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COMPARATIVE STUDY OF TIME COST OPTIMIZATION- A CASE STUDY
Ritesh D.Hallikerimath
1, V.Srinivasa Raghavan
21Student,Construction Engineering & Management, Manipal Institute of Technology, Manipal, (India)
2Associate Professor, Civil Department, Manipal Institute of Technology, Manipal, (India)
ABSTRACT
Construction Planning is the most important phase of the project. Planning regarding crew size, materials, equipments etc., should be done in the planning phase. Construction planners use Critical path Method & PERT analysis to reduce the duration of the project. As the duration of the project reduces, the cost of the project increases i.e. there is a Trade-off between Time & Cost. The Study shows comparative results of the two techniques used to achieve Time-Cost Optimization. The two techniques used were Critical Path Method- Project Crashing Method & Linear Programming Method (LP). To compare the results a case study was considered i.e. construction of a Water Treatment Plant. Hence this paper presents: 1) Application of Project Crashing Method & 2) Application of Linear Programming (LP).
Keywords: CPM, Linear Programming Method, Project Crashing Method, Time-Cost Trade-off.
I. INTRODUCTION
Project Management has proved to be an important tool for construction industry. Project Planning, Project execution, Project control etc. have been successfully used in the construction of the Project. Various methods &
Algorithms used to achieve the desired outcome have proved to be very useful. Among this Project Management Methods Time-Cost Trade-off has been one of the most important methods of analysis. Contractor may choose different crew size, equipments & construction methods to complete the activities of the project. Hence the above decision made by the contractor will ultimately decide the time & cost of the project. The main objective of the Time-cost trade-off analysis is to find the shortest duration & minimum cost required to complete the activities of the Project [1]. The total cost function of a project has two components: direct and indirect costs.
Direct costs are incurred because of the performance of project activities, while indirect costs include those items that are not directly related to individual project activities and thus can be assessed for the entire project.
In general, indirect cost increases almost linearly with the increase of project duration and usually assumed as a percentage of project direct cost [2]. If time is of secondary concern, each activity could be performed at its lowest possible normal cost. If cost is of secondary importance, each activity could be speeded up to be completed in the least crash time. Between these two limits lies the best or optimal solution. However, to find such solution, it requires the consideration of complex collection of concurrent, interrelated, and overlapping activities [2].
91 | P a g e 1.1 Existing Techniques
In the past years 2 types of techniques were developed. The 2 types of techniques were heuristic methods &
Mathematical programming models.
Heuristic Methods: These methods are based on basic thumb rules & have no mathematical rigor. They provide good solutions but achieving optimality is difficult. Hence they cannot be relied. They consider the relationship between time-cost is linear within the activities. They also do not a provide range of possible solutions which makes it difficult to adapt to the scenarios[3].
Mathematical programming models: Mathematical programming methods convert the TCTP to mathematical models and utilize linear programming, integer programming, or dynamic programming to solve them[3]. . Kelly (1961) formulated time-cost optimization by assuming linear time-cost relationships within activities. Other approaches such as those by Hendrickson & Au (1989) & Pagoni (1990) also used linear programming as the tool to solve the Time cost trade-off. The main disadvantage of mathematical Model is they require great computational efforts. In case of large scale projects (100 or more activities) these methods are very tedious to perform[3].
Judging from the state of research done no case study was taken into consideration. In this paper, a case study was taken into consideration. The Time-cost trade-off analysis was done on the case study & results were obtained. The Project Crashing Method & Linear Programming method was applied to a on-going live project.
The case study was a construction of the Water Treatment Plant in Belgaum District, Karnataka, India.
II. METHODOLOGY
In this paper, two techniques of Time-cost Trade-off analysis were used. The Work Breakdown Structure of the Project was done. The activities required to complete the Project were finalized. TABLE 1 shows the activities of the Project. The Cost estimates & quantity estimates of the Project were collected. The Normal Duration (Tn)
& Crash Durations (Tc) were calculated for each activity using the quantity estimates. The Normal Cost (Cn) &
Crash Cost (Cc) of the activities was also calculated using the cost estimates. Crash costs were calculated from the cost estimates and added to the extra labor cost & labor overtime cost. TABLE 2 shows the Durations &
Costs for each activity. Hence the initial Data required to perform the analysis was calculated. The time-cost slope of each activity was also calculated. Below are the formulas used to calculate the above durations & cost.
Normal Durations = (Work Quantity * Productivity Rate) (1)
(Peak Man-power * 0.65) Considering 25% overtime,
Crash Durations = (Work Quantity * Productivity Rate) (2)
(Peak Man-power * 0.65 * 1.25)
Cost Slope= (Cc-Cn)/(Tn-Tc) (3)
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Table 1: List of all the ActivitiesActivity Description Activity ID
Design, Drawing & Surveying A
Providing Line out for all the components of the water treatment plant B
Earthwork in Excavation and leveling of all the components C
Providing raft foundation for the Aerator D
Providing isolated footing & column for raw water channel E
Providing raft foundation for the Flash Mixer F
Providing Raft foundation for the Clari-floculator G
Providing Foundation for Filter House and Filter Bed H
Providing Foundation for Wash water Tanks I
Providing External walls for the Aerator J
Providing Shaft & Thrust blocks or cascade to the aerator K
Construction of collection chamber L
Providing RCC slab for raw water channel M
Providing Electrical and mechanical components for Raw water channel and Aerator N
Providing framework and RCC external walls for the Flash Mixer O
Installation of Spun pipes for the Flash Mixer P
Installation of inlet valves & scour valves for the flash Mixer Q
Installation of HP motor & gear box & turbine paddles R
Installation of MS ladder S
Carrying out electrical & Mechanical works T
Providing framework and RCC external & internal baffle walls for the Clari-floculator U
Providing pipes as per the design for the Clari-floculator V
Making provisions for inlet valve & sludge removal equipment W
Carrying out electrical & Mechanical works X
Providing Framework and RCC external walls and internal walls for Filter House and Bed Y Providing plinth beams and slab for the external filter house building Z
Providing Electrical and mechanical components for Filter bed A1
Construction of chlorine house B1
Construction of chemical house C1
Finishing works for the chlorine house and chemical house D1
Finishing works for the filter bed and external building E1
Providing external walls for the wash water tanks F1
Providing inlet pipes and collection chamber for the wash water tank G1
Providing MS ladder and inspection door H1
Providing Hp motor gear box for the wash water tanks I1
Finishing works of wash water tank J1
Earthwork in Excavation and leveling for the external compound walls K1
Providing foundation for the external walls L1
Carrying out masonry works for the external compound walls O1
Finishing works for the compound walls N1
Finishing works for the Aerator & Raw water channel M1
Testing of water treatment plants P1
Commissioning & handing over to the client after applying necessary corrections Q1 Activity ID Normal Cost Normal Duration Crash Cost Crash Duration Cost Slope
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Table 2: Duration & Cost of the activitesINDIRECT COST= Rs. 3000/-
An AON network diagram was created to show the precedence relationship between the activities. Fig.1 shows the network diagram for the activities of the project. The Critical path of the network is represented in the dashed line. The Critical Path duration was found to be 368 days.
1-2 150000 35 150000 35 0
2-3 3000 3 4000 1 500
3-4 2000 2 2800 1 800
3-5 1339 2 2000 1 700
3-6 202 27 400 14 15.38
4-7 8250 11 9000 10 375
5-8 2000 9 3500 8 1500
6-9 1670000 46 1750000 32 5714
7-10 119000 12 150000 10 5167
8-11 22000 8 25000 6 1500
9-12 1036000 59 1200000 39 8200
10-13 64000 18 70000 12 1000
10-15 4800 8 5500 4 175
11-13 15000 6 16500 4 750
12-14 29000 33 32000 21 250
13-16 620000 46 750000 31 8667
14-18 5900 3 6000 2 100
15-17 48000 15 50000 5 200
16-22 3480000 88 3600000 61 4444
17-21 8000 3 8000 3 0
18-19 37500 5 37500 5 0
18-20 10500 27 12000 16 136
19-20 550000 32 570000 26 3333
20-22 1040000 27 1100000 25 30000
21-23 10000 10 10000 10 0
22-23 420000 32 450000 22 3000
22-24 600000 10 600000 10 0
22-25 28000 10 30000 8 1000
23-26 350000 26 380000 18 3750
24-26 48000 22 50000 14 250
24-27 21000 5 21000 5 0
25-28 24500 25 27000 20 500
26-29 630000 20 630000 20 0
27-29 70000 15 70000 15 0
28-29 200000 7 200000 7 0
29-30 201000 8 201000 8 0
29-31 0 10 0 10 0
30-32 0 10 0 10 0
31-32 0 10 0 10 0
94 | P a g e Fig.1: Network Diagram of all the activities
III. RESULTS & DISCUSION
3.1 Project Crashing Method
Project crashing method is one of the analytical method used to perform time-cost trade-off analysis.
The Optimal cost & minimal durations are achieved by crashing of the durations. The optimization is achieved when further crashing of the durations is not possible. A number of iterations have to be performed to achieve optimization. In this paper, the optimization was achieved in 14 iterations.
Initially, the time scale network for the above AON network diagram was calculated. The crashing of the durations was done along the critical path. TABLE. 3 shows the direct cost, indirect cost & total cost of the project. Fig.2 shows the Graph of Time v/s Total cost of the Project. According to the analysis perform
ed, the results are as follows:Minimum Duration of the Project= 331 days Critical Path Duration= 368 Days
Optimum Cost of the Project= Rs. 12574291/- Initial Cost of the Project= Rs. 12650991/-
Table 3: Direct Cost, Indirect Cost & Total Cost
STAGE 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Time 368 355 354 342 341 331 325 317 303 294 293 292 286 283 281
Direct Cost 11546991 11547191 11547291 11550291 11551291 11581291 11601289 11631289 11711285 11785085 11793660 11802660 11857860 11895792 11902680
Indirect Cost 1104000 1065000 1062000 1026000 1023000 993000 975000 951000 909000 882000 879000 876000 858000 849000 843000
Total Cost 12650991 12612191 12609291 12576291 12574291 12574291 12576289 12582289 12620285 12667085 12672660 12678660 12715860 12744792 12745680
29
30
31
32
95 | P a g e Fig.2: Time v/s Cost Graph
3.2 Linear Programming Method
Linear Programming deals with minimizing or maximizing the objective function subjected to some constraints in those situations where the objective function & constraints can be slated in terms of linear expression.
Activities of the project are also known as decision variables such x1, x2…..xn (all are non-negative). Each LP model must have an Objective Function subjected minimization or maximization so that optimal value can be achieved. TABLE .4 shows the results of the LP analysis conducted in LINDO. In this paper, the objective function & constraints are as follows:
Min X0= min x0= -0y1-500y2-800y3-700y4-15.38y5-375y6-1500y7-5714y8- 5167y9-1500y10-8200y11- 1000y12-175y13-750y14-250y15-8667y16-100y17-200y18-4444y19-0y20-0y21-136y22-3333y23-30000y24-
0y25-3000y26-0y27-1000y28-3750y29-0y30-500y31-0y32-0y33-0y34-0y35-0y36-0y37-0y38+3000x32 (4)
Subject to: x1=0,x2-y1>=0,y1=35,x3-x2-y2>=0,y2<=3,y2>=1,x4-x3-y3>=0,y3<=2,y3>=1,x5-x3- y4>=0,y4<=2,y4>=1,x6-x3-y5>=0,y5<=27,y5>=14,x7-x4-y6>=0,y6<=11,y6>=10,x8-x5-
y7>=0,y7<=9,y7>=8,x9-x6-y8>=0,y8<=46,y8>=32,x10-x7-y9>=0,y9<=12,y9>=10,x11-x8-
y10>=0,y10<=8,y10>=6,x12-x9-y11>=0,y11<=59,y11>=39,x13-x10-y12>=0,y12<=18,y12>=12,x15-x10- y13>=0,y13<=8,y13>=4,x13-x11-y14>=0,y14<=6,y14>=4,x14-x12-y15>=0,y15<=33,y15>=21,x16-x13- y16>=0,y16<=46,y16>=31,x18-x14-y17>=0,y17<=3,y17>=2,x17-x15-y18>=0,y18<=15,y18>=5,x22-x16- y19>=0,y19<=88,y19>=61,x21-x17-y20>=0,y20=3,x19-x18-y21>=0,y21=5,x20-x18-
y22>=0,y22<=27,y22>=16,x20-x19-y23>=0,y23<=32,y23>=26,x22-x20-y24>=0,y24<=27,y24>=25,x23-x21-
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y25>=0,y25=10,x23-x22-y26>=0,y26<=32,y26>=22,x24-x22-y27>=0,y27=10,x25-x22-y28>=0,y28<=10,y28>=8,x26-x23-y29>=0,y29<=26,y29>=18,x27-x24-y30>=0,y30=5,x28-x25-
y31>=0,y31<=25,y31>=20,x29-x26-y32>=0,y32=20,x29-x27-y33>=0,y33=15,x29-x28-y34>=0,y34=7,x30-
x29-y35>=0,y35=8,x31-x29-y36>=0,y36=10,x32-x30-y37>=0,y37=10,x32-x31-y38>=0,y38=10 (5)-(25)
To perform the analysis, LINDO software was used. The objective function & the constraints were fed into the software to get the optimized results. The results are as follows:
Objective Function Value = Rs. 11070571/- at a duration of 320 days.
Optimum Cost= Rs. 11070571/- Minimum Duration = 320 days
Table 4: Results obtained from LINDO
IV. CONCLUSION
In this paper, the main objective was to achieve the Time-Cost Optimization of a case study. The optimization was achieved by two analysis techniques- 1) Project Crashing Method & 2) Linear Programming Method.
Initially, Work-break down structure, Activities, Network diagram, Normal Duration, Normal Cost, Crash Duration & Crash cost were calculated. The results of the analysis indicate that Linear Programming Method better than Project crashing method cost and Time wise. Computational wise Project Crashing Method was VARIABLE VALUE REDUCED
COST
Y1 35.000000 0.000000 Y2 1.000000 0.000000 Y3 2.000000 0.000000 Y4 2.000000 0.000000 Y5 14.000000 0.000000 Y6 11.000000 0.000000 Y7 9.000000 0.000000 Y8 46.000000 0.000000 Y9 12.000000 0.000000 Y10 8.000000 0.000000 Y11 59.000000 0.000000 Y12 18.000000 0.000000 Y13 8.000000 0.000000 Y14 6.000000 0.000000 Y15 21.000000 0.000000 Y16 46.000000 0.000000 Y17 2.000000 0.000000 Y18 15.000000 0.000000 Y20 3.000000 0.000000 Y21 5.000000 0.000000 Y22 27.000000 0.000000 Y23 32.000000 0.000000 Y24 27.000000 0.000000
VARIABLE VALUE REDUCED COST
Y25 10.000000 0.000000 Y26 22.000000 0.000000 Y27 10.000000 0.000000 Y28 10.000000 0.000000 Y29 26.000000 0.000000 Y30 5.000000 0.000000 Y31 25.000000 0.000000 Y32 20.000000 0.000000 Y33 15.000000 0.000000 Y34 7.000000 0.000000 Y35 8.000000 0.000000 Y36 10.000000 0.000000 Y37 10.000000 0.000000 Y38 10.000000 0.000000 X32 330.000000 0.000000 X1 0.000000 0.000000 X2 35.000000 0.000000 X3 36.000000 0.000000 X4 38.000000 0.000000 X5 56.000000 0.000000 X6 50.000000 0.000000 X7 49.000000 0.000000 X8 65.000000 0.000000 X9 96.000000 0.000000
VARIABLE VALUE REDUCED COST
X10 61.000000 0.000000 X11 73.000000 0.000000 X12 155.000000 0.000000 X13 79.000000 0.000000 X15 236.000000 0.000000 X14 176.000000 0.000000 X16 154.000000 0.000000 X18 178.000000 0.000000 X17 251.000000 0.000000 X22 242.000000 0.000000 X21 254.000000 0.000000 X19 183.000000 0.000000 X20 215.000000 0.000000 X23 264.000000 0.000000 X24 290.000000 0.000000 X25 252.000000 0.000000 X26 290.000000 0.000000 X27 295.000000 0.000000 X28 303.000000 0.000000 X29 310.000000 0.000000 X30 320.000000 0.000000 X31 320.000000 0.000000
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difficult to perform even though the network had just 45 activities. Linear Programming Method was easy to compute as software was available to perform the analysis. The results from Project Crashing Method were obtained in 14 iterations & that of linear programming method was obtained in 81 iterations. Hence we can conclude that Linear Programming method was robust, easy to compute & software was available to perform the analysis. The comparative results of the two analyses are given below in TABLE. 5:Table 5: Comparative results
Method Project Crashing Method Linear Programming Method
Optimum Duration(days) 331 325
Minimum Cost(Rs.) 12574291/- 11070571/-
REFERENCES Journal Paper:
[1]. Liang Liu, Scott A. Burns and Chung-Wei Fung, Construction Time-Cost Trade-Off Analysis Using LP/IP Hybrid Method, Journal of Construction Engineering & Management, vol. 121,1995, pp. 0733-9364.
[2]. Mohammad A. Ammar, Optimization of Time-Cost Trade-Off Problem with Discounted Cash Flows, Journal of Construction Engineering & Management, vol.137, 2011, pp. 0773-9364.
[3]. Chung-Wei Fung, Liang Liu and Scott A. Burns, Using Genetic Algorithm To Solve Time-Cost Trade-Off Problems, Journal of Computing in Civil Engineering, vol.11, 1997, pp. 0887-3801.
[4]. Chan, W.-T., Chua, D. K. H., and Kannan, G., Construction resource scheduling with genetic algorithms, Journal of Construction Engineering & Management, 1995, pp. ASCE 0733-9364/125-132.
Books:
1. Chitkara, Construction Project Management (Tata McGraw-Hill Education,1998)
Proceedings:
1. Omar M. Elmabrouk, Scheduling Project Crashing Time using Linear Programming Technique: Proc.
International Conference on Industrial Engineering and Operations Management, Istanbul, Turkey, 2012.
