MARTÍN DARIO ARANGO SERNA
Escuela de Ingeniería de la Organización. Facultad de Minas. Universidad Nacional, mdarango@unalmed.edu.co

WILSON ADARME JAIMES
Facultad de Ingeniería, Universidad Nacional de Colombia, Sede Bogotá. wiadarme@unal.edu.co

JULIAN ANDRES ZAPATA CORTES
Facultad de Ingeniería, Universidad Nacional de Colombia, Sede Bogotá. wiadarme@unal.edu.co

Received for review February 1^th, 2010, accepted May 13^th, 2010, final version June, 12^th, 2010

ABSTRACT: This paper describes an algorithm for solving the micro-routing problem when it is used modes like bicycle and motorcycle with different capacities and serving different routes which are associated to a set of customers and also considering time windows restrictions. From the research in the Palmira bread industry, in which the bakeries were geo-referenced and in which the warehouse and the transport of goods management systems were characterized, was possible to establish the set of parameters needed for the numerical routing algorithm that seeks serving the daily customer requirements. The numerical results show the importance and how incident can be this methodology for SME´s which do not have a formal structure, resources neither appropriated information systems to compete in the actual market.

KEYWORDS: Vehicle routing, bicycle, time windows restriction, perishable goods.

RESUMEN: Este artículo describe un algoritmo para resolver el problema de micro ruteo cuando se dispone de medios como la Bicicleta y la motocicleta, con diferentes capacidades que cubren distintas rutas asociadas al conjunto de clientes con restricciones de tiempo. A partir de la investigación en el sector panificador de Palmira, donde se geo-referenciaron las panaderías y se caracterizaron los sistemas de gestión de inventarios y transporte de insumos, se establecieron los parámetros necesarios para los ejemplos numéricos del algoritmo de ruteo que busca atender las necesidades diarias de los demandantes. Los resultados numéricos muestran la importancia e incidencia que tendría esta metodología en microempresas que no disponen de organización, recursos ni de sistemas de información apropiados para competir en los escenarios actuales.

PALABRAS CLAVE: Ruteo de vehículos, bicicletas, ventanas de tiempo, producto perecedero.

The vehicle routing problem (VRP) has gained high attention both in academic research as in practice. VRP has been studied from several scopes and in this paper it is focused in the bicycles and motorcycles routing problem (BMRP) using daily planning for the bread industry in Palmira, Valle del Cauca, Colombia; integrated by costumers with higher demands than their stock capabilities as can be study in [1]. The results reported have considered a variation of the vehicle routing problem with time windows (VRPTM), where the same vehicle can serve several routs in an working day.[2] Besides, types of transportation as bicycle and motorcycle have not been deeply studied in applied works to delivered express of perishable goods. This problem will be really important in the near future due to the technology advance and the change in the customer customs.

There are different studies about VRP without time windows [3] using heuristics in their solution, as well as several with time windows using exact models as the solving tool [2], what illustrates the advance in this subject. Studies as the one presented in [4] show heuristics of insertion that can handle different types of restrictions, including time windows and multiple vehicles. In [5] the authors present delivered Express problem, which is really similar to the one faced in this work.. Others studies [6] consider logistics and socio-economical issues about several types of the delivery express problem.

The present article rooted by the studies made in [2], shows an exact algorithm for solving the VRP with different vehicle capacities, time windows and multiple routes. In the second part of the article it is presented the mathematical model formulation. In the third part it is shown the characterization and the parameterization of the bread sub-industry. Later it is presented the results of a numerical example using the Clarke and Wright heuristic dealing with the number of nodes that is included in the evaluation and finally it is presented the conclusions and further works in this field.

Due to the difficulties presented on the bread industry, related to the quality level and to the asepsis of the raw material and finish goods, it have been detected through several studies, that the central main causes are related to the way of care, handle, storage and transport those elements. One of the biggest concerning about those problems is how to fulfill all the customers' requirements but following a continue supply politic, without affecting the quality and service, and taking into account ways of transport as bicycle, motorcycle and moto-trailer.

The problem description is:

The symbol N⁺ is used to N Ụ {0, n+1} and A⁺ Ụ {0, n+1}, where {0, n+1} is a fictitious arc with a distance d_0,n+1 = 0 and a travel time of t_0,n+1 =0. The starting time σ^r for load a vehicle is associated which each route r Є K. The goal of the model is to minimize the total distance to serve all the customers, while satisfying the capacity, time windows and limited deliver constrains. The problem is formulated as following, using M as a big arbitrary constant.

Operative/Productive Sub-system.
In small bakeries (SB) the most common area is 60-80 m² (40%), in medium bakeries (MB) the most common area is 120-160 m2 (60%) and in the big bakeries the most common area is 150-200 m² (60%). 40% of SB have areas until 120 m², but still present problems related to the space for production, due to this space is common used for both living and operate the bakery.

Around the 80% of Palmira Bakeries are open from 5:00 a.m.to 12:00 pm. Salespeople work from 6:00 am to 10:00 pm divided in two shifts, the first one from 6:00 a.m. to 2:00 p.m and the second one from 2:00 p.m. to 10:00 p.m. Administrative people work from 8:00 a.m. to 12:00 pm and from 2:00 p.m. to 6:00. Those working times apply to big bakeries; in medium and small bakeries there is not a strong distinction between those roles and it is common that the manager is in charge of production, sales, cleaning, and this person works from the opening and the closing of the store, e.g. from 6:00 am to 10:00 pm.

Small Bakeries have between 11 to 15 machines, 80% of medium bakeries have between 16 to 20 machines and all the big bakeries have between 16 to 20 machines, which are flexible regarding to the volume and variability of products. Preventive Maintenance is used in big bakeries and reactive maintenance is used in Medium and small bakeries; the more common equipments are ovens, rotative oven, wetting machines, cylinders and mixers.

Supply goods demand
A research made for the 76 evaluated bakeries valued that in 2007 the consumption of flour is 1.490 tons, sugar 352 tons, cheese 293 tons salt 63 tons, yeast 196 tons and butter 401 tons. Annual aggregate consumption of the main goods (flour, sugar, cheese, salt, yeast and butter) differentiated by size of the bakery is: 102 tons/year for big bakeries, 40 tons/year for medium bakeries, 20 tons/year for small bakeries. The average labor resource occupancy for big, medium and small bakeries are 21, 6 and 3; the studied bakeries generated 496 directed jobs in 2007.

Routing design and tecnical characteristics of the transportations modes
Moto-trailer Model Enduro TS 124; cubic Capacity: 125 C.C.; weight in empty: 85 Kg; performance: 85 km/gallon; fuel: normal gasoline. Cargo Capacity: 350 Kg. Figure i shows a view of this moto-trailer. This type of transport is accepted by law for the colombian goverment in the law number 769 of 2002 in the Código Nacional de Tránsito Terrestre, Chapter V, articles 94 y 96. This type of transportations is a good option when there are too much traffic, the investmen is low, technical specifications, easy of drive, fast, performance, capacity, industrial safety, road safety and also conservation and asepsis in transport of supply goods. In a technical evaluation of the volumetric capacity of the trailer (1.24m x 0.80m x 0.72m = 0,714m³); there was found that the constrains are related only to the weight and not to the volume due to the density of the goods (goods (flour, sugar, cheese, salt, yeast and butter)

Figure 1: Moto-Trailer

An in situ evaluation was made in order to evaluate the traffic, vehicular and pedestrian flux, roads characterization and normal distances in routing and also there was found the average speed for this type of transport, which was determined in 7,336 km./hr (2,037 m/s) in downtown area and 27.692 km./hr. (7.692 m/s) in residential neighborhoods.

Non-motorized types of transportation
This tipe of transportation is allowed by the law number 769 of 2002 "Código Nacional de Tránsito Terrestre, Chapter V, article 94";. Distribution centers consider double-rack bicycle as the favorite type of trasport due to its low cost, capacity, fast operation (considering traffic jams), easy-driving, cheap mantenience and operation. Figure 2 shows this type of transport.

For the cheese industry moto-trailers are the preferred type of transportation since this devices allow moving up to 300 Kg. The research presents for reference, discussions and analysis the possibility of implementing the double-rack bicycle in this business but taking into account that the maximum load for this device is 100 Kg, which is 200 kg less than the original one in some cases. The limit time is 5 hour in every labor shift, and considering the flat topographic of Palmira, where a GPS study showed that the lower point in the city is at 993 meters above sea level (MASL) and the highest point is at 1047 MASL.

For those conditions it was made a field evaluation and was determine an average speed for bicycles of 6,746 km/hr. (1,874 m/s) in downtown and 8,235 km/hr (2,287 m/s) in residence areas.

Costs
The transportation cost structure for the evaluated types of transport was determined as fixed costs and variable costs for both moto-trailer and bicycle. Daily Fixed cost are shown in Table 2, assuming 25 working days every month.

Table 2: Daily fixed costs

Unit variable cost (UVC) are those costs that vary depending on the operation conditions and are established in pesos/Kilometer (Pesos is the Colombian currency, COP).

This cost considers the fuel, tires, lubricants, filters, maintenance, repair, unforeseen events and others. Table 2 shows the variable cost per km of every type of transportation.

Table 3. Variable cost per kilometer

Table 4. Aggregated Consumption

Based on this information was developed the programming and the vehicle routing design using moto-trailer and bicycle for a load capacity of lower than 100 kg/bicycle.

The programming and route design consider the following considerations:

The model have been run using the Clarke and Wright method with the aim of minimizing the total travel distance fot all the vehicles needed to serve all the stops. Recently this routing programming is made with a frequency of two times per week, what is really different to the six times proposed before this study which were using a FIFO systems (first in, First out) without product consolidation. This method was made using Excel™ for the data analysis and Logware 5.0, ROUTER module for the routing design.

4.2. Results of the Clarke and Wright method application
With the aim of fully apply the Clarke and Wright method, the bakeries and the distribution centers were Geo-reference through GPS seeking to find the geographical and Cartesian (IGAC) coordinates. This information is an input for the software Logware. There were taken 142 points (Cheese company and bakeries), finally in the de-codification and processing activity there were selected 124 points including the cheese company and the 35 bakeries.

The demand level of the time windows in each bakery was determined. This demand levels were acquired by analyzing 76 small Family-companies, 35 of those were selected due that those are cheese company suppliers.

The evaluation has been made by using the types of transport totally independent and through the combinations of these. Results are expressed in the standard software units (Miles as a length measure, to meters). The correction factor used in Logware for the distances was obtained through a field evaluation of the real distances, and it is equal to 1.26, 1.27, 1.29 and 1.3 for some routes, due to the short distances evaluated in this work, and because that the going and returning trip is made through different roads and streets. In the evaluation the correction factor was 1.3. The speed used for the moto-Trailer was established in 8859 meters/hour (weighted average between jammed urban zone 78% and non-jammed area 22%) and for the bicycle it was used a speed of 6745 meters/hour (speed achieved in the jammed urban zone)

Results indicates that there must be 39 routes for all the bakeries that have a daily demand higher that the vehicle (bicycle) capacity which is 100kg, the amount should be divided considering an utilization of 100% of the vehicle capacity and the remaining amount should be used for a new route. For example, the bakery number 30 has a daily demand of 549, so it is equal to 4 demands of 100 kg plus one of 49 Kg, seeking optimizing the vehicle use. This was made with all the bakeries that have a daily demand higher than 100 kg, which causes that of the 35 bakeries the model has 54 stops with a lower demand than 100Kg/day.

Figure 3 shows how the bicycle routing should be to serve all the stops. It is important to notice that some points could not appear due to some points superimpose others as happens in bakeries 22 and 30 in which it is necessary make several trips to the same point. Table 7 is a summary of the main indicators of the process.

Figure 3: Clarke and Wright method - double-rack Bicycle

The total time of 26.6 hour allows programming the routing with 4 bicycles (8 hours daily) but considering that this is not a motorized transportation type it was made a programming using an average working time of 5 hours daily and with the constrain that each bicycle should not travel more than 10 km (1000m). It means that it is necessary programming 6 double-rack Bicycle, with a initial investment of 4.5 millions of pesos. (6 x 750.000).

With those six bicycles it is necessary 39 routes (Logware Results) which have an average time of 23 minutes an occupancy level of 93.7% of the vehicle capacity.