A model for optimizing time and costs of collection pallets

(1)

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 8, August 2012)

13

A model for optimizing time and costs of collection pallets

Eduardo Gutiérrez González

1

, Olga Vladimirovna Panteleeva

2

, Víctor Manuel Córdoba Lobo

3 1 _{National Polytechnic Institute, Section of graduate studies and research, Av. Te 950, 08400, Mexico}

2_{Colegio de Postgraduados, Posgrado en Socioeconomía, Estadística e informática Campus Montecillo México-Texcoco56230}

Montecillo, México

3 _{National Polytechnic Institute, math academies, Av. Te 950, 08400, Mexico}

Abstract - This paper combines a linear optimization model, the Traveling Salesman Problem and economic evaluation projected to 5 years that minimizes the collection time and the costs for a pallet leasing company when the pallets are empty in supermarkets. The model is based on optimizing an adequate fleet to collect the pallets. The plan is to verify the model and compare the collection times used with the current company work scheme, and that obtained with the proposed solution to collect 357,687 pallets located at 55 institutions located in 46 cities, municipalities and delegations of central, south, and western Mexico, which should supply the Metropolitan area reservoir. With the proposed model the pallet time collection is reduced by 27%. So the proposal would result in a 5 years saving of $781,195.1 usd.

Mathematics Subject Classification (2010): 90B06, 90C10

Keywords - Model optimization, Traveling Salesman Problem, NP hard time, tour

I. INTRODUCTION

The routing of vehicles on road transportation networks is an area of great importance to transportation planners within scientific literature. This field includes well known and studied problems like those of traveling salesman or TSP (Rodriguez & Ruiz, 2012). This problem has been extensively studied since its processing time is NP hard and different metaheuristics algorithms have been proposed (Erdogan et al., 2012), heuristics for the TSP (Rego et al., 2011), the parallelized genetic ant colony system (Chen & Chien, 2011), the Lin-Kernighan heuristic (Karapetyan & Gutin, 2011), the genetic algorithm for the one-commodity pickup-and-delivery TSP (Zhao et al., 2009), etc..

The TSP is that it has large logistic applications in companies that deliver or collect their products, e.g. Bonomo et al. propose a TSP method to optimize vehicles routes to collect trash bins all over the south of Buenos Aires, Argentina. With this application there were real reductions in travel distances ranging from 10% to 40% with savings of $200,000 per year (Bonomo et al., 2012).

Mladenovic et al. present another application of TSP characterized by a set of customers such that each customer either supplies (pickup customers) or demands (delivery customers) a given amount of a single product (Mladenovic et al., 2012). Laporte et al. showed a generalization of the TSP problems (Laporte et al., 1996). These problems include location-routing, material flow system design, post-box collection, stochastic and vehicle routing. The TSP applications can be in different media tours, e.g. Rakke et at. show a maritime problem to determine an optimal sequence of visits of the cargo ships to ports (Rakke et at., 2012). There are also generalizations of the TSP where it is not necessary to visit all nodes, as shown by (Feillet et al., 2005) or the work of (Baldacci et al., 2003). They described new integer programming formulation for the TSP with mixed deliveries and collections based on a two-commodity network flow approach. Butler et al. described a new extension to the Symmetric Travelling Salesman Problem in which some nodes are visited in 2 periods and the remaining nodes visited in 1 period (Butler et al., 1997).

The generalized traveling salesman problem (GTSP) is a well-known combinatorial optimization problem with a host of applications. It is an extension of the TSP where the set of cities is partitioned into so-called clusters, and the salesman has to visit every cluster only once. This article applies the GTSP (Bontoux, Artigues & Feillet, 2010), (Karapetyan & Gutin, 2012) to the strategic supply chain company CHEP Mexico, because it has presented problems in recent years of a shortage of pallets to fill orders for its customers during the year, mainly in the last quarter. The season of the year-end cost to the company is very high, so that some clients choose to go with the competition as a result of incurred delivery delays. This situation is exacerbated mainly in the central region of the country and is due to the following:

(2)

International Journal of Emerging Technology and Advanced Engineering

14

• Time taken in return to depot.

• Time taken for a pallet to be inspected, and / or repaired.

• Unavailability of transport units.

Undoubtedly, this causes an increase in the collection time which has been found to be up to three weeks after the date Mexico CHEP reported the existence of empty pallets. Therefore, the company needs a better method of collection and distribution of pallets to optimize time in peak periods and does not require the purchase or erection of new pallets, making a saving that can be used for the purchase of unit transport, to formalize its own fleet of staff to collect and distribute pallets.

This article will design and evaluate a TSP model for the company to reduce collection times of pallets in 55 supermarket entities located in 12 states of Mexico and Mexico City and taken to the warehouse located in Tultitlan MX16 Mexico State. The evaluation period takes place during the year-end collection (covering the months October, November and December), considered as one of the most critical seasons.

The solution of the linear programming model includes: 1. The optimal number of truck types collecting the

weekly volumes of pallets.

2. The assignment of each of these trucks in the collection of the 46 locations, based on the number of trips taken per week.

3. Planning optimal routes provided by the GTSP. II. MATERIALS AND METHODS

Finding the optimal quantity of pallets to collect and organize a typical Traveling Salesman Problem to optimize the collection time.

2.1 Model for calculating the optimum number of transport units to collect pallets

Suppose we have m types of collection trucks of pallets with capacitiesc1,c2,,c_m, such thatc1c2cm

.

On the other hand, C_T is the total number of pallets to collect on the net per week denoted by x₁,x₂,,x_m decision variables are defined as: x_j decision variables are defined

as c_j to collect C_T pallets, with j1,2,3,,m.

Namely, _T

m

j j jx C c





 1

, then the problem is:

m j x c C x c C x c x Z j T m j j j T m j j j m j j , , 2 , 1 integer negative -non 1 min 1 1 1 1        



  

This model indicates the number of trips each truck required to make with capacities c1,c2,,c_m, so that they can carry all C_T network pallets. In the case of n tours a model is applied to each, for the number of trucks that are required for each type. To this we must know how many trips each truck can make a week.

Let q₁,q₂,,q_m be the number of trips a week that the truck can make with capacity c₁,c₂,,c_m respectively. In case of necessity to restrict the number of trucks used for

each type, the constraint added is _j _j n

i

ij q r x 



1

, where, r_j

represents the availability of trucks type j.

Objective function for the entire network:



   n i m j ij x Z 1 1 min . With restrictions m j n i j i x r q x c C x c C x c ij j j n i ij T m j ij j T m j ij j i i , , 2 , 1 ; , , 2 , 1 ; , integer negative -non 1 1 1 1 1          



  

Where c_j  Truck capacity j, x_ij Number of truck trips j in tour i, 

i T

C Number of pallets to collect on the

tour I and r_j  Availability of trucks j for collection.

2.2 Calculation of the optimal number of trucks

(3)

International Journal of Emerging Technology and Advanced Engineering

15

3 ½ pickup (collection capacity:

c

1



72

pallets)

Torton 24 feet (collection capacity:

c

₂



252

pallets)

Platform 40 feet (collection capacity:

c

₃



432

pallets)

c

4



504

pallets)

c

₅



540

pallets)

The network of pallet collection by the company is bounded by a storage center that caters to 55 institutions in 12 states and Federal District. The geographical network position is decomposed into n=8 tours, denoted by the letters A, B,…, H.

Tour A:Neza (AN), Ixtapaluca (AI), Puebla (AP), San Martín Texmelucan (AS), Tehuacán (ATH), Teziutlan (ATZ) y Oaxaca (AO).

Tour B: Metepec (BM), Toluca (BT), Zinacantepec (BZ), Morelia (BMO), Uruapan (BU), Guadalajara (BG), Tlaquepaque (BTL), Tonalá (BTO) y Zapopan (BZA).

Tour C (Metropolitan zone): Atizapán (CA), Cuautitlán (CC), Ecatepec (CE), Huixquilucan (CH), Tecamac (CT), Tlalnepantla (CTL) y Naucalpan de Juárez (CN).

Tour D (Federal District): Álvaro Obregón (DA), Benito Juárez (DB), Coyoacán (DC), Gustavo A. Madero (DG), Miguel Hidalgo (DM), Tlalpan (DT), y Venustiano Carranza (DV).

Tour E: Querétaro (EQ) y San Juan del Río (ES).

Tour F:Ciudad Valles (FC), San Luis Potosí (FS), Fresnillo (FF) y Zacatecas (FZ).

Tour G:Celaya (GC), Irapuato (GI), León (GL) y Aguascalientes (GA).

Tour H:Cuernavaca (HC), Jiutepec (HJ), Acapulco de Juárez (HA), Iguala de la Independencia (HI) y Zihuatanejo (HZ).

To collect the weekly demand for two years, the maximum value is chosen, being spread over 36.929 pallets tours set, see Table 2.1. Gutierrez et al 2012.

For tours C and D larger capacity trucks (48 and 53 feet) are excluded because these tours only involve locations within the Metropolitan area and Federal District, by the type of large trucks moving with traffic on Metropolitan area avenues which will be extremely slow and in some cases impossible.

Then the model:

5 4 3 2 1 5 4 3 2 1 5 4 3 2 1 5 4 3 2 1 3 2 1 3 2 1 5 4 3 2 1 5 4 3 2 1 min xh xh xh xh xh xg xg xg xg xg xf xf xf xf xf xe xe xe xe xe xd xd xd xc xc xc xb xb xb xb xb xa xa xa xa xa z                                     Subject to 5233 432 252 72 3136 432 252 72 4372 540 504 432 252 72 2212 540 504 432 252 72 1715 540 504 432 252 72 3708 540 504 432 252 72 1902 540 504 432 252 72 15006 540 504 432 252 72 5304 432 252 72 3136 432 252 72 4443 540 504 432 252 72 2283 540 504 432 252 72 3 2 1 3 2 1 5 4 3 2 1 5 4 3 2 1 5 4 3 2 1 5 4 3 2 1 5 4 3 2 1 5 4 3 2 1 3 2 1 3 2 1 5 4 3 2 1 5 4 3 2 1                                                     xd xd xd xc xc xc xb xb xb xb xb xa xa xa xa xa xh xh xh xh xh xg xg xg xg xg xf xf xf xf xf xe xe xe xe xe xd xd xd xc xc xc xb xb xb xb xb xa xa xa xa xa 1644 540 504 432 252 72 3637 540 504 432 252 72 1831 540 504 432 252 72 14935 540 504 432 252 72 5 4 3 2 1 5 4 3 2 1 5 4 3 2 1 5 4 3 2 1                     xh xh xh xh xh xg xg xg xg xg xf xf xf xf xf xe xe xe xe xe

(4)

International Journal of Emerging Technology and Advanced Engineering

[image:4.612.74.529.157.305.2]

16

Table 1

Preliminary amount of trips required for each type of truck on tours

Tour / amount to be collected

Travel required per week

Pallets to be collected

Type of truck A B C D E F G H

2,212 4,372 3,065 5,233 14,935 1,831 3,637 1,644

1 (72) 1 1 0 0 0 0 0 1 3 360

2 (252) 0 0 2 2 0 1 0 0 5 0

3 (432) 0 0 6 11 0 0 0 0 17 8,640

4 (504) 0 0 0 0 4 0 2 1 7 5,040

5 (540) 4 8 0 0 24 3 5 2 46 24,300

Collection

capacity 2,232 4,392 3,096 5,256 14,976 1,872 3,708 1,656 78 38,340

leftovers 20 20 31 23 41 41 71 12 259

There are a total of 78 collecting trips: 3 trips of truck type-1, 5 type-2 trips, 17 trips of type-3, 7 trips of 4 and 46 type 5. Contemplating 5 days a week of work and assuming that each truck can make one trip per day on each tour, with the exception of tours C and D, where it can take up to two trips per day times because the shop receiving time is 06:00 to 19:00. The optimum amount of trucks that are recommended is 14, distributed as follows: One type 1, two type 3, two type 4 and nine type 5.

In the last column to the right of Table 1 can be seen the maximum amount collected from truck types 1 and 3 is 9.000 pallets. Then the maximum amount a week tours C and D is bounded by 9.000 pallets.

2.3 Weekly collection pallets

After determining the optimal quantity of trucks types the company intends to require a trial period in which the model is verified and the collection of pallets made during this period.

For the routes of the proposed truck fleet in the collection of pallets the Traveling Salesman Problem (TSP) is used. Thus, from a starting node called in this case depot,

1 

n sites in c_ij distance between nodes i, j must be visited. The aim therefore is to create a program for the minimum route distance.

Variables: For each possible distance the graph defines a variable,

 

 



otherwise ,

0

, , 2 , 1 , cycle in the this ) , ( distance the

if ,

1 i j i j n

xij



The objective function is to minimize travel distances:



  



1 1

min i

i j j

ij ijx c

Z .

Restrictions: It is necessary to visit all the locations where there are pallets and this requires two types of restrictions, arrival and departure, respectively

n j

x

j i i

ij 1, 1,2,... 1

 



 

,

n j

x

j i

j

ij 1, 1,2,... 1

 





III. RESULTS

For verification, the model is applied for a 13 week period of pallet collection.

3.1 Weekly collection of pallets with the model proposed With the use of 14 trucks 357.687 pallets are to be collected distributed in 8 tours during the 13 weeks, see Table 2.

(5)

International Journal of Emerging Technology and Advanced Engineering

[image:5.612.72.531.157.321.2]

17

Table 2

Pallet volumes to collect on each tour for 13-week trial

Tour 1 2 3 4 5 6 7 8 9 10 11 12 13

A 1,609 1,164 1,256 1,345 1,446 1,158 1,107 2,216 2,212 1,204 824 666 674 B 1,916 4,933 2,168 3,008 854 1,350 2,474 418 4,372 2,240 2,535 2,180 2,526 C 2,322 2,452 2,949 1,491 2,811 1,527 2,112 628 3,065 252 301 305 586 D 3,304 4,081 2,732 3,954 3,678 3,372 4,063 714 5,233 1,094 572 774 845 E 8,016 13,559 17,039 9,080 13,713 11,797 15,164 12,417 14,935 10,456 12,046 12,021 14,598 F 2,602 2,778 3,124 1,424 412 1,164 2,939 1,295 1,831 2,917 2,017 2,560 795 G 3,992 4,678 4,816 2,340 4,028 2,002 5,263 4,535 3,637 1,813 3,254 3,234 2,856 H 2,828 1,758 2,363 450 1,934 2,320 859 0 1,644 686 812 1,180 634

26,589 35,403 36,447 23,092 28,876 24,690 33,981 22,223 36,929 20,662 22,361 22,920 23,514

Total to collect 357,687

3.2 Design of the route

Tables A.2 show the distances between each site and collection depot for each of the eight tours. To determine the routes that transport units will make assuming that demand is available prior to the allocation of collection, the TSP algorithm is using.

Finally, the collection is done in reverse, i.e. the truck starts at the most distant store until arriving at the depot.

Tables A.3 show the routes taken during the thirteen weeks analyzed for each of the seven constructed tours. They also show the distance per week and the total. How to interpret tables A.4 is very easy, each row shows the week and a sequence of numbers shows the route, which begins in the depot and ends in the same denoted by D. Tour E is very easy and shows separately their optimal tours.

Tour E: In weeks 1-5, 7 and 9-13 the tour is from Queretaro depot then to San Juan del Rio depot for a total of 346 km. For weeks 6 and 8, the tour is Queretaro depot and return with a distance of 346 km also. The total in thirteen weeks is 4.498 Km.

IV. DISCUSSIONS

Collection with this model requires 57 days in total to collect 357,687 pallets, the amount collected by carriers engaged in 78 days, working 6 days a week for 13 weeks. Then, there is a saving of 21 days, with respect to the number of days employed by external carriers. This proposal would have to initially invest in the purchase of trucks, but making an economic study, after several years it is more profitable to invest in the purchase of fourteen trucks.

a. Costs with external transport: $675,943.93usd

The approximate annual cost of collections with external transport during the year is calculated by extrapolating the three months that were taken as the study period, resulting in:

 Average number of pallet collection : 27,514  Average amount per day of collection: 4,586  Approximate cost per pallet collection: $0.47 US  Annual cost of collection is $675,943.93 US b.Costs with the proposed improvements: $1,017,952.75

USD

The act of acquiring transport units involves a very strong initial outlay for the company. However we must take into account that the company must acquire them without losing capital.

Breakdown of approximate annual costs:

1.- Purchase of trucks: $584,251.96usd

Type of truck Collection

capacity amount

Cost Approx.

per unit (US) Total cost Truck 3 1/2 72 1 $20,472.44 $20,472.44 Platform 40 feet 432 2 $31,496.06 $62,992.12 platform 48 feet 504 2 $37,795.27 $75,590.55 Platform 53 feet 540 9 $47,244.09 $425,196.85

(6)

International Journal of Emerging Technology and Advanced Engineering

18

2.- Salaries: $173,858.26usd

Post Staff Monthly

Salary Total Monthly _Salary

Annual Salary drivers 14 $866.14 $12,125.98 $145,511.81 supervisors 2 $1,181.10 $2,362.20 $28,346.45

$173,858.26

3.- Annual fixed costs: $259.84 usd

Parts and Maintenance: $55,118.11 usd. Booths: $47,244.09 usd.

Communication: $7,874.01 usd.

Insurance and Holdings: $39,370.07 usd. Fuel: $47,244.09 usd.

Other costs: $62,992.12 usd.

Without the purchase of trucks the cost would be $433,700.78usd representing a saving of $242,243.15 usd.

The purchase of trucks shows that the proposed improvement would be spending $342,008.82usd more than that spent on external transport. To actually measure the economic impact that the current work scheme is costing the company an economic assessment of 5 years must be made.

c. Economic evaluation

To compare the money adequately Present Value (PV) is used. To evaluate both options a minimum acceptable rate of 30% return annually will be handled.

With external transport:

if i30%; C_T  $675,905.51 usd; n5years

58 . 308 , 646 , 1 ) 30 . 1 ( 51 . 951 , 675 ) 30 . 1 ( 51 . 951 , 675 ) 30 . 1 ( 51 . 951 , 675 ) 30 . 1 ( 51 . 951 , 675 ) 30 . 1 ( 51 . 951 , 675 5 4 3 2 1       PV

Proposal for improvement:

5 5 2 4 2 3 2 2 2 1 2 1 ) 1 ( . . ) 1 ( ) 1 ( ) 1 ( ) . 1 ( ) 1 ( i V S i C i C i C i C i C C PV             

Where i 30%; C₁$584,251.96 usd ; C_T  $1,017,952.75 usd, n5years.

Other costs = $433,700.78 usd Depreciation = 25% annual

Depreciation recovery = $584,251.96 usd. times 0.25= $146,063 usd.

 2

C other costs -depreciation recovery = $287,637.8usd. 

. .V

S Salvage value of vehicles = C₁50%C₁

= $292,125.98 usd (the vehicles to be sold at the end of fifth year) 85 . 135 , 206 , 1 ) 30 . 1 ( 98 . 292125 ) 30 . 1 ( 80 . 287637 ) 30 . 1 ( 80 . 287637 ) 30 . 1 ( 80 . 287637 ) 30 . 1 ( 80 . 287637 ) 30 . 1 ( 80 . 287637 96 . 584251 5 5 4 3 2 1         PV

The option to be selected is the least cost incurred in present value. Therefore the best choice of costs is the proposed improvement.

Finally, in 5 years the approximate cost for collection in the Metropolitan area, assuming a 6% annual work scheme, the approximate cost would be $3,810,358.82usd in the same period. So the proposal would result in a 5 years saving of $781,295.2 usd.

V. CONCLUSIONS AND FUTURE WORK

The proposed model allows in the first instance, establishing the optimum number of trucks needed in terms of the existing collection volumes. To this was considered the collection volume of 13 weeks (3 months). Based on this, it was determined that the fleet should consist of: one 3 ½ pickup, two trucks of 40 feet, two trucks of 48 feet and nine trucks of 53 feet.

The results showed that while foreign carriers took 78 days to collect 357.687 pallets and supply them to MX16 depot located in the Metropolitan area, that same volume fleet consisting of 14 trucks would take 57 days. The collection area to cater to this depot included shops and/or distribution centers located in 46 cities, municipalities and/or delegations from 13 central, south and west Federal states.

To achieve such a time reduction the collection area was segmented into 8 tours, where transport units had to make the tours every week. To design the routes it was necessary to plan them based on the Traveling Salesman Algorithm, making them always represent the shortest path.

(7)

International Journal of Emerging Technology and Advanced Engineering

19

ACKNOWLEDGMENTS

The authors thank the company Voltran S.A. of C.V. for

their support in obtaining the information used in development work, and we thank the anonymous referees whose comments led to an improved presentation of our work.

REFERENCES

[1] Baldacci, R., Hadjiconstantinou, E., & Mingozzi, A. (2003). An exact algorithm for the traveling salesman problem with deliveries and collections. Networks, 42(1), 26-41.

[2] Bonomo, F., Duran, G., & Larumbe, F. (2012). A method for optimizing waste collection using mathematical programming: a Buenos Aires case study. Waste Management & Research, 30(3), 311-324.

[3] Bountox, B., Artiques, C., & Feillet, D. (2010). A Memetic Algorithm with a large neighborhood crossover operator for the Generalized Traveling Salesman Problem,. Computers & Operations Research, 37(11 Special Issue: SI), 1844-1852. [4] Butler, M., Williams, H., & Yarrow, L. (1997). The two-period

travelling salesman problem applied to milk collection in Ireland. Computational Optimization and Applications, 7(3), 291-306. [5] Chen, S.-M., & Chien, C.-Y. (2011). Parallelized genetic ant colony

systems for solving the traveling salesman problem. Expert Systems with Applications, 38(4), 3873-3883.

[6] Erdogan, G., Battarra, M., & Laporte, G. (2012). Metaheuristics for the traveling salesman problem with pickups, deliveries and handling costs,. Computers & Operations Research, 39(5), 1074-1086.

[7] Feillet, D., Dejax, P., & Genderau, M. (2005). Traveling salesman problems with profits. Transportation Science, 39(2), 188-205. [8] Glomvik, R. J., Christiansen, M., & Fagerholt, K. (2012). The

Traveling Salesman Problem with Draft Limits,. Computers & Operations Research, 39(9), 2161-2167.

[9] Gutiérrez G.E., Cordoba L.V.M & Panteleeva O.V. (2012). A method for optimizing the collection of pallets using the Traveling Salesman Problem, Advances and Applications in Discrete Mathematics, pp. (0)

[10] Karapetyan, D., & Gutin, G. (2011). Lin-Kernighan heuristic adaptations for the generalized traveling salesman problem,. European journal of Operation Research, 208(3), 221-232. [11] Karapetyan, D., & Gutin, G. (2012). Efficient local search

algorithms for known and new neighborhoods for the generalized traveling salesman problem,. European journal of Operation Research, 219(2), 234-251.

[12] Laporte, G., AsefVaziri, A., & Sriskandarajah, C. (1996). Some applications of the generalized travelling salesman problem,. Journal of The Operational Research Society, 47(12), 1461-1467. [13] M, B., HP, W., & LA., Y. (1997). The two-period travelling

salesman problem applied to milk collection in Ireland,. Computational Optimization and Applications, 7(3), 291-306. [14] Mladenovic, N., Urosevic, D., & Hanafi, S. (2012). A general

variable neighborhood search for the one-commodity pickup-and-delivery travelling salesman problem,. European Journal of Operational Research, 220(1), 270-285.

[15] Rakke, J. G., Christiansen, M., & Fagerholt, K. (2012). The Traveling Salesman Problem with Draft Limits. Computers & Operations Research, 39(9), 2161-2167.

[16] Rego, C., Gamboa, D., & Glover, F. (2011). Traveling salesman problem heuristics: Leading methods, implementations and latest advances,. European journal of Operation Research, 211(3), 427-441.

[17] Rodriguez, A., & Ruiz, R. (2012). The effect of the asymmetry of road transportation networks on the traveling salesman problem. Computers & Operations Research, 39(7), 1566-1576.

[18] Shyi-Ming, C., & Chih-Yao, C. (2011). Parallelized genetic ant colony systems for solving the traveling salesman problem,. Expert Systems with Applications, 38(4), 3873-3883.

[19] Shyi-Ming, C., & Chih-Yao, C. (2011). Parallelized genetic ant colony systems for solving the traveling salesman problem,. Expert Systems with Applications, 38(4), 3873-3883.

(8)

International Journal of Emerging Technology and Advanced Engineering

[image:8.612.66.548.173.695.2]

20

VI. APPENDIX

Table A.1

Number of trips required for collection by tour for each of the 4 trucks types.

Week Type of

truck

Tour 1 2 3 4 5 6 7 8 9 10 11 12 13

A

2 2 1 1 2 2 2 1

3 2 4 2 3

1 3 1 1 4

3 1 1 1 1 4 1 2 1 1 5

B

1 1

1 4 4 2 2 2 1 4 4 1 3

3 1 5 4 1 4

6 1 1 3 5 1 1 4 3 5

C 3 5 5 3 4 4 3 1 4 5 5 3 1

5 5 6 3 6 3 5 1 7 1 3

D 4 3 2 1 4 5 3 4 1 4 2 5 1

7 9 6 9 8 7 9 1 12 2 1 1 2 3

E

1 1 1 1

3 3 3 1 3

2 5 1 1 4 7 4

13 25 27 16 23 21 28 23 24 17 20 15 27 5

F

2 1

4 1 1 2 3 2 2 2 3

1 2 1 1 2 1 3 3 4

4 2 4 1 2 3 3 1 2 5

G

1 1

2 3 2 1 3 3 3 3

3 4 1 1 4 2 4

3 5 9 1 5 9 6 5 1 6 6 3 5

H

1 3 2 2 1

3 3 2 1 3 2 2 3

1 1 2 1 2 1 1 4

2 1 1 2 2 1 2 5

Totals 60 81 79 51 66 61 72 50 78 53 51 57 53

Savings in

(9)

International Journal of Emerging Technology and Advanced Engineering

21

Tables A.2

Distances in kilometers between the depot and each tour location.

Tour A Depot AN AI AP AS ATH ATZ AO

Depot 64 66 149 111 263 296 482

AN 64 22 110 72 224 257 443

AI 66 22 94 56 208 241 427

AP 149 110 94 38 114 147 333

AS 111 72 56 38 152 185 371

ATH 263 224 208 114 152 261 283

ATZ 296 257 241 147 185 261 480

AO 482 443 427 333 371 283 480

Tour B Depot BM BT BZ BMO BU BG BTL BTO BZA

Depot 80 78 88 329 438 565 552 569 570

BM 80 11 21 249 358 498 485 502 503

BT 78 11 10 238 347 487 474 491 492

BZ 88 21 10 248 357 497 484 501 502

BMO 329 249 238 248 109 276 263 280 281

BU 438 358 347 357 109 303 290 299 308

BG 565 498 487 497 276 303 13 4 5

BTL 552 485 474 484 263 290 13 7 18

BTO 569 502 491 501 280 299 4 7 9

BZA 570 503 492 502 281 308 5 18 9

Tour C Depot CA CC CE CH CT CTL CN

Depot 20 13 22 42 30 16 32

CA 20 13 26 33 41 7 17

CC 13 13 30 38 40 20 27

CE 22 26 30 50 15 23 38

CH 42 33 38 50 65 28 13

CT 30 41 40 15 65 38 53

CTL 16 7 20 23 28 38 16

CN 32 17 27 38 13 53 16

Tour D Depot DA DB DC DG DM DT DV Tour E Dep EQ ES

Depot 43 36 45 31 29 56 40 Depot 173 131

DA 43 14 20 18 16 31 27 EQ 173 42

DB 36 14 12 13 16 22 18 ES 131 42

DC 45 20 12 26 21 15 19

DG 31 18 13 26 16 31 11

DM 29 16 16 21 16 30 16

DT 56 31 22 15 31 30 40

DV 40 27 18 19 11 16 40

Tour F Depot FS FC FF FZ Tour G Depot GC GI GL GA

Depot 418 461 654 594 Depot 236 283 353 501

FS 418 297 278 218 GC 236 49 119 267

FC 461 297 547 489 GI 283 49 70 218

FF 654 278 547 60 GL 353 119 70 148

FZ 594 218 489 60 GA 501 267 218 148

Tour H Depot HC HJ HA HI HZ

Depot 127 138 413 230 655

HC 127 11 286 103 528

HJ 138 11 297 114 539

HA 413 286 297 183 242

HI 230 103 114 183 436

(10)

International Journal of Emerging Technology and Advanced Engineering

22

Tables A.3.

Seven routes for tours during the 13 weeks analyzed, with the distance traveled.

A AO ATZ ATH AP AS AI AN D Totals BZA BTO BTL BG BU BMO BZ BT BM D Totals

1 1 2 3 4 5 820 1 2 4 3 6 7 5 8 1190

2 1 2 3 4 5 617 1 3 2 4 6 7 5 8 1209

3 1 2 3 4 5 6 557 1 3 2 5 6 4 7 1182

4 1 2 3 4 5 617 1 3 2 5 6 4 7 1182

5 1 2 3 4 5 551 1 3 2 4 667

6 1 2 3 4 5 551 1 2 3 176

7 1 2 3 4 5 6 584 2 1 4 5 3 6 1172

8 1 2 3 4 1258 1 2 3 4 1167

9 1 2 3 4 526 1 2 3 4 5 1190

10 1 2 3 4 5 617 1 2 4 5 3 6 1173

11 1 2 3 4 526 1 2 4 5 3 6 1173

12 1 2 3 323 1 2 3 5 4 6 1325

13 1 2 3 323 1 2 3 4 665

Totalkm traveledtour A 7870 Total km traveled tour B 13471

C CN CTL CT CH CE CC CA D Total DV DT DM DG DC DB DA D Total

1 3 4 2 1 6 5 7 134 1 4 7 2 5 6 3 8 172

2 4 5 1 3 2 6 7 157 3 1 6 4 2 5 7 164

3 4 5 1 3 2 7 6 8 157 3 6 1 4 5 2 7 152

4 4 5 1 3 2 6 7 157 3 1 6 4 2 5 7 164

5 3 4 2 1 5 6 84 3 1 5 4 2 6 188

6 4 5 1 3 2 7 6 8 157 3 1 6 4 2 5 7 164

7 3 5 2 1 4 6 125 2 1 5 4 3 6 182

8 2 1 3 43 1 2 86

9 4 6 1 3 2 5 7 148 3 1 5 2 4 6 164

10 1 2 3 68 1 2 5 3 4 6 152

11 2 1 3 137 3 1 2 4 130

12 1 3 2 4 5 148 1 4 2 3 5 109

13 4 1 3 2 5 139 1 4 2 3 5 116

Totalkm traveledtour C 1654 Totalkm traveledtour D 1943

F FZ FF FC FS D Total GA GL GI GC D Total HZ HI HA HJ HC D Total

1 3 2 1 4 5 1704 1 2 3 4 5 1004 1 3 2 4 5 1310

2 2 1 3 4 1350 1 2 3 4 5 1004 1 3 2 4 5 6 1332

3 1 2 836 1 2 3 4 5 1004 1 3 2 4 5 6 1332

4 2 1 3 4 1350 1 2 3 4 1004 1 2 3 826

5 1 2 3 1176 1 2 3 4 708 1 2 3 4 5 1343

6 1 2 3 1176 1 2 3 4 5 1004 1 3 2 4 5 6 1332

7 1 2 3 1230 1 2 3 4 5 1004 1 2 3 4 482

8 2 1 3 4 1586 1 2 3 4 5 1004 0

9 1 2 836 1 2 3 4 5 1004 2 1 3 4 5 848

10 2 1 3 4 1350 1 2 3 706 2 1 3 826

11 2 1 3 4 1586 1 2 3 706 1 2 3 4 482

12 1 2 3 1230 1 2 3 4 1002 1 3 2 4 5 1332

13 2 1 3 4 1586 1 2 3 706 1 2 3 460