Designing salespeople's routes with multiple visits of customers: A case study

ARTICLE IN PRESS Int. J. Production Economics 119 (2009) 46–54

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Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe

Designing salespeople’s routes with multiple visits of customers: A case study Laia Ferrer, Rafael Pastor , Alberto Garcı´a-Villoria IOC Research Institute, Technical University of Catalonia, Avenue Diagonal 647, p11, 08028 Barcelona, Spain

a r t i c l e i n f o

abstract

Article history: Received 15 November 2006 Accepted 19 December 2008 Available online 29 January 2009

In this paper, we present a procedure for designing the routes for the salespeople that work for the commercial network of a home video company. The commercial area is divided into different zones and each salesperson is assigned to one of them, working and having independent routes from the others. The company wanted to generate salespeople’s routes for a commercial cycle, which is usually one month long. The commercial cycle has the following main features: each customer has to be visited a specific number of times (from 1 to 5); and two-day routes are possible when the customers are located far from the salesperson’s residence. After the set of routes is generated, each salesperson assigns his/her own routes to each working day, taking into account local holidays, national meetings and other unforeseen events. The objective function is to minimize the weighted sum of the travel time plus overtime penalties. In less than one month, an ad hoc procedure was designed, coded and calibrated. The company has qualified the routes as very satisfactory and the tool is now used by the commercial network management to: reassign customers, assign new customers and vary the number of visits to the customers. & 2009 Elsevier B.V. All rights reserved.

Keywords: OR application Vehicle routing problem Traveling salesman problem Case study

1. Introduction This paper studies the case of the commercial department of a multinational entertainment industry company (unidentified here for data confidentiality reasons) which sells its products (DVDs, etc.) to department stores. Its commercial network was originally organized as follows. The commercial area (Spain) is divided into provinces (a province is an administrative division that groups towns and villages) grouped into eight geographical zones, that has around 450 customers in total. Each salesperson is assigned to a zone and most of the customers in the zone. The salespeople visit their customers an established number of times (from 1 to 5) in a commercial cycle,

 Corresponding author. Tel.: +34 93 40117 01; fax: +34 93 401 66 05.

E-mail addresses: [email protected] (L. Ferrer), [email protected] (R. Pastor), [email protected] (A. Garcı´a-Villoria). 0925-5273/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2008.12.017

which is usually one month long. Each salespeople design their own routes and decide when in the commercial cycle to undertake them, taken into account that some days are set aside for national meetings, settlements and unforeseen events. The organization outlined the following main problems: (i) the routes and the assignment of customers to salespeople need to be improved (so that the management costs of the commercial network can be reduced); (ii) the salespeople’s self-organization inhibited any increase in the number of their visits (either to current or new customers), as they always said that they were overloaded (i.e., the current workload of the salespeople cannot be controlled). To solve these problems, the company needed a tool that generates routes given a set of assigned customers and the number of available days within the commercial cycle. This tool could then be used to manage the commercial network: to evaluate the different scenarios in which customers can be reassigned to salespeople; to assign new customers to salespeople;

ARTICLE IN PRESS L. Ferrer et al. / Int. J. Production Economics 119 (2009) 46–54

and/or to vary the number of visits to customers. This last point is especially important, as the company has already evaluated the expected increase in sales for each additional visit made to a customer. In this context, our work consisted of designing, codifying and calibrating the algorithm used to generate the routes. This work had to be completed in 28 calendar days. Two types of routes can be designed: single and double routes. Single routes take one working day, up to 8 h, in which the salespeople leave and return to their residences. In addition, double routes (two working days, up to 16 h) are also allowed, as may be needed to visit customers located far from the salesperson’s residence (in this case, the salesperson sleeps one night outside his/her residence). As a specific consideration to avoid long driving at the end of the working day, the company and the salespeople agreed that the last customer on a route should not be situated far from the salesperson’s residence. However, when this cannot be achieved, a double route should be designed and the salesperson should stay overnight in the customer’s city. The objective to minimize was multiple. Firstly, it considered the total time needed for traveling and visiting the customers. Secondly, when possible, a single route should take less than 7 h and a double route less than 14 h, to allow the salespeople to spend their remaining working day on administrative work. Moreover, to evaluate the scenarios proposed in the design and/or the management of the commercial network, the company decided to allow the generation (although penalizing it) of routes that lasted longer than a working day (i.e. 8 h in a single route and 16 h in double routes). In addition to the commercial network selling home videos, the company has an independent rental network. This has around 400 customers, 6 salespeople, and the same problems. The company also wanted to use the tool to manage the rental network effectively. In short, the company wanted the following specifications to be incorporated into the design of the routegenerating procedure:

 Each salesperson should be studied independently, 

 

since he/she has a set of different preassigned customers. The following data are known: the travel time between any pair of customers or between the salesperson’s residence and a customer; the length of the visit to each customer; the number of times that each customer is to be visited; and the number of available working days in a commercial cycle (usually 17 or 18 days in a month). The salespeople will assign the routes obtained by the tool to the available working days. If the last customer on a route is located far from the salesperson’s residence, a double route has to be designed. There is a limit to the number of double routes that can be undertaken in each cycle (for instance, 4). Moreover, the travel time which indicates that a customer is far from the salesperson’s residence is known (for instance, 2 h).

47

 The objective function consists of minimizing the sum

 

(weighted by the user) of: (i) the time taken on the routes (which includes the journey time and visit times to customers); (ii) the number of single routes that take longer than 7 h and double routes that take over 14 h; (iii) the time taken on the single routes over 8 h and double routes over 16 h. The computing time has to be brief, as this tool will be used to evaluate different scenarios, in order to manage the commercial and the rental network. Finally, the tool should be developed and operative in less than one month.

To solve this problem we developed an ad hoc heuristic procedure in phases. The rest of this paper is organized as follows: some related research is summarized in Section 2; the algorithm is detailed in Section 3; and the results and conclusions are presented in Sections 4 and 5, respectively. 2. Related literature The problem presented in this paper is a variation of the classic vehicle routing problem (VRP). The VRP involves determining a set of routes to minimize the total travel time or distance for a fleet of vehicles serving a set of customers with known demands or supplies. The routes obtained for each vehicle in the VRP are equivalent to the routes for each salesperson in the case presented in this paper. The number of visits to each real customer could be introduced as different virtual customers located at the same point, as long as the routes do not contain two virtual customers that represent the same real one in the same route. The length of the working day could correspond to the maximum time of a route, and the violation of this maximum time is allowed subject to a penalty. The VRP has been studied extensively (see, for instance: Laporte 1992; Laporte and Osman 1995; Laporte et al. 2000; Toth and Vigo 2002 for recent reviews). Many variations of the problem have attracted attention, including: time windows (Hong and Park, 1999); backhauls (Osman and Wassan, 2002); pickups and deliveries (Nagy and Salhi, 2005); multiple depots (Wasner and Za¨pfel, 2004); asymmetric distances (Toth and Vigo, 1999); or split deliveries (Jin et al., 2007). In the case described in this paper, the length of the routes corresponds to one or two working days. There is no predefined number of routes of each type. The possible variation in route length can be related to the heterogeneous vehicle routing problem (HVRP), in which vehicles that have different capacities and costs are used. The number of available vehicles of each type is usually unlimited (Gendreau et al, 1999; Salhi and Sari, 1997). However, problems with a maximum number of vehicles of each type have also been studied (Taillard, 1999). In our case, the numbers of single and double routes are limited and interrelated: the number of single routes plus twice the number of double ones cannot be more than the number of available days in the commercial cycle.

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Moreover, there is a maximum number allowed of double routes. The difference between the heterogeneity in vehicle capacity and route length is that it is acceptable to exceed the route length in this problem, although it is penalized. In contrast, the vehicle capacity can never be exceeded. Our problem is also related to the periodic vehicle routing problem (PVRP), which generates the routes for a set of vehicles during a time horizon in which each customer has to be visited a certain number of times. The PVRP has been used in refuse collection (Baptista et al., 2002) or in the collection of components for auto parts manufacturers (Delgado et al., 2005). The periodicity of visits is important in these types of problems (Francis and Smilowitz, 2006). The definition of periodicity may be given in the following ways (Paletta and Triki, 2004): all the sets of possible combinations to a customer can be explicitly stated (Russell and Gribbin, 1991); or the distance in days between two visits to each customer can be specified (Chao et al., 1995; Cordeau et al., 1997). Paletta (1992) introduced additional constraints that determine the maximum and minimum number of days that can elapse between two successive visits. However, in our application, the periodicity does not condition the route generation. It is only considered when each salesperson assigns a route to a particular day in the commercial cycle. Thus, this particular problem is not a PVRP. Instead, it is a variation of the basic VRP.

3. The procedure The problem to be solved for each salesperson can be briefly formalized as follows. Let N ði; j ¼ 1; . . . ; NÞ be the number of customers to be visited by a salesperson during D ðd ¼ 1; . . . ; DÞ available working days within the commercial cycle. Let T be the matrix with the traveling time between any pair of customers, and between the residence of the salesperson and any customer: t ij is the traveling time between customers i and j; t 0i is the traveling time between the salesperson’s residence and customer i. Moreover, the distances are considered symmetrical: tij ¼ t ji and t0i ¼ t i0 . Let vt i , pi and vi be the length of the salesperson’s visit to customer i, the province in which customer i is located and the number of visits (from 1 to 5) to customer i during D working days, respectively. A normal working day for a single route is 7 working hours long. This limit can be exceeded, up to a maximum of 8 h; however, such cases will be penalized. Working days that are longer than 8 h are penalized more heavily. The reference values for double routes are 14 and 16 h, respectively. According to the company’s conditions, daily routes cannot finish far from the salesperson’s residence (distance, dt max , to be defined by the user in time units). The values dt max and t0i are used to determine whether customer i is a close or a distant customer. It is assumed that a maximum number of double routes DR can be made. The maximum number of single routes is D  2f , where f (0pf pDR) is the number of double routes done. Finally, a customer cannot be visited more than one time during the same route.

The aim is to assign all the visits to be made to N customers to the routes in such a way that the value of the following non-linear objective function is minimized: ½MINZ ¼ a

D X d¼1

þ

X

Rd þ g1 X

8d2DjRSd 47 h

8d2DjRSd 48 h

RS6d þ

X

1 þ g2 X

1

8d2DjRDd 414 h

RD6d

8d2DjRDd 416 h

where Rd is the time of the route d, RSd is the time of the single route d, and RDd is the time of the double route d. This weighted objective function considers: (i) the time taken for all the routes; (ii) the number of single routes with a duration of over 7 h and double routes over 14 h; (iii) the time taken on single routes that are longer than 8 h and on double routes longer than 16 h. The weight of each aspect has to be defined by the user, by adjusting the value of the factors a, g1 and g2 . The exponent of 6 in the last two summands was fixed after some initial tests. This penalization function, mutually agreed upon with the company, only considers the number of routes over 7 h (so it adds the same penalization for either a 7 h 1 min or a 7 h 59 min route); moreover, it penalizes the whole journey time for routes over 8 h. However, other functions could also be possible; for instance, considering a continuous and increasing penalization, depending on X X a minð7; RSd Þ þ g1 minð1; RSd  7Þ 8d 8djRSd 47 h X ðRSd  8Þb ðb41Þ. þ 8d2DjRSd 48 h An equivalent consideration could also be done for double routes. An ad hoc heuristic algorithm was developed to solve the problem. This algorithm generates the routes in three main phases, which are divided into six stages in total. In the first phase, an initial solution is generated; in the second phase, a local optimization is carried out; and in the third phase, a metaheuristic procedure is applied to improve the solutions obtained up to that point. Fig. 1 illustrates the structure of the algorithm that generates the routes; this procedure is detailed in the following sections. 3.1. Phase 1: initial route generation In this first phase, which is divided into four stages, an initial solution is generated. 3.1.1. Stage 1: designing double routes The main objective of this first stage is to design the double routes, if needed. These routes will include customers who are located far from the salesperson’s residence (distant customers), but close to each other. Let the distance of a province be defined as the sum, for all the distant customers i, of the distance between the salesperson’s residence and the customer i, t 0i , multiplied by the number of visits to i still to be made. Let us consider customers i and j close to each other, those in the

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Phase 1: Initial route generation

49

Stage 1: Designing double routes Design of double routes, which include customers who are far from the salesperson’s residence, but close to each other. Single routes with customers who are far from the salesperson’s residence are also generated.

Stage 2: Mending and completing single routes that end far away The single routes that finish with distant customers are completed by inserting some close customers between the salesman’s residence and the last distant customer.

Stage 3: Completing the double routes The double routes are completedwith close costumers.

Stage 4: Designing the remaining routes The remaining single routes are designed.

Phase 2: Local optimization

Stage 5: Local optimization A local optimization is carried out in three steps to improve the solution obtained in the first phase.

Phase 3: Metaheuristic procedure

Stage 6: Metaheuristic procedure A tabu search is implemented to improve the solution obtained in the second phase.

Fig. 1. The structure of the algorithm.

same province, and those in different provinces when t ij p0:5ðt 0i þ t 0j Þ. Let b1 and b2 , respectively, be the number of double and single routes generated so far.

finishes far from the salesperson’s residence (b2 ¼ b2 þ 1). If there are any other distant customers and b1 oDR and b2 oD  2b1  1, then go to 1.

1. Identify the farthest province, ML , in which remains some customer to be visited. 2. Taking the distant customers still to be visited in ML and the salesperson’s residence, solve a traveling salesman problem (TSP) considering the partial sequence of visits established so far and a maximum time of 16 h (i.e., the TSP is solved while the time constraint of the tour is fulfilled). Use the sequential savings algorithm (Clarke and Wright, 1964), taking into account the length of the visits to the customers. 3. If all the distant customers still to be visited in ML have already been included in routes, check whether any distant customers in other provinces still have to be visited that are close to either the first or the last customer of the partial route obtained so far. If such customer exist, choose the closest one t and go to 2; now the province of customer t, pt , become ML . Otherwise, go to 4. If not all the distant customers in ML have been included in routes yet, go to 4. 4. End of the route in construction. The route obtained is a double route (b1 ¼ b1 þ 1) or a single distant route (when its duration is less than 7 h) that starts and

3.1.2. Stage 2: mending and completing the single routes that end far away The b2 single routes, that both start and finish with distant customers, should be turned into double routes. The aim of this stage is to insert at least one close customer between the residence of the salesperson and either the first or the last customer of the route designed so far. When this stage is completed, the b2 single routes finish at a close customer, except when there are not enough close customers. Let i and j be the first and the last customer on a partial route. The most central customer k is the one who has the minimum value minðt 0k þ vt k þ tki ; t 0k þ vt k þ t kj Þ. 1. Rank the b2 simple routes according to their duration, in decreasing order. 2. If there is any route which has not yet been dealt with and there are close customers still to be visited, select the next route; otherwise go to 5. 3. Keep the partial route that has already been designed for distant customers and solve a TSP using sequential savings, incorporating the close customers that still have to be visited and considering a maximum

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duration of 7 h. If any close customers have been included in the route, go to 2; otherwise go to 4. 4. Insert a close customer even when the maximum time (7 h) is exceeded. Such customer is the most central customer still to be visited in the trajectory between the first or the last customer on the partial route and the salesperson’s residence. Go to 2. 5. Finally, since the distances are considered symmetrical, if a route starts with a close customer but finishes at a distant one, the order of the route is swapped around (thus, it is not necessary to become a double route).

3.1.3. Stage 3: completing the double routes The objective of this stage is to complete the b1 double routes designed so far. 1. Rank the b1 routes according to their duration, in decreasing order. 2. If there is any route which has not yet been dealt with and there are close customers still to be visited, select the next route; otherwise, end of this stage. 3. If the duration of the route is more than 14 h, go to 2. 4. Keeping the partial route that has already been designed for distant customers, solve a TSP using sequential savings with maximum duration of 14 h and including close customers who still have to be visited. Go to 2.

3.1.4. Stage 4: designing the remaining routes In this stage, up to H ¼ D  2b1  b2 new single routes may be designed. At the end of this stage, all visits to the customers are already in a route and an initial solution is obtained. 1. 2. 3. 4.

k ¼ 1. If there are no customers left to visit, end of this stage. If k4H , go to 5. Solve a TSP using sequential savings, by incorporating all the customers who still have to be visited and considering a maximum duration of 7 h. k ¼ k þ 1, go to 2. 5. Rank the customers i who still have to be visited, according to t 0i . 6. If there are any customers who still have to be visited, select the next customer t from the list; otherwise end of this stage. 7. Insert t in the route and in the position with the minimum value of the objective function Z if the customer has not already been included in the route. Go to 6.

Phase 2 carries out the following three steps of local optimization sequentially, in order to improve the solution obtained in phase 1. Step 1. Initial local optimization of each route. If the number of customers on the route, n, is np6, a complete exploration of all the possible routes is carried out and the best one is chosen. If n46, an exhaustive local optimization (ELO) is carried out. An ELO generates all the neighboring routes to the current route and takes the best one as the new current route, until the best neighboring route is not better than the current route. Neighboring routes are generated by swapping the position of the visit to each pair of customers of the route. One customer in the pair is real, while the other may be real, in which case the position of the customers is swapped, or be a dummy one, in which case a real customer is inserted between two other real ones (or between a real customer and the salesperson’s residence). Step 2. This second step involves a new ELO. Neighboring solutions are generated by swapping the visit of a customer i assigned to the route k with the visit of a customer j assigned to the route l. This is done for all the pairs of k  l routes. Only one of the two customers can be a dummy. If one of the customers is dummy, a new customer is attracted or taken out from a route. The empty routes with one dummy customer should also be considered in the swaps. Step 3. The third step involves the ELO procedure explained in step 2, but with two differences. First, the generation of the neighborhood is modified. When customer i of route k is swapped with customer j of route l, all of the possible positions in route l are examined. Secondly, the procedure stops when the improvement in the value of the objective function is less than 0.5% in two successive iterations. 3.3. Phase 3 (stage 6): metaheuristic procedure In the third phase, a metaheuristic procedure is carried out to improve the quality of the solution obtained so far. A tabu search (TS) was chosen for this phase. TS is a strategy for guiding heuristic improvement procedures to overcome local optimality and it has been used in other VRP studies, such as: Semet and Taillard (1993), Osman and Wassan (2002), and Archetti et al. (2006). According to Laporte et al. (2000), the TS ‘‘has proved to be the most successful metaheuristic approach’’ in VRP studies. The features of the designed TS procedure are detailed briefly below (see, for example, Glover (1989, 1990) or Glover and Laguna (1997) for a more detailed description of TS).

 Objective: to minimize the value of the objective 3.2. Phase 2 (stage 5): local optimization In the second and third phases, single or double routes that finish with a distant customer should be avoided. From this stage on, a new term MM is added to the objective function Z, which adds a big value to each route that both starts and finishes with a distant customer.

function Z.

 Initial solution: the one obtained in phase 2.  Neighborhood: as in step 3 of phase 2.  Aspiration level: the best neighboring solution is better than the best solution obtained so far.

 Tabu rules: customer-i/route-k: customer i was in the route k in the abandoned solution.

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 Length of the tabu list: 10 sets tabu.  End condition: the computational time introduced by

To check the improvements that phases 2 and 3 make to the initial solution, a brief analysis was carried out using real data from the different zones. When phase 2 was applied, the reductions in the objective value ranged from 4% to 82%. The average reduction was 32%. A reduction of 100% was obtained in one case, excluded from the previous average, in which a route with a duration of over 8 h was eliminated. The application of the third phase obtained slightly better solutions than the second phase. This indicates the power of phase 2. Nevertheless, the third phase was kept in the procedure, although the tabu search should only take a short computing time. Finally, the results of analyzing one of the zones before customers were reassigned to salespeople are presented in Appendix. The tool has been used by the company to: manage its commercial and rental network of home videos; evaluate the different scenarios of reassigning customers to salespeople; assign new customers to salespeople; and/or to vary the number of visits to customers. This last aspect is very important, as the company has already found that each additional visit to a customer produces an increase in sales. Initially, the workload of the different salespeople was verified. This analysis was used to reassign each zone’s customers to the salespeople in a more rational way. This tool has also been used to negotiate the annual sales objectives and to manage the single and/or double routes to be undertaken in each commercial cycle. The advantages of the work developed are listed below. These improvements were arrived at jointly by our research team and the company:

the user. The third phase finishes with the individual optimization of each route. This is the same optimization as that used in the first step of phase 2. 4. Results The algorithm was designed, coded and calibrated in less than one month. The company that commissioned the study has tested and validated the tool with real data. Company managers qualified the routes obtained using the tool as very satisfactory; both for the commercial and rental network (about 450 customers and 8 salespeople, and about 400 customers and 6 salespeople, respectively). The values of the parameters used in these tests were the following ones: each salesperson had between 35 and 85 customers; customers were located in all areas of Spain; each customer had from 1 to 5 visits; there were 16 to 21 available days in a commercial cycle; there was a maximum of 0 to 4 double routes; working days had 7 and 14 recommended hours and maximum lengths of 8 and 16 h for single and double routes, respectively; a distant customer was defined as between 100 and 150 min away; and the maximum computing time for the metaheuristic procedure was between 30 and 120 s. The computing time to solve the problem in a zone, directly depends on the maximum computing time permitted for the metaheuristic procedure (stage 6), since the computing time needed for the first 5 stages was just a few seconds, in the worst cases.

Lugo

month.

Álava Navarra Burgos La Rioja Palencia

León

Pontevedra

 The tool was developed and operative in less than one

Vizcaya Cantabria Guipúzcoa

Asturias

La Coruña

Orense

Zamora Valladolid Salamanca

Soria

Huesca Lérida

Gerona

Barcelona

Zaragoza

Tarragona

Segovia Guadalajara

Ávila Madrid Cáceres

51

Toledo

Teruel Castellón

Cuenca Valencia Baleares

Ciudad Real

Badajoz

Albacete Alicante

Córdoba Huelva

Sevilla Málaga

Jaén

Murcia

Granada Almería

Cádiz Fig. 2. The analyzed zone.

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Table 1 Results of the different stages of the procedure.

Phase 1 Stage 1

Route {customers} value of Z

No. of customers

Duration (travel/visit)

R1 {20, 21} R2 {20}

2 1

8.40 7.40

Stage 2

Same results as stage 1

Stage 3

R1 {30, 26, 31, 28, 20, 21, 25, 24, 15} R2 {27, 32, 29, 22, 20, 26}

9 6

13.95 13.83

Stage 4

R1 {30, 26, 31, 28, 20, 21, 25, 24, 15} R2 {27, 32, 29, 22, 20, 26} R3 {2, 5, 3} R4 {27, 4, 5} R5 {33, 34, 35, 9} R6 {23, 32, 10, 14} R7 {1, 23} R8 {6, 13, 11, 19, 14} R9 {11, 13, 6, 12} R10 {8, 16, 18, 12, 17} R11 {7, 16} Z ¼ 303 000

9 6 3 3 4 4 2 5 4 5 2

13.95 13.83 7.00 6.87 6.38 6.62 6.58 6.85 6.42 6.80 2.87

R1 {14, 30, 26, 31, 28, 20, 25, 24, 11} R2 {10, 27, 29, 32, 22, 23, 20, 21, 6} R3 {2, 5, 3} R4 {27, 4, 5, 1} R5 {12, 34, 33, 35, 9} R6 {23, 32, 26, 14} R7 {7, 11, 19, 13, 16} R8 {15, 17, 13} R9 {8, 16, 18, 12, 6} Z ¼ 279 840

9 9 3 4 5 4 5 3 5

14.78 15.60 7.00 7.97 7.50 7.82 6.53 3.53 6.83

Phase 2 Stage 5

Phase 3 Stage 6

(8.28/6.50) (7.10/8.50) (4.00/3.00) (3.97/4.00) (4.50/3.00) (3.32/4.50) (1.53/5.00) (1.03/2.50) (1.33/5.50)

Same results as stage 5

 Significant savings are achieved in the time dedicated

 The overload reduction of the salespeople is taken into

to management the commercial and rental network. Is it possible to control the workload of the salespeople, and conflicts depending on work overload are avoided. The potential capacity of the salespeople is better used: overtimes decrease (so, costs of the commercial network), and reassignations of customers, incorporations of new customers, and increase in the number of visits to current customers are possible. The route-generating procedure fulfils all the requirements of the problem: the number of visits to the customers to be made, the generation and the maximum number of double routes, etc. The computing time is brief; then different scenarios can be simulated in order to organize and control the commercial and rental network.

account as a goal using a penalized non-linear objective function.

 





5. Conclusions In this paper, we present a procedure for generating salespeople’s routes for a home video company’s commercial network. The commercial area is divided into different zones and each salesperson is assigned to one of them, working and having independent routes from the others. The company wanted to generate salespeople’s routes for a commercial cycle, which is usually one month long. The commercial cycle has the following main features: each customer has to be visited a specific number of times; and two-day routes are possible to visit the customers located far from the

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Route1

53

Route3

Route2

Albacete (3) 183

San Juan (1) Orihuela (1) 13 Murcia 25 41 Alicante (5)

Elche (1)

Molina Segura 41 (1) 52

Lorca(1) 105

175

24

Alicante Torrevieja (1) San Javier 27 (1) Cartagena (3) 42

25

Almería (1)

Route4

Almería (2)

Valencia (2) Carcaixent (1)

50

55

120

Finestrat (1) 33

Alicante

50

Route6

39

88

Albacete (3)

Murcia (1)

Alicante

150

Route5

59

120

Alcoi (1)

Alicante

Murcia (2) 47

49 42 54

Alicante Torrevieja (1)

Cartagena (1)

Route7

Route8

Petrer 31 (1) 27 22

Route9

Petrer 30 (1)

Alicante (2)

Alicante (2)

San Juan (1) 13

28

Finestrat(1) 26

Alicante (3)

Fig. 3. The routes obtained.

salesperson’s residence. The objective function is to minimize the weighted sum of the travel time plus overtime penalties. In less than one month, an ad hoc procedure was designed, coded and calibrated: first, a set of initial routes are generated, then, a local optimization and a metaheuristic procedure improve them. The company has qualified the routes as very satisfactory and the tool is now used by

the commercial network management to: reassign customers, assign new customers and vary the number of visits to the customers. Although the developed algorithm was for a specific client, the authors think similar problems could arise for many companies. Hence, the paper has applicability and it is relevant to possible problems in the logistics industry.

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Further research could focus on two aspects: enhancing the developed algorithm to solve the faced problem; and generalizing its applicability.

Acknowledgments The authors are very grateful to the anonymous reviewers for their valuable comments, which have helped to enhance this paper. Appendix The results of analyzing one of the zones before customers were reassigned to salespeople are presented below. The values of the parameters used were: 35 customers; 12 had to be visited twice (customers 5, 6, 11, 12, 13, 14, 16, 20, 23, 26, 27 and 32); the commercial cycle had 17 available days; there was a maximum number of 4 double routes; the recommended working days were 7 and 14 h long; the maximum working day was 8 and 16 h long for single and double routes, respectively; a distant customer was defined as being 150 min away (so there were two distant customers: 20 and 21); the maximum computing time of phase 3 (metaheuristic procedure) was 30 s; a ¼ 60 and g1 ¼ g2 ¼ 120; the unit of time used in the objective function was in minutes. Fig. 2 shows the zone and the salesperson’s town of residence (Alicante). Table 1 shows the partial results obtained when the different stages of the algorithm were carried out. It also shows the final solution. The routes are presented by indicating the assigned customers, the duration of the routes and the value of the objective function at the end of each phase. Moreover, for each of the routes in the final solution, the travel time and the time of the visits are shown. The table also shows how each stage fulfils its function, according to the design. The final solution consists of 9 routes (7 single and 2 double ones) to be undertaken in 11 available days. None of the routes exceeds the maximum established working day of 8 or 16 h. Five routes have a duration over the recommended 7 and 14 h, because of the objective function and the values of g1 and g2 used: the routes that exceed the recommended duration of the working day are penalized with 120 min in a binary way. The final value Z ¼ 279 840 is obtained as follows: 60n (60n(14.7833+15.60+7.00+7.9667+7.50+7.8167+6.5333+ 3.5333+6.8333))+120n3+120n2. Fig. 3 shows the routes graphically. The number of customers to be visited is given below the name of each of the towns. The travel time from the last customer visited in a town to the first one visited in the next town is shown in minutes next to the bows.

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