Construction Management: "Time Cost Trade-off"

Time Cost Trade-off

Time and cost are the two most important resources that a project manager deals with. Both of these resources have constrained and the job of a project manager is to have a judicious balance between them. The judicious balance between time and cost is called Time-Cost trade-off and it can be achieved by studying the availability and demand of these resources for the given project.

Time - Cost Trade-off Analysis, Why do project managers require this?

Total project costs include both direct costs and indirect costs of performing the activities of the project. 

If each activity of the project is scheduled for the duration that results in the minimum direct cost (normal duration) then the time to complete the entire project might be too long and substantial penalties associated with the late project completion might be incurred.

At the other extreme, a manager might choose to complete the activity in the minimum possible time, called crash duration, but at a maximum cost.

Thus, planners perform what is called time cost trade-off analysis to shorten the project duration.

This can be done by selecting some activities on the critical path to shorten their duration. As the direct cost for the project equals the sum of the direct costs of its activities, then the project direct cost will increase by decreasing its duration. On the other hand, the indirect cost will decrease by decreasing the project duration, as the indirect cost are almost a linear function with the project duration.  

To analyze the time and cost trade-off, we have to first go through some theory and definition of terminologies as follows: 

Project cost: 

For any project total expenditure incurred in terms of manpower, equipment, machinery, materials and time to achieve a particular goal is known as the total cost of the project. The total cost of the project is the sum of two distinct costs. Direct and Indirect costs.

Direct cost:

The cost of materials, equipment, and money spent on manpower form the direct cost. The direct costs of a project are of major concern and behavior pattern of direct costs with time is of importance. The direct cost of a project depends on the completion time of the project, but the variation is not linear (Refer figure 'a' below).

Figure 'a' : Direct cost curve (Direct cost behavior with time)

Indirect cost

The expenditures which cannot be allotted clearly to the individual activities of the project, but are assessed as a whole are called indirect costs. The indirect cost includes overhead charges, administrative and establishment charges, supervision charges, loss of revenue, loss in profit and penalty, etc. Indirect cost is directly proportional to the time. (Refer figure 'b' below) 

                    Figure 'b' : Indirect cost curve (Indirect cost behavior with time)

Total Cost:

The combinations of direct cost and indirect cost is the total cost. The sum of direct costs and indirect costs is known as the total costs of the project. The relationship between indirect costs, direct cost and total cost with duration is given in the figure 'c' below. From the figure, it is seen that the curve of direct costs does not have a linear relationship with time. Thus the combination of direct and indirect cost gives a curvilinear relationship with time as shown.

Figure 'c' : Total cost curve (Total cost behavior with time)

From the total cost curve ACB, it is seen that the total cost of a project is minimum for a certain time duration. This duration is known as optimum duration (To) for the corresponding minimum cost (Cm). Further, if the project duration is increased, the total cost will also increase. On the other hand, if the project duration is decreased to the crash value, the project cost will be the highest. The optimum duration is less than the normal duration (Tn) corresponding to the direct costs.

This is because both direct and indirect cost increases beyond the normal duration, whereas below the normal duration indirect cost decrease, but the direct cost increases.

Direct and Indirect cost in Rate Analysis 

The direct cost and indirect cost of any activity are calculated for the each items during the analysis of rates. Quantity Estimator calculates the direct cost of an item which includes the cost of materials, cost of labor and cost of equipment. Similarly, the indirect cost (contractor's overhead/ profit) of that activity is calculated as 15 % of direct cost in flat. For example see figure 'd' below.  

Figure 'd' : Direct and Indirect cost in Analysis of rate

Definition of terminologies for Time-Cost Trade-off

Normal time (Tn)

The time usually allowed for an activity by the estimator is known as normal time. It is the standard time for that activity and is denoted by Tn.

Crash time (Tc)

The minimum possible time in which an activity can be completed by deploying extra resources is known as crash time. Beyond the crash time, the duration of an activity cannot be reduced or shortened by any amount of increase in mobilization. It is denoted by Tc.

Normal cost (Cn)

The direct cost required to complete the activity in the normal duration is called normal cost and is denoted by C_n.

Crash cost (Cc)

The direct cost corresponding to the crash time of completing an activity is known as crash cost and is denoted by Cc.

Cost slope (C.S)

The direct cost curve generally is a curve as shown below. But this curve can be approximated by the straight line or more than one straight lines depending upon the flatness of the curve. Thus the slope of this straight line is cost slope. (Refer figure 'e' below)

Note: To crash the activity duration from  normal time (Tn) to Crash time (Tc), we have to expense an extra resources in addition to Normal cost (Cn). without spending extra resources in direct cost, an activity cannot be reduced to crash duration. 

Cost optimization through network (Steps in Optimization of Cost)

1. First, draw the network diagram and find out all the possible paths. Critical path and non-critical paths. Critical path duration is the normal project duration for that project
2. Calculate the direct cost by adding the normal cost of all activities. Calculate indirect cost by multiplying project duration with the indirect cost (overhead or indirect expense) per unit time
3. Calculate the total cost by adding direct cost and indirect cost. This total cost is the baseline cost for the calculation. Similarly, the longest duration (critical path duration) is the project duration as a baseline duration for the calculation.
4. Calculate the cost slopes of all activities and rank them in ascending order of cost slopes.
5. Start crashing the project from a critical path (since this is the project duration) by selecting such critical activity having least cost slope as per ranking to the maximum possible extent (observe other non-critical paths which are likely to be critical)
6. Calculate the direct cost by adding the extra cost of crashing (cost slope * (Tn-Tc)) required for crashing that activity to the direct cost calculated in previous steps and corresponding indirect cost of reduced project duration.

After step six, three possible cases may arise:

Case I: If the same path remains critical (longest) then crashing is considered for the critical activity serially in the ascending order to their cost slope as far as possible. 

Case II: While crashing the critical path, there may emerge a new critical path/s then crashing is carried out for the common activity. If there is more than one common activity, activity with the least cost slope is selected first and continues serially as per the ranking of cost slopes.

Case III:   After crashing as per the case II, if there is a possibility of crashing then, one activity from each path is selected at one time for crashing. This crashing is known as simultaneous crashing or parallel crashing. The process is continued until the project shortening is possible.

Note:

Every time after crashing, the direct cost and indirect cost is calculated to find out the total cost. If the total cost calculated after crashing is greater than the earlier step, we come to the end of optimization.

Whereas, for the determination of minimum possible time (Tm), crashing should be continued until the possibility. 

Solved Example:

The following table shows the costs and duration of each activity of a project. The network is shown. The indirect cost may be taken as Rs. 3150.- per week. Determine the optimum duration of the project and the corresponding minimum cost.

Activity	Normal duration (Tn)	Normal cost (Cn)	Crash duration (Tc)	Crash cost (Cc)
A B C D E	7 9 5 6 6	8000 5000 7000 9000 6000	4 6 2 4 4	15500 9500 10000 16000 12000

Solution: First of all drawing the network diagram. Finding possible paths, project duration and total cost

Figure: Network Diagram for above question

The possible paths of the network are:

Here, The path A – C – E is critical path with the project duration equals to 18 Weeks 

Project Cost: 

Direct cost of the project = Summation of Normal cost of all activities = å normal cost = 35000

Indirect project cost of project = project duration * Indirect cost per week = 18 × 3150 = 5670

\ Total project cost = 91700

This shows that the total project cost is Cn= 91700 for Tn = 18 weeks of normal project duration 

Calculation of Cost Slopes: 

Activity	Difference in cost (Crash cost- normal cost)	Difference in duration (Normal duration - crash duration)	Cost slope = Cc-Cn/Tn-Tc	Ranking in ascending order
A B C D E	7500 4500 3000 7000 6000	3 3 3 2 2	2500 1500 1000 3500 3000	3 2 1 5 4

Crashing:

From Critical path A-C-E, select activity C for crashing [activity 'C'  has been selected due to the minimum cost slope amongst critical activities by 5-2 weeks = 3 weeks (maximum possible extent) and revised duration be: 

Revised Project Duration

Revised Cost

Extra cost of crashing activity C by 3 weeks  = 3 × 1000 = 3000

Increment in Direct cost         = 35000 + 3000 = 38000

Decrement in Indirect cost    = 15 × 3150 = 47250

Total Cost of project        = 85250

Note: Remember, every time after crashing, we have to calculate the revised project duration and revised project cost. The revised cost needs to be compared with the total cost calculated in earlier step.

If revised cost is less than the previous cost, crashing shall be continued. 

Here, the revised cost amounting 85250 is less than the previous cost amounting 91700. Therefore, further crashing (optimization) is required. 

Crashing continuation:

After Crashing activity C, here we find that there are two critical paths A – C – E and B – E with duration equal to 15 weeks. this shows the emergence of case -II (i.e. common activity crashing).  So, crashing must be carried out by selecting common activity to both of the paths for an immediate result.

Common activity crashing:  As we see in both paths activity 'E' is common. So selecting activity E for crashing  by 6 – 4 =  2 weeks and revised duration be:

 Revised Project duration 

Revised Project Cost 

Extra cost of crashing activity E by 2 weeks  = 2 × 3000 = 6000

Direct cost                = 6000 + 38000 = 44000

Indirect cost              = 13 × 3150 = 40950

Total cost                  = 84950

Note: Here the total project cost is less than the total cost calculated in earlier step, hence further optimization is required.                                    

After Crashing activity E, here we find that all paths become critical at this stage with project duration\

equal to 13weeks. This emerge the case - III

Simultaneous crashing: Now, Selecting 'A' from path A – D and A – C – E to crash and simultaneously

selecting activity  'B' from path B – E. Activity 'A' is selected due to common for both paths A – D and

A – C – E. So, crashing activities 'A' and 'B' by 3 weeks simultaneously we get. (Note: always select

equal duration for crashing while crashing activities simultaneously) 

Revised Project duration

Revised Project Cost

Extra cost of crashing activity A and B by 3 weeks = 3 × (2500 + 1500) = 12000

Direct cost  12000 + 44000 = 56000

Indirect cost = 10 × 3150 = 31500

Total Cost = 87500
Note: Here the total cost amounting 87500 is greater than the cost amounting 84950 calculated in earlier step. This shows the end of cost optimization. 

Remember, once the minimum cost has been achieved or cost optimization is done, there shall be no chance of decreasing the cost even crashing is done for small amount of time period. For example in this case even if we crash the project duration by only 1 week each for activity A and B, the total cost won't be decreased below the 84950.

Therefore the conclusion is:

Minimum cost of the project (Cm): 84950

Optimum duration of the project (To): 13 Weeks, and

Minimum project duration (Tm): 10 weeks

References: 

1. Subash Kumar Bhattarai et. al. ''Text Book of Construction Management" Heritage Publishers and Distributors Pvt. Ltd., Bhotahity, Kathmandu.

2. Chitkara, K. K, Construction Project Management, Planning Scheduling, and Controlling; Second Edition, Tata McGraw Hill Education Private Limited, New Delhi, India

3. Saleh Mubarak “ Construction Project Scheduling and Control” Second Edition, Wiley India-Edition

4. Gupta, B.L, Gupta, Amit; Construction Management and Machinery; Standard Publishers Distributors

5. https://www.researchgate.net/figure/Non-linear-time-and-cost-trade-off-for-an-activity-Project-time-cost-relationship