OPTIMIZED ALLOCATION OF EQUIPMENT FOR EARTHWORK PROJECTS ACCORDING TO COST AND TIME CRITERIA
Abbas Rashidi 1, Hossein Rajaie 2, Ayoub Hazrati 3
1 Lecturer, Civil Engineering Department, Islamic Azad University, Semnan Branch, Iran E-mail: [email protected]
2Assistant Professor, Civil Engineering Department, AmirKabir University of Technology, Tehran, Iran
3 Graduate Student, Civil Engineering Department, AmirKabir University of Technology, Tehran, Iran
Abstract. In earthwork projects, especially in developing countries, major amounts of costs are spent on equipment. Therefore in such projects, the procedures in which proper equipment are allocated are of great importance to contractors. Furthermore, using mathematical methods in optimum allocation of equipment can cause cost decrease. At the same time the mathematical methods can play the role of a communal basis for contractors and owners. In this paper using a relatively large scale earthwork project as a case study, considering the project time schedule, equipment allocation will be done. The Queue theory will be used as the controlling tool for checking the accuracy of number of allocated equipment. Finally, the effect of any changes in number of equipment on the project duration will be assessed and the project sensitivity to any type of equipment will be studied. Comparison between calculation results and field observations and actual records shows a good compatibility between calculations and actual project conditions.
Key Words: Transportation, Working cycle, Equipment, Queue Theory, Optimized Allocation
1 INTRODUCTION The history of Queue theory goes back to 1909. In this year A. K.Erlang, a Danish engineer, issued a paper on the Queue systems and waiting time in telecommunication systems, for the first time. In 1953 David G. Kendall formulated the Queue theory to the form known today and showed the empirical applications of this theory for different problems. The first book discussing the basis and applications of Queue theory issued in 1958 by Philip M. Morse. After that, the Queue theory has been used extensively in solving different problems related to giving better service to telecommunication customers, transportation network users and hospital clients and production procedure in factories and technological complexes. The main components of Queue system are interiors (customers) and service suppliers. When a customer refers to a system for receiving service, two different cases may happen. If one of the service suppliers is free, then giving the service to the customer begins immediately. On the other hand if all service suppliers are busy, then the customers should wait and thus the queue will be made. Figure 1 shows the main components of Queue system 1. The objective of solving the Queue system problem is twofold: finding the time percentage that the total system is free due to the absence of customers and waiting time of any customer in the queue and simultaneously optimizing these times. In most of the Queue systems, the Abbas Rashidi , Hossein Rajaie and Ayoub Hazrati
mentioned times are accidental in nature. For this reason the probability theory is used extensively in solving Queue problems.
Service Canals
Queue Customer's Entry Customers Exit
Figure 1: Components of Queue system
Depending on the choice of statistical process for the mode of customer entering into the system and customer exit rate, several models has been proposed for simulating the Queue problem. The most common of which is Classic or model. In this model it is assumed that the time interval between entering of two subsequent customers and the time of providing service for them is following exponential functions. Service providers' number for each separate system is equal to 1, the number of customers is potential and the queue capacity is limitless. The entering process of customers is similar to Poison's model. The entering rate of customers is assumed to be independent of the number of customers in the system 2. In the truck filling and refilling problem, the trucks are assumed as the customers of Queue system. Loaders are known as service providers in this system. One loader along with specified number of trucks is known as a Queue system. The objective of solving this problem is to determine the number of loaders and trucks in a manner to increase the probability of truck existence for one loader as much as possible. In other words, by allocating appropriate number of trucks the loader will be always busy. Since the duration required for each loading & traveling operation is independent and is not related to the previous or next loading & traveling operations, the Poisons statistical distribution is used and the probability of existence of at least one truck for each loader, using Queue theory can be determined with following equations 3:
−1 n n! i (1) P0 = ∑ (r) i=0 (n − i)!
Pt = 1− P0 (2) t r = 1 (3) Tt n : Number of trucks for each loader
P:0 The probability of non-existence of at least one truck for one loader in a specified time interval
Pt : The probability of existence of at least one truck for one loader in a specified time interval r : Loading duration to travel duration t1 : required Time for truck loading by loader,
Tt : Travel duration
Abbas Rashidi , Hossein Rajaie and Ayoub Hazrati
2 PROJECT DESCRIPTION
The under study project includes excavation, compaction and grading of a relatively large site, located in the Southern Pars Especial Economical Zone in the south of Iran. Figure 2 shows the site plan and the coordinates of some corner points (The north direction is not exact). The site area is about 23500 m2. On the western side of the site (left side of 6-1 axis) and eastern side of the site (right side of 3-2 axis) neighbor sites are located. The suggested way for transferring soil is using 5-4-3 axis which has good accessibility to the main road and lower elevation than the northern axis (1-2 axis). D-8 bulldozers have the role of earth excavation and soil transformation to the southern side of the site, where the loaders fill the trucks. If the high number of machineries in the southern side decreases the efficiency and cause wasting of time, the northern side can alternatively be used for soil loading and transformation. According to topographical maps and soil specification of site (20% composed of rock and 80% of hard compacted soil) the excavation volume is equal to 205500 m3 loose soils. The area considered for soil depot is located in 4 to 5 kilometers distance from the main site. Considering the site broadness, after finishing the excavation, the final surface should be graded by graders and compacted by rollers. Considering the project time schedule, total time considered for the excavation and grading is 60 working days. Daily working hours are 12 hours which has 10 efficient working hours.
Figure 2: Site plan and corner point coordinates
3 GRADER AND ROLLER SELECTION
Usually in earthwork projects the following operations are done: - Soil excavation by different type of bulldozers with various engine powers related to soil type - Soil loading with different type of loaders - Transformation of loaded soil and unloading in depot by truck - Grading of final surface by grader, Watering and compaction by different type of rollers 4. As the duration of grading and compaction operations' are related to the surface of the site and as the time of this operations have no relationship with the volume of excavation, firstly, time and machinery required for these operations will be determined.
Abbas Rashidi , Hossein Rajaie and Ayoub Hazrati
If a G-14 model grader with 1200 m2 per hour efficiency or a similar model is used, the total area of the site will be graded in two working days (20 efficient working hours). Two different methods have been used in calculation of work performance of different equipment: - Work performance calculation based on machinery catalogs while considering project conditions 5. - Data collection from 5 different earth moving contractors which execute projects in the same area and calculating the average work performance of equipment. If a significant difference exists between two mentioned methods' calculations, the lower value will be used in calculations. Considering the allocation of two 12-14 tons static rollers with the average efficiency of 420 m2 per hour, the total area of site will be compacted in 3 working days. To provide sufficient humidity for the compaction operations a water tank is used during the operation period . Considering the above calculations, 5 days of the total project duration is allocated for grading and compaction operations and 55 days will remain for excavation operation.
4 NUMBER OF BULLDOZERS CALCULATION Considering that 55 days are allocated for excavation and the amount of loose soil is equal to 205500 m3 and at the same time considering the average work performance of 130 m3 loose soil for D-8 type bulldozer, the number of required bulldozer for the mentioned project can be determined as follows [5]: 205500 Number of required bulldozer: = 2.87 55×10 ×130 Therefore 3 bulldozers are needed.
5 COMPLETE WORKING CYCLE A complete working cycle includes following parts 6: - Time required for loading a truck by loader (t1) - Time required for truck traveling to depot (t2) - Time required for truck unloading (t3) - Time required for returning of truck (t4) - Time required for truck to get to appropriate direction to be loaded by loader (t5) If the travel duration is shown by Tt and complete working cycle duration is shown by TC then:
Tt = t2 + t3 + t4 + t5 (4)
Tc = t1 + t2 + t3 + t4 + t5 (5)
Among the above mentioned times, duration of traveling to and retuning from depot (t2 and t4) are related to truck speed and traveling distance and are unequal for different projects. But, the loading duration (t1), unloading (t3) and getting the truck to the right direction (t5) are discrete from the nature of project and are calculated depending on the type of equipment used in the project. In this project, using the field observations and pervious recorded data, the total
Abbas Rashidi , Hossein Rajaie and Ayoub Hazrati
duration of unloading and getting the truck to the right direction (t3+t5) is considered equal to 2 minutes In determining the truck speed, the road situation, the road's traffic level and number of lines of roads should be considered. In this project, considering that some other projects in neighborhood are under construction, the main road has a high traffic. Due to recorded data the average speed in the forward path is about 15-25 km/h and in the backward path is about 25-45 km/h 7. Table 1 shows the calculation of travel duration for different travel distances and different truck speeds in both forward and backward paths. During the estimation of the required time for loading the truck by loader, it should be noted that according to economical considerations, the truck capacity is better to be an integer multiple of loader filler capacity 8.In this project the loader bucket capacity was considered about 2.5 m3 and consequently each truck with 8 m3 capacity can be filled in 3 times of loading. According to recorded field data in this project the time required for complete loading of a truck is about 5 minutes. This time has been used as the basis of the calculations. With regard to the fact that the traveling distance is roughly 4-5 km, the average travel duration is obtained around 24.05 minutes form Table 1. Consequently, the duration equal to 25 minutes is assumed for the basis of calculations. (Tt = 25). Considering 5 minutes for loading, the complete working cycle obtains equal to 30 minutes. (TC = 30)
Travel Departure and Duration of unloading Average Travel Total velocity in returning and getting to appropriate travel velocity in duration backward duration direction distance forward path Tt path ( t 2 + t4 ) ( t3 + t5 ) Km Km/hour Km/hour Minute Minute Minute 25 25.6 2.0 27.6 15 30 24.0 2.0 26.0 35 22.9 2.0 24.9 30 20.0 2.0 22.0 4 20 35 18.9 2.0 20.9 40 18.0 2.0 20.0 35 16.5 2.0 18.5 25 40 15.6 2.0 17.6 45 14.9 2.0 16.9 25 32.0 2.0 34.0 15 30 30.0 2.0 32.0 35 28.6 2.0 30.6 30 25.0 2.0 27.0 5 20 35 23.6 2.0 25.6 40 22.5 2.0 24.5 35 20.6 2.0 22.6 25 40 19.5 2.0 21.5 45 18.7 2.0 20.7 Average Value 24.05
Table 1: Duration time changes for departure and returning of trucks for different travel distances and velocities
Abbas Rashidi , Hossein Rajaie and Ayoub Hazrati
6 OPTIMUM SELECTION OF LOADERS AND TRUCKS NUMBER If the capacity of a truck assumed to be 8 m3 , considering different truck loading durations by loaders (4 to 6 minutes) and the duration of a complete working cycle in minutes, the number of required trucks and loaders can be determined according to Table 2.
Duration of complete loading by loaders A 4 5 6 (min) Loading numbers in B=60/A 15 12 10 1 hour Loaded volume in 1 C=8× B 120 96 80 working hour Loaded volume in 1 D=C× 10 1200 960 800 working day Number of required loaders according to E=(205500/55)/D 3.11 3.89 4.67 loose soil volume a complete working F 20 25 30 35 20 25 30 35 20 25 30 35 cycle duration (min)
Total required trucks G=F*E/A 16 20 24 28 16 20 24 28 16 20 24 28
Table 2: Number of required loaders and trucks
If the duration of complete loading equals to 5 minutes then complete working cycle duration equals to 30 minutes, and the number of required loaders is 4 and number of required trucks is 24.
7 EXISTENCE PROBABILITY OF AT LEAST ONE TRUCK FOR EACH LOADER To control the sufficiency of number of allocated trucks for each loader, the Queue theory can be used as an efficient solution. The calculations of probability of presence of at least one truck for each loader have been done by equations 1 to 3 and are shown in Table 3. It should be noted that the calculations are based on the number of trucks and loaders mentioned in Table 2 and the ratio of number of trucks to one loader (n) has been rounded to the nearest integer number. Table 3 shows that the minimum probability of the existence of at least one truck in the loading operation for one loader, The loading duration equal to 5 minutes and a complete working cycle equal to 30 minutes is 0.81, which could be acceptable from project managers point of view. If this probability is considered to be low, then the number of trucks would have been required to increase accordingly. Figure 3 shows the probability of the existence of at least one truck in the case of having 4 loaders for various numbers of trucks and different r values.
Abbas Rashidi , Hossein Rajaie and Ayoub Hazrati
duration of complete loading by loaders 4 5 6 t (min) 1 a complete working T 20 25 30 35 20 25 30 35 20 25 30 35 cycle duration (min) c Travel duration Tt 16 21 26 31 15 20 25 30 14 19 24 29 (min) Number of trucks to n 5 6 8 9 4 5 6 7 3 4 5 6 each loader
r factor r 0.25 0.19 0.15 0.13 0.33 0.25 0.20 0.17 0.43 0.32 0.25 0.21 probability of existence of at least P 0.80 0.79 0.84 0.84 0.79 0.80 0.81 0.82 0.74 0.78 0.80 0.83 1 truck in loading t operation
Table 3: Calculation of probability of existence of at least one truck for each loader in loading operation
r=0.15 r=0.2 r=0.25 r=0.3
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10 Existence probability of at least one truck 0.00 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 Truck numbers
Figure 3: probability of existence of at least one truck for various numbers of trucks and different r values
Figure 3 shows that as the curve slope decreases at high percentages (bigger than 90%), the probability of presence of at least one truck for one loader increases, which is not economical as it requires high increase in the number of trucks. Table 4 shows the final equipment allocation to this project.
Abbas Rashidi , Hossein Rajaie and Ayoub Hazrati
Work Number of Number of Total work Type of equipment performance Unit required working performance Per hour machines days D-8 Bulldozer 130 m3/hour 3 55 214500 L-120 loader 96 m3/hour 4 55 211200 8 m3 truck 16 m3/hour 26 55 - G-14 grader 1200 m3/hour 1 2 24000 12-14 ton static roller 420 m3/hour 2 3 25200
Table 4: Final heavy equipment allocation to project
8 EQUIPMENT DECREASE EFFECT ON THE PROJECT DURATION
If by any reason, the equipment allocation process encounters to any problems, the calculated duration changes. Figures 4 through 6 show the equipment decrease effect on the project duration.
6 Number of D-8 bulldozers
5
4
3
2
Number of machinary of Number 1
0 30 35 40 45 50 55 60 65 70 75 80 85 Total excavation duration (day)
Figure 4: Project Duration changes against number of bulldozers changes
9 3 minutes loading duration 8 4 minutes loading duration
7 5 minutes loading duration
6 6 minutes loading duration
5
4
3
Number of macinary of Number 2
1
0 30 35 40 45 50 55 60 65 70 75 80 85 Total excavation duration (day)
Figure 5: Project Duration changes against number of loaders changes
Abbas Rashidi , Hossein Rajaie and Ayoub Hazrati
60 20 minutes complete cycle duration
50 25 minutes complete cycle duration 30 minutes complete cycle duration 40 35 minutes complete cycle duration
30
20 Number of machinary of Number 10
0 30 35 40 45 50 55 60 65 70 75 80 85 Total Excavation duration (day)
Figure 6: Project Duration changes against number of trucks changes
Some conclusions can be made from the above charts which are as follows: - If any problem occurs and number of bulldozers decreases from 3 to 2, the operation duration will increase from 55 days to 80 days. - If the loader number decreases from 4 to 3, the excavation operation duration will increase from 55 days to 70 days. - The truck decrease by 1 number, does not affect project duration significantly. The above conclusions show that the project duration is sensitive to number of bulldozers, loaders and trucks respectively form the highest effect to the lowest effect.
9 CONCLUSIONS & RECOMMENDATIONS - Considering the similarity between operations in most earthwork projects, the equipment allocation process can be presented as a step by step procedure. This procedure can be used in different projects 9, 10. - Considering several uncertainties like, exact transportation distance, loading duration, truck speed in both ways, modification factors on efficiency, compatibility between bulldozers, loader and trucks systems and … the above calculations should be reassessed after operation commencement and any essential changes should be made on the assumptions considering field recorded data 11. - During loading operation, special cares should be taken in order that trucks loaded near to their nominal capacity (8 m3 in this project). Extra loading may cause technical problems for trucks or be fined by police and minor loading leads to economical disadvantages. During project operation the roads should be in good conditions. Bad road conditions cause that the trucks could not transport their loads with optimum speed and capacity 12. - The assumed working period in every day (10 working hours in a day) is obtained for normal conditions during operation. With any changes in the assumed duration (like lunch time, leisure time, spares working hours and …) new calculated efficient working hours should be used in calculations. In order to omit any unwanted delay in earthwork operations due to technical problems in equipment, stand by equipment should be considered for the project.
Abbas Rashidi , Hossein Rajaie and Ayoub Hazrati
- The calculations are based on this assumption that all loaders have equal efficiency and capacity and all trucks are with equal capacity. Certainly, using trucks with several levels of capacities produce some errors in calculations 13. - The basis of calculations in machinery allocation in every earth moving project is soil volume in the project. Consequently, more attention in soil stratification detection and soil volume determination in loose state, make the calculations and real state conditions, more close to each other. - Machinery allocation in this project has been done with these assumptions that the operation works has a specified duration and at the same time, there is no limitation in equipment preparation for the project. Certainly, any limitation in equipment supplement and its effect on the project duration should be considered in calculations.
10 REFERENCES
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