\}$
${\mathit{Order}}_i$	Order of oil tank i
Parameters
$L_i^0$	The initial inventory of the oil tank i (L)
$v_i$	$\mathrm{The} \mathrm{sales} \mathrm{rate} \mathrm{of} \mathrm{oil} \mathrm{tank} i$(L/h)
$v_c$	The driving speed of the truck (km/h)
$G_i$	$\mathrm{The} \mathrm{capacity} \mathrm{of} \mathrm{oil} \mathrm{tank} i$(L)
${\alpha }_k$	The unit transportation cost of the truck of type k (CNY)
β	The unit employment cost for additional driver (CNY)
${\gamma }_k$	The fixed cost of the truck of type k (CNY)
$C_p$	The unit penalty cost (CNY)
$w_i^{\mathit{kvr}}$	Number of available compartments while the truck arrives at node i in trip r
$Q^k$	The capacity of each compartment (L)
$a_i$	$\mathrm{The} \mathrm{number} \mathrm{of} \mathrm{compartments} \mathrm{required} \mathrm{of} {\mathit{Order}}_{\mathit{i}}$
$D_i$	Demand of petrol product p of oil tank i (L)
$d_{\mathit{ij}}$	The distance between node i and node j (km)
$Q_i^s$	Safety stock of the oil tank i
$t_{\mathit{ij}}$	Transportation time of nodes i to j (h)
$\phi$	Discharge velocity of petrol (L/h)
T	The length of the study horizon (h)
$T_{\mathit{work}}$	The maximum working time of a driver (h)
$N_t$	Number of trucks used in the distribution plan
e	The fixed loading time of a truck in the depot (h)
$[{\mathit{ET}}_i,{\mathit{LT}}_i]$	The time window of oil tank i (h)
$t_r^{\mathit{kv}}$	The loading time of the truck at the oil depot in trip r.
$t_i^a$	The time of truck arrives at oil tank i
$t_i^w$	The waiting time for the truck at the oil tank i (h)
$t_i^u$	Duration of out of stock at oil tank i (h)
$t_i^s$	The beginning time of unloading at oil tank i (h)
$t_i^l$	Duration of unloading at oil tank i (h)
$M_0$	The sufficiently large number.
Decision variables
$x_{\mathit{ij}}^{\mathit{kvr}}$	Binary variable, which equals to 1 if truck k of type v travels from i to j in trip r and otherwise, 0.
$y_i^{\mathit{kvr}}$	$\mathrm{Binary} \mathrm{variable}, \mathrm{which} \mathrm{equals} \mathrm{to} 1 \mathrm{if} {\mathit{Order}}_i$ is served by truck k of type v in trip r.
$z_{\mathit{kv}}$	Binary variable, which equals to 1 if truck k of type v is used and otherwise, 0.

.

The FRP-IDM considering heterogeneous fleet distribution can be expressed as Formula (1). It takes the minimum total cost of distribution as the objective function. The model is expressed as follows.

$$min C{=C}_1+C_2+C_3$$

(1)

3.3. Objective Function

(1)

Transport cost

The transport cost is the fuel consumption cost associated with the transport distance, as shown in Formula (2).

$$C_1=\sum _{i\in O}\sum _{j\in O}\sum _{k\in K}\sum _{v\in V}\sum _{r\in R}{d}_{\mathit{ij}}{x}_{\mathit{ij}}^{\mathit{kvr}}{\alpha }_k$$

(2)

(2)

Employment cost

The employment cost of a truck includes the driver’s salary, depreciation cost, insurance premium, and other expenses related to truck running. If the truck’s working hours are not exceeded, only once will the employment cost be considered. If the maximum working hours are exceeded, hire another driver and pay additional employment costs. The calculation is shown in the following Formulas (3) and (4).

$$T_{\mathit{kv}}={\displaystyle \frac{\sum _{i\in O}\sum _{j\in O}\sum _{r\in R}{d}_{\mathit{ij}}{x}_{\mathit{ij}}^{\mathit{kvr}}}{v_c}}+\sum _{i\in I}\sum _{r\in R}{y}_i^{\mathit{kvr}}(t_i^w+t_i^l)+\sum _{r\in R}{t}_r^{\mathit{kv}}, k\in K, v\in V$$

(3)

$$C_2=\mathit{max}\{0,\sum _{k\in K, v\in V}({T_{\mathit{kv}}-T}_{\mathit{work}}\left)\right\}\beta +{\gamma }_k\sum _{k\in K}\sum _{v\in V}{z}_{\mathit{kv}}$$

(4)

(3)

Penalty cost

The penalty cost is the product of the sum of the out-stock time of all tanks and the unit penalty cost, as shown in Formula (5).

$$C_3=C_p\sum _{i\in I}{t}_i^u$$

(5)

A time window determining method based on IDM is proposed as follows.

The demand for the oil tank is the minimum integer multiple of the delivery units that makes the oil tank larger than the safety stock $Q_i^s$ at the end of the study horizon. ${\mathit{ET}}_i$ is the earliest time that can accommodate the demand and ${\mathit{LT}}_i$ is the time when the inventory is sold out. Therefore, formulas of [${\mathit{ET}}_i$,${\mathit{LT}}_i$] are given in (6) to (9). In addition, time of different states of the ${\mathit{Order}}_i$ are calculated according to Formulas (10) to (13).

$$G_i=L_i^0+D_i-v_i{\mathit{ET}}_i,\forall i\in I$$

(6)

$${\mathit{ET}}_i={\displaystyle \frac{L_i^0+D_i-G_i}{v_i}},\forall i\in I$$

(7)

$$L_i^0-v_i{\mathit{LT}}_i=0,\forall i\in I$$

(8)

$${\mathit{LT}}_i={\displaystyle \frac{L_i^0}{v_i}},\forall i\in I$$

(9)

$$t_i^w=\mathit{max}\left\{0,{\mathit{ET}}_i-t_i^a\right\},\forall i\in I$$

(10)

$$t_i^u=\mathit{max}\left\{0,t_i^a-{\mathit{LT}}_i\right\},\forall i\in I$$

(11)

$$t_i^s=\mathit{max}\left\{{\mathit{ET}}_i,t_i^a\right\},\forall i\in I$$

(12)

$$t_i^l={\displaystyle \frac{D_i}{\phi }},\forall i\in I$$

(13)

3.4. Constraints

$$\sum _{i\in O}\sum _{\forall k\in K}\sum _{v\in V}\sum _{ r\in R}{x}_{\mathit{ij}}^{\mathit{kvr}}=\sum _{i\in O}\sum _{k\in K}\sum _{v\in V}\sum _{r\in R}{x}_{\mathit{ji}}^{\mathit{kvr}}, \forall j \in I$$

(14)

$$\sum _{k\in K}\sum _{v\in V}\sum _{r\in R}{y}_i^{\mathit{kvr}}\le 1, \forall i\in I$$

(15)

$$\sum _{i\in O}{x}_{0i}^{\mathit{kv}1}=1,\forall k\in K, v\in V$$

(16)

$$\sum _{i\in M}\sum _{j\in I}{x}_{\mathit{ij}}^{\mathit{kvr}}=\sum _{j\in I}\sum _{i\in M}{x}_{\mathit{ji}}^{\mathit{kvr}}\le 1, \forall k\in K, v\in V, r\in R$$

(17)

$$x_{\mathit{ij}}^{\mathit{kvr}}\ge x_{\mathrm{hy}}^{\mathit{kv}\left(r+1\right)}, \forall i,j,h,y\in O, r,r+1\in R, \forall k\in K, v\in V$$

(18)

$$\sum _{k\in K}\sum _{v\in V}{x}_{\mathit{ii}}^{\mathit{kvr}}=0,\forall i\in O, r\in R$$

(19)

$$\sum _{j\in O}\sum _{k\in K}\sum _{v\in V}{x}_{\mathit{ij}}^{\mathit{kvr}} \le 1, \forall i\in O, r\in R$$

(20)

$$\sum _{i\in O}\sum _{k\in K}\sum _{v\in V}{x}_{\mathit{ij}}^{\mathit{kvr}}\le 1, \forall j\in O, r\in R$$

(21)

$${Q^k\le D}_i<\mathit{min}\{G_i,Q^k{m}_k\},\forall i\in I, k\in K$$

(22)

$$t_r^{\mathit{kv}}=e,k\in K, v\in V,r\in R$$

(23)

$$w_i^{\mathit{kvr}}=u_k,\forall i\in M, r\in R$$

(24)

$$w_i^{\mathit{kvr}} -a_i+M_0(1-x_{\mathit{ij}}^{\mathit{kvr}})\ge w_j^{\mathit{kvr}}\ge 0,\forall i\in I, j\in O, k\in K, v\in V, r\in R$$

(25)

$$0\le t_i^a\le T,\forall i\in I$$

(26)

$$t_i^a+t_i^w+t_i^l+t_{\mathit{ij}}\le t_j^a+M_0\left(1-x_{\mathit{ij}}^{\mathit{kvr}}\right),\forall i, j\in O, k\in K, v\in V, r\in R$$

(27)

Constraint (14) indicates node traffic balance. Constraint (15) indicates that oil tank i will be served at least once. Constraint (16) defines the first trip of each tanker truck starting from the oil depot. Constraints (17) to (19) are the visit order constraints. In trip r, the truck should start from the oil depot, visit a number of oil tanks then return to the oil depot, and start from the oil depot again in the trip r + 1 or terminate the trip. This means that if the truck does not have the trip r, then the trip r + 1 must not exist. Constraints (20) and (21) state that a truck can only arrive at the oil tank once and leave the tank once at most. Constraint (22) ensures that the demand of the order is at least the capacity of the compartment and must be less than the oil tank’s capacity and the maximum capacity of the tanker truck. Constraint (23) represents the fixed loading time of all tank trucks at the oil depot, that is, the average loading time for trucks to load all the oil products required for delivery in this trip at the oil depot at the beginning of each trip, which is taken as a fixed value e. Constraint (24) indicates the number of available compartments of each tanker truck at the beginning of each trip is $u_k$. Constraint (25) represents the relationship between the number of available compartments of the two oil tanks. Constraint (26) ensures that tankers are working within the study horizon. Constraint (27) indicates the time relationship between the arrival at oil tanks i and j.

4. Solution

Because the FRP-IDM is NP-hard, exact methods for large-scale problems take a prolonged computational time. Most approaches to FRP are heuristic-based [13]. Based on Fisher’s first application of the two-stage algorithm to solve the VRP problem [18], this paper proposes a two-stage heuristic algorithm K-IBKA based on improved BKA and time-space clustering. The core idea is to cluster customers first, then arrange vehicles for grouped customers and optimize routes. The procedure is as follows.

(1)

Time-space clustering

Since each petrol station has multiple oil product demands, that is, multiple orders. The similarity of order delivery categories is quantified based on the order time-space distance so that the average driving distance between nodes in the entire class is kept short, thereby reducing transportation costs. This clustering method is also a heuristic planning process or a sub-path construction process. Therefore, we group the oil tanks with spatio-temporal similarity by K-means and use the same truck for delivery, which helps to reduce the occurrence of repeated routes significantly. First, normalize the temporal and spatial distance, and then take the weighted average between the two points as Formula (28).

$$D_{\mathit{ij}}^{\mathit{ST}}={\omega }_1(D_{\mathit{ij}}^S-\mathit{min}{D}_{\mathit{ij}}^S)/(\mathit{max}{D}_{\mathit{ij}}^S-\mathit{min}{D}_{\mathit{ij}}^S)+{\omega }_2{(D}_{\mathit{ij}}^T-\mathit{min}{D}_{\mathit{ij}}^T)/(\mathit{max}{D}_{\mathit{ij}}^T-\mathit{min}{D}_{\mathit{ij}}^T)$$

(28)

$D_{\mathit{ij}}^S$ is the spatial distance between oil tanks i and j, $D_{\mathit{ij}}^T$ is the temporal distance between tanks i and j. ${\omega }_1$ and ${\omega }_2$ are the weights of temporal and spatial distances. Temporal distance is used to measure the convenience of oil tank j being served after tank i. This paper designs a temporal distance measurement method to measure the difference between the time windows of two oil tanks.

Assume that a truck serves oil tank i and oil tank j in turn, and their time windows are [a, b] and [c, d]. Respectively, let a ≤ c. If $t_i^a\in \left[a,b\right]$, the unloading time at oil tank i is $t_i^l$, and the transportation time from oil tank i to oil tank j is $t_{\mathit{ij}}$, then the time to reach oil tank j is $t_j^a\in [a+t_i^l+t_{\mathit{ij}}, b+t_i^l+t_{\mathit{ij}}\}$, recorded as $a^{\prime }=a+t_i^l+t_{\mathit{ij}}, b^{\prime }=b+t_i^l+t_{\mathit{ij}}$.

• If $a\prime <d$, that is, the tanker truck arrives at tank j later than ${\mathit{LT}}_j$, so the truck cannot serve tanks i and j at the same time, and the temporal distance between tanks i and j is set to infinity.
• If $c\le a\prime <d$ or $a\prime <c<b\prime$, that is, $t_j^a$ is within the time window of tank j, then the temporal distance between tanks i and j is ${t_j^a-t}_i^a$.
• If $b\prime <c$, that is, the tanker truck must wait for a period of time before unloading the petrol after arriving at tank j, then the temporal distance between tanks i and j is $c-b\prime$.

In summary, the calculation formula for temporal distance is shown as Formula (29):

$$D_{\mathit{ij}}^{\mathit{ST}}=\left\{\begin{array}{c}\infty ,if a^{\prime }>d\\ t_j^a -t_i^a,if {c\le a}^{\prime }<d or a\prime <c<b\prime \\ {c-b}^{\prime },if b\prime <c\end{array}\right.$$

(29)

(2)

Optimization based on I-BKA

In recent years, swarm intelligence heuristic algorithms have performed well in solving VRP problems, and their optimization results are better than those of ALNS and GA. In this stage, the improved black kite algorithm is used to optimize the route of clustered customers. The black kite algorithm (BKA) was proposed by Wang [19]. It is a swarm intelligence optimization algorithm based on the attack and migration behavior of black kites. Since BKA has poor global search and convergence capabilities, this paper uses the butterfly optimization algorithm and hierarchical mutation strategy to improve it. The detailed steps of I-BKA are as follows.

Step 1: Encoding and decoding. Create a set of random solutions with a matrix representing the position of each black-winged kite, which is shown in Formula (30). Where pop is the number of potential solutions, dim is the size of the given problem’s dimension, which is equal to the number of customers, and $y_{i,j}$ is the position of the dimension j of the black-winged kite i. Based on the greedy algorithm, the initial solution is built on the principle of prioritizing filling the oil tank and meeting the tank’s working time window. Take a black-winged kite as an instance, a random number with a lower bound of 0 and an upper bound of 1 is generated. By sorting, the order of the tanks to be served is obtained, and each trip is scheduled according to the capacity of the random tank truck. The coding structure for FRP-IDM is depicted in Figure 2.

$$\mathit{BK}=\left[\begin{array}{ccc}\begin{array}{cc}{y}_{1,1}& y_{12}\\ y_{2,1}& y_{22}\end{array}& \cdots & \begin{array}{c}{y}_{1,\mathit{dim}}\\ y_{2,\mathit{dim}}\end{array}\\ ⋮& \ddots & ⋮\\ \begin{array}{cc}\begin{array}{c}{y}_{\mathit{pop}-1,1}\\ y_{\mathit{pop},1}\end{array}& \begin{array}{c}{y}_{\mathit{pop}-1,2}\\ y_{\mathit{pop},2}\end{array}\end{array}& \cdots & \begin{array}{c}{y}_{\mathit{pop}-1,1}\\ y_{\mathit{pop},\mathit{dim}}\end{array}\end{array}\right]$$

(30)

Step 2: Attack behavior

In the attack behavior of the original BKA algorithm, although different attack behaviors of the search are included, the influence of different prey on the pheromone exchange of black kites is ignored, so the algorithm fails to find the global optimal solution and easily causes the algorithm to fall into the local optimum. In addition, the algorithm is limited in search strength due to the lack of information exchange during the attack phase. Therefore, referring to the odor search phase in the butterfly optimization algorithm (BOA), the pheromone concentration of different prey on black kites is obtained so that the current individual and all prey individuals in its domain can exchange information, thereby improving the convergence speed of the algorithm and preventing the algorithm from falling into the local optimum.

BOA is a group intelligence algorithm inspired by the sense of direction of butterflies during foraging and courtship. At iteration, when the butterfly can smell odor from other butterflies, it moves toward the strongest odor, which is called the global search phase [20]. Applying this process to the attack behavior of BKA, the improved calculation formulas are shown in (31) and (32). $y_{\mathit{best}}$ is the global optimal solution of the current iteration. r is a random number between 0 and 1, $p_c$ is a constant value of 0.9. $T_m$ is the total number of iterations, and t is the number of iterations that have been completed so far. $f_i$ is related to the fitness value of the black-winged kite i.

$$y_{t+1}^{i,j}=\left\{\begin{array}{c}{y}_t^{i,j}+\left(r^2\times y_{best}-y_t^{i,j}\right)\times f_i p_c<r\\ y_t^{i,j}+n\left(2r-1\right)\times y_t^{i,j} else\end{array}\right.$$

(31)

$$n=0.05\times e^{-2\times {\left({\displaystyle \frac{t}{T_m}}\right)}^2}$$

(32)

Step 3: Migration behavior

Many classic heuristic algorithms (such as simulated annealing and the variable neighborhood algorithm) rely on local information for optimization and have no global guidance strategy. Therefore, the individual searches may be limited to a certain local area and cannot quickly jump out of the local optimum, resulting in slow convergence. In order to avoid these defects, the BKA algorithm introduces a leader strategy. By selecting individuals with high fitness values as leaders, others learn and improve the solution through interaction with the leader [19]. If the fitness value of the current population is less than that of the random population, then the leader will give up leadership and join the migration population. Conversely, if the fitness value of the current population is greater than that of the random population, the population is guided to its destination. $F_i$ is the current fitness of black-winged kite i in the t iteration. $F_{\mathit{ri}}$ is a random fitness of a black-winged kite in the t iteration. The improved method is to use the hierarchical mutation strategy to control the search range of black-winged kites in different periods of algorithm iteration, which improves the population diversity, the convergence rate, and enhances the search ability of the algorithm. $y_{t+1}^{i,j}$ is calculated by Formulas (33) and (34). $\overline{y_t^{i,j}}$ is the average fitness of all black-winged kites in the dimension j at the iteration t. L is the maximum diagonal distance of the search space, and W represents the weight of the hierarchical mutation strategy, which is calculated by Formulas (35) to (37).

$$y_{t+1}^{i,j}=\left\{\begin{array}{c}{y}_t^{i,j}+W\times \left(y_t^{i,j}-y_{\mathit{best}}\right) F_i<F_{\mathit{ri}}\\ y_t^{i,j}+W\times \left(y_{best}-{m\times y}_t^{i,j}\right) else\end{array}\right.$$

(33)

$$m=2\times \mathit{sin}(r+{\displaystyle \frac{\pi }{2}})$$

(34)

$$W=e^C·(1-{\displaystyle \frac{t}{T_m}})$$

(35)

$$C={\displaystyle \frac{1}{N·L}}·{\sum }_{i=1}^N\sqrt{{{\sum }_{j=1}^{\mathit{dim}}(y_t^{i,j}-\overline{y_t^{i,j}})}^2}$$

(36)

$$L=\sqrt{{{\sum }_i^{\mathit{dim}}\left({\mathit{BK}}_{j,\mathit{ub}}-{\mathit{BK}}_{j,\mathit{lb}}\right)}^2}$$

(37)

Step 4: Select the best individual

Repeat steps 3 and 4 until the maximum number of iterations is reached or the termination cases are satisfied and then get the optimal solution.

In summary, compared with the original BKA, IBKA includes improvements in attack behavior combined with BOA and migration behavior combined with the hierarchical mutation strategy. These two improvements can optimize the search process, accelerate convergence, and prevent falling into the local optimum. Then, IBKA is combined with K-means based on time-space clustering to obtain a more reasonable distribution solution for the FRP problem. Figure 3 describes the procedure of K-IBKA.

5. Instance Analysis

In order to verify the effectiveness of the algorithm, we construct small-scale and large-scale instances and conduct numerical experiments based on the actual sales data of a large petrochemical company in China. The small-scale case includes 10 petrol stations and 30 oil tanks as the research objects, and the large-scale case includes 60 petrol stations and 180 oil tanks. The study includes three petrol products: nos. 92, 95, and 98. Then, the case data of the petrol distribution network will be generated. Safety stock is 10% of tank capacity. The tanker truck information and other parameters in the small-scale instance are shown in

Table 3. Tanker truck information.

Truck Type	Compartment						$\mathbf{Total} \mathbf{Capacity} (\mathbf{k}\mathbf{L}$)	Fixed Cost (CNY)	Unit Transportation Cost (CNY/km)
	1	2	3	4	5	6
1	5	5	5	5	-	-	20	100	15
2	5	5	5	5	5	-	25	180	20
3	5	5	5	5	5	5	30	250	25

,

Table 4. Other parameters.

Symbol	Definition	Value
$\phi$	Discharge velocity of petrol (L/h)	60,000
T	The length of the study horizon. (h)	16
$T_{\mathit{work}}$	The maximum working time of driver (h)	8
$C_p$	The unit penalty cost (CNY)	2000
$v_c$	The driving speed of the truck (km/h)	50
e	The fixed loading time of a truck in the depot (h)	0.16
${\omega }_1,$ ${\omega }_2$	Weights of temporal and spatial distances	0.5
$p_c$	Attack behavior parameters of black-winged kites	0.9

and

Table 5. Information of depot and petrol station in small-scale instance.

Petrol Station	Oil Tank	Type	Initial Inventory (L)	Capacity of Tank (L)	Sales Rate (L/h)
1	1	I	6072	20,000	1042
	2	II	2220	20,000	695
	3	III	13,165	20,000	1506
2	4	I	5588	25,000	884
	5	II	16,423	25,000	797
	6	III	8304	25,000	667
3	7	I	4267	20,000	698
	8	II	14,203	20,000	1210
	9	III	8000	20,000	370
4	10	I	15,696	20,000	599
	11	II	12,375	20,000	1253
	12	III	12,137	20,000	851
5	13	I	6822	15,000	802
	14	II	7730	15,000	860
	15	III	7356	15,000	568
6	16	I	12,437	20,000	895
	17	II	8844	20,000	979
	18	III	13,689	20,000	567
7	19	I	8807	15,000	552
	20	II	7055	15,000	477
	21	III	8071	15,000	515
8	22	I	3162	40,000	844
	23	II	13,623	40,000	332
	24	III	4905	40,000	585
9	25	I	6304	20,000	239
	26	II	13,335	20,000	680
	27	III	10,185	20,000	1088
10	28	I	12,675	15,000	795
	29	II	4953	15,000	840
	30	III	11,234	15,000	704

.

5.1. The Analysis of Small-Scale Instance

Figure 4 shows the detailed distribution plan of the best solution of the K-IBKA algorithm for the small-scale instance. According to preliminary calculations, all oil tanks need to be replenished during this period. Green represents the tanker truck in motion, yellow represents the tanker truck waiting at the petrol station, orange represents the tanker truck loading petrol at the oil depot, blue represents the tanker truck unloading petrol at the oil tank, and gray represents the tanker truck resting. Figure 5 shows the inventory variation of each tank in the solution. Two trucks are used in the case of IDM, and each has four trips. The total cost of the distribution plan is 13,278 CNY, of which the transportation cost is 12,499 CNY, the employment cost is 779 CNY, and there is no penalty cost. The solution shows that orders belonging to the same petrol station tank will be delivered by carpooling. It is worth noting that the initial inventory of oil tank 4 is small and will soon be out of stock. In order to avoid being out of stock, tanker 2 will make its first trip to oil tank 4 for delivery. This arrangement reflects the characteristic of prioritizing the delivery of oil tanks with a high urgency of being out of stock. The first trip of truck 2 arrives at tank 7 at 2.56, but the tank cannot accommodate the delivery volume at this time. The truck needs to wait. As the sales proceed, the liquid level in the tank drops. It waits until 3.84 when the remaining space in the tank reaches 10,000 L. The yellow color in Figure 4 indicates that the tanker is waiting at tank 7 at [2.56, 3.84]. The variation in the inventory of petrol stations is depicted in Figure 5.

To demonstrate the advantages of the IDM, we also give a detailed distribution plan for passive distribution. The company’s original plan can be seen as a classic example of PDM. All petrol stations only propose delivery of the ordered product and do not require a specific delivery time. It only requires that the product arrive on the same day [13]. Figure 6 shows the detailed distribution plan of the best solution for PDM. Two trucks are used in this solution, and tanker truck 1 and truck 2 each have four trips. The total cost of the distribution plan of PDM is 14,924 CNY, of which the transportation cost is 13,354 CNY, the employment cost is 764 CNY, and the penalty cost is 805 CNY. By comparison, the total cost of the distribution plan obtained by IDM is 11.03% lower than that of PDM. In addition, as shown in Figure 7, oil tanks 2 and 17 were temporarily out of stock. Oil tank 22 was out of stock for 1.32 h at 4.00, and the truck did not arrive to unload until 5.32.

5.2. The Analysis of Large-Scale Instance

Similar to Section 5.1, we continue to add oil tanks and construct a large-scale instance with 60 petrol stations and 180 oil tanks. According to preliminary calculations, 143 oil tanks need to be replenished during this period. A depot with a full range of petrol types and sufficient inventory is responsible for distributing three types of petrol. There are also enough tanker trucks of different types in the depot, as shown in Table 3. At the same time, the results are compared by changing the number of trucks. We find that the best solution was obtained when the number of tanker trucks is 10; see

Table 6. Cost of distribution plans with different numbers of tanker trucks.

${\mathbf{N}}_{\mathbf{t}}$	Transportation Cost (CNY)	Employment Cost (CNY)	Penalty Cost (CNY)	Total Cost (CNY)
+2	88,224	4776	836	93,836
+1	83,063	4544	0	87,607
10	79,421	4121	0	83,546
−1	82,691	4949	847	88,487
−2	86,506	5126	280	91,912

.

The best distribution plan obtained by the K-IBKA algorithm uses 10 trucks, including four trucks of the first type, one truck of the second type, and five trucks of the third type. The total cost is 83,546 CNY, including transportation costs of 79,421 CNY, employment costs of 4121 CNY, and no penalty cost. The detailed distribution plan of a large-scale instance is shown in Figure 8.

To verify the feasibility of heterogeneous truck delivery, we will compare it with the homogeneous distribution plan and use 10 tanker trucks of a single type. The detailed distribution plan is shown in Figure 9, with a total cost of 91,364 CNY, including transportation cost of 84,240 CNY, employment cost of 5683 CNY, and penalty cost of 1451 CNY. By comparison, the total cost of the distribution plan obtained by heterogeneous trucks is 13.07% lower than that of homogeneous trucks.

The result is compared with the unimproved BKA, ALNS, and GA to verify the effectiveness of K-IBKA. The above algorithms are used to run 10 times respectively. The results of each run are shown in

Table 7. Comparison of results from different algorithms.

Sequence	Cost (CNY)	K-IBKA	K-IBKA (PDM)	BKA	GA	ALNS
1	${\mathrm{C}}_1$	79,421	134,144	127,048	123,314	104,296
	${\mathrm{C}}_2$	4124	6826	5418	3675	6526
	${\mathrm{C}}_3$	0	7152	4132	2135	4704
	$\mathrm{C}$	83,545	148,122	136,598	126,989	115,526
2	${\mathrm{C}}_1$	80,284	127,643	126,223	120,345	102,253
	${\mathrm{C}}_2$	4996	6843	5328	5969	6874
	${\mathrm{C}}_3$	1111	6742	2017	1300	5965
	$\mathrm{C}$	86,391	141,228	133,568	127,614	115,092
3	${\mathrm{C}}_1$	79,947	134,923	126,111	129,124	105,055
	${\mathrm{C}}_2$	5200	7593	5448	5408	6307
	${\mathrm{C}}_3$	368	8654	3624	1420	3907
	$\mathrm{C}$	85,515	151,170	135,183	135,952	115,269
4	${\mathrm{C}}_1$	78,296	140,463	119,650	118,837	104,277
	${\mathrm{C}}_2$	5059	6331	4263	5135	6925
	${\mathrm{C}}_3$	0	5477	2589	2726	4430
	$\mathrm{C}$	83,355	152,271	126,502	126,698	115,632
5	${\mathrm{C}}_1$	77,762	127,464	134,328	126,401	107,882
	${\mathrm{C}}_2$	4859	6854	5736	5059	7073
	${\mathrm{C}}_3$	0	5834	5065	2663	6473
	$\mathrm{C}$	82,621	140,152	145,129	131,460	121,428
6	${\mathrm{C}}_1$	77,596	118,474	124,804	121,224	106,836
	${\mathrm{C}}_2$	0	6994	5471	3585	6988
	${\mathrm{C}}_3$	4882	5734	3059	1605	4726
	$\mathrm{C}$	82,478	131,202	133,334	126,414	118,550
7	${\mathrm{C}}_1$	78,814	130,378	130,039	121,613	102,419
	${\mathrm{C}}_2$	5015	6050	5436	4078	6582
	${\mathrm{C}}_3$	0	7968	1002	2877	4680
	$\mathrm{C}$	83,829	144,396	136,477	128,568	113,681
8	${\mathrm{C}}_1$	76,533	137,129	128,072	120,344	101,507
	${\mathrm{C}}_2$	4876	6860	4797	4895	6554
	${\mathrm{C}}_3$	650	9640	4326	1450	3017
	$\mathrm{C}$	82,059	153,629	137,195	126,689	111,078
9	${\mathrm{C}}_1$	76,986	117,754	121,651	123,485	107,020
	${\mathrm{C}}_2$	4801	7639	4937	4887	6953
	${\mathrm{C}}_3$	88	5643	3125	1338	4819
	$\mathrm{C}$	81,875	131,036	129,713	129,710	118,792
10	${\mathrm{C}}_1$	80,460	142,476	130,270	128,985	96,679
	${\mathrm{C}}_2$	4995	6153	6261	6166	6474
	${\mathrm{C}}_3$	936	6672	1975	1743	3905
	$\mathrm{C}$	86,391	155,301	138,506	135,151	107,058
Average total cost		83,806	143,998	134,475	129,525	115,211

. It can be seen that K-IBKA has apparent advantages. Then, we calculate the average total cost, which is reduced by 37.68%, 35.30%, and 27.26%, respectively, compared with the unimproved BKA, GA, and ALNS. Compared with the passive distribution model, the total cost is reduced by 41.80%.

6. Conclusions

In view of the high cost of the fuel replenishment problem in PDM and the unreasonable route and time arrangement, this paper proposes a petrol distribution route planning based on a heterogeneous fleet and IDM. We also consider multi-trip and multi-compartment distribution. The optimal route with the lowest sum of transportation, employment, and penalty costs is finally determined. In order to reduce repeated routes and reasonably allocate trucks for each trip to distribute oil products, this paper proposes a two-stage heuristic algorithm K-IBKA. In the first stage, considering the spatial and temporal characteristics of oil tanks, the tanks with spatio-temporal similarities are first clustered into one category and distributed using different trips of the same tanker truck, this stage greatly reduces the search efficiency for the same type of orders. In the second stage, the I-BKA algorithm, combining BOA and a hierarchical mutation strategy, is used to allocate routes and oil products to each truck in detail. Finally, through small-scale and large-scale case calculations, the effectiveness of the IBKA and the feasibility of K-IBKA combined with time-space clustering are verified based on the total cost comparison of the solutions.

(1)

The FRP-IDM proposed in this paper ensures the punctuality of petrol delivery and reduces the distribution cost. Compared to PDM, the distribution cost calculated by IDM is reduced by 11.03% and 41.80%, respectively, in small and large-scale instances.

(2)

K-IBKA has a high advantage in solving FRP-IDM, not only because the centralized distribution of oil tanks with similar space and time makes the solution accuracy higher, but because IBKA has superior global search capabilities. Compared with the unimproved BKA, ALNS, and GA, the total cost solved by K-IBKA was reduced by 37.68%, 35.30%, and 27.26%, respectively. Compared with the distribution by homogeneous trucks, the total cost of distributing by heterogeneous trucks is reduced by 13.07%.

However, the fuel replenishment problem is still a practical and complex problem. In future work, considering variable sales, multi-cycle, and multi-warehouse will solve the problem better.