Profitable Vehicle Routing Problem with Multiple Trips: modeling and constructive heuristics

CHBICHIB Ahlem PhD Student, FSEG

University of Sfax, G.I.A.D. FSEG Sfax 3018 – Tunisia

ahlemchbichib@yahoo.fr

MELLOULI Racem Associate Professor, ESC

University of Sfax, G.I.A.D. FSEG Sfax 3018 - Tunisia

racem.mellouli@yahoo.fr

CHABCHOUB Habib Professor, IHEC

University of Sfax, G.I.A.D. FSEG Sfax 3018 - Tunisia

habib.chabchoub@fsegs.rnu.tn

Abstract— In this paper, we study a new variant of the Vehicle Routing Problem (VRP) which combines two known variants: the Profitable VRP (PVRP) and the VRP with Multiple Trips (VRPMT). This problem may be called as (PVRPMT). The problem’s objective is to maximize the sum of collected profit prior to the customers visit minus the transportation costs. This problem has a very practical interest especially for daily distribution schedules with a short course transportation network and a limited vehicle fleet. A discursive method for profit quantification is suggested so as to make it more realistic. A mixed integer programming formulation of the problem is proposed. In addition, two constructive heuristics and a set of benchmark instances which include small and large size data are provided. Finally, a computational study evaluates and compares the performance of the different proposed methods.

Keywords-Vehicle Routing Problem; Mutiple Trips; Profitable VRP; Mixed Integer Programming; Heuristics.

I. INTRODUCTION Nowadays, effective management of the supply chain is recognized as a determinant key of competitiveness and success for most manufacturing organizations. The problems of distributing goods from depots to final consumers or routing goods between different logistical sites are very important, and their adequate programming may produce significant savings for many distribution systems. The Vehicle Routing Problem (VRP) is a well-known problem studied by researchers in Operational Research. It deals with finding the optimal routes of delivery or collection from one or several depots to a number of cities or customers by a fleet of vehicles, while satisfying some constraints. The solution of a VRP is a set of minimum cost routes which fulfill the customers’ requirements. Several operational requirements and constraints are considered in practical applications of the VRP.

In this paper, we focus on two variants which did not gain, in our view, much of interest in the literature especially the second one: the Vehicle Routing Problem with Multiple Trips or with multiple uses of vehicles (VRPMT or VRPM) and the Profitable Vehicle Routing Problem (PVRP). The VRPMT, with is NP-hard [11], is an extension of the

classical VRP in which each vehicle may perform several routes in the same planning period. Solving this problem not only implies the design of a set of routes but also the assignment of those routes to the available vehicles. This makes the VRPMT very practical especially at an operational context where daily drivers schedules must be achieved wit a fixed vehicle fleet and short distances distribution networks. The first research that explicitly addresses the multiple trips case was made by Salhi [1]. Limited to double-trips, a matching algorithm is proposed to allocate routes to vehicles within a refinement process. Fleischmann [2] proposed a modification of the algorithm of Clarke and Wright [3] and used the bin packing heuristic Best Fit Decreasing (BFD) [4] to assign the routes to the vehicles. Taillard et al. [5] proposed a three-phase algorithm to solve the problem. In the first phase, a set of routes satisfying capacity constraints is constructed from a population of routes generated using a tabu search heuristic. Next, these routes are combined into complete VRP solutions. Finally, the routes of each solution are assigned to the vehicles solving a bin packing problem. A constructive and improvement heuristics were proposed by Brandao and Mercer [6]. They tackled multiple trips case as part of a more extensive problem involving delivery time windows and vehicles with both volume and weight and driving time capacities. To compare with the benchmark of Taillard et al. [5], Brandao and Mercer [7] modified their heuristic to solve the classical VRPMT. Their approach is based on the nearest neighbor rule and an insertion criterion to assign customers to routes within vehicles. An improvement phase attempts to remove overtime before reducing the routing cost using insert and swap moves. Golden et al. [8] adopted the approach of Taillard et al. [5] to solve a VRPMT using the minimax objective. Petch and Salhi [9] developed a multi-phase constructive heuristic. This heuristic integrates in part the approach of Taillard et al. [5] as initial solution and that of Brandao and Mercer [6] as an improvement phase. Lin et al. [10] studied a variant of a multi-objective problem that integrates the multiple uses of vehicles and the location of deposits. The goal is to satisfy all customers and to achieve the objectives of the managers. The proposed heuristic is composed of a tracking phase and a routing phase. Olivera and Viera [11] put forward an interesting implementation of adaptive memory

978-1-61284-4577-0324-9/11/$26.00 ©2011 IEEE 500

search for the VRPMT. This is based on the same principle of Taillard et al. [5] with some incorporated enhancements. Initial VRP solutions are constructed by a sweep-based algorithm. The memory is then constructed by the top solutions up to a certain memory size. The routes are selected randomly from the memory and a bin packing procedure is adopted to pack the routes into vehicles while introducing some refinements based on reducing the driver overtime. The new routes are then fed back into the memory and those routes belonging to the poorest solutions are discarded. The whole process is executed for five runs. Excellent results are obtained. Recently, Battar et al. [12] consider the PTVMT with time window. A two-phase heuristic is proposed. The first step is to determine a list of roads by vehicle routing problem with time window heuristics. During the second phase a set of routes are aggregated into multiple routes based on a bin packing heuristic.

While the literature is rich with research which considers problems of vehicle routing where all the customers are served, only few papers approaches the contrary case. In many practical situations, it is not possible to satisfy the entire costumers request for lack of means or for insufficiency of the offer. It is thus necessary to privilege the most important customers according to criteria of potentiality in the long terms or recorded effective sales turnover. Obviously, the profit brought by a satisfied and faithful customer can be quantified according to various manners. In addition, the economic value of such a profit must be projected with an appropriate manner with the used time horizon for optimization. It is clear that a customer can be described as profitable or advantageous if it allows a direct and punctual benefit relating to the current sales turnover, i.e. realized in the temporal horizon of required optimization, or to a still fictitious sales turnover which can be potentially carried out in the future with a probable commercial reinforcement. The number of papers on vehicle routing problems with profits is much more limited comparing to papers on VRPMT. The interest in these problems is growing recently (see Toth and Vigo [13]). Dell’Amico et al. [14] are the first to prepare the study of the PVRP. They presented the problem which they called the Profitable Tour Problem (PTP) as a variant of the Traveling Salesman Problem. If a vertex (costumer) is left unvisited a given penalty has to be paid. The objective function is to find a balance between these penalties and the cost of the tour. In addition, they considered the Prize-collecting Traveling Salesman Problem (PCTSP) where each vertex is associated with a prize and there is a knapsack constraint which guarantees that a sufficiently large prize must to be collected. They proposed two lower bounds: the first is obtained by relaxing the assignment constraint. The second bound is obtained by relaxing the sub-tour elimination constraint. Bienstock et al. [15] developed a polynomial algorithm with 5/2 worst case bound for the undirected PVRP. Based on this work, Goemans and Williamson [16] developed a purely combinatorial algorithm which improves the worst case bound to (2 - 1/(n-1)). In the context of a scheduling problem on m identical machines with sequence-depending

time to change tools, Helimberg [17] defines a new problem. The author explains that by focusing on the m-asymmetric TSP, that the machine scheduling problem is a PVRP. Feillet et al. [18] were the first to gather and propose a classification of the routing problems which authorize to not serve some customers with the consideration of the profit brought by the visit of each one. The survey carried out by Feillet et al. [18] focused on Traveling Salesman Problems (TSP) with profits using a single available vehicle. The objective may be either the maximization of the collected total profit (problem called Orienteering Problem shortened to OP), the minimization of the total traveling costs and penalties associated to unselected costumers (PTP), or the optimization of a combination of both. Feillet et al. [19] add the capacity constraint at PTP and Team Orienteering Problem (TOP). For each problem a lower bound is determined via a column generation procedure. Then two tabu search heuristics are developed. Recently, Feillet et al. [20] studied the Undirected Capacitated Arc Routing Problem with profits (UCARPP). The objective is to find a set of routes that satisfy the constraints about the duration of routes and about the vehicle capacity and to maximize the total collected profit. They propose a branch-and-price algorithm and several heuristics.

In this paper, we propose to combine the profitable vehicle routing problem and the vehicle routing problem with multiple trips to propose a new variant of VRP. The work is organized as follows. In Section 2, we give a detailed theoretical graph description of PVRPMT and we introduce the corresponding notations. In Section 3, we provide mathematical models of the problem including MILP formulations. Section 4 describes two constructive heuristics. Computational study compares the performance of different heuristics with the linear bound (for large size instances) and with the optimal value (for small size instances) on a set of benchmark instances that we propose in Section 6. Finally, in Section 7, we put some conclusions and discuss future directions of the research.

II. PROBLEM DEFINITION AND NOTATION: We consider a complete undirected graph G = (V,A), where V={0,…,n} is the set of vertices and A is the set of edges. Vertex 0 represents a depot where a fleet k = {1,…,m} of identical vehicles is based. Each vehicle has a limited capacity Q and a maximum number of trips L. An edge ( ),i j A∈ represents the possibility to travel from customer i to customer j. A non-negative demand qi, profit pi, and time service Si, are associated with each customer i (p0=q0=0). A symmetric travel time tij and cost cij are associated with each edge ( ),i j A∈ . Each vehicle starts and ends its tour at vertex 0, and can visit any subset of customers with a total demand that does not exceed the capacity Q. The profit of each customer can be collected by one vehicle at most, and it does not change with the time. In addition, there exists a time horizon denoted by Tmax which establishes the duration of a working day. It is assumed that all parameters are nonnegative integers. The studied problem in this paper, called PVRPMT, consists on

501

determining a set of routes and to assign each route to one vehicle where each vehicle can be used by several routes while respecting the time horizon capacity. The objective is to maximize the difference between the total collected profit and the cost of the total traveled distance. Note that the following properties:

• The optimal solution may be composed by a sub set of customer.

• The number of trips in the optimal solution does not exceed L*m

Figure 1 represents an illustrative example of the MTPVRP with n=19; m=3; L=2

Non served customer vehicle 1

Served customer vehicle 2

The depot vehicle 3

Figure 1. illustrative example

III. A MATHEMATICAL MODELS FOR THE PVRPMT

A. First formulation (MILP1) We consider the following binary decision variables:

( )1 ,

0klij

if the edge i j A assigned to the vehicle k during the trip lx

Otherwise⎧ ∈⎪= ⎨⎪⎩

10i

if the custom er i is visitedy

O therw ise⎧

= ⎨⎩

The formulation is the following:

( )1.1klj j ij ij

j i j i k l

Max p y c x≠

−∑ ∑∑∑∑

Subject to:

( )1..., 1.2klij i

j i k l

x y i n≠

= ∀ =∑∑∑( )

i j1,..., 1.3kl

ij jk l

x y j n≠

= ∀ =∑∑∑( )

i1,..., ; 1,..., 1.4kl

i ijj i

q x Q k n l L≠

≤ ∀ = =∑∑

( )maxi

1,..., 1.5klij ij i i

j i l i lt x S y T k m

≠

+ ≤ ∀ =∑∑∑ ∑∑( ) ( )1 1,..., ; 1,..., 1.6k lkl

ij iji j i i j i

x x k m l L−

≠ ≠

≤ ∀ = =∑∑ ∑∑( )1

00

1 1,..., 1.7kj

jx k m

≠

≤ ∀ =∑( )0 0,..., ; 1,..., ; 1,..., 1.8kl kl

ih hji h j h

x x h n k m l L≠ ≠

− = ∀ = = =∑ ∑{ } ( ) ( )1 1,..., ;2 1 1.9kl

iji S j i

x S S n S n∈ ≠

≤ − ⊂ ≤ ≤ −∑∑{ } ( )0,1 ; , 0,..., ; 1,..., ; 1,..., 1.10kl

ijx i j i j n k m l L∈ ∀ ≠ = = =

{ } ( )0,1 1,..., 1.11iy i n∈ ∀ =In this formulation, the objective is to maximize the overall collected profit minus the cost of transportation. The constraint (1.2) and (1.3) guarantee that the customer i is visited at most ones. (1.4) represent the capacity constraint. The horizon of the service time is restricted by (1.5). A vehicle k can be used during the trip l if and only if it has been used during l-1 trip (1.6). The constraint (1.7) guarantees that the vehicle k may be used at most ones during the first trip. (1.8) and (1.9) represent the fleet constraint and the sub tours elimination constraint respectively. (1.10) and (1.11) are the integrity constraint. The major limitation of this formulation is at the level of sub tour elimination constraint, it is an exponential constraint group.

B. Second formulation (MILP2) An improved formulation is given in this subsection. It in addition overcomes the exponential limitation of the precedent formulation and uses different binary variables. We use the following additional variables:

10

kl i f th e veh ic le k is u sed d u rin g th e tr ip lO th erw ise

δ ⎧= ⎨⎩

Ui an integer variable associated to customer i used to reformulate the sub-tour elimination constraints (see Miller et al. [21] and Kara et al. [22]).

The formulation is the following:

( )Max ( ) 2.1kli ij ij

i j i k l

p c x≠

−∑∑∑∑subject to:

( )j i

1 1,..., 2.2klij

k l

x i n≠

≤ ∀ =∑∑∑( )1 1,..., 2.3kl

iji j k l

x j n≠

≤ ∀ =∑∑∑( )

i1,..., ; 1,..., 2.4kl

i ijj i

q x Q k m l L≠

≤ ∀ = =∑∑( )max

i( ) 1,..., 2.5kl

ij i ijj i l

t S x T k m≠

+ ≤ ∀ =∑∑∑( ) ( )1 1,..., ; 1,..., 2.6k lkl

ij iji j i i j i

x x k m l L−

≠ ≠≤ ∀ = =∑∑ ∑∑

( )10

01 1,..., 2.7k

jj

x k m≠

≤ ∀ =∑

502

( )0 1,..., ; 1,..., ; 1,..., 2.8kl klih hj

i h j hx x h n k m l L

≠ ≠− = ∀ = = =∑ ∑

( )

( )

1,..., ; 1,..., ; 1,..., 2.9

kl kli j ij i j ji j

k l k l

U U Q x Q q q x Q q

i n k m l L

− + + − − ≤ −

∀ = = =

∑∑ ∑∑

( )0 0 2* 1,..., ; 1,..., 2.10n n

kl kl kli j

i j ix x k n l Lδ

≠

+ = ∀ = =∑ ∑

( )1,..., ; 1,..., 2.11n

kl klij

i j ix k m l Lδ

≠

≤ ∀ = =∑ ∑

( )* 1,..., ; 1,..., 2.12n n

kl klij

i j ix M k m l Lδ

≠

≤ ∀ = =∑ ∑( )0 0,..., 2.13iU Q i n≤ ≤ ∀ =

{ } ( ), 0,1 0,..., ; 1,..., ; 1,..., 1.14kl klijx i j n k m l Lδ ∈ ∀ ≠ = = =

{ } ( )0,1 1,..., 1.15iy i n∈ ∀ =Note that we eliminate in MILP2 the variable yi used in MILP1 by its expression. It is easy to remark that:

(2.16)kl

i ijj k l

y x=∑∑∑

The constraint (1.2), (1.3) and (1.5) are replaced by (2.2), (2.3) and (2.5). The sub tours elimination constraint (1.9) is replaced by the constraints (2.9) to (2.13).

C. Third formulation ( MILP3):heteregenious fleet In this subsection, we proposed an extended formulation to solve the special case of PVRPMT with heterogeneous fleet. So, each vehicle k has its own capacity Qk. Then, some modifications on MILP2 are done. The constraint (2.4) is replaced by (3.4). (2.9) and (2.10) are replaced by (3.9) and (3.10):

( )3.4n n

kli ij k

i j iq x Q

≠

≤∑ ∑

( )

( * ( ) )

( )( ) 3.9

kl kli j k ij k i j ji

k lkl kl

k j ij jik l

U U Q x Q q q x

Q q x x

− + + − −

≤ − +

∑∑

∑∑

( )* * 3.10kl kli ij i k ij

k l j i k l j iq x U Q x

≠ ≠≤ ≤∑∑∑ ∑∑∑

IV. CONSTRUCTIVE HEURISTICS FOR THE MTPVRP The goal of this section is to construct good feasible solutions for the problem. We propose two gready constructive heuristics . These heuristics use some local optimalies in certains steps of the heuristics algorithms.

A. Heuristic H1 The main idea of this heuristic is to construct the trips by selecting constumors one by and choosing the most locally profitable (optimal) unvisited costumer, with the option to keep the same vehicle used in the last iteration. A customer i is locally optimal, leaving from costomer (i-1), if its local contribution (pi-ci-1,i) in the objectif function is the best. The route is considered as completed if the vehicle can not

receive any more any other costumer due to its physical capacity. Then, the vehicle returns to the depot and a new trip starts with the same used vehicle in the last iteration. The vehicle is changed only if the daily time horizon Tmax will be violated by the addition of a new trip with the same vehicle. In this case, the next trip is done by an other one (see Figure 2).

Remarq 1: in the case of a heterogeneous fleet, the vehicles are sorted in the decreasing order of the physical capacities. This order is used as priority rule for the vehicle choices.

B. Heuristic H2 This heuristic H2 is almost identical with heuristic H1 with the following basic difference : to construct the trip, we use the same previous procedure, but at each iteration, the vehice is not necessary kept in prior. We choose the vehicle which has the longest remaining time service by breaking ties with the largest capacity order (see Figure 3).

Remarq 2: in the case of heterogeneous fleet, we use the same priority rule for choices used in H1.

V. LINEAR BOUND FOR THE PVRPMT: We are mainly interested in seeking bounds (lower with heuristics and upper by relaxation) with good quality. Then, it is important to calculate upper bounds in order to evaluate empirically our heuristics, and vice versa. We would like to have a certain guarantee on the quality of the generated bounds to prepare possible faster exact resolution of the problem.

VI. COMPUTATIONAL RESULTS The tests will be applied first on our own benchmarks and then on the benchmarks of Taillard et al. [5], taken from the VRP library, with some extended data. In our benchmark, twenty small size instances are generated randomly with adaptation to obtain interesting instance values according to the reality and the practice. For each instance, we indicate the number of vertices n which ranges from 6 to 20, the number of vehicle m equal to 2 or 3, the maximum number of trip L equal to 2 or 3 and the vehicle physical capacity limit Q ranging from 1000 to 3000 kgs.

The horizon time limit for all the small size instances is equal to 480 munites (i.e. 8 hours of a daily work). The profit pi of customer i depends on the three terms: cons, h and the customer demand qi, where h is a random ratio number uniformly generated in the interval [0,1] and cons is a constant factor that measures the profit according to a sale turnover. In his article “the Capacitated Team Orienteering Problem and Profitable Tour Problem”, Feillet et al. [19] consider pi = (cons + h)qi and suppose that cons=0.5 indifferently with the level of greatness of the other values used for the instances data. This implies a lack of guarantee of a necessary coherence between the different values while the choices of the units and the level of greatness of the used for data are unambiguous: costomers demands qi distances dij transportation times tij and transportation costs cij.

503

Figure 2. Heuristic H1

Figure 3. Heuristic H2

It is important to obtain meaningful and significant proportions of the profits according to the transportation costs. In our case,are assimilated to dij Concerning the profit calculation, we take as reference a general realistic model where the logistical cost represents between 5 and 10 per cent of the sale turnover, and the gross benefit may represent between 20 and 50 per cent of the sale turnover. So, it is possible to have a global profit which ranges almost between 3 and 5 times the gobal transportation cost. The distribution of this global profit on customers may respect the demand quantity qi of each customer i. qmax and qmin be respectively the grestest and the smallest value of all

No All customers are visited?

Stop

Step3: return to the depot and update the vehicle

loading

Step4: Add the customer to the trip and update the current vehicle loading and the used vehicle time

Yes

No

Yes

No

Yes

Start

It exists a vehicle with a “remaining time”

sufficient to be used (vehicle availability)

Step1: select the available vehicle (according to the priority list) and take the depot position.

Step2: select the next customer according to the local optimality rule

Feasibility of insertion to the current vehicle

trip

Vehicle remaining time capacity will be

violated

Vehicle physical capacity will be violated

No

Yes

No

Yes

Start

It exists a vehicle with a “remaining time”

sufficient to be used (vehicle availability)

Step1: select the available vehicle (according to the priority list) and take the depot position.

Step2: select the next customer according to the local optimality rule

Feasibility of insertion to the current vehicle

trip

All customers are visited?

Stop

Step3: return to the depot and update the vehicle

loading

Step4: Add the customer to the trip and update the current vehicle loading and the used vehicle time

Yes

504

demand quantities. qmax and qmin can be associated respectively to αmax =5 and αmax =3 factors. We use a projetion to calculated the factor αi of the customer i according to its demande qi .

In order to estimate the transportation cost, we calculate firstly the average distance:

( )21

iji j i

dd

n≠=

−

∑∑

So, pi is used to be equal to .)1( dhi +α

Note that tij are also assimilated to dij and the level of greatness of the vertices coordinates values is choosen adequately to obtain significant tij values suitable with Tmax.

In addition, we point out that L, which is used in MILP formulations, represents the total number of trip which can be made by a vehicle during Tmax . Its value can be estimated as :

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥

⎤⎢⎢

⎡⎥⎥

⎤⎢⎢

⎡ ∑∑dn

TQ

QQ

Q

k

k

k

k max

min;

minmin

The algorithm was tested on a Intel Pentium 4 CPU 1.70 GHZ and 504 Mo RAM. The codes are written in C++ and we use in order to test the proposed models, the commercial solver CEPLEX7.0. of ILOG®. All the algorithm was stopped after one hour of computational time. The results are shown in Table 1.

TABLE I. RESULTS FOR THE SMALL SIZE INSTANCES

Instance Exact Relaxation H1 H2

Inst_n_m_L_Q f* CPU F CPU % LB1 (%)* (%) LB2 (%)* (%)

Inst_6_2_2_1000 631,70 0,47 724,71 0,03 14,72 619,99 16,89 1,85 619,99 16,89 1,85

Inst_7_2_2_2000 963,37 0,63 1076,18 0,02 11,71 934,36 15,18 3,01 934,36 15,18 3,01

Inst_7_2_3_3000 1945,21 3,03 2351,45 0,03 20,88 1581,65 48,67 18,69 1581,65 48,67 18,69

Inst_9_2_2_2000 695,49 16,06 772,47 0,17 11,07 681,39 13,37 2,03 681,39 13,37 2,03

Inst_9_2_3_2000 2266,98 41,34 3072,99 0,06 35,55 2237,53 37,34 1,30 1687,25 82,13 25,57

Inst_10_2_3_2000 2445,22 30,39 2765,93 0,09 13,12 2424,20 14,10 0,86 2247,61 23,06 8,08

Inst_11_2_2_2000 2623,9 506,33 3275,25 0,09 24,82 1998,02 63,92 23,85 2006,60 63,22 23,53

Inst_12_2_3_3000 2536,3 144,91 2656,69 0,19 4,75 2477,46 7,23 2,32 2477,46 7,23 2,32

Inst_13_2_3_1000 _ _ 4213,11 0,30 _ 2477,46 70,06 _ 2477,46 70,06 _

Inst_13_3_2_2000 _ _ 3397,86 0,31 _ 2875,57 18,16 _ 2991,24 13,59 _

Inst_14_2_2_3000 4046,89 984,14 5348,35 0,19 _ 3569,96 49,82 11,79 3581,62 49,33 11,50

Inst_15_3_2_3000 _ _ 6209,17 0,25 _ 4212,19 47,41 3785,42 64,03 _

Inst_16_2_3_3000 4588,65 1093,72 6857,59 0,38 49,45 2960,99 131,60 35,47 2971,23 130,80 35,25

Inst_16_3_3_3000 _ _ 6183,25 0,28 _ 3535,87 74,87 _ 3535,87 74,87 _

Inst_17_3_2_2000 _ _ 3544,39 0,27 _ 2982,45 18,84 _ 2951,16 20,10 _

Inst_13_3_2_2000 _ _ 5770 0,30 _ 3301,45 74,77 _ 3833,96 50,50 _

Inst_18_3_3_3000 _ _ 4477,41 0,77 _ 3910,65 14,49 _ 3954,98 13,21 _

Inst_19_3_2_2000 _ _ 3099,70 0,34 _ 2600,83 19,18 _ 2835,14 9,33 _

Inst_20_2_3_3000 _ _ 4974,4 0,13 _ 3175,78 56,64 _ 3163,64 57,24 _

Inst_20_2_2_3000 _ _ 7485,19 0,5 _ 5414,09 38,25 _ 5320,73 40,68 _

505

TABLE II. RESULTS FOR THE LARGE SIZE INSTANCES

Instance Relaxation H1 H2

CMT_n_Q_m_D1_L F CPU LB1 (%)* LB2 (%)*

CMT_51_Q_160_m_1_D1_L2 3574,84 0,56 2365,28 51,14 2365,28 51,14

CMT_51_Q_160_m_2_D1_L2 4118,55 1,08 2511,11 64,01 2228,04 84,85

CMT_51_Q_160_m_3_D1_L2 4132,09 1,98 2634,89 56,82 2634,89 56,82

CMT_51_Q_160_m_4_D1_L2 4132,09 2,75 2388,58 72,99 2388,58 72,99

CMT_76_Q_140_m_1_D1_L2 3634,90 1,42 3045,77 19,34 2354,73 54,37

CMT_76_Q_140_m_2_D1_L2 5714,12 9,66 3914,18 45,99 3784,21 51,00

CMT_76_Q_140_m_3_D1_L2 7019,43 25,00 3964,94 77,04 3650,34 92,30

CMT_76_Q_140_m_4_D1_L2 7388,38 13,55 3691,53 100,14 3644,75 102,71

CMT_76_Q_140_m_5_D1_L2 7381,15 21,16 3961,40 86,33 3952,34 86,75

CMT_76_Q_140_m_6_D1_L2 7388,39 32,11 3819,88 93,42 3819,88 93,42

CMT_76_Q_140_m_7_D1_L2 7381,12 63,36 3846,63 91,89 3846,63 91,89

CMT_101_Q_200_m_1_D1_L2 4686,47 4,63 4361,22 7,46 3052,25 53,54

CMT_101_Q_200_m_2_D1_L2 7215,88 18,56 4508,87 60,04 4392,70 64,27

CMT_101_Q_200_m_3_D1_L2 8061,15 16,36 4678,01 72,32 4397,59 83,31

CMT_101_Q_200_m_4_D1_L2 8053,05 25,80 4157,09 93,72 4157,09 93,72

CMT_101_Q_200_m_5_D1_L2 8053,05 36,33 4337,91 85,64 4337,91 85,64

CMT_101_Q_200_m_6_D1_L2 8036,85 57,36 4046,90 98,59 4046,90 98,59

CMT_121_Q_200_m_1_D1_L2 5177,53 6,09 4535,53 14,15 3392,72 52,61

CMT_121_Q_200_m_2_D1_L2 8369,66 157,31 6206,85 34,85 5879,51 42,35

CMT_121_Q_200_m_3_D1_L2 10188,05 98,41 6158,75 65,42 5768,51 76,61

CMT_121_Q_200_m_4_D1_L2 10651,39 60,2 5611,15 89,83 5711,72 86,48

CMT_121_Q_200_m_5_D1_L2 10643,03 87,55 5882,53 80,93 5891,95 80,64

CMT_151_Q_200_m_1_D1_L3 7869,747 46,38 5819,96 35,22 4546,97 73,08

CMT_151_Q_200_m_2_D1_L3 10769,16 99,25 5496,98 95,91 5956,69 80,79

CMT_151_Q_200_m_4_D1_L3 11249,91 232,77 5364,34 109,72 5677,48 98,15

CMT_151_Q_200_m_5_D1_L3 11249,91 328,14 5975,11 88,28 5975,11 88,28

CMT_151_Q_200_m_6_D1_L3 11249,91 518,59 6082,53 84,95 6082,53 84,95

CMT_151_Q_200_m_7_D1_L3 11231,19 597,58 5548,32 102,43 5548,32 102,43

CMT_151_Q_200_m_8_D1_L3 11249,09 772,83 5553,75 102,55 5553,75 102,55

CMT_200_Q_200_m_1_D1_L3 15137,99 3822,06 7060,04 114,42 7060,04 114,42

CMT_200_Q_200_m_2_D1_L3 9457,07 97,83 7621,22 24,09 3348,91 182,39

CMT_200_Q_200_m_3_D1_L3 13354,73 303,3 7195,74 85,59 7095,39 88,22

CMT_200_Q_200_m_4_D1_L3 14959,67 463,06 7355,56 103,38 7355,56 103,38

CMT_200_Q_200_m_5_D1_L3 15098,45 390,01 6876,72 119,56 7204,40 109,57

CMT_200_Q_200_m_6_D1_L3 26074,35 233,69 6778,04 284,69 7020,47 271,40

CMT_200_Q_200_m_7_D1_L3 15098,45 1032,28 7226,56 108,93 7226,56 108,93

CMT_200_Q_200_m_8_D1_L3 26933,152 20,03 7921,17 240,01 7921,17 240,01

CMT_200_Q_200_m_9_D1_L3 15137,98 3166,45 7664,81 97,50 7664,81 97,50

CMT_200_Q_200_m_10_D1_L3 15128,11 3093,13 7074,27 113,85 7074,27 113,85

506

The optimal solution obtained by CPLEX is indicated under column f *.. The “_” means that we can not obtain the optimal solution before the time limit. The column F represent the solution of the linear relaxed problem (the upper bound). The column LB1 and LB2 represent the results of the heuristics H1 and H2 respectively. In addition, (%) = (f*-bound)/bound × 100 and (%)*= (f*-LB)/LB× 100. We observe on Table 1 that the optimal value can be obtained within the time limit just for small size instances. For the instances where the number of vertices is superior to 16 the optimal solution cannot be determined. The two constructive heuristics produce acceptable solutions close to the optimal ones for small size instances. Table 2 gives the results of the large size instances case. For each instance the upper and the lower bound and their deviation are determined. The performances can be deduced in future works after some enhancements.

VII. CONCLUSION In this paper, we describe a new variant of the vehicle routing problem namely the Profitable Vehicle Routing Problem with Multiple Trips. MILP formulations and two constructive heuristics are proposed. A more practical quantification of the customer profit is suggested. Experimental study shows some promising results to be enhanced in the future with further techniques and compared to some adaptations of known performed heuristics in literature.

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