4.12 Construction Processes
The previous sections described the primary inputs of labor, material and equipment to the construction process. At varying levels of detail, a project manager must insure that these inputs are effectively coordinated to achieve an efficient construction process. This coordination involves both strategic decisions and tactical management in the field. For example, strategic decisions about appropriate technologies or site layout are often made during the process of construction planning. During the course of construction, foremen and site managers will make decisions about work to be undertaken at particular times of the day based upon the availability of the necessary resources of labor, materials and equipment. Without coordination among these necessary inputs, the construction process will be inefficient or stop altogether.
Example 4-13: Steel erection
Erection of structural steel for buildings, bridges or other facilities is an example of a construction process requiring considerable coordination. Fabricated steel pieces must arrive on site in the correct order and quantity for the planned effort during a day. Crews of steelworkers must be available to fit pieces together, bolt joints, and perform any necessary welding. Cranes and crane operators may be required to lift fabricated components into place; other activities on a job site may also be competing for use of particular cranes. Welding equipment, wrenches and other hand tools must be readily available. Finally, ancillary materials such as bolts of the correct size must be provided.
In coordinating a process such as steel erection, it is common to assign different tasks to specific crews. For example, one crew may place members in place and insert a few bolts in joints in a specific area. A following crew would be assigned to finish bolting, and a third crew might perform necessary welds or attachment of brackets for items such as curtain walls.
With the required coordination among these resources, it is easy to see how poor management or other problems can result in considerable inefficiency. For example, if a shipment of fabricated steel is improperly prepared, the crews and equipment on site may have to wait for new deliveries.
Example 4-14: Construction process simulation models
Computer based simulation of construction operations can be a useful although laborious tool in analyzing the efficiency of particular processes or technologies. These tools tend to be either oriented toward modeling resource processes or towards representation of spatial constraints and resource movements. Later chapters will describe simulation in more detail, but a small example of a construction operation model can be described here. The process involved placing concrete within existing formwork for the columns of a new structure. A crane-and-bucket combination with one cubic yard capacity and a flexible "elephant trunk" was assumed for placement. Concrete was delivered in trucks with a capacity of eight cubic yards. Because of site constraints, only one truck could be moved into the delivery position at a time. Construction workers and electric immersion-type concrete vibrators were also assumed for the process.
The simulation model of this process is illustrated in Figure 4-5. Node 2 signals the availability of a concrete truck arriving from the batch plant. As with other circular nodes in Figure 4-5, the availability of a truck may result in a resource waiting or queueing for use. If a truck (node 2) and the crane (node 3) are both available, then the crane can load and hoist a bucket of concrete (node 4). As with other rectangular nodes in the model, this operation will require an appreciable period of time. On the completion of the load and hoist operations, the bucket (node 5) is available for concrete placement. Placement is accomplished by having a worker guide the bucket's elephant trunk between the concrete forms and having a second worker operate the bucket release lever. A third laborer operates a vibrator in the concrete while the bucket (node 8) moves back to receive a new load. Once the concrete placement is complete, the crew becomes available to place a new bucket load (node 7). After two buckets are placed, then the column is complete (node 9) and the equipment and crew can move to the next column (node 10). After the movement to the new column is complete, placement in the new column can begin (node 11). Finally, after a truck is emptied (nodes 12 and 13), the truck departs and a new truck can enter the delivery stall (node 14) if one is waiting.
Figure 4-5: Illustration of a Concrete-Placing Simulation Model
Application of the simulation model consists of tracing through the time required for these various operations. Events are also simulated such as the arrival times of concrete trucks. If random elements are introduced, numerous simulations are required to estimate the actual productivity and resource requirements of the process. For example, one simulation of this process using four concrete trucks found that a truck was waiting 83% of the time with an average wait at the site of 14 minutes. This type of simulation can be used to estimate the various productivity adjustment factors described in the previous section.
4.13 Queues and Resource Bottlenecks
A project manager needs to insure that resources required for and/or shared by numerous activities are adequate. Problems in this area can be indicated in part by the existence of queues of resource demands during construction operations. A queue can be a waiting line for service. One can imagine a queue as an orderly line of customers waiting for a stationary server such as a ticket seller. However, the demands for service might not be so neatly arranged. For example, we can speak of the queue of welds on a building site waiting for inspection. In this case, demands do not come to the server, but a roving inspector travels among the waiting service points. Waiting for resources such as a particular piece of equipment or a particular individual is an endemic problem on construction sites. If workers spend appreciable portions of time waiting for particular tools, materials or an inspector, costs increase and productivity declines. Insuring adequate resources to serve expected demands is an important problem during construction planning and field management.
In general, there is a trade-off between waiting times and utilization of resources. Utilization is the proportion of time a particular resource is in productive use. Higher amounts of resource utilization will be beneficial as long as it does not impose undue costs on the entire operation. For example, a welding inspector might have one hundred percent utilization, but workers throughout the jobsite might be wasting inordinate time waiting for inspections. Providing additional inspectors may be cost effective, even if they are not utilized at all times.
A few conceptual models of queueing systems may be helpful to construction planners in considering the level of adequate resources to provide. First, we shall consider the case of time-varying demands and a server with a constant service rate. This might be the situation for an elevator in which large demands for transportation occur during the morning or at a shift change. Second, we shall consider the situation of randomly arriving demands for service and constant service rates. Finally, we shall consider briefly the problems involving multiple serving stations.
Single-Server with Deterministic Arrivals and Services
Suppose that the cumulative number of demands for service or "customers" at any time t is known and equal to the value of the function A(t). These "customers" might be crane loads, weld inspections, or any other defined group of items to be serviced. Suppose further that a single server is available to handle these demands, such as a single crane or a single inspector. For this model of queueing, we assume that the server can handle customers at some constant, maximum rate denoted as x "customers" per unit of time. This is a maximum rate since the server may be idle for periods of time if no customers are waiting. This system is deterministic in the sense that both the arrival function and the service process are assumed to have no random or unknown component.
Figure 4-6: Cumulative Arrivals and Departures in a Deterministic Queue
A cumulative arrival function of customers, A(t), is shown in Figure 4-6 in which the vertical axis represents the cumulative number of customers, while the horizontal axis represents the passage of time. The arrival of individual customers to the queue would actually represent a unit step in the arrival function A(t), but these small steps are approximated by a continuous curve in the figure. The rate of arrivals for a unit time interval t from t-1 to t is given by:
While an hour or a minute is a natural choice as a unit time interval, other time periods may also be used as long as the passage of time is expressed as multiples of such time periods. For instance, if half an hour is used as unit time interval for a process involving ten hours, then the arrivals should be represented by 20 steps of half hour each. Hence, the unit time interval between t-1 and t is t = t - (t-1) = 1, and the slope of the cumulative arrival function in the interval is given by:
The cumulative number of customers served over time is represented by the cumulative departure function D(t). While the maximum service rate is x per unit time, the actual service rate for a unit time interval t from t-1 to t is:
The slope of the cumulative departure function is:
Any time that the rate of arrivals to the queue exceeds the maximum service rate, then a queue begins to form and the cumulative departures will occur at the maximum service rate. The cumulative departures from the queue will proceed at the maximum service rate of x "customers" per unit of time, so that the slope of D(t) is x during this period. The cumulative departure function D(t) can be readily constructed graphically by running a ruler with a slope of x along the cumulative arrival function A(t). As soon as the function A(t) climbs above the ruler, a queue begins to form. The maximum service rate will continue until the queue disappears, which is represented by the convergence of the cumulative arrival and departure functions A(t) and D(t).
With the cumulative arrivals and cumulative departure functions represented graphically, a variety of service indicators can be readily obtained as shown in Figure 4-6. Let A'(t) and D'(t) denote the derivatives of A(t) and D(t) with respect to t, respectively. For 0 t ti in which A'(t) x, there is no queue. At t = ti, when A'(t) > D'(t), a queue is formed. Then D'(t) = x in the interval ti t tk. As A'(t) continues to increase with increasing t, the queue becomes longer since the service rate D'(t) = x cannot catch up with the arrivals. However, when again A'(t) D'(t) as t increases, the queue becomes shorter until it reaches 0 at t = tk. At any given time t, the queue length is
For example, suppose a queue begins to form at time ti and is dispersed by time tk. The maximum number of customers waiting or queue length is represented by the maximum difference between the cumulative arrival and cumulative departure functions between ti and tk, i.e. the maximum value of Q(t). The total waiting time for service is indicated by the total area between the cumulative arrival and cumulative departure functions.
Generally, the arrival rates At = 1, 2, . . ., n periods of a process as well as the maximum service rate x are known. Then the cumulative arrival function and the cumulative departure function can be constructed systematically together with other pertinent quantities as follows:
1. Starting with the initial conditions D(t-1)=0 and Q(t-1)=0 at t=1, find the actual service rate at t=1:
2. Starting with A(t-1)=0 at t=1, find the cumulative arrival function for t=2,3,. . .,n accordingly:
3. Compute the queue length for t=1,2, . . .,n.
4. Compute Dt for t=2,3,. . .,n after Q(t-1) is found first for each t:
5. If A'(t) > x, find the cumulative departure function in the time period between ti where a queue is formed and tk where the queue dissipates:
6. Compute the waiting time w for the arrivals which are waiting for service in interval t:
7. Compute the total waiting time W over the time period between ti and tk.
8. Compute the average waiting time w for arrivals which are waiting for service in the process.
This simple, deterministic model has a number of implications for operations planning. First, an increase in the maximum service rate will result in reductions in waiting time and the maximum queue length. Such increases might be obtained by speeding up the service rate such as introducing shorter inspection procedures or installing faster cranes on a site. Second, altering the pattern of cumulative arrivals can result in changes in total waiting time and in the maximum queue length. In particular, if the maximum arrival rate never exceeds the maximum service rate, no queue will form, or if the arrival rate always exceeds the maximum service rate, the bottleneck cannot be dispersed. Both cases are shown in Figure 4-7.
Figure 4-7: Cases of No Queue and Permanent Bottleneck
A practical means to alter the arrival function and obtain these benefits is to inaugurate a reservation system for customers. Even without drawing a graph such as Figure 4-6, good operations planners should consider the effects of different operation or service rates on the flow of work. Clearly, service rates less than the expected arrival rate of work will result in resource bottlenecks on a job.
Single-Server with Random Arrivals and Constant Service Rate
Suppose that arrivals of "customers" to a queue are not deterministic or known as in Figure 4-6. In particular, suppose that "customers" such as joints are completed or crane loads arrive at random intervals. What are the implications for the smooth flow of work? Unfortunately, bottlenecks and queues may arise in this situation even if the maximum service rate is larger than the average or expected arrival rate of customers. This occurs because random arrivals will often bunch together, thereby temporarily exceeding the capacity of the system. While the average arrival rate may not change over time, temporary resource shortages can occur in this circumstance.
Let w be the average waiting time, a be the average arrival rate of customers, and x be the deterministic constant service rate (in customers per unit of time). Then, the expected average time for a customer in this situation is given by:
If the average utilization rate of the service is defined as the ratio of the average arrival rate and the constant service rate, i.e.,
Then, Eq. (4.27) becomes:
In this equation, the ratio u of arrival rate to service rate is very important: if the average arrival rate approaches the service rate, the waiting time can be very long. If a x, then the queue expands indefinitely. Resource bottlenecks will occur with random arrivals unless a measure of extra service capacity is available to accommodate sudden bunches in the arrival stream. Figure 4-8 illustrates the waiting time resulting from different combinations of arrival rates and service times.
Figure 4-8: Illustrative Waiting Times for Different Average Arrival Rates and Service Times
Both of the simple models of service performance described above are limited to single servers. In operations planning, it is commonly the case that numerous operators are available and numerous stages of operations exist. In these circumstances, a planner typically attempts to match the service rates occurring at different stages in the process. For example, construction of a high rise building involves a series of operations on each floor, including erection of structural elements, pouring or assembling a floor, construction of walls, installation of HVAC (Heating, ventilating and air conditioning) equipment, installation of plumbing and electric wiring, etc. A smooth construction process would have each of these various activities occurring at different floors at the same time without large time gaps between activities on any particular floor. Thus, floors would be installed soon after erection of structural elements, walls would follow subsequently, and so on. From the standpoint of a queueing system, the planning problem is to insure that the productivity or service rate per floor of these different activities are approximately equal, so that one crew is not continually waiting on the completion of a preceding activity or interfering with a following activity. In the realm of manufacturing systems, creating this balance among operations is called assembly line balancing.
Figure 4-9: Arrivals and Services of Crane Loads with a Crane Breakdown
Example 4-15: Effect of a crane breakdown
Suppose that loads for a crane are arriving at a steady rate of one every ten minutes. The crane has the capacity to handle one load every five minutes. Suppose further that the crane breaks down for ninety minutes. How many loads are delayed, what is the total delay, and how long will be required before the crane can catch up with the backlog of loads?
The cumulative arrival and service functions are graphed in Figure 4-9. Starting with the breakdown at time zero, nine loads arrive during the ninety minute repair time. From Figure 4-9, an additional nine loads arrive before the entire queue is served. Algebraically, the required time for service, t, can be calculated by noted that the number of arrivals must equal the number of loads served. Thus:
A queue is formed at t = 0 because of the breakdown, but it dissipates at A(t) = D2(t). Let
from which we obtain t = 180 min. Hence
The total waiting time W can be calculated as the area between the cumulative arrival and service functions in Figure 4-9. Algebraically, this is conveniently calculated as the difference in the areas of two triangles:
so the average delay per load is w = 810/18 = 45 minutes.
Example 4-16: Waiting time with random arrivals
Suppose that material loads to be inspected arrive randomly but with an average of 5 arrivals per hour. Each load requires ten minutes for an inspection, so an inspector can handle six loads per hour. Inspections must be completed before the material can be unloaded from a truck. The cost per hour of holding a material load in waiting is $30, representing the cost of a driver and a truck. In this example, the arrival rate, a, equals 5 arrivals per hour and the service rate, x, equals 6 material loads per hour. Then, the average waiting time of any material load for u = 5/6 is:
At a resource cost of $30.00 per hour, this waiting would represent a cost of (30)(0.4)(5) = $60.00 per hour on the project.
In contrast, if the possible service rate is x = 10 material loads per hour, then the expected waiting time of any material load for u = 5/10 = 0.5 is:
which has only a cost of (30)(0.05)(5) = $7.50 per hour.
Example 4-17: Delay of lift loads on a building site
Suppose that a single crane is available on a building site and that each lift requires three minutes including the time for attaching loads. Suppose further that the cumulative arrivals of lift loads at different time periods are as follows:
4 per hour
8 per hour
15 per hour
4 per hour
25 per hour
0 per hour
5 per hour
Using the above information of arrival and service rates
- Find the cumulative arrivals and cumulative number of loads served as a function of time, beginning with 6:00 AM.
- Estimate the maximum queue length of loads waiting for service. What time does the maximum queue occur?
- Estimate the total waiting time for loads.
- Graph the cumulative arrival and departure functions.
The maximum service rate x = 60 min/3 min per lift = 20 lifts per minute. The detailed computation can be carried out in the Table 4-2, and the graph of A(t) and D(t) is given in Figure 4-10.
Table 4-2 Computation of queue length and waiting time
- Bourdon, C.C., and R.W. Levitt, Union and Open Shop Construction, Lexington Books, D.C. Heath and Co., Lexington, MA, 1980.
- Caterpillar Performance Handbook, 18@+(th) Edition, Caterpillar, Inc., Peoria, IL, 1987.
- Cordell, R.H., "Construction Productivity Management," Cost Engineering, Vol. 28, No. 2, February 1986, pp. 14-23.
- Lange, J.E., and D.Q. Mills, The Construction Industry, Lexington Books, D.C. Heath and Co., Lexington, MA, 1979.
- Nunnally, S.W., Construction Methods and Management, Prentice-Hall, Englewoood Cliffs, NJ, 2nd Ed., 1987.
- Peurifoy, R.L., Construction Planning, Equipment and Methods, 2nd Edition, McGraw-Hill, New York, 1970.
- Tersine, R.J., Principles of Inventory and Materials Management, North Holland, New York, 1982.