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MILP formulations for highway petrol station replenishment in initiative distribution mode

MILP formulations for highway petrol station replenishment in initiative distribution mode To investigate highway petrol station replenishment in initiative distribution mode, this paper develops a mixed-integer linear programming (MILP) model with minimal operational costs that includes loading costs, unloading costs, transport costs and the costs caused by unpunctual distribution. Based on discrete representation, the working day is divided into equal time intervals, and the truck distribution process is decomposed into a pair of tasks including driving, standby, rest, loading and unloading. Each truck must execute one task during a single interval, and the currently executing task is closely related to the preceding and subsequent tasks. By accounting for predictive time-varying sales at petrol stations, real-time road con- gestion and a series of operational constraints, the proposed model produces the optimal truck dispatch, namely, a detailed task assignment for all trucks during each time interval. The model is tested on a real-world case of a replenishment system comprising eight highway petrol stations, one depot, one garage and eight trucks to demonstrate its applicability and accuracy. Keywords MILP · Initiative distribution · Petrol station · Replenishment · Discrete representation Sets and indices Continuous parameters b ∈ B = 1,… , bm The set of tasks. bm is the maxi- c The unit cost of the loading opera- i i i L mum numbering of truck i’s task tion, CNY i ∈ I={1, … , im} The set of trucks. im is the maxi- c The unit cost of unpunctual distri- UO mum numbering of trucks bution, CNY/(m  h) j, j ∈ J = {1, … , jm} The set of nodes. jm is the maxi- c The transport costs per km, CNY/ UT mum numbering of nodes km k ∈ K = {1, … , km} The set of products. km is the c The unit cost of the unloading maximum numbering of products operation, CNY t, t ∈ T = {1, … , tm} The set of time nodes. tm is the h The number of loading arms for Lmaxj,k maximum numbering of time product k at node j (if node j is a nodes depot) l The distance between nodes j and j,j′ j′, km M The sufficiently large number Mi The sufficiently small number r The average driving speed of each truck, km/h r The loading flow rate, m /h Edited by Xiu-Qiu Peng r The unloading flow rate, m /h * Yong-Tu Liang v The maximum capacity of product Dmaxj,k liangyt21st@163.com k at node j (if node j is a depot), Beijing Key Laboratory of Urban Oil and Gas Distribution v The minimal capacity of product k Technology, Beijing 102249, China Dminj,k at node j (if node j is a depot), m Center for Spatial Information Science, The University v The capacity of truck I, m of Tokyo, 5-1-5 Kashiwanoha, Kashiwa-shi, Chiba 277-8568, NUi Japan Vol:.(1234567890) 1 3 Petroleum Science (2021) 18:994–1010 995 v The received volume of product Binary variables Oj,k,t k at node j from refineries during E The binary variable indicating BTi,b,t time interval (t, t + 1) (if node j is that truck i’s task b starts at time a depot), m /h node t v The sales rate of product k at node E The binary variable indicating Sj,k,t DVi,j,b,k,t k during time interval (t, t + 1) (if that truck i’s task b is loading node j is petrol station), m /h product k to node j during time v The maximum capacity of product interval (t, t + 1) Pmaxj,k k at node j (if node j is petrol sta- E The binary variable indicating DVBi,j,b,k,t tion), m that truck i’s task b is to start t The maximal rest time, h loading product k from node j at Rmax t The minimal rest time, h time node t Rmin t The maximum standby time, h E The binary variable indicating Smax DVNi,j,b,k,t t The minimum standby time, h that truck i’s task b is to complete Smin Z The congestion coefficient loading product k from node j at t,j,j′ between node j and j′ during time time node t interval (t, t + 1) E The binary variable indicating PVi,j,b,k,t Δτ The length of the time interval, h that truck i’s task b is to unload τm The length of the study horizon, h product k at node j and is finished α The coefficient for the transport at time node t costs of trucks with full of product N The binary variable indicating Ci,b,j k that truck i is staying at node j when its task b starts Binary parameters P The binary variable indicating Li,b,k f The binary parameter indicating Li,k that truck i is filled with product k that truck i can load product k during its task b f The binary parameter indicating PDj S The binary variable indicating Di,b that node j is a depot that truck i’s task b is to drive to f The binary parameter indicating PGj another node that node j is the garage S The binary variable indicating Li,b,k f The binary parameter indicating PPj that truck i’s task b is to load that node j is a petrol station product k at the depot f The binary parameter indicating PSj S The binary variable indicating Ri,b that the standby task is permissi- that truck i’s task b is to rest at the ble at node j garage Positive continuous variables S The binary variable indicating Si,b C The cost caused by the unpunctual that truck i’s task b is to standby Ot,j,k distribution of node j ’s product k S The binary variable indicating Ui,b at time node t, CNY that truck i’s task b is to unload C The transport cost of truck i’s task product at a petrol station Tt,j,k b, CNY T The total rest time of truck i from Ri,b the beginning of the study horizon 1 Introduction to the start time of its task b, h V The inventory of product k at 1.1 Background Pj,k,t node j at time node t (if node j is a petrol station), m As necessary facilities for urban transportation, petrol sta- V The inventory of product k at tions play a critical role in the steady supply of refined prod- Dj,k,t node j at time node t (if node j is a ucts (Fernandes et al. 2013), and efficient distribution of depot), m refined products is also important in the smooth operation of cities. To daily replenish petrol stations, the transport company must simultaneously determine detailed routes and product assignments for each truck (Cattaruzza et al. 1 3 996 Petroleum Science (2021) 18:994–1010 2014; Liu and Jiang 2012). As we know, product oil distri- They developed an integer programming model to address bution between refineries and oil depots is usually carried the problem of a nationwide bulk delivery fleet of petroleum by pipelines (Fernandes et al. 2013), and there are distribu- tanker trucks. The objective function was to minimize trans- tion stations along the pipelines. Researchers often focus on port costs under the constraints of equitable manpower and the formulation of scheduling plans (Zhang et al. 2016) and equipment workload distributions, safety, customer service, transmission plans (Fernandes et  al. 2013). Zhang et  al. and equipment compatibility restrictions. The model was (2017) established a MILP model to solve problems, and applied to a scale of more than 80 bulk terminals on a fleet some researchers (Moradi et al. 2019) had also proposed exceeding 300 vehicles with approximately 2600 loads per efficient algorithm to solve scheduling studies. Liang et al. day. Brown et al. (1987) extended the work of Brown and (2012) considered regional differences in electricity prices Graves (1981) for a real-time, transaction-driven information along the pipeline, established an optimization model for a management system that could solve the problem for 120 multi-product pipeline which has a known delivery demand bulk terminals and more than 430 vehicles. and operation plan for each off-take station. Different from Apart from mathematical programming methods, heu- the distribution between refineries and oil depots, the replen- ristic algorithms (Azad et al. 2017; Kasivisvanathan et al. ishment of petrol stations takes truck transportation as the 2014; Nambirajan et  al. 2016), intelligent optimization mode of transportation, which is easier to be affected by (Diaby et al. 2016; Marinakis et al. 2013; Soto et al. 2017) the condition of road congestion. The replenishment of and hybrid algorithm (Alinaghian and Shokouhi 2018; Shi petrol stations is mainly classified into two types: passive et al. 2017) had also been widely used in dealing with the distribution and initiative distribution. For passive distribu- scheduling optimization issue. For example, focusing on the tion, petrol stations order products before the next work- truck assignment and routing problem of a Hong Kong pet- ing day, specifying the minimum and maximum quantity rol distribution company, Ng et al. (2008) proposed a deci- to be delivered for each ordered product. The minimum sion support system that combines heuristic clustering and quantity is often determined by the stations’ average daily optimal routing to determine the optimal fleet assignment sales, while the maximum quantity is determined by the dif- and routing. Multiple objective functions included the mini- ference between the capacity of the station’s storage tanks mal number of used trucks, the minimal number of drops and an estimation of the remaining stock before the next in each trip, the maximal profits in terms of total products replenishment. Generally, petrol stations specify that prod- delivered and the maximal utilization of resources. For the ucts be replenished during periods when sales are low and multi-depot petrol station replenishment problem, Cornil- not during peak periods. However, passive distribution has lier et al. (2008) presented a heuristic that contained route the following problems. First, the specified replenishment construction and truck loading procedures, a route packing time of different petrol stations may be highly similar, so procedure and two procedures enabling the anticipation or that trucks are extremely busy during these periods and idle the postponement of deliveries. Next, Cornillier et al. (2009) when the petrol stations are busy and unable to replenish studied the filling station replenishment problem with time stock. Second, a single petrol station may receive a variety intervals (PSRPTW). The introduction of time intervals of products several times from transportation companies in a leads to a great increase in model scale and computational single day, and it is difficult to estimate and coordinate each effort, so they proposed an MILP model and two heuristics distribution time. By contrast, with initiative distribution, for this problem to determine the quantity of each product to petrol stations do not order products but instead forecast the deliver, the assignment of products to truck compartments, sales for each station, while the transport company assigns delivery routes and schedules. In fact, petroleum products the specific time to deliver the products. The advantage of are stored in a number of different depots, and they are deliv - initiative distribution is that it treats the whole replenishment ered to a set of petrol distribution stations. Each depot has process as an integrated system, and the replenishment time its own fleet of heterogeneous and compartmented tanker is determined by the system instead of the petrol stations. trucks. Therefore, in their later work, Cornillier et al. (2012) This approach allows the working time of trucks in a day to developed a heuristic procedure to generate a restricted set be more balanced, thereby avoiding the inefficiency created of promising feasible trips beforehand for the multi-depot by concentrated periods of demand. Overall, initiative dis- petrol station replenishment problem with time intervals tribution improves efficiency in the dispatch of trucks and (MPSRPTW). Wang et  al. (2018) established a mixed- therefore in the running of the entire replenishment system. integer linear model and proposed an improved heuristics algorithm to solve the petrol truck scheduling problem which 1.2 Related work considered multi-path selection and congestion. Boctor et al. (2011) firstly introduced a generalized trip packing One of the first articles addressing the issue of petrol station problem, in which trucks have different capacities and trip replenishment was published by Brown and Graves (1981). net revenues depend on the trucks used. Meanwhile, some 1 3 Petroleum Science (2021) 18:994–1010 997 construction, improvement and neighborhood search solu- 1.3 Contributions of this work tion heuristics are proposed to solve a real-life case. In addition to petrol station replenishment issues, a large • With a focus on highway petrol station replenishment number of studies have also focused on the vehicle routing in the initiative distribution mode, an MILP approach problem (VRP), which focuses on the use of efficient algo- is proposed to provide a more reasonable schedule for rithms (Li et al. 2012; Goodson et al. 2012; Schyns 2015). trucks and improve efficiency for the entire replenish- Brandão and Mercer (1997) presented a novel tabu search ment system. heuristic algorithm to solve the multi-trip routing and sched- • Various factors appearing in practice, such as time-var- uling problem (VRSP) with respect to constraints in which ying sales, task time limitations, real-time road conges- a vehicle can make more than one trip each day and the cus- tion, different unit transport costs under full-load or no- tomers impose delivery time intervals. Based on the adaptive load conditions, road capacity, the detailed operational memory procedure, Olivera and Viera (2007) described a mode (namely, loading, unloading, rest and standby) and heuristic method to solve the vehicle routing problem with operational limitations, are all taken into consideration. multiple trips. To solve a wide range of different types of • The optimal distribution plan, including the quantities, VRPs, Gromicho et al. (2012) presented a highly flexible specific time and replenishment object of each product framework accommodating various real-life restrictions. to be delivered by each truck, is solved by the proposed Wang et al. (2019) established a mathematical model to model. solve the petrol truck routing problem and proposed a heu- • A real case study is given to demonstrate the practicality ristic algorithm based on a local branch-and-bound search of this method. with a tabu list, several inequalities are added to accelerate the solving process. A comparison with state-of-the-art local 1.4 Paper organization search methods indicates the superiority of this new frame- work in practical applications. The adopted methodology and details of the mathematical This paper studies on highway petrol station replenish- model are given in Sects. 2 and 3, respectively. Section 4 ment in the initiative distribution mode. Different from clas- provides a real case study in China and demonstrates the sical petrol station VRP problem, highways have a large and model’s practicability. Finally, conclusions are presented in various demand for refined products in this study, so a single Sect. 5. petrol station on a highway may be replenished more than once during a working day. Moreover, highway petrol sta- tions receive a variety of refined products transported from 2 Methodology depots, and a reasonable replenishment time for each type of product is difficult to predict only considering each petrol 2.1 Problem description station. In fact, the replenishment mode studied in previ- ous relevant work belongs to the passive distribution type, Figure  1 illustrates the highway petrol station replenish- in which the quantity and delivery timing of each product ment system studied in this paper. There are three types of are given by petrol stations. Compared with initiative dis- tribution, passive distribution is dominated by human fac- Garage tors and often results in impractical truck allocation and Petrol station low distribution efficiency. With the increasing popularity of Internet technology, the shift from passive distribution to initiative distribution is a compelling trend among petro- leum enterprises looking to accomplish the daily informa- Unload Drive tion-based management of oil resources. However, to the Rest best of our knowledge, this paper is the first research work that addresses this replenishment problem in an initiative Depot distribution mode rather than a passive distribution mode. None of the abovementioned papers have simultaneously Petrol station determined the quantities, timing and replenishment object Standby of each product to be delivered by each truck through a com- Drive prehensive consideration of time-varying sales at petrol sta- tions, real-time road congestion and the state monitoring of Load all trucks. Fig. 1 Work flow for highway petrol station replenishment 1 3 998 Petroleum Science (2021) 18:994–1010 destinations, namely garages, depots and petrol stations. At The congestion condition during each time interval. • The basic information of trucks. the initial time of one working day, trucks departure from garages to depots and then cover refined products to the pet- Determine: rol stations which are short of oil. Considering the limited number of loading arms, an upper limit is set for the number • The detailed scheduling of trucks: the executed task dur- of trucks that are loading simultaneously. Therefore, trucks need to wait when there is no free loading arm. Additionally, ing each time interval. • The departure point and the corresponding destination of all trucks are stored in the garage at the end of the work- ing day. Since the petrol stations are sparsely distributed each transport task. • The quantity and type of product being loaded (unloaded) along the highway, the distance between any petrol stations is rather long and an oil depot can serve only a small number of each load (unload) task. • The inventory variations of depots and petrol stations. of petrol stations. It usually takes a long time for a truck to complete a single delivery task, which brings higher-level The total operational costs. demands for truck scheduling from oil tankers. Distribut- ing trucks according to the reports from petrol stations may Objective: The objective is to create a distribution plan with minimal result in higher demand in some busy time periods and no demand in other times, thereby greatly reducing the effi- operational costs while respecting a variety of operational and technical constraints. To establish and solve the model ciency of the dispatch system. In order to overcome the lack of passive distribution effectively, assumptions are made as follows. model stated above and provide a more reasonable schedule for trucks and reduce cost, this paper develops a discrete- The trucks drive at a constant speed and can only load one type of specified product during each time interval. time MILP model in the initiative distribution mode and incorporates the constraint of sales predictions as an addition After the truck arrives at the depot or petrol station, it needs time to load or unload the petrol products, and the to other constraints. In the proposed model, the truck distri- bution process is decomposed into a pair of tasks, includ- speed of loading and unloading is constant. • During a single workday, all trucks start from the garage ing driving, standby, rest, loading and unloading, and each task is closely related to the preceding and subsequent task. and return to the garage after completing all tasks. • Petroleum companies predict the daily sales plan of the For any truck, a complete replenishment process involves driving to a depot, loading, driving to a petrol station and petrol stations and a daily delivery plan for the depots beforehand. unloading. In practical engineering, a truck can replenish the oil supply of multiple petrol stations or of a single pet- The standby times and rest times of each vehicle during the working day should be within the allowed limits. rol station repeatedly for 1 day. Therefore, there are several tasks that must be completed at specified destinations for The transport time between any two destinations is closely related to their real distance and real-time con- each truck. Moreover, the entire scheduling time horizon is divided into several equal time intervals, and each truck gestion conditions. must execute one task during each interval. 3 Mathematical model 2.2 Model requirements This paper develops the MILP model based on discrete rep- The model is formulated as MILP, and MATLAB R2014a is resentation (Shaik and Bhat 2014). The studied system is made up of a set of nodes j ∈ J, including depots (f = 1), employed to calculate the detailed truck routing and schedul- PDj ing of petrol station replenishment. petrol stations (f = 1) and garages (f = 1). Time nodes PPj PGj t ∈ T is used to divide the working day into several equal time Given: intervals, and the truck distribution process is decomposed into a pair of tasks. Each truck must execute one task during The time horizon of the working day. a single interval, and the currently executing task is closely • related to the preceding and subsequent task. By accounting The locations of garages, depots and petrol stations. The delivery plan of the depot (delivery volume during for predictive time-varying sales at petrol stations, real-time road congestion and a series of operational constraints, the each time interval) and the sales plan of the petrol sta- tions (sales volume during each time interval). proposed model produces the optimal truck dispatch, i.e., • the detailed task assignment of all trucks during each time The inventory of the depot and petrol stations for each product. interval. 1 3 Petroleum Science (2021) 18:994–1010 999 Overall, the model involves five sets: the set of trucks (I ); P ≤ f i ∈ I, k ∈ K L i,b,k L i,k (4) the set of tasks (B ); the set of nodes (J); the set of prod- i b∈B uct types (K); and the set of time nodes (T) chronologically The executable operations of all trucks include load, arranged. Moreover, this model comprises of positive con- unload, rest, drive and standby, only one of which can be tinuous variables controlling the product inventory at pet- executed during a single task: rol stations (V ) and depots (V ), the accumulated rest Pj,k,t Dj,k,t time of trucks (T ), transport costs (C ) and the costs for Ri,b Ti,b S + S + S + S + S = 1 i ∈ I, b ∈ B L i,b,k U i,b R i,b D i,b S i,b i unpunctual distribution (C ). Beyond that, a series of key Ot,j,k k∈K binary variables are also introduced to identify if truck i are (5) loading/unloading/resting/driving/standing by during its task b (in case S , S , S , S , S = 1), if truck i is filled Li,b,k Ui,b Ri,b Di,b Ri,b 3.3 Loading constraints with product k during its task b (in case P = 1), if truck i’s Li,b,k task b starts at time node t(in case E = 1), and if truck i BTi,b,t For any truck i, the loading operation can only be executed is staying at node j when its task b starts (in case N = 1). Ci,b,j at depots. In other words, the binary variable of the loading operation (S ) can equal one when truck i is located at node Li,b,k 3.1 Objective function j, with node j being a depot (i.e., N = 1, f = 1). Ci,b,j PDj The objective function is the minimum opera - S ≤ N f i ∈ I, b ∈ B L i,b,k C i,b,j PD j i (6) tional cost, including the loading operation cost k∈K j∈J ∑ ∑ ∑ ( c S ), the unloading operation cost i∈I b∈B k∈K L L i,b,k ∑ ∑ ∑ ∑ Constraint (7) states that only no-load trucks can execute ( c S ), the transport cost ( C ) U Ui,b Ti,b i∈I b∈B i∈I b∈B i i loading operations. If truck i loads product k during its task b, and the cost caused by the late distribution for petrol sta- ∑ ∑ ∑ it will be filled with product k at the start time of its next task, tions ( C ). Ot,j,k t∈T j∈J k∈K that is to say, P would be one if S is equal to 1 [see Li,b+1,k Li,b,k constraint (8)]. Otherwise, the truck without product k will min F = c S + c S + C L L i,b,k U U i,b T i,b remain unchanged. As seen in constraint (9), if P is zero Li,b,k i∈I b∈B k∈K P must be zero. Li,b+1,k + C O t,j,k S ≤ 1 − P i ∈ I, b ∈ B , k ∈ K (7) L i,b,k L i,b,k i t∈T j∈J k∈K P (1) S ≤ P i ∈ I, b < bm , k ∈ K L i,b,k L i,b+1,k i (8) 3.2 T rucks’ state constraints P ≤ P + S i ∈ I, b < bm , k ∈ K L i,b+1,k L i,b,k L i,b,k i (9) This paper introduces binary variables to indicate the posi- tion of all trucks. N equals 1 when truck i is located at Ci,b,j node j at the start time of task b. During any task, a truck 3.4 Unloading constraints can only stay at one node. For any truck i, the unloading operation can only be executed N = 1 i ∈ I, b ∈ B C i,b,j i (2) at a petrol station. Specifically, the binary variable of the j∈J unloading operation (S ) equals one when truck i is located Ui,b There are two states for all trucks: one is full-load, and at node j, while node j is a petrol station. the other is no-load. If truck i is filled with product k at the S ≤ N f i ∈ I, b ∈ B start time of its task b, P = 1, and otherwise P = 0. U i,b C i,b,j PP j i Li,b,k Li,b,k (10) j∈J In this way, P = 1 stands for full-load while Li,b,k k∈K P = 0 stands for no-load. Furthermore, the trucks k∈K Li,b,k Constraint (11) states that only full-load trucks can execute are also divided into two types: one stores gasoline, and the the unloading operation. If truck i unloads oil during its task b, other stores diesel oil. Equations (3) and (4) ensure that each it will be empty at the start time of its next task, that is to say, truck only transports one type of specified oil during a single P must be zero as long as S is equal to one [see Li,b+1,k Ui,b k∈K task. Note that the binary parameter f in Eq. (3) is used to Li,k constraint (12)]. If it does not unload oil, the truck that is filled indicate if truck i can load product k. with product k will remain unchanged. As seen in constraint (13), if P is one P must be one. Li,b,k Li,b+1,k P ≤ 1 i ∈ I, b ∈ B L i,b,k i (3) k∈K 1 3 1000 Petroleum Science (2021) 18:994–1010 If truck i’s position at the start time of task b is different S ≤ P i ∈ I, b ∈ B U i,b L i,b,k i (11) from that at task b + 1, then we can conclude that truck i’s k∈K task b is driving. As constraint (19) states, S is bounded Di,b by the maximum value of N − N . Ci,b+1,j Ci,b,j S ≤ 1 − P i ∈ I, b < bm U i,b L i,b+1,k i (12) k∈K N − N ≤ S i ∈ I, b < bm , j ∈ J C i,b+1,j C i,b,j D i,b i (19) P ≤ P + S i ∈ I, b < bm , k ∈ K (13) L i,b,k L i,b+1,k U i,b i 3.6 Time constraints 3.5 Rest, standby and driving constraints To develop the constraints between time and tasks, this paper introduces a binary variable (E ), which equals 1 when BTi,b,t Only when the truck stops at the garage at the work start truck i’s task b starts at time node t. By constraint (20), each time can it rest. By constraint (14), S can be one only Ri,b task must start at one time node. Constraint (21) determines when N f = 1. Ci,b,j PGj that no task start be earlier than the start time of the preced- j∈J ing task. Furthermore, the last task of any truck must start S ≤ N f i ∈ I, b ∈ B R i,b C i,b,j PG j i (14) before the study horizon, which is stated in constraint (22). j∈J E = 1 i ∈ I, b ∈ B BT i,b,t i Due to safety concerns, full-load trucks cannot rest. This (20) t∈T operation condition is enforced by constraint (15). The binary variable of rest operation (S ) must be zeros if Ri,b tΔE ≤ tΔE i ∈ I, b ∈ B P = 1. BT i,b,t BT i,b+1,t i Li,b,k k∈K (21) t∈T t∈T P ≤ 1 − S i ∈ I, b ∈ B L i,b,k R i,b i (15) k∈K tΔE ≤ mi ∈ I BT i,bm ,t i (22) t∈T To ensure safety, trucks can only stand by at some par- ticular nodes. By constraint (16), S can be one only when Si,b If the truck is loading products during its task b (i.e., N f = 1. j∈J C i,b,j PS j S = 1 ), the time consumed by this task is related to k∈K Li,b,k the truck’s capacity and the loading flow rate. If the truck is S ≤ N f i ∈ I, b ∈ B S i,b C i,b,j PS j i (16) not loading products, constraints (23)–(24) are redundant, and j∈J the interval duration can then match the value computed from the time constraints of another unload, rest, drive and standby. At the end of the weekday, all trucks must be at a rest station, and the last task must be rest, as seen in Eq. (17). NU i (23) tΔ𝜏 E − tΔ𝜏 E ≤ +Δ𝜏 + 1 − S 𝜏 mi ∈ I, b < bm BT i,b+1,t BT i,b,t L i,b,k i t∈T t∈T k∈K NU i (24) tΔ𝜏 E − tΔ𝜏 E ≥ + S − 1 𝜏 mi ∈ I, b < bm BT i,b+1,t BT i,b,t L i,b,k i t∈T t∈T k∈K Similarly, if truck i is unloading products during its task b (i.e., S = 1), the time consumed by this task is related N f = 1 i ∈ I, j ∈ J Ui,b C i,bm ,j PG j (17) j∈J to the truck’s capacity and unloading flowrate. Otherwise, constraints (25)–(26) are redundant and the interval duration All trucks cannot standby twice continuously. In con- is bounded by other operations. straint (18), only one of two adjacent tasks b and b + 1 can be standby. NU i tΔ𝜏 E − tΔ𝜏 E ≤ BT i,b+1,t BT i,b,t t∈T t∈T S +S ≤ 1 i ∈ I, b < bm (18) S i,b S i,b+1 i +Δ𝜏 + 1 − S 𝜏 mi ∈ I, b < bm U i,b i (25) 1 3 Petroleum Science (2021) 18:994–1010 1001 If truck i is standing by during its task b (i.e., S = 1), the NU i Si,b tΔ𝜏 E − tΔ𝜏 E ≥ BT i,b+1,t BT i,b,t r standby time should not be too long. Constraints (34)–(35) t∈T t∈T impose the lower and upper bounds on the standby time. + S − 1 𝜏 mi ∈ I, b < bm U i,b i (26) tΔ𝜏 E − tΔ𝜏 E ≤ t BT i,b+1,t BT i,b,t Smax If truck i is resting during its task b (i.e., S = 1), the rest Ri,b t∈T t∈T time of this truck should be equal to the accumulated rest + 1 − S 𝜏 mi ∈ I, b < bm S i,b i (34) time before task b plus the duration of task b [i.e., constraints (27)–(28)]. If not, the rest time of this truck should be equal to the accumulated rest time before task b [i.e., constraints tΔ𝜏 E − tΔ𝜏 E ≥ t BT i,b+1,t BT i,b,t Smin (29)–(30)]. The total rest time of each truck should be lim- t∈T t∈T ited to a range [i.e., constraint (31)]. + S − 1 𝜏 mi ∈ I, b < bm S i,b i (35) T ≤ T + tΔ𝜏 E − tΔ𝜏 E R i,b+1 R i,b BT i,b+1,t BT i,b,t t∈T t∈T 3.7 Petrol station constraints + 1 − S 𝜏 mi ∈ I, b < bm R i,b i (27) The binary variable E denotes the completed status PVi,j,b,k,t of replenishment and equals one when a full-load truck i T ≥ T + tΔ𝜏 E − tΔ𝜏 E R i,b+1 R i,b BT i,b+1,t BT i,b,t finishes unloading product k to node j at time node t. By t∈T t∈T constraints (36)–(37), if any value of binary variables S Ui,b + S − 1 𝜏 mi ∈ I, b < bm R i,b i (28) E , N and P is equal to zero, E must BTi,b+1,t Ci,b+1,j Li,b,k PVi,j,b,k,t be zero. Conversely, if these variables are all equal to one, T ≤ T + S 𝜏 mi ∈ I, b < bm R i,b+1 R i,b R i,b i (29) E must be one. PVi,j,b,k,t S + E + N + P U i,b BT i,b+1,t C i,b+1,j L i,b,k T ≥ T − S 𝜏 mi ∈ I, b < bm E ≥ Mi + R i,b+1 R i,b R i,b i (30) PV i,j,b,k,t − 1 i ∈ I, j ∈ J, b < bm , k ∈ K, t ∈ T t ≤ T ≤ t i ∈ I R min R i,bm R max (31) (36) S + E + N + P U i,b BT i,b+1,t C i,b+1,j L i,b,k If truck i is driving during its task b (i.e., S = 1), the Di,b E ≤ 1 − 1 − Mi PV i,j,b,k,t time consumed must be related to the average of the driving distance, the speed of the truck and the coefficient of road i ∈ I, j ∈ J, b < bm , k ∈ K, t ∈ T congestion during this time period. Specifically, if truck i ’s (37) ∑ ∑ ∑ task b is to move from node j to node j′ during time interval If E is more than one, the PVi,j,b,k,t i∈I b∈B k∈K t (i.e., S , N , N = 1), inequation constraints Di,b Ci,b+1,j′ Ci,b+1,j received volume of node j during time interval (t, t + 1) (32)–(33) work as an equation constraint. By doing so, the equals the sum of the capacities of the corresponding trucks. transport time is equal to the driving distance between node Otherwise, the received oil volume is zero. The inventory j and node j′(l ) being divided by the truck speed (r ) and j,j′ D of node j equals the inventory at the last time node plus the then being multiplied by the congestion coefficient during received oil volume and minus the sales volume during the interval t(Z ). t,j,j′ last time interval. ∑ ∑ z E + z E � � ⎛ t,j,j BT i,b+1,t t,j,j BT i,b,t ⎞ � � l � j,j t∈T t∈T ⎜ ⎟ tΔ𝜏 E − tΔ𝜏 E ≤ +Δ𝜏 BT i,b+1,t BT i,b,t r ⎜ 2 ⎟ t∈T t∈T (32) ⎝ ⎠ � � N + N � � � C i,b+1,j C i,b,j + 1 − S 𝜏 m+ 1 − 𝜏 mi ∈ I, b < bm , j, j ∈ J D i,b i ∑ ∑ z �E + z �E ⎛ t,j,j BT i,b+1,t t,j,j BT i,b,t ⎞ � � l � j,j t∈T t∈T ⎜ ⎟ tΔ𝜏 E − tΔ𝜏 E ≥ BT i,b+1,t BT i,b,t r ⎜ 2 ⎟ t∈T t∈T ⎝ ⎠ (33) � � N + N � � � C i,b+1,j C i,b,j + S − 1 𝜏 m + − 1 𝜏 mi ∈ I, b < bm , j, j ∈ J D i,b i 1 3 1002 Petroleum Science (2021) 18:994–1010 V = V + E v − v j ∈ J, k ∈ K, t > 1 P j,k,t P j,k,t−1 PV i,j,b,k,t NU i S j,k,t−1 E ≤ h j ∈ J , k ∈ K, t ∈ T DV i,j,b,k,t L max j,k i (45) i∈I b∈B i∈I b∈B (38) ∑ ∑ ∑ If E is more than one, the dis- During any time interval t, the inventory of the petrol sta- i∈I b∈B k∈K DVi,j,b,k,t charged volume of node j during time interval (t, t + 1) is equal tion should meet certain limits. To obtain a feasible solution to the aggregate capacities of the trucks that are loading. Oth- when faced with stockouts caused by falling product prices, erwise, the discharged volume is zero. Hence, the inventory at there is no lower limit for the inventory of the petrol station. node j is equal to the inventory of the last time node plus the However, this model has added the costs caused by unpunc- received volume from refineries and minus the discharged oil tual distribution into the objective function to ensure that each volume during the last time interval. petrol station keeps a stock of petrol products. V = V + v − E v j ∈ J, k ∈ K, t ∈ T V ≤ v j ∈ J, k ∈ K, t ∈ T D j,k,t D j,k,t−1 Oj,k,t−1 DVN i,j,b,k,t NU i P j,k,t Pmax j,k (39) i∈I b∈B (46) During any time interval t, the inventory of all depots 3.8 Depot constraints should meet certain limits. This paper introduces three types of binary variables to rep- v ≤ V ≤ v j ∈ J, k ∈ K, t ∈ T Dmin j,k D j,k,t Dmax j,k (47) resent the start, finish and process of the loading operation. Specifically, E equals one when a no-load truck i starts DVBi,j,b,k,t loading product k from node j at time node t [i.e., constraints 3.9 Cost constraints (40)–(41)], while E equals one when the truck has just DVNi,j,b,k,t finished loading oil [i.e., constraints (42)–(43)]. As constraint For a given transport route, fuel consumption increases (44) shows, E is bounded by E and E , DVi,j,b,k,t DVBi,j,b,k,t DVNi,j,b,k,t with the load. Hence, the transport costs are calculated and it equals one when truck i is loading product k from node by transportation distance and truck status, as shown in j during time interval (t, t + 1). constraints (48)–(49). More specifically, transport costs for S + E + N L i,b,k BT i,b,t C i,b,j all empty trucks only depend on the product of unit cost E ≥ Mi + DVB i,j,b,k,t 3 and transportation distance. However, if truck i is full of − 1 i ∈ I, j ∈ J, b < bm , k ∈ K, t ∈ T product k, the transport costs must be a bitter higher than i (40) the costs of empty trucks. Therefore, the parameter α in constraints (48)–(49) is used to compute the extra costs for S + E + N L i,b,k BT i,b,t C i,b,j E ≤ 1 − 1 − Mi carrying v m of product k. DVB i,j,b,k,t NUi N + N C i,b+1,j C i,b,j i ∈ I, j ∈ J, b < bm , k ∈ K, t ∈ T i (41) C ≤ c l 1 + P v 𝛼 + 1 − M T i,b UT j,j L i,b,k NU i k k∈K i ∈ I, j, j ∈ J, b < bm , t ∈ T S + E + N L i,b,k BT i,b+1,t C i,b+1,j E ≥ Mi + DVN i,j,b,k,t (48) N + N − 1 i ∈ I, j ∈ J, b < bm , k ∈ K, t ∈ T � C i,b+1,j C i,b,j i (42) C ≥ c l 1 + P v 𝛼 + − 1 M T i,b UL j,j L i,b,k NU i k k∈K S + E + P i ∈ I, j, j ∈ J, b < bm , t ∈ T L i,b,k BT i,b+1,t C i,b+1,j i E ≤ 1 − 1 − Mi DVN i,j,b,k,t (49) It is forbidden for the truck to be delivering oil if its i ∈ I, j ∈ J, b < bm , k ∈ K, t ∈ T location remains unchanged. If this occurs, then the cor- (43) responding costs are at the maximum level. By constraint E = E + E DV i,j,b,k,t DV i,j,b,k,t−1 DV Bi,j,b,k,t (50), transport costs ( C ) would be greater than a suf- T i,b − E i ∈ I, j ∈ J, b < bm , k ∈ K, t > 1 ficiently large value (1/2M ) when binary variables S , DV Ni,j,b,k,t i (44) Di,b N , N are all equal to one. Ci,b+1,j Ci,b,j Due to the limited number of loading arms, a depot can N + N only permit a limited number of trucks to load oil during a C i,b+1,j C i,b,j C ≥ M + S − 1 M + − 1 M T i,b D i,b single time interval. 2 2 i ∈ I, j ∈ J, b < bm (50) 1 3 Petroleum Science (2021) 18:994–1010 1003 As for any petrol station, a negative inventory stands for are usable for P2 and P3. However, each truck can only unpunctual distribution, which leads to economic losses. The load one type of product at a time. The oil tank capacity right term of constraint (51) is a positive value when the of each truck is 20 m , and the inventory of the petrol sta- inventory V is negative, and thus imposes a lower bound tion is 30 m . There are 2, 1 and 1 loading arms for P1, Pj,k,t for the costs of unpunctual distribution C . P2 and P3 in the depot, respectively. The unit labor cost Ot,j,k and wastage cost of loading or unloading oil each time is C ≥ −c V Δ j ∈ J, k ∈ K, t ∈ T O t,j,k UO P j,k,t (51) 50 CNY, and the unit cost caused by unpunctual distribu- tion is 600 CNY/(m  h) During the transport task, the unit transport cost for an empty truck is 10.9 CNY per kilo- meter, while it is 16.7 CNY per kilometer for a full-load 4 Results and discussion truck. The trucks’ speeds when loading, unloading and 3 3 driving are 60 m /h, 60 m /h and 60 km/h, respectively. 4.1 Basic data And the replenishment speeds of P1, P2 and P3 of the 3 3 3 depot are 71.67 m /h, 56.67 m /h and 60 m /h, respec- This section is to validate our model and the superiority tively. Within the study horizon of the study, the sales rate of initiative distribution by using a real-world case of a of petrol station will change several times and is shown highway in Beijing, China. Figure 2 illustrates a real-world in Fig. 3. Standby time should be less than 1.5 h, while block where there are eight petrol stations (S1, S2, S3, the total rest time for each driver should be less than 3 h. S4, S5, S6, S7 and S8), a garage and a depot. The depot is located at the center of the block and is responsible for the 4.2 Real‑world scheduling plan product supplies of these petrol stations. A daily distribu- tion plan starts at 6:00 and finishes at 24:00. During the To demonstrate our advantages in the following part, we workday, all trucks start from the garage and return to the first give an overview of the actual scheduling plan premade garage after completing all tasks. For this replenishment by the company, which can be seen as a classic example of system, there are a total of eight available trucks (C1, C2, passive distribution. All the petrol station only puts forward C3, C4, C5, C6, C7 and C8) to transport three types of the ordered products capacity and has no specific require- products (P1, P2 and P3). To avoid contamination, C1, ments on the delivery time. It only requires the products C2 and C3 can only load P1, while C4, C5, C6, C7 and C8 050 100 150 300 300 250 250 200 200 150 150 100 100 50 50 0 0 050 100 150 Fig. 2 Positions of the depot, garage and petrol stations 1 3 1004 Petroleum Science (2021) 18:994–1010 2.5 2.5 P1 P1 P2 P2 2.0 2.0 P3 P3 1.5 1.5 1.0 1.0 0.5 0.5 0 0 6:00 12:00 18:00 0:00 6:00 12:00 18:00 0:00 Time Time (a) S1 (b) S2 2.5 2.5 P1 P1 P2 P2 2.0 2.0 P3 P3 1.5 1.5 1.0 1.0 0.5 0.5 0 0 6:00 12:00 18:00 0:00 6:00 12:00 18:00 0:00 Time Time (c) S3 (d) S4 2.5 2.5 P1 P1 P2 P2 2.0 2.0 P3 P3 1.5 1.5 1.0 1.0 0.5 0.5 0 0 6:00 12:00 18:00 0:00 6:00 12:00 18:00 0:00 Time Time (e) S5 (f) S6 2.5 2.5 P1 P1 P2 P2 2.0 2.0 P3 P3 1.5 1.5 1.0 1.0 0.5 0.5 0 0 6:00 12:00 18:00 0:00 6:00 12:00 18:00 0:00 Time Time (g) S7 (h) S8 Fig. 3 Sales rates of petrol stations 1 3 3 3 3 3 Sale rate, m /h Sale rate, m /h Sale rate, m /h Sale rate, m /h 3 3 3 3 Sale rate, m /h Sale rate, m /h Sale rate, m /h Sale rate, m /h Petroleum Science (2021) 18:994–1010 1005 Driving Standby Rest Loading Unloading C1 68.7 P1 72.3 S7 72.3 P1 90.4 S8 90.4 P1 112.11 S2 76.3 C2 68.7 P1 113.1 S6 113.1 P1 94.8 S5 94.8 P1 72.3 S7 72.3 P1 79.3 S1 36.2 C3 68.7 P1 79.3 S1 79.3 P1 148.1 S3 148.1 P1 121.7 S4 76.7 C4 68.7 P2 72.3 S7 72.3 P2 148.1 S3 148.1 P2 112.1 S2 176.3 P3 C5 68.7 P2 90.4 S8 90.4 P2 121.7 S4 121.7 P3 72.3 S7 72.3 P3 79.3 S1 36.2 C6 68.7 P3 112.1 S2 112.1 P3 90.4 S8 90.4 P3 148.1 S3 132.9 C7 68.7 P3 113.1 S6 113.1 P2 79.3 S1 79.3 P2 94.8 S5 132.8 C8 68.7 P3 94.8 S5 94.8 P2 113.1 S6 113.1 P3 121.7 S4 76.7 6:00 7:00 8:00 9:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00 24:00 Fig. 4 Detailed distribution plan of passive distribution to be delivered on today. The total cost of this real-world The model is solved by the Gurobi Optimizer under the hard- scheduling plan is 82,706 CNY, including 2600 CNY for ware environment of the Intel Core i7-4770k (3.50 GHz). loading and unloading operations, 78,775 CNY for trans- The Gurobi Optimizer is a state-of-the-art mathematical porting operations and 1331 CNY for the costs caused by programming solver able to handle all large-scale problem the late distribution for petrol stations. types. This solver was designed from the ground up to exploit Figure 4 shows the detailed distribution plan. Green rep- modern architectures and multicore processors by using the resents the transportation task, yellow represents the standby latest implementations of the latest algorithms. Furthermore, task, orange represents the loading task, gray represents the the solver supports matrix-oriented interfaces for MATLAB. rest task and blue represents the unloading task. The orange Given the same MILP model, Gurobi can solve it faster and mark represents the type of product that needs to be loaded use less memory than the nested solver of MATLAB. The and unloaded, and the blue mark represents the petrol station optimal distribution plan of trucks is shown in Fig. 7. Its total that needs to be replenished. One complete replenishment cost during the research time horizon is 79,254 CNY, includ- plan for a truck, including the loaded product type and the ing 2600 CNY for loading and unloading operations and replenishing object, is closely associated with the current 76,654 CNY for transportation. Each petrol station is replen- inventory of all petrol stations, one complete replenishment ished before its inventory is low; thus, there is no cost caused plan for a truck, including the loaded product type and the by unpunctual distribution. In contrast to the real-world result, replenishing object. At the start time, all trucks go from the the cost of transportation and out of stock is greatly saved. garage to the depot and then perform the loading operation As is known from Fig. 7, C2 and C6 replenish petrol sta- under the constraints of the number of loading arms and the tions four times a day, while the others replenish petrol sta- demanded oil types. Afterward, each truck drives to the pet- tions three times a day. The loading order depends on the rol station to unload oil and finally back to the garage to rest. loading arm constraint and the truck’s destination. Taking Overall, C2 and C5 replenish petrol stations four times a day, the 7th and 8th time intervals (i.e., 7:10–7:30) as an exam- and the others replenish petrol stations three times a day. ple, all trucks except C6 arrive at the depot, while only C2, Figures 5 and 6 depict the inventory variation of petrol C3, C4 and C5 are loaded because there are 2, 1 and 1 load- stations and depots. It can be seen from Fig. 8 that some- ing arms for P1, P2 and P3 in the depot, respectively. Moreo- times oil shortage occurs in a few petrol stations. Specifi- ver, there are three trucks that need to load P3 during the first cally, P1 is out of stock in S1 at 20:40, P2 is out of stock in loading operation, but the loading order is C4 (replenish for S2 at 16:00, P2 is out of stock in S5 at 18:30, and P3 is out of S2), C7 (replenish for S6) and C8 (replenish for S5). From stock in S8 at 13:40. The stock out of S2 is the most severe Fig. 5, it can be seen that the first station to reach the lower and lasts almost 3 h. limitation is S2, followed by S5 and S6. Considering that the transportation distance of S6 is longer than that of S5, C7 4.3 Optimal scheduling plan loads P3 ahead of C8. With a comprehensive consideration of the transportation time and out-of-stock time, initiative To obtain the optimal scheduling plan of initiative distribu- distribution is able to optimize the delivery sequence for all tion, we develop a mathematical model according to Sect. 3. petrol stations. After the optimization, the time to replenish 1 3 1006 Petroleum Science (2021) 18:994–1010 30 30 P1 P1 25 25 P2 P2 P3 P3 20 20 15 15 10 10 5 5 0 0 6:00 12:00 18:00 0:00 6:00 12:00 18:00 0:00 -5 -5 Time Time (a) S1 (b) S2 30 30 P1 P1 25 P2 25 P2 P3 P3 20 20 15 15 10 10 5 5 0 0 6:00 12:00 18:00 0:00 6:00 12:00 18:00 0:00 -5 -5 Time Time (c) S3 (d) S4 P1 P1 25 P2 25 P2 P3 P3 20 20 15 15 10 10 5 5 6:00 12:00 18:00 0:00 6:00 12:00 18:00 0:00 -5 -5 Time Time (e) S5 (f) S6 30 30 P1 P1 25 P2 25 P2 P3 P3 20 20 15 15 10 10 5 5 0 0 6:00 12:00 18:00 0:00 6:00 12:00 18:00 0:00 -5 -5 Time Time (g) S7 (h) S8 Fig. 5 Inventory variation of petrol stations P1 at S1, P2 at S2, P2 at S5 and P3 at S8 is brought forward Besides, most trucks’ final task is to replenish the petrol to 20:00, 13:10, 18:10 and 13:10, thus avoiding the out of stations that are relatively close to the garage, such as S1 stock occurred under the mode of passive distribution. and S4, and then to return back to the garage to rest after 1 3 3 3 3 3 Inventory, m Inventory, m Inventory, m Inventory, m 3 3 3 3 Inventory, m Inventory, m Inventory, m Inventory, m Petroleum Science (2021) 18:994–1010 1007 objective function and considers the detailed truck status P1 P2 and a series of operational constraints. In the cost compo- P3 nent, loading and unloading operating cost, transportation cost and cost caused by out of stock are taken into account. One step further than previous studies is that the proposed model is capable of dealing with a variety of products, time-varying sales rates and congestion conditions. For a real-world case containing eight petrol stations, one depot with three types of oil products and one garage with eight trucks, the detailed truck dispatch plan is optimized using this model. During the 18 h of the study period, the sales 6:00 12:00 18:00 0:00 rate of each petrol station varies several times and every Time truck has a detailed task assignment in each time interval. And it also visualizes the inventory change of each pet- Fig. 6 Inventory variation of depots rol station and depot. Compared to the actual scheduling plan in passive mode premade by the company, initiative completing the loading task. From Fig. 7, C1 and C3 are the mode leads to less operational cost while controlling the first trucks to arrive at the garage (at 21:00), and they have a inventory of all the petrol stations at low but safe level. In 3-h rest time. Figure 8 illustrates that all the petrol stations the initiative mode, inventory is predicted scientifically would receive timely replenishments when their inventories and the products are sent to the petrol station in a suitable are close to their lower limits, and the inventory during the time, which effectively improves the efficiency of product study horizon stays within the allowable range. This also oil resource allocation and enhances the stability of the indicates that initiative distribution enables petrol stations to supply of petrol station. control their inventory at a relatively low but safe level and However, the truck scheduling problem is an NP-hard gives a potential advantage of reducing inventory handling problem with strict demands on computational time and cost when compared to passive distribution. The inventory space. For a mathematical model, it is suitable for obtain- variation of the depot is depicted in Fig. 9 and meets the ing the optimal solution for a relatively small problem capacity constraints during the entire study horizon. because the solving efficiency is closely related to the model’s scale. Therefore, developing a highly efficient meta-heuristic algorithm to solve increasingly complex 5 Conclusions truck scheduling problems within an acceptable time is important for future works. In order to improve efficiency for the entire replenishment Acknowledgements This work was part of the Program of “Study on system of highway petrol stations, this paper establishes a Optimization and Supply side Reliability of Oil Product Supply Chain MILP model which sets the minimum operating cost as the Driving Standby Rest Loading Unloading C1 68.7 P1 79.3 S1 79.3 P1 148.1 S3 148.1 P1 79.3 S1 36.2 C2 68.7 P1 72.3 S7 S7 72.3 P1 90.4 S8 90.4 P1 112.1 S2 112.1 P1 72.3 S7 80.7 C3 68.7 P1 113.1 S6 113.1 P1 94.8 S5 94.8 P1 121.7 S4 76.7 C4 68.7 P3 112.1 S2 112.1 P2 148.1 S3 148.1 P3 121.7 S4 76.7 C5 68.7 P2 90.4 S8 90.4 P3 90.4 S8 90.4 P3 148.1 S3 132.9 C6 68.7 P2 72.3 S7 72.3 P2 112.1 S2 112.1 P2 94.8 S5 94.8 P3 79.3 S1 36.2 C7 68.7 P3 113.1 S6 113.1 P2 79.3 S1 79.3 P3 72.3 S7 80.7 C8 68.7 P3 94.8 S5 94.8 P2 113.1 S6 113.1 P2 121.7 S4 76.7 6:00 7:00 8:00 9:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00 24:00 Fig. 7 Detailed distribution plan solved by the proposed method 1 3 Inventory, m 1008 Petroleum Science (2021) 18:994–1010 30 30 P1 P1 P2 P2 25 25 P3 P3 20 20 15 15 10 10 5 5 0 0 6:00 12:00 18:00 0:00 6:00 12:00 18:00 0:00 Time Time (a) S1 (b) S2 30 30 P1 P1 P2 P2 25 25 P3 P3 20 20 15 15 10 10 5 5 0 0 6:00 12:00 18:00 0:00 6:00 12:00 18:00 0:00 Time Time (c) S3 (d) S4 30 30 P1 P1 P2 P2 25 25 P3 P3 20 20 15 15 10 10 5 5 0 0 6:00 12:00 18:00 0:00 6:00 12:00 18:00 0:00 Time Time (e) S5 (f) S6 30 30 P1 P1 P2 P2 25 25 P3 P3 20 20 15 15 10 10 5 5 0 0 6:00 12:00 18:00 0:00 6:00 12:00 18:00 0:00 Time Time (g) S7 (h) S8 Fig. 8 Inventory variation of petrol stations 1 3 3 3 3 3 Inventory, m Inventory, m Inventory, m Inventory, m 3 3 3 3 Inventory, m Inventory, m Inventory, m Inventory, m Petroleum Science (2021) 18:994–1010 1009 200 Cornillier F, Laporte G, Boctor FF, Renaud J. 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MILP formulations for highway petrol station replenishment in initiative distribution mode

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1672-5107
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1995-8226
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10.1007/s12182-021-00551-4
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Abstract

To investigate highway petrol station replenishment in initiative distribution mode, this paper develops a mixed-integer linear programming (MILP) model with minimal operational costs that includes loading costs, unloading costs, transport costs and the costs caused by unpunctual distribution. Based on discrete representation, the working day is divided into equal time intervals, and the truck distribution process is decomposed into a pair of tasks including driving, standby, rest, loading and unloading. Each truck must execute one task during a single interval, and the currently executing task is closely related to the preceding and subsequent tasks. By accounting for predictive time-varying sales at petrol stations, real-time road con- gestion and a series of operational constraints, the proposed model produces the optimal truck dispatch, namely, a detailed task assignment for all trucks during each time interval. The model is tested on a real-world case of a replenishment system comprising eight highway petrol stations, one depot, one garage and eight trucks to demonstrate its applicability and accuracy. Keywords MILP · Initiative distribution · Petrol station · Replenishment · Discrete representation Sets and indices Continuous parameters b ∈ B = 1,… , bm The set of tasks. bm is the maxi- c The unit cost of the loading opera- i i i L mum numbering of truck i’s task tion, CNY i ∈ I={1, … , im} The set of trucks. im is the maxi- c The unit cost of unpunctual distri- UO mum numbering of trucks bution, CNY/(m  h) j, j ∈ J = {1, … , jm} The set of nodes. jm is the maxi- c The transport costs per km, CNY/ UT mum numbering of nodes km k ∈ K = {1, … , km} The set of products. km is the c The unit cost of the unloading maximum numbering of products operation, CNY t, t ∈ T = {1, … , tm} The set of time nodes. tm is the h The number of loading arms for Lmaxj,k maximum numbering of time product k at node j (if node j is a nodes depot) l The distance between nodes j and j,j′ j′, km M The sufficiently large number Mi The sufficiently small number r The average driving speed of each truck, km/h r The loading flow rate, m /h Edited by Xiu-Qiu Peng r The unloading flow rate, m /h * Yong-Tu Liang v The maximum capacity of product Dmaxj,k liangyt21st@163.com k at node j (if node j is a depot), Beijing Key Laboratory of Urban Oil and Gas Distribution v The minimal capacity of product k Technology, Beijing 102249, China Dminj,k at node j (if node j is a depot), m Center for Spatial Information Science, The University v The capacity of truck I, m of Tokyo, 5-1-5 Kashiwanoha, Kashiwa-shi, Chiba 277-8568, NUi Japan Vol:.(1234567890) 1 3 Petroleum Science (2021) 18:994–1010 995 v The received volume of product Binary variables Oj,k,t k at node j from refineries during E The binary variable indicating BTi,b,t time interval (t, t + 1) (if node j is that truck i’s task b starts at time a depot), m /h node t v The sales rate of product k at node E The binary variable indicating Sj,k,t DVi,j,b,k,t k during time interval (t, t + 1) (if that truck i’s task b is loading node j is petrol station), m /h product k to node j during time v The maximum capacity of product interval (t, t + 1) Pmaxj,k k at node j (if node j is petrol sta- E The binary variable indicating DVBi,j,b,k,t tion), m that truck i’s task b is to start t The maximal rest time, h loading product k from node j at Rmax t The minimal rest time, h time node t Rmin t The maximum standby time, h E The binary variable indicating Smax DVNi,j,b,k,t t The minimum standby time, h that truck i’s task b is to complete Smin Z The congestion coefficient loading product k from node j at t,j,j′ between node j and j′ during time time node t interval (t, t + 1) E The binary variable indicating PVi,j,b,k,t Δτ The length of the time interval, h that truck i’s task b is to unload τm The length of the study horizon, h product k at node j and is finished α The coefficient for the transport at time node t costs of trucks with full of product N The binary variable indicating Ci,b,j k that truck i is staying at node j when its task b starts Binary parameters P The binary variable indicating Li,b,k f The binary parameter indicating Li,k that truck i is filled with product k that truck i can load product k during its task b f The binary parameter indicating PDj S The binary variable indicating Di,b that node j is a depot that truck i’s task b is to drive to f The binary parameter indicating PGj another node that node j is the garage S The binary variable indicating Li,b,k f The binary parameter indicating PPj that truck i’s task b is to load that node j is a petrol station product k at the depot f The binary parameter indicating PSj S The binary variable indicating Ri,b that the standby task is permissi- that truck i’s task b is to rest at the ble at node j garage Positive continuous variables S The binary variable indicating Si,b C The cost caused by the unpunctual that truck i’s task b is to standby Ot,j,k distribution of node j ’s product k S The binary variable indicating Ui,b at time node t, CNY that truck i’s task b is to unload C The transport cost of truck i’s task product at a petrol station Tt,j,k b, CNY T The total rest time of truck i from Ri,b the beginning of the study horizon 1 Introduction to the start time of its task b, h V The inventory of product k at 1.1 Background Pj,k,t node j at time node t (if node j is a petrol station), m As necessary facilities for urban transportation, petrol sta- V The inventory of product k at tions play a critical role in the steady supply of refined prod- Dj,k,t node j at time node t (if node j is a ucts (Fernandes et al. 2013), and efficient distribution of depot), m refined products is also important in the smooth operation of cities. To daily replenish petrol stations, the transport company must simultaneously determine detailed routes and product assignments for each truck (Cattaruzza et al. 1 3 996 Petroleum Science (2021) 18:994–1010 2014; Liu and Jiang 2012). As we know, product oil distri- They developed an integer programming model to address bution between refineries and oil depots is usually carried the problem of a nationwide bulk delivery fleet of petroleum by pipelines (Fernandes et al. 2013), and there are distribu- tanker trucks. The objective function was to minimize trans- tion stations along the pipelines. Researchers often focus on port costs under the constraints of equitable manpower and the formulation of scheduling plans (Zhang et al. 2016) and equipment workload distributions, safety, customer service, transmission plans (Fernandes et  al. 2013). Zhang et  al. and equipment compatibility restrictions. The model was (2017) established a MILP model to solve problems, and applied to a scale of more than 80 bulk terminals on a fleet some researchers (Moradi et al. 2019) had also proposed exceeding 300 vehicles with approximately 2600 loads per efficient algorithm to solve scheduling studies. Liang et al. day. Brown et al. (1987) extended the work of Brown and (2012) considered regional differences in electricity prices Graves (1981) for a real-time, transaction-driven information along the pipeline, established an optimization model for a management system that could solve the problem for 120 multi-product pipeline which has a known delivery demand bulk terminals and more than 430 vehicles. and operation plan for each off-take station. Different from Apart from mathematical programming methods, heu- the distribution between refineries and oil depots, the replen- ristic algorithms (Azad et al. 2017; Kasivisvanathan et al. ishment of petrol stations takes truck transportation as the 2014; Nambirajan et  al. 2016), intelligent optimization mode of transportation, which is easier to be affected by (Diaby et al. 2016; Marinakis et al. 2013; Soto et al. 2017) the condition of road congestion. The replenishment of and hybrid algorithm (Alinaghian and Shokouhi 2018; Shi petrol stations is mainly classified into two types: passive et al. 2017) had also been widely used in dealing with the distribution and initiative distribution. For passive distribu- scheduling optimization issue. For example, focusing on the tion, petrol stations order products before the next work- truck assignment and routing problem of a Hong Kong pet- ing day, specifying the minimum and maximum quantity rol distribution company, Ng et al. (2008) proposed a deci- to be delivered for each ordered product. The minimum sion support system that combines heuristic clustering and quantity is often determined by the stations’ average daily optimal routing to determine the optimal fleet assignment sales, while the maximum quantity is determined by the dif- and routing. Multiple objective functions included the mini- ference between the capacity of the station’s storage tanks mal number of used trucks, the minimal number of drops and an estimation of the remaining stock before the next in each trip, the maximal profits in terms of total products replenishment. Generally, petrol stations specify that prod- delivered and the maximal utilization of resources. For the ucts be replenished during periods when sales are low and multi-depot petrol station replenishment problem, Cornil- not during peak periods. However, passive distribution has lier et al. (2008) presented a heuristic that contained route the following problems. First, the specified replenishment construction and truck loading procedures, a route packing time of different petrol stations may be highly similar, so procedure and two procedures enabling the anticipation or that trucks are extremely busy during these periods and idle the postponement of deliveries. Next, Cornillier et al. (2009) when the petrol stations are busy and unable to replenish studied the filling station replenishment problem with time stock. Second, a single petrol station may receive a variety intervals (PSRPTW). The introduction of time intervals of products several times from transportation companies in a leads to a great increase in model scale and computational single day, and it is difficult to estimate and coordinate each effort, so they proposed an MILP model and two heuristics distribution time. By contrast, with initiative distribution, for this problem to determine the quantity of each product to petrol stations do not order products but instead forecast the deliver, the assignment of products to truck compartments, sales for each station, while the transport company assigns delivery routes and schedules. In fact, petroleum products the specific time to deliver the products. The advantage of are stored in a number of different depots, and they are deliv - initiative distribution is that it treats the whole replenishment ered to a set of petrol distribution stations. Each depot has process as an integrated system, and the replenishment time its own fleet of heterogeneous and compartmented tanker is determined by the system instead of the petrol stations. trucks. Therefore, in their later work, Cornillier et al. (2012) This approach allows the working time of trucks in a day to developed a heuristic procedure to generate a restricted set be more balanced, thereby avoiding the inefficiency created of promising feasible trips beforehand for the multi-depot by concentrated periods of demand. Overall, initiative dis- petrol station replenishment problem with time intervals tribution improves efficiency in the dispatch of trucks and (MPSRPTW). Wang et  al. (2018) established a mixed- therefore in the running of the entire replenishment system. integer linear model and proposed an improved heuristics algorithm to solve the petrol truck scheduling problem which 1.2 Related work considered multi-path selection and congestion. Boctor et al. (2011) firstly introduced a generalized trip packing One of the first articles addressing the issue of petrol station problem, in which trucks have different capacities and trip replenishment was published by Brown and Graves (1981). net revenues depend on the trucks used. Meanwhile, some 1 3 Petroleum Science (2021) 18:994–1010 997 construction, improvement and neighborhood search solu- 1.3 Contributions of this work tion heuristics are proposed to solve a real-life case. In addition to petrol station replenishment issues, a large • With a focus on highway petrol station replenishment number of studies have also focused on the vehicle routing in the initiative distribution mode, an MILP approach problem (VRP), which focuses on the use of efficient algo- is proposed to provide a more reasonable schedule for rithms (Li et al. 2012; Goodson et al. 2012; Schyns 2015). trucks and improve efficiency for the entire replenish- Brandão and Mercer (1997) presented a novel tabu search ment system. heuristic algorithm to solve the multi-trip routing and sched- • Various factors appearing in practice, such as time-var- uling problem (VRSP) with respect to constraints in which ying sales, task time limitations, real-time road conges- a vehicle can make more than one trip each day and the cus- tion, different unit transport costs under full-load or no- tomers impose delivery time intervals. Based on the adaptive load conditions, road capacity, the detailed operational memory procedure, Olivera and Viera (2007) described a mode (namely, loading, unloading, rest and standby) and heuristic method to solve the vehicle routing problem with operational limitations, are all taken into consideration. multiple trips. To solve a wide range of different types of • The optimal distribution plan, including the quantities, VRPs, Gromicho et al. (2012) presented a highly flexible specific time and replenishment object of each product framework accommodating various real-life restrictions. to be delivered by each truck, is solved by the proposed Wang et al. (2019) established a mathematical model to model. solve the petrol truck routing problem and proposed a heu- • A real case study is given to demonstrate the practicality ristic algorithm based on a local branch-and-bound search of this method. with a tabu list, several inequalities are added to accelerate the solving process. A comparison with state-of-the-art local 1.4 Paper organization search methods indicates the superiority of this new frame- work in practical applications. The adopted methodology and details of the mathematical This paper studies on highway petrol station replenish- model are given in Sects. 2 and 3, respectively. Section 4 ment in the initiative distribution mode. Different from clas- provides a real case study in China and demonstrates the sical petrol station VRP problem, highways have a large and model’s practicability. Finally, conclusions are presented in various demand for refined products in this study, so a single Sect. 5. petrol station on a highway may be replenished more than once during a working day. Moreover, highway petrol sta- tions receive a variety of refined products transported from 2 Methodology depots, and a reasonable replenishment time for each type of product is difficult to predict only considering each petrol 2.1 Problem description station. In fact, the replenishment mode studied in previ- ous relevant work belongs to the passive distribution type, Figure  1 illustrates the highway petrol station replenish- in which the quantity and delivery timing of each product ment system studied in this paper. There are three types of are given by petrol stations. Compared with initiative dis- tribution, passive distribution is dominated by human fac- Garage tors and often results in impractical truck allocation and Petrol station low distribution efficiency. With the increasing popularity of Internet technology, the shift from passive distribution to initiative distribution is a compelling trend among petro- leum enterprises looking to accomplish the daily informa- Unload Drive tion-based management of oil resources. However, to the Rest best of our knowledge, this paper is the first research work that addresses this replenishment problem in an initiative Depot distribution mode rather than a passive distribution mode. None of the abovementioned papers have simultaneously Petrol station determined the quantities, timing and replenishment object Standby of each product to be delivered by each truck through a com- Drive prehensive consideration of time-varying sales at petrol sta- tions, real-time road congestion and the state monitoring of Load all trucks. Fig. 1 Work flow for highway petrol station replenishment 1 3 998 Petroleum Science (2021) 18:994–1010 destinations, namely garages, depots and petrol stations. At The congestion condition during each time interval. • The basic information of trucks. the initial time of one working day, trucks departure from garages to depots and then cover refined products to the pet- Determine: rol stations which are short of oil. Considering the limited number of loading arms, an upper limit is set for the number • The detailed scheduling of trucks: the executed task dur- of trucks that are loading simultaneously. Therefore, trucks need to wait when there is no free loading arm. Additionally, ing each time interval. • The departure point and the corresponding destination of all trucks are stored in the garage at the end of the work- ing day. Since the petrol stations are sparsely distributed each transport task. • The quantity and type of product being loaded (unloaded) along the highway, the distance between any petrol stations is rather long and an oil depot can serve only a small number of each load (unload) task. • The inventory variations of depots and petrol stations. of petrol stations. It usually takes a long time for a truck to complete a single delivery task, which brings higher-level The total operational costs. demands for truck scheduling from oil tankers. Distribut- ing trucks according to the reports from petrol stations may Objective: The objective is to create a distribution plan with minimal result in higher demand in some busy time periods and no demand in other times, thereby greatly reducing the effi- operational costs while respecting a variety of operational and technical constraints. To establish and solve the model ciency of the dispatch system. In order to overcome the lack of passive distribution effectively, assumptions are made as follows. model stated above and provide a more reasonable schedule for trucks and reduce cost, this paper develops a discrete- The trucks drive at a constant speed and can only load one type of specified product during each time interval. time MILP model in the initiative distribution mode and incorporates the constraint of sales predictions as an addition After the truck arrives at the depot or petrol station, it needs time to load or unload the petrol products, and the to other constraints. In the proposed model, the truck distri- bution process is decomposed into a pair of tasks, includ- speed of loading and unloading is constant. • During a single workday, all trucks start from the garage ing driving, standby, rest, loading and unloading, and each task is closely related to the preceding and subsequent task. and return to the garage after completing all tasks. • Petroleum companies predict the daily sales plan of the For any truck, a complete replenishment process involves driving to a depot, loading, driving to a petrol station and petrol stations and a daily delivery plan for the depots beforehand. unloading. In practical engineering, a truck can replenish the oil supply of multiple petrol stations or of a single pet- The standby times and rest times of each vehicle during the working day should be within the allowed limits. rol station repeatedly for 1 day. Therefore, there are several tasks that must be completed at specified destinations for The transport time between any two destinations is closely related to their real distance and real-time con- each truck. Moreover, the entire scheduling time horizon is divided into several equal time intervals, and each truck gestion conditions. must execute one task during each interval. 3 Mathematical model 2.2 Model requirements This paper develops the MILP model based on discrete rep- The model is formulated as MILP, and MATLAB R2014a is resentation (Shaik and Bhat 2014). The studied system is made up of a set of nodes j ∈ J, including depots (f = 1), employed to calculate the detailed truck routing and schedul- PDj ing of petrol station replenishment. petrol stations (f = 1) and garages (f = 1). Time nodes PPj PGj t ∈ T is used to divide the working day into several equal time Given: intervals, and the truck distribution process is decomposed into a pair of tasks. Each truck must execute one task during The time horizon of the working day. a single interval, and the currently executing task is closely • related to the preceding and subsequent task. By accounting The locations of garages, depots and petrol stations. The delivery plan of the depot (delivery volume during for predictive time-varying sales at petrol stations, real-time road congestion and a series of operational constraints, the each time interval) and the sales plan of the petrol sta- tions (sales volume during each time interval). proposed model produces the optimal truck dispatch, i.e., • the detailed task assignment of all trucks during each time The inventory of the depot and petrol stations for each product. interval. 1 3 Petroleum Science (2021) 18:994–1010 999 Overall, the model involves five sets: the set of trucks (I ); P ≤ f i ∈ I, k ∈ K L i,b,k L i,k (4) the set of tasks (B ); the set of nodes (J); the set of prod- i b∈B uct types (K); and the set of time nodes (T) chronologically The executable operations of all trucks include load, arranged. Moreover, this model comprises of positive con- unload, rest, drive and standby, only one of which can be tinuous variables controlling the product inventory at pet- executed during a single task: rol stations (V ) and depots (V ), the accumulated rest Pj,k,t Dj,k,t time of trucks (T ), transport costs (C ) and the costs for Ri,b Ti,b S + S + S + S + S = 1 i ∈ I, b ∈ B L i,b,k U i,b R i,b D i,b S i,b i unpunctual distribution (C ). Beyond that, a series of key Ot,j,k k∈K binary variables are also introduced to identify if truck i are (5) loading/unloading/resting/driving/standing by during its task b (in case S , S , S , S , S = 1), if truck i is filled Li,b,k Ui,b Ri,b Di,b Ri,b 3.3 Loading constraints with product k during its task b (in case P = 1), if truck i’s Li,b,k task b starts at time node t(in case E = 1), and if truck i BTi,b,t For any truck i, the loading operation can only be executed is staying at node j when its task b starts (in case N = 1). Ci,b,j at depots. In other words, the binary variable of the loading operation (S ) can equal one when truck i is located at node Li,b,k 3.1 Objective function j, with node j being a depot (i.e., N = 1, f = 1). Ci,b,j PDj The objective function is the minimum opera - S ≤ N f i ∈ I, b ∈ B L i,b,k C i,b,j PD j i (6) tional cost, including the loading operation cost k∈K j∈J ∑ ∑ ∑ ( c S ), the unloading operation cost i∈I b∈B k∈K L L i,b,k ∑ ∑ ∑ ∑ Constraint (7) states that only no-load trucks can execute ( c S ), the transport cost ( C ) U Ui,b Ti,b i∈I b∈B i∈I b∈B i i loading operations. If truck i loads product k during its task b, and the cost caused by the late distribution for petrol sta- ∑ ∑ ∑ it will be filled with product k at the start time of its next task, tions ( C ). Ot,j,k t∈T j∈J k∈K that is to say, P would be one if S is equal to 1 [see Li,b+1,k Li,b,k constraint (8)]. Otherwise, the truck without product k will min F = c S + c S + C L L i,b,k U U i,b T i,b remain unchanged. As seen in constraint (9), if P is zero Li,b,k i∈I b∈B k∈K P must be zero. Li,b+1,k + C O t,j,k S ≤ 1 − P i ∈ I, b ∈ B , k ∈ K (7) L i,b,k L i,b,k i t∈T j∈J k∈K P (1) S ≤ P i ∈ I, b < bm , k ∈ K L i,b,k L i,b+1,k i (8) 3.2 T rucks’ state constraints P ≤ P + S i ∈ I, b < bm , k ∈ K L i,b+1,k L i,b,k L i,b,k i (9) This paper introduces binary variables to indicate the posi- tion of all trucks. N equals 1 when truck i is located at Ci,b,j node j at the start time of task b. During any task, a truck 3.4 Unloading constraints can only stay at one node. For any truck i, the unloading operation can only be executed N = 1 i ∈ I, b ∈ B C i,b,j i (2) at a petrol station. Specifically, the binary variable of the j∈J unloading operation (S ) equals one when truck i is located Ui,b There are two states for all trucks: one is full-load, and at node j, while node j is a petrol station. the other is no-load. If truck i is filled with product k at the S ≤ N f i ∈ I, b ∈ B start time of its task b, P = 1, and otherwise P = 0. U i,b C i,b,j PP j i Li,b,k Li,b,k (10) j∈J In this way, P = 1 stands for full-load while Li,b,k k∈K P = 0 stands for no-load. Furthermore, the trucks k∈K Li,b,k Constraint (11) states that only full-load trucks can execute are also divided into two types: one stores gasoline, and the the unloading operation. If truck i unloads oil during its task b, other stores diesel oil. Equations (3) and (4) ensure that each it will be empty at the start time of its next task, that is to say, truck only transports one type of specified oil during a single P must be zero as long as S is equal to one [see Li,b+1,k Ui,b k∈K task. Note that the binary parameter f in Eq. (3) is used to Li,k constraint (12)]. If it does not unload oil, the truck that is filled indicate if truck i can load product k. with product k will remain unchanged. As seen in constraint (13), if P is one P must be one. Li,b,k Li,b+1,k P ≤ 1 i ∈ I, b ∈ B L i,b,k i (3) k∈K 1 3 1000 Petroleum Science (2021) 18:994–1010 If truck i’s position at the start time of task b is different S ≤ P i ∈ I, b ∈ B U i,b L i,b,k i (11) from that at task b + 1, then we can conclude that truck i’s k∈K task b is driving. As constraint (19) states, S is bounded Di,b by the maximum value of N − N . Ci,b+1,j Ci,b,j S ≤ 1 − P i ∈ I, b < bm U i,b L i,b+1,k i (12) k∈K N − N ≤ S i ∈ I, b < bm , j ∈ J C i,b+1,j C i,b,j D i,b i (19) P ≤ P + S i ∈ I, b < bm , k ∈ K (13) L i,b,k L i,b+1,k U i,b i 3.6 Time constraints 3.5 Rest, standby and driving constraints To develop the constraints between time and tasks, this paper introduces a binary variable (E ), which equals 1 when BTi,b,t Only when the truck stops at the garage at the work start truck i’s task b starts at time node t. By constraint (20), each time can it rest. By constraint (14), S can be one only Ri,b task must start at one time node. Constraint (21) determines when N f = 1. Ci,b,j PGj that no task start be earlier than the start time of the preced- j∈J ing task. Furthermore, the last task of any truck must start S ≤ N f i ∈ I, b ∈ B R i,b C i,b,j PG j i (14) before the study horizon, which is stated in constraint (22). j∈J E = 1 i ∈ I, b ∈ B BT i,b,t i Due to safety concerns, full-load trucks cannot rest. This (20) t∈T operation condition is enforced by constraint (15). The binary variable of rest operation (S ) must be zeros if Ri,b tΔE ≤ tΔE i ∈ I, b ∈ B P = 1. BT i,b,t BT i,b+1,t i Li,b,k k∈K (21) t∈T t∈T P ≤ 1 − S i ∈ I, b ∈ B L i,b,k R i,b i (15) k∈K tΔE ≤ mi ∈ I BT i,bm ,t i (22) t∈T To ensure safety, trucks can only stand by at some par- ticular nodes. By constraint (16), S can be one only when Si,b If the truck is loading products during its task b (i.e., N f = 1. j∈J C i,b,j PS j S = 1 ), the time consumed by this task is related to k∈K Li,b,k the truck’s capacity and the loading flow rate. If the truck is S ≤ N f i ∈ I, b ∈ B S i,b C i,b,j PS j i (16) not loading products, constraints (23)–(24) are redundant, and j∈J the interval duration can then match the value computed from the time constraints of another unload, rest, drive and standby. At the end of the weekday, all trucks must be at a rest station, and the last task must be rest, as seen in Eq. (17). NU i (23) tΔ𝜏 E − tΔ𝜏 E ≤ +Δ𝜏 + 1 − S 𝜏 mi ∈ I, b < bm BT i,b+1,t BT i,b,t L i,b,k i t∈T t∈T k∈K NU i (24) tΔ𝜏 E − tΔ𝜏 E ≥ + S − 1 𝜏 mi ∈ I, b < bm BT i,b+1,t BT i,b,t L i,b,k i t∈T t∈T k∈K Similarly, if truck i is unloading products during its task b (i.e., S = 1), the time consumed by this task is related N f = 1 i ∈ I, j ∈ J Ui,b C i,bm ,j PG j (17) j∈J to the truck’s capacity and unloading flowrate. Otherwise, constraints (25)–(26) are redundant and the interval duration All trucks cannot standby twice continuously. In con- is bounded by other operations. straint (18), only one of two adjacent tasks b and b + 1 can be standby. NU i tΔ𝜏 E − tΔ𝜏 E ≤ BT i,b+1,t BT i,b,t t∈T t∈T S +S ≤ 1 i ∈ I, b < bm (18) S i,b S i,b+1 i +Δ𝜏 + 1 − S 𝜏 mi ∈ I, b < bm U i,b i (25) 1 3 Petroleum Science (2021) 18:994–1010 1001 If truck i is standing by during its task b (i.e., S = 1), the NU i Si,b tΔ𝜏 E − tΔ𝜏 E ≥ BT i,b+1,t BT i,b,t r standby time should not be too long. Constraints (34)–(35) t∈T t∈T impose the lower and upper bounds on the standby time. + S − 1 𝜏 mi ∈ I, b < bm U i,b i (26) tΔ𝜏 E − tΔ𝜏 E ≤ t BT i,b+1,t BT i,b,t Smax If truck i is resting during its task b (i.e., S = 1), the rest Ri,b t∈T t∈T time of this truck should be equal to the accumulated rest + 1 − S 𝜏 mi ∈ I, b < bm S i,b i (34) time before task b plus the duration of task b [i.e., constraints (27)–(28)]. If not, the rest time of this truck should be equal to the accumulated rest time before task b [i.e., constraints tΔ𝜏 E − tΔ𝜏 E ≥ t BT i,b+1,t BT i,b,t Smin (29)–(30)]. The total rest time of each truck should be lim- t∈T t∈T ited to a range [i.e., constraint (31)]. + S − 1 𝜏 mi ∈ I, b < bm S i,b i (35) T ≤ T + tΔ𝜏 E − tΔ𝜏 E R i,b+1 R i,b BT i,b+1,t BT i,b,t t∈T t∈T 3.7 Petrol station constraints + 1 − S 𝜏 mi ∈ I, b < bm R i,b i (27) The binary variable E denotes the completed status PVi,j,b,k,t of replenishment and equals one when a full-load truck i T ≥ T + tΔ𝜏 E − tΔ𝜏 E R i,b+1 R i,b BT i,b+1,t BT i,b,t finishes unloading product k to node j at time node t. By t∈T t∈T constraints (36)–(37), if any value of binary variables S Ui,b + S − 1 𝜏 mi ∈ I, b < bm R i,b i (28) E , N and P is equal to zero, E must BTi,b+1,t Ci,b+1,j Li,b,k PVi,j,b,k,t be zero. Conversely, if these variables are all equal to one, T ≤ T + S 𝜏 mi ∈ I, b < bm R i,b+1 R i,b R i,b i (29) E must be one. PVi,j,b,k,t S + E + N + P U i,b BT i,b+1,t C i,b+1,j L i,b,k T ≥ T − S 𝜏 mi ∈ I, b < bm E ≥ Mi + R i,b+1 R i,b R i,b i (30) PV i,j,b,k,t − 1 i ∈ I, j ∈ J, b < bm , k ∈ K, t ∈ T t ≤ T ≤ t i ∈ I R min R i,bm R max (31) (36) S + E + N + P U i,b BT i,b+1,t C i,b+1,j L i,b,k If truck i is driving during its task b (i.e., S = 1), the Di,b E ≤ 1 − 1 − Mi PV i,j,b,k,t time consumed must be related to the average of the driving distance, the speed of the truck and the coefficient of road i ∈ I, j ∈ J, b < bm , k ∈ K, t ∈ T congestion during this time period. Specifically, if truck i ’s (37) ∑ ∑ ∑ task b is to move from node j to node j′ during time interval If E is more than one, the PVi,j,b,k,t i∈I b∈B k∈K t (i.e., S , N , N = 1), inequation constraints Di,b Ci,b+1,j′ Ci,b+1,j received volume of node j during time interval (t, t + 1) (32)–(33) work as an equation constraint. By doing so, the equals the sum of the capacities of the corresponding trucks. transport time is equal to the driving distance between node Otherwise, the received oil volume is zero. The inventory j and node j′(l ) being divided by the truck speed (r ) and j,j′ D of node j equals the inventory at the last time node plus the then being multiplied by the congestion coefficient during received oil volume and minus the sales volume during the interval t(Z ). t,j,j′ last time interval. ∑ ∑ z E + z E � � ⎛ t,j,j BT i,b+1,t t,j,j BT i,b,t ⎞ � � l � j,j t∈T t∈T ⎜ ⎟ tΔ𝜏 E − tΔ𝜏 E ≤ +Δ𝜏 BT i,b+1,t BT i,b,t r ⎜ 2 ⎟ t∈T t∈T (32) ⎝ ⎠ � � N + N � � � C i,b+1,j C i,b,j + 1 − S 𝜏 m+ 1 − 𝜏 mi ∈ I, b < bm , j, j ∈ J D i,b i ∑ ∑ z �E + z �E ⎛ t,j,j BT i,b+1,t t,j,j BT i,b,t ⎞ � � l � j,j t∈T t∈T ⎜ ⎟ tΔ𝜏 E − tΔ𝜏 E ≥ BT i,b+1,t BT i,b,t r ⎜ 2 ⎟ t∈T t∈T ⎝ ⎠ (33) � � N + N � � � C i,b+1,j C i,b,j + S − 1 𝜏 m + − 1 𝜏 mi ∈ I, b < bm , j, j ∈ J D i,b i 1 3 1002 Petroleum Science (2021) 18:994–1010 V = V + E v − v j ∈ J, k ∈ K, t > 1 P j,k,t P j,k,t−1 PV i,j,b,k,t NU i S j,k,t−1 E ≤ h j ∈ J , k ∈ K, t ∈ T DV i,j,b,k,t L max j,k i (45) i∈I b∈B i∈I b∈B (38) ∑ ∑ ∑ If E is more than one, the dis- During any time interval t, the inventory of the petrol sta- i∈I b∈B k∈K DVi,j,b,k,t charged volume of node j during time interval (t, t + 1) is equal tion should meet certain limits. To obtain a feasible solution to the aggregate capacities of the trucks that are loading. Oth- when faced with stockouts caused by falling product prices, erwise, the discharged volume is zero. Hence, the inventory at there is no lower limit for the inventory of the petrol station. node j is equal to the inventory of the last time node plus the However, this model has added the costs caused by unpunc- received volume from refineries and minus the discharged oil tual distribution into the objective function to ensure that each volume during the last time interval. petrol station keeps a stock of petrol products. V = V + v − E v j ∈ J, k ∈ K, t ∈ T V ≤ v j ∈ J, k ∈ K, t ∈ T D j,k,t D j,k,t−1 Oj,k,t−1 DVN i,j,b,k,t NU i P j,k,t Pmax j,k (39) i∈I b∈B (46) During any time interval t, the inventory of all depots 3.8 Depot constraints should meet certain limits. This paper introduces three types of binary variables to rep- v ≤ V ≤ v j ∈ J, k ∈ K, t ∈ T Dmin j,k D j,k,t Dmax j,k (47) resent the start, finish and process of the loading operation. Specifically, E equals one when a no-load truck i starts DVBi,j,b,k,t loading product k from node j at time node t [i.e., constraints 3.9 Cost constraints (40)–(41)], while E equals one when the truck has just DVNi,j,b,k,t finished loading oil [i.e., constraints (42)–(43)]. As constraint For a given transport route, fuel consumption increases (44) shows, E is bounded by E and E , DVi,j,b,k,t DVBi,j,b,k,t DVNi,j,b,k,t with the load. Hence, the transport costs are calculated and it equals one when truck i is loading product k from node by transportation distance and truck status, as shown in j during time interval (t, t + 1). constraints (48)–(49). More specifically, transport costs for S + E + N L i,b,k BT i,b,t C i,b,j all empty trucks only depend on the product of unit cost E ≥ Mi + DVB i,j,b,k,t 3 and transportation distance. However, if truck i is full of − 1 i ∈ I, j ∈ J, b < bm , k ∈ K, t ∈ T product k, the transport costs must be a bitter higher than i (40) the costs of empty trucks. Therefore, the parameter α in constraints (48)–(49) is used to compute the extra costs for S + E + N L i,b,k BT i,b,t C i,b,j E ≤ 1 − 1 − Mi carrying v m of product k. DVB i,j,b,k,t NUi N + N C i,b+1,j C i,b,j i ∈ I, j ∈ J, b < bm , k ∈ K, t ∈ T i (41) C ≤ c l 1 + P v 𝛼 + 1 − M T i,b UT j,j L i,b,k NU i k k∈K i ∈ I, j, j ∈ J, b < bm , t ∈ T S + E + N L i,b,k BT i,b+1,t C i,b+1,j E ≥ Mi + DVN i,j,b,k,t (48) N + N − 1 i ∈ I, j ∈ J, b < bm , k ∈ K, t ∈ T � C i,b+1,j C i,b,j i (42) C ≥ c l 1 + P v 𝛼 + − 1 M T i,b UL j,j L i,b,k NU i k k∈K S + E + P i ∈ I, j, j ∈ J, b < bm , t ∈ T L i,b,k BT i,b+1,t C i,b+1,j i E ≤ 1 − 1 − Mi DVN i,j,b,k,t (49) It is forbidden for the truck to be delivering oil if its i ∈ I, j ∈ J, b < bm , k ∈ K, t ∈ T location remains unchanged. If this occurs, then the cor- (43) responding costs are at the maximum level. By constraint E = E + E DV i,j,b,k,t DV i,j,b,k,t−1 DV Bi,j,b,k,t (50), transport costs ( C ) would be greater than a suf- T i,b − E i ∈ I, j ∈ J, b < bm , k ∈ K, t > 1 ficiently large value (1/2M ) when binary variables S , DV Ni,j,b,k,t i (44) Di,b N , N are all equal to one. Ci,b+1,j Ci,b,j Due to the limited number of loading arms, a depot can N + N only permit a limited number of trucks to load oil during a C i,b+1,j C i,b,j C ≥ M + S − 1 M + − 1 M T i,b D i,b single time interval. 2 2 i ∈ I, j ∈ J, b < bm (50) 1 3 Petroleum Science (2021) 18:994–1010 1003 As for any petrol station, a negative inventory stands for are usable for P2 and P3. However, each truck can only unpunctual distribution, which leads to economic losses. The load one type of product at a time. The oil tank capacity right term of constraint (51) is a positive value when the of each truck is 20 m , and the inventory of the petrol sta- inventory V is negative, and thus imposes a lower bound tion is 30 m . There are 2, 1 and 1 loading arms for P1, Pj,k,t for the costs of unpunctual distribution C . P2 and P3 in the depot, respectively. The unit labor cost Ot,j,k and wastage cost of loading or unloading oil each time is C ≥ −c V Δ j ∈ J, k ∈ K, t ∈ T O t,j,k UO P j,k,t (51) 50 CNY, and the unit cost caused by unpunctual distribu- tion is 600 CNY/(m  h) During the transport task, the unit transport cost for an empty truck is 10.9 CNY per kilo- meter, while it is 16.7 CNY per kilometer for a full-load 4 Results and discussion truck. The trucks’ speeds when loading, unloading and 3 3 driving are 60 m /h, 60 m /h and 60 km/h, respectively. 4.1 Basic data And the replenishment speeds of P1, P2 and P3 of the 3 3 3 depot are 71.67 m /h, 56.67 m /h and 60 m /h, respec- This section is to validate our model and the superiority tively. Within the study horizon of the study, the sales rate of initiative distribution by using a real-world case of a of petrol station will change several times and is shown highway in Beijing, China. Figure 2 illustrates a real-world in Fig. 3. Standby time should be less than 1.5 h, while block where there are eight petrol stations (S1, S2, S3, the total rest time for each driver should be less than 3 h. S4, S5, S6, S7 and S8), a garage and a depot. The depot is located at the center of the block and is responsible for the 4.2 Real‑world scheduling plan product supplies of these petrol stations. A daily distribu- tion plan starts at 6:00 and finishes at 24:00. During the To demonstrate our advantages in the following part, we workday, all trucks start from the garage and return to the first give an overview of the actual scheduling plan premade garage after completing all tasks. For this replenishment by the company, which can be seen as a classic example of system, there are a total of eight available trucks (C1, C2, passive distribution. All the petrol station only puts forward C3, C4, C5, C6, C7 and C8) to transport three types of the ordered products capacity and has no specific require- products (P1, P2 and P3). To avoid contamination, C1, ments on the delivery time. It only requires the products C2 and C3 can only load P1, while C4, C5, C6, C7 and C8 050 100 150 300 300 250 250 200 200 150 150 100 100 50 50 0 0 050 100 150 Fig. 2 Positions of the depot, garage and petrol stations 1 3 1004 Petroleum Science (2021) 18:994–1010 2.5 2.5 P1 P1 P2 P2 2.0 2.0 P3 P3 1.5 1.5 1.0 1.0 0.5 0.5 0 0 6:00 12:00 18:00 0:00 6:00 12:00 18:00 0:00 Time Time (a) S1 (b) S2 2.5 2.5 P1 P1 P2 P2 2.0 2.0 P3 P3 1.5 1.5 1.0 1.0 0.5 0.5 0 0 6:00 12:00 18:00 0:00 6:00 12:00 18:00 0:00 Time Time (c) S3 (d) S4 2.5 2.5 P1 P1 P2 P2 2.0 2.0 P3 P3 1.5 1.5 1.0 1.0 0.5 0.5 0 0 6:00 12:00 18:00 0:00 6:00 12:00 18:00 0:00 Time Time (e) S5 (f) S6 2.5 2.5 P1 P1 P2 P2 2.0 2.0 P3 P3 1.5 1.5 1.0 1.0 0.5 0.5 0 0 6:00 12:00 18:00 0:00 6:00 12:00 18:00 0:00 Time Time (g) S7 (h) S8 Fig. 3 Sales rates of petrol stations 1 3 3 3 3 3 Sale rate, m /h Sale rate, m /h Sale rate, m /h Sale rate, m /h 3 3 3 3 Sale rate, m /h Sale rate, m /h Sale rate, m /h Sale rate, m /h Petroleum Science (2021) 18:994–1010 1005 Driving Standby Rest Loading Unloading C1 68.7 P1 72.3 S7 72.3 P1 90.4 S8 90.4 P1 112.11 S2 76.3 C2 68.7 P1 113.1 S6 113.1 P1 94.8 S5 94.8 P1 72.3 S7 72.3 P1 79.3 S1 36.2 C3 68.7 P1 79.3 S1 79.3 P1 148.1 S3 148.1 P1 121.7 S4 76.7 C4 68.7 P2 72.3 S7 72.3 P2 148.1 S3 148.1 P2 112.1 S2 176.3 P3 C5 68.7 P2 90.4 S8 90.4 P2 121.7 S4 121.7 P3 72.3 S7 72.3 P3 79.3 S1 36.2 C6 68.7 P3 112.1 S2 112.1 P3 90.4 S8 90.4 P3 148.1 S3 132.9 C7 68.7 P3 113.1 S6 113.1 P2 79.3 S1 79.3 P2 94.8 S5 132.8 C8 68.7 P3 94.8 S5 94.8 P2 113.1 S6 113.1 P3 121.7 S4 76.7 6:00 7:00 8:00 9:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00 24:00 Fig. 4 Detailed distribution plan of passive distribution to be delivered on today. The total cost of this real-world The model is solved by the Gurobi Optimizer under the hard- scheduling plan is 82,706 CNY, including 2600 CNY for ware environment of the Intel Core i7-4770k (3.50 GHz). loading and unloading operations, 78,775 CNY for trans- The Gurobi Optimizer is a state-of-the-art mathematical porting operations and 1331 CNY for the costs caused by programming solver able to handle all large-scale problem the late distribution for petrol stations. types. This solver was designed from the ground up to exploit Figure 4 shows the detailed distribution plan. Green rep- modern architectures and multicore processors by using the resents the transportation task, yellow represents the standby latest implementations of the latest algorithms. Furthermore, task, orange represents the loading task, gray represents the the solver supports matrix-oriented interfaces for MATLAB. rest task and blue represents the unloading task. The orange Given the same MILP model, Gurobi can solve it faster and mark represents the type of product that needs to be loaded use less memory than the nested solver of MATLAB. The and unloaded, and the blue mark represents the petrol station optimal distribution plan of trucks is shown in Fig. 7. Its total that needs to be replenished. One complete replenishment cost during the research time horizon is 79,254 CNY, includ- plan for a truck, including the loaded product type and the ing 2600 CNY for loading and unloading operations and replenishing object, is closely associated with the current 76,654 CNY for transportation. Each petrol station is replen- inventory of all petrol stations, one complete replenishment ished before its inventory is low; thus, there is no cost caused plan for a truck, including the loaded product type and the by unpunctual distribution. In contrast to the real-world result, replenishing object. At the start time, all trucks go from the the cost of transportation and out of stock is greatly saved. garage to the depot and then perform the loading operation As is known from Fig. 7, C2 and C6 replenish petrol sta- under the constraints of the number of loading arms and the tions four times a day, while the others replenish petrol sta- demanded oil types. Afterward, each truck drives to the pet- tions three times a day. The loading order depends on the rol station to unload oil and finally back to the garage to rest. loading arm constraint and the truck’s destination. Taking Overall, C2 and C5 replenish petrol stations four times a day, the 7th and 8th time intervals (i.e., 7:10–7:30) as an exam- and the others replenish petrol stations three times a day. ple, all trucks except C6 arrive at the depot, while only C2, Figures 5 and 6 depict the inventory variation of petrol C3, C4 and C5 are loaded because there are 2, 1 and 1 load- stations and depots. It can be seen from Fig. 8 that some- ing arms for P1, P2 and P3 in the depot, respectively. Moreo- times oil shortage occurs in a few petrol stations. Specifi- ver, there are three trucks that need to load P3 during the first cally, P1 is out of stock in S1 at 20:40, P2 is out of stock in loading operation, but the loading order is C4 (replenish for S2 at 16:00, P2 is out of stock in S5 at 18:30, and P3 is out of S2), C7 (replenish for S6) and C8 (replenish for S5). From stock in S8 at 13:40. The stock out of S2 is the most severe Fig. 5, it can be seen that the first station to reach the lower and lasts almost 3 h. limitation is S2, followed by S5 and S6. Considering that the transportation distance of S6 is longer than that of S5, C7 4.3 Optimal scheduling plan loads P3 ahead of C8. With a comprehensive consideration of the transportation time and out-of-stock time, initiative To obtain the optimal scheduling plan of initiative distribu- distribution is able to optimize the delivery sequence for all tion, we develop a mathematical model according to Sect. 3. petrol stations. After the optimization, the time to replenish 1 3 1006 Petroleum Science (2021) 18:994–1010 30 30 P1 P1 25 25 P2 P2 P3 P3 20 20 15 15 10 10 5 5 0 0 6:00 12:00 18:00 0:00 6:00 12:00 18:00 0:00 -5 -5 Time Time (a) S1 (b) S2 30 30 P1 P1 25 P2 25 P2 P3 P3 20 20 15 15 10 10 5 5 0 0 6:00 12:00 18:00 0:00 6:00 12:00 18:00 0:00 -5 -5 Time Time (c) S3 (d) S4 P1 P1 25 P2 25 P2 P3 P3 20 20 15 15 10 10 5 5 6:00 12:00 18:00 0:00 6:00 12:00 18:00 0:00 -5 -5 Time Time (e) S5 (f) S6 30 30 P1 P1 25 P2 25 P2 P3 P3 20 20 15 15 10 10 5 5 0 0 6:00 12:00 18:00 0:00 6:00 12:00 18:00 0:00 -5 -5 Time Time (g) S7 (h) S8 Fig. 5 Inventory variation of petrol stations P1 at S1, P2 at S2, P2 at S5 and P3 at S8 is brought forward Besides, most trucks’ final task is to replenish the petrol to 20:00, 13:10, 18:10 and 13:10, thus avoiding the out of stations that are relatively close to the garage, such as S1 stock occurred under the mode of passive distribution. and S4, and then to return back to the garage to rest after 1 3 3 3 3 3 Inventory, m Inventory, m Inventory, m Inventory, m 3 3 3 3 Inventory, m Inventory, m Inventory, m Inventory, m Petroleum Science (2021) 18:994–1010 1007 objective function and considers the detailed truck status P1 P2 and a series of operational constraints. In the cost compo- P3 nent, loading and unloading operating cost, transportation cost and cost caused by out of stock are taken into account. One step further than previous studies is that the proposed model is capable of dealing with a variety of products, time-varying sales rates and congestion conditions. For a real-world case containing eight petrol stations, one depot with three types of oil products and one garage with eight trucks, the detailed truck dispatch plan is optimized using this model. During the 18 h of the study period, the sales 6:00 12:00 18:00 0:00 rate of each petrol station varies several times and every Time truck has a detailed task assignment in each time interval. And it also visualizes the inventory change of each pet- Fig. 6 Inventory variation of depots rol station and depot. Compared to the actual scheduling plan in passive mode premade by the company, initiative completing the loading task. From Fig. 7, C1 and C3 are the mode leads to less operational cost while controlling the first trucks to arrive at the garage (at 21:00), and they have a inventory of all the petrol stations at low but safe level. In 3-h rest time. Figure 8 illustrates that all the petrol stations the initiative mode, inventory is predicted scientifically would receive timely replenishments when their inventories and the products are sent to the petrol station in a suitable are close to their lower limits, and the inventory during the time, which effectively improves the efficiency of product study horizon stays within the allowable range. This also oil resource allocation and enhances the stability of the indicates that initiative distribution enables petrol stations to supply of petrol station. control their inventory at a relatively low but safe level and However, the truck scheduling problem is an NP-hard gives a potential advantage of reducing inventory handling problem with strict demands on computational time and cost when compared to passive distribution. The inventory space. For a mathematical model, it is suitable for obtain- variation of the depot is depicted in Fig. 9 and meets the ing the optimal solution for a relatively small problem capacity constraints during the entire study horizon. because the solving efficiency is closely related to the model’s scale. Therefore, developing a highly efficient meta-heuristic algorithm to solve increasingly complex 5 Conclusions truck scheduling problems within an acceptable time is important for future works. In order to improve efficiency for the entire replenishment Acknowledgements This work was part of the Program of “Study on system of highway petrol stations, this paper establishes a Optimization and Supply side Reliability of Oil Product Supply Chain MILP model which sets the minimum operating cost as the Driving Standby Rest Loading Unloading C1 68.7 P1 79.3 S1 79.3 P1 148.1 S3 148.1 P1 79.3 S1 36.2 C2 68.7 P1 72.3 S7 S7 72.3 P1 90.4 S8 90.4 P1 112.1 S2 112.1 P1 72.3 S7 80.7 C3 68.7 P1 113.1 S6 113.1 P1 94.8 S5 94.8 P1 121.7 S4 76.7 C4 68.7 P3 112.1 S2 112.1 P2 148.1 S3 148.1 P3 121.7 S4 76.7 C5 68.7 P2 90.4 S8 90.4 P3 90.4 S8 90.4 P3 148.1 S3 132.9 C6 68.7 P2 72.3 S7 72.3 P2 112.1 S2 112.1 P2 94.8 S5 94.8 P3 79.3 S1 36.2 C7 68.7 P3 113.1 S6 113.1 P2 79.3 S1 79.3 P3 72.3 S7 80.7 C8 68.7 P3 94.8 S5 94.8 P2 113.1 S6 113.1 P2 121.7 S4 76.7 6:00 7:00 8:00 9:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00 24:00 Fig. 7 Detailed distribution plan solved by the proposed method 1 3 Inventory, m 1008 Petroleum Science (2021) 18:994–1010 30 30 P1 P1 P2 P2 25 25 P3 P3 20 20 15 15 10 10 5 5 0 0 6:00 12:00 18:00 0:00 6:00 12:00 18:00 0:00 Time Time (a) S1 (b) S2 30 30 P1 P1 P2 P2 25 25 P3 P3 20 20 15 15 10 10 5 5 0 0 6:00 12:00 18:00 0:00 6:00 12:00 18:00 0:00 Time Time (c) S3 (d) S4 30 30 P1 P1 P2 P2 25 25 P3 P3 20 20 15 15 10 10 5 5 0 0 6:00 12:00 18:00 0:00 6:00 12:00 18:00 0:00 Time Time (e) S5 (f) S6 30 30 P1 P1 P2 P2 25 25 P3 P3 20 20 15 15 10 10 5 5 0 0 6:00 12:00 18:00 0:00 6:00 12:00 18:00 0:00 Time Time (g) S7 (h) S8 Fig. 8 Inventory variation of petrol stations 1 3 3 3 3 3 Inventory, m Inventory, m Inventory, m Inventory, m 3 3 3 3 Inventory, m Inventory, m Inventory, m Inventory, m Petroleum Science (2021) 18:994–1010 1009 200 Cornillier F, Laporte G, Boctor FF, Renaud J. 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Journal

Petroleum ScienceSpringer Journals

Published: Feb 9, 2021

Keywords: MILP; Initiative distribution; Petrol station; Replenishment; Discrete representation

References