Green Vehicle Routing and Scheduling Problem with Split Delivery

Green Vehicle Routing and Scheduling Problem with Split Delivery

Available online at www.sciencedirect.com Electronic Notes in Discrete Mathematics 69 (2018) 13–20 www.elsevier.com/locate/endm Green Vehicle Routi...

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Available online at www.sciencedirect.com

Electronic Notes in Discrete Mathematics 69 (2018) 13–20

www.elsevier.com/locate/endm

Green Vehicle Routing and Scheduling Problem with Split Delivery Marcos Raylan Sousa Matos 1 Yuri Frota 2 Luiz Satoru Ochi 2 Computer Institute - Federal Fluminense University Av. Gal. Milton Tavares de Souza, s/n - Computer Institute S˜ ao Domingos, Niter´ oi - RJ - Zip code: 24210-346 Phones: (+055 21) 2629-5629 / 5628 / 5627 (fax). Abstract The Green Vehicle Routing and Scheduling Problem (GVRSP) aims to minimize carbon Dioxide Emissions (CO2 ) in logistics systems through better planning of deliveries made by a fleet of vehicles. In this paper, the GVRSP considers heterogeneous vehicles, time-varying traffic congestion, capacity constraints and the delivery can be fractionated. The GVRSP also allows vehicles to stop on arcs, so vehicle is allowed to travel an arc in multiple time periods. We present a mathematical formulation to describe the problem and a hybrid algorithm that combines the metaheuristic Iterated Local Search (ILS), the Random Variable Neighborhood Descent (RVND) procedure and an exact Set Covering model. The results obtained are compared with the state of the art and we show that the proposed approach has a strong performance. Keywords: Vehicle Routing, CO2 Emission, Traffic Congestion, Matheuristic.

1

Introduction

The classical Vehicle Routing Problem (VRP) was first treated in [2], this work addressed the problem of fuel distribution and aimed to determine the lowest cost of transportation to meet a set of resellers. In this scenario, the Green Vehicle Routing Problem (GVRP) can be defined as a VRP where the 1 2

Email: [email protected] Email: {yuri,satoru}@ic.uff.br

https://doi.org/10.1016/j.endm.2018.07.003 1571-0653/© 2018 Elsevier B.V. All rights reserved.

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objective function is, in addition to minimizing transport costs, minimizing emissions of carbon emitted by the fleet of vehicles used in the routing. The term green logistics has been a subject undergoing intense study and attracted the attention of many Operations Research professionals [1], [5], [9], [3,4], [10], [7], [12,11] and [6]. In these works, the GVRSP is formally defined by the objective of balancing the environmental and economic costs by implementing effective vehicle routes and schedules. Routes are usually considered time-dependent, in order to better simulate reality. With this approach, the time required for a vehicle to travel along a road varies according to the period of time the trip is taken. Besides that, vehicles are allowed to stop at arcs, so it can traverse an arc in multiple time periods. More recently, Xiao and Konak [13] presented a formulation for the GVRSP with time-dependent and scheduling constraints for a heterogeneous fleet of vehicles. The authors proposed a hybrid method based in the concepts of Variable Neighborhood Search (VNS) and Mixed Integer Linear Programming (MILP) optimization. Following their lead, the research reported in this paper tackles the same GVRSP but considering now split delivery constraints, allowing each customer to be visited by more than one vehicle, in order to achieve a reduction in the CO2 emissions costs. We approach the problem with a hybrid method, based now on a multi-start Iterated Local Search (ILS) [8] using a RVND [8] procedure as local search. Furthermore, we use an exact Set Covering (SC) model, which determines the best combination of routes generated during the execution of the ILS. In the next section, we define GVRSP with split delivery (GVRSP-Split).

2

Problem Definition

The problem is formulated on a complete network G = (N, A) where N is the set of nodes representing the customers (and depot) and A = {(i, j) : ∀i ∈ N, j ∈ N, i = j} is the set of arcs representing the roads between the nodes, where Dij defines the distance for each pair of two customers. Moreover, we define K as the set of periods of time, and M as the fleet of heterogeneous vehicles. Each vehicle has payload and fuel tank constraints in addition to time-window usage constraints. Similarly, each customer has a non-negative demand, an attendance time-window, and a service time that has to be respected. Furthermore, every vehicle v has a fuel-to-CO2 conversion rate cv , a vehicle coefficient ϕv , and a CO2 estimated fuel consumption rate v gijk , as a function of travel speed and travel load, defined in each arc (i, j) and each period k (for more details, see [13]).

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In this context, the GVRSP-Split can be defined as a VRP with split delivery and time-dependent routes, trying to minimize the CO2 cost function defined emissions induced by vehicles routes, defined  by two  sums:  (i) the v by (i,j)∈A k∈K v∈M cv gijk dvijk , where variable dvijk indicates the traveled distance of arc (i, j) in period k byvehicle  v and (ii) the emissions induced by the vehicles’ loads, defined by (i,j)∈A v∈M cv ϕv Dij fijv , where variable fijv indicates vehicles k load on the arc (i, j). The described scenario can be formulated as an MILP problem, named GVRSP-Split-IP, where the CO2 cost function is presented next:     v CO2 = cv gijk dvijk + cv ϕv Dij fijv (i,j)∈A k∈K v∈M

(i,j)∈A v∈M

Unfortunately, due to space limitations, we do not present a formal description of the formulation, we only briefly comment the GVRSP-Split-IP constraints. The formulation must ensure that every route starts and ends at the depot. Moreover, all client demands must be satisfied in the respective time-window, while vehicle payload and usage periods are not exceeded.

3

A Hybrid Multi-Start ILS-SC method

The proposed hybrid method, denoted as MS-ILS-SC, is composed of two main stages: first, a multi-start ILS heuristic with RVND local search is executed, and later, a SC (Set Covering) formulation is solved as an intensification phase. The pseudo code of the MS-ILS-SC algorithm is given in Algorithm 1. Algorithm 1 Multi-Start ILS-SC algorithm 1: procedure MS-ILS SC(maxIT, maxP ) 2: i = 0,S ∗ ← ∅,ES ← ∅ 3: while i ≤ maxIT do 4: S ← InitialSolution() 5: j=0 6: while j ≤ maxP do 7: S ← RN V D(S) 8: InsertEliteSet(ES, S) 9: if f (S) < f (S ∗ ) then 10: S∗ ← S 11: j=0 12: end if 13: S ← P erturbation(S) 14: j++ 15: end while 16: i++ 17: end while ∗ 18: S ← min(setCovering(ES), S ∗ ) 19: return S ∗ 20: end procedure

 Add solution S to elite set

 Solve Set Covering and put to S ∗  The best solution

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3.1

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The Multi-Start ILS

The multi-start ILS iteratively executes in sequence three phases: (i) a random/greed heuristic in a constructive phase, (ii) a RVND in the local search phase and (iii) perturbation moves as a diversification mechanism. After each successful perturbation move, the ILS metaheuristic returns to phase (ii) until no more improvement is found. The heuristic executes maxIT iterations and it returns the best solution S ∗ among all iterations. In the remainder section we describe these three phases. Construction phase. At each iteration, the heuristic first creates a new route with a random customer, and then uses some criteria to select the best customer to be inserted into the best location in the current partial route (see Alg. 1 line 4), the process is repeated until the capacity of the route is reached and the method starts a new iteration. The heuristic uses three selection criteria based on the algorithm reported in [8], randomly chosen at each iteration: Parallel Insertion Strategy (PIS), Economical Insertion Viable Criterion (EIVC) and Nearest Insertion Viable Criterion (NIVC). RVND. The local search method (line 7) is composed by the following traditional routing neighborhoods: The Exchange switches the positioning of two clients in the same route while 2-opt removes two non-adjacent arcs (i1 , j1 ) and (i2 , j2 ) of a route, then two new arcs (i1 , i2 ) and (j1 , j2 ) are inserted. The Swap (1,1) swaps two clients from different routes and Shift (1,0) removes a customer from one route and insert into another. In the Merge neighbourhood, two routes are concatenated into a new one. Finally, K-Split splits the load of a customer between several vehicles. Our method uses a random neighborhood ordering (RVND) [8], i.e., whenever a given neighborhood fails to improve the incumbent solution, the RVND randomly chooses another neighborhood to continue the search throughout the solution space. Perturbation Moves. We perform multiple Swap (1,1) and Shift (1,0) movements randomly chosen (line 13) in such a way that the resulting modification is sufficient to escape from local optima. Moreover, the parameter maxP represents the maximum number of consecutive perturbations allowed without improvements. 3.2

Set Covering Formulation

The intensification phase of the proposed hybrid method (line 18) involves the exact resolution of a set partitioning problem. Let S ∈ ES be a solution (set of routes) of the family of elite solutions ES (line 8) and Sv ⊆ S be the subset of routes that contain customer v ∈ N in solution S. We define the binary

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variable Xr associated to each route r ∈ S, and Cr as the cost of that route. The SC formulation can be expressed as follows. M in

 

=

C r Xr

(1)

S∈ES r∈S

Subject to:

 

Xr ≥ 1,

∀v ∈ N

S∈ES r∈Sv

Xr = Xs ,

Xr ∈ {0, 1},

∀S ∈ ES;

∀r ∈ S;

∀r, s ∈ S, such routes r and s share a client

∀S ∈ ES

(2)

(3)

(4)

The objective function (1) finds the set of minimum cost routes given that constraint (2) ensures that at least one route pass through each client. The feasibility of the solution is guaranteed by constraint (3) that ensures that routes with split demand will always be in the same solution.

4

Computational experiments

The proposed method was developed in the C++ language using the g++ 4.8.2 compiler and run on a single-thread, with 8.0 GB RAM, processor TM R Intel Core i7-870 CPU @ 2.93 GHz and under the Linux operating system platform (Ubuntu 14.04 LTS x64). The GVRSP-Split-IP and SC formulation were solved by CPLEX 12.1 (using the default parameters). Since the GVRSP-Split defined in this paper has not been studied in prior research, we cannot directly compare the performance of the MS-ILS-SP using results from the literature. Thus, we chose to compare our method with the best results for the GVRSP with no split delivery ([13]) in order to show the advantages of the proposed approach. The test problem set presented in [13] includes seven groups (10 × 5, 15 × 5, . . . , 75 × 5 and 100 × 5), with each group containing 20 instances. For each of the 140 instances, five runs were performed with parameters empirically calibrated of maxIT =| N | and maxP =| N | + | M |. First, we analyze the strength of the heuristic method by presenting a comparison between the optimal solution for the GVRSP-Split, obtained by

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the formulation GVRSP-Split-IP, and our heuristic MS-ILS-SC for a set of small instances that could be exactly solved by the formulation within a time limit of 2 hours (Table 1). The first column indicates the name of the instances. The next two columns indicate the dioxide emission cost CO2IP for the GVRSP-Split-IP, and the execution time in minutes (timeIP ), respectively. Furthermore, the next two columns CO2ILS and timeILS , present the same information for the MS-ILS-SC method. The last column displays the CO2 cost gap between the two methods, which is in accordance with the following 100∗COILS definition: gap = COIP2 − 100. 2 These preliminary results indicate that the heuristic is robust, since it presented an average gap of only 0.32% for the instances of size 10. Moreover, we want to point out that instance UK-10x5-11 has no feasible solution. GVRSP-Split-IP Instances UK-10x5-1 UK-10x5-2 UK-10x5-3 UK-10x5-4 UK-10x5-5 UK-10x5-6 UK-10x5-7 UK-10x5-8 UK-10x5-9 UK-10x5-10 UK-10x5-11 UK-10x5-12 UK-10x5-13 UK-10x5-14 UK-10x5-15 UK-10x5-16 UK-10x5-17 UK-10x5-18 UK-10x5-19 UK-10x5-20 Average

MS-ILS-SC

CO2IP timeIP CO2ILS timeILS 471.44 752.34 601.28 581.50 511.31 778.55 582.33 733.06 502.30 579.47 510.25 569.20 537.04 361.31 532.97 516.03 441.29 439.99 423.18

31.64 24.73 32.73 26.51 28.37 41.13 26.28 37.42 37.42 22.26 27.01 29.41 49.53 75.97 105.46 34.87 44.65 31.25 46.56

548.67 39.87

gap

472.2 752.4 602.89 582.4 511.54 780.05 583.69 735.22 502.9 580.1 512.3 569.87 540.4 364.6 535.18 519 443.48 441.06 426.31

25.11 16.72 22.87 17.32 21.06 33.18 19.45 24.17 25.94 14.05 19.24 21.09 38.1 59.21 78.82 25.29 35.12 24.81 38.13

0.16% 0.00% 0.26% 0.15% 0.04% 0.19% 0.23% 0.29% 0.11% 0.10% 0.40% 0.11% 0.62% 0.91% 0.41% 0.57% 0.49% 0.24% 0.73%

550.29

28.6

0.32%

Table 1 Comparison of results for problem group 10 × 5.

Next, Table 2 presents the comparison between the best results achieved by the hybrid heuristic presented in [13] for the GVRSP problem, our new heuristic without the split neighborhoods Merge and K-Split (GVRSP), and the complete method with all neighborhoods (GVRSP-Split). Similar to Table 1, we present now the average information (per group of 20 instances) of CO2 emission cost, time and gap between the methods, which column gap

M.R. Sousa Matos et al. / Electronic Notes in Discrete Mathematics 69 (2018) 13–20

is in accordance with the following definition: gap = Hybrid (GVRSP) [13] a CO2HY

timeHY CO2ILS timeILS

10 × 5 15 × 5 20 × 5 25 × 5 50 × 5 75 × 5 100 × 5

557.9 746.1 958.7 936 1710.2 2483.5 3149.8

7.73 12.91 17.98 31.06 59 89 98

Average 1506.02 45.09 a

MS-ILS-SP (GVRSP)

Groups

572.87 716.18 1017.25 1013.01 1997.02 2684.14 3672.07

6.62 9.77 15.18 19.06 35.41 48.19 52.06

1667.51 26.61

gap 2.68% -4.01% 6.11% 8.23% 16.77% 8.08% 16.58% 7.78%

100∗CO2ILS CO2HY

19

− 100.

MS-ILS-SP (GVRSP-Split) CO2ILS timeILS

gap

28.6 29.27 33.02 39.91 45.29 53.52 68.14

-3.10% -6.55% -1.22% -0.84% -0.72% -0.81% -0.69%

1486.05 42.53

-1.99%

540.6 697.2 946.98 928.17 1697.88 2463.39 3128.17

TM i5-2400S CPU @ 2.50 GHz R IntelCore

Table 2 Comparison of results for seven problem groups.

We first notice that the proposed method applied for the GVRSP (without the split neighborhoods) has a poor performance compared with the hybrid method proposed in [13]. Our heuristic was tailored specifically to deal with the split characteristics of the GVRSP-Split and was unable to reach the same performance of the original method. Therefore, the MS-ILS-SP for GVRSP attained an average gap of 7.78% compared to the best solution presented by the Hybrid method. On the other hand, the complete method overcame the results for all group of instances. Note that the average CO2 cost presented by [13] was 1506.02 while the new approach reached an average CO2 cost of only 1490.46. This behavior is due to the fact that GVRSP with split delivery is able to find better solutions by splitting the customers demands into more then one route. The split feature of the problem can rapidly lead to a combinatorial explosion in the solution space, but fortunately, the new method was able to reach better results in all instances of all groups (average gap of −1.40%) requiring less average time (42.53 minutes) than [13], with a machine 1.23x slower (based on Passmark benchmark)

5

Conclusions

This paper presented a new approach based on Iterated Local Search (ILS) to solve the GVRSP with split delivery. The tests carried out proved that the proposed methodology was able to find good quality solutions for the problem in a reasonable time. Actually, the heuristic was able to reduce the CO2 cost in all 140 instances tested when compared with the GVRSP with no split delivery presented in [13], maintaining a competitive time.

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