Dynamic minimum lag (DML) relationship

Abstract

The dynamic minimum lag (DML) is a new logical relationship in critical path method (CPM) networks. It maintains a minimum lag between the successor and its predecessor during the entire duration of the two activities, thus not allowing the successor to get ahead of (or closer than the assigned lag with) the successor at any point. This is fundamentally different from the well-known and used start-to-start (SS) and finish-to-finish (FF) relationships. These two relationships focus on either the starting point or finishing point of the activity, not on any other point in between. Even when we assign a combination SS + FF relationship with lag(s) between two activities, the successor may get ahead of (or closer than the assigned lag with) the predecessor due to different productivity rates. The DML relationship eliminates this possibility, thereby eliminating the need for a combination SS+FF relationship.

Introduction

When the critical path method (CPM) was introduced in the late 1950s as a tool for scheduling projects, activity on arrow (AOA) networks were used for depicting the logic and performing the calculations. At this time, there was only one type of logic relationship: finish-to-start (FS). Shortly after, activity on node (AON) networks were introduced, but these also had the FS relationship only. During the early 1960s, Professor John Fondahl introduced precedence diagrams exhibiting four types of relationships: FS, start-to-start (SS), finish-to-finish (FF), and start-to-finish (SF). Precedence diagrams revolutionized project scheduling because they accommodated overlapping activities for the first time in CPM networks without the need to split them.

The dynamic minimum lag (DML) is a new type of relationship: Unlike the SS and FF relationships, the DML relationship is, as the name suggests, dynamic. It moves through the activities along with work progress. Its main advantage is that it does not allow the successor to get ahead of (or undesirably close to) the predecessor. It maintains a minimum lag between the successor and the predecessor during the entire course of work progress (Exhibit 1).

Exhibit 1--The Dynamic Minimum Lag (DML) Relationship

Why the Dynamic Minimum Lag?

Traditionally in precedence networks, overlapping activities are connected with an SS, an FF, or a combination SS + FF relationship. If productivity is assumed to be linear in both the predecessor and the successor and if actual work progress conforms to the estimated productivities and durations, then DML works exactly like the SS and/or FF relationship (Exhibit 2).

Exhibit 2--Two Overlapping Activities Represented in Three Different Ways (SS, FF, SS + FF), All Leading
to the Same Dates (Activities Assumed to be Contiguous)

However, as we all know, things rarely go as planned. Many scenarios exist in which activity B can get ahead of activity A (or within less than the 1-day lag) (see Exhibit 3, below):

Exhibit 3--Cases Where the Successor (Activity B) Has Gotten Ahead of the Predecessor (Activity A) at Some
Point Although There Was No Violation to the Traditional SS and FF Relationships

Dynamic Minimum Lag and the Linear Scheduling Method

Although the DML relationship is similar to the linear scheduling method (LSM) in that both keep a distance or time buffer (as in Exhibit 2a), they are not the same. There are two major differences: DML works with linear and nonlinear productivities for the predecessor (the successor is assumed to be linear), whereas the LSM (as its name implies) works strictly for linear activities. Secondly, the LSM is a different scheduling method from CPM and cannot be implemented in CPM network scheduling. DML is a logical relationship that is fully compatible with CPM calculations and network scheduling.

How Does the Dynamic Minimum Lag Work?

We start with the percent complete (PC) of both predecessor and successor activities:

PC_A > PC_B

The successor is still assumed to be linear, while the predecessor can follow any productivity distribution (the cases shown in Exhibit 3b and c can be implemented by the DML relationship, but the case in Exhibit 3a cannot). If we assign a minimum lag L, then:

PC_A ≥ PC_B + L (days)

However, the equation above is not homogeneous; percent complete is unitless while the lag has a time unit (e.g., day). In order to correct the equation, let’s analyze the percent complete to its basic components:

Where:

All of the four variables in Equation (2) are user’s defined; however, the lag (L) is set usually with the logic. The predecessor’s percent complete is assumed to be given (reported /estimated by the field team). This leaves us with the successor’s durations AD_B and RD_B: if one is given by the user (AD_B), the other one (RD_B) is calculated by the equation:

Because this is an “inequality,” RD_B has to be equal or greater than the right-hand side. So, in a software program case, the program calculates the minimum amount for RDB, but the user can override it with a larger, not a smaller, amount.

Although unlikely, it is also possible to allow the user to give the RD_B. In this case, Equation 3 will solve for AD_B:

Similarly, the software program calculates the maximum amount for AD_B, but the user can override it with a smaller, not a larger, amount.

When using the DML relationship, the software program should be set on the default rule to have AD of the successor input by the user, and calculate the minimum RD, with the ability of the user to reverse this rule.

Rounding Numbers

Because most schedulers use “day” as the unit of time, software programs need to round up RD_B to the nearest larger day (or unit of time) and truncate AD_B to the nearest smaller day (or unit of time).

Note of Caution

Percent complete may be interpreted differently by different people in different situations (see “So, What’s my Percent Complete Anyway?” a presentation by the author at the 3rd Annual PMI College of Scheduling Conference, April 2006, Orlando, FL.). Software programs have numerous types of activity percent complete. For DML calculations, percent complete is strictly defined in Equation (1) above.

Dynamic Minimum Lag Relationship in the Critical Path Method Calculations

When creating a schedule (i.e., project is 0% complete), the DML relationship is treated---in the forward and backward passes---like an SS + FF combination relationship: make certain that the successor’s start is equal to or greater than the predecessor’s start plus the specified lag, and the successor’s finish is equal to or greater than the predecessor’s finish plus the specified lag. This is sufficient until the successor starts, and then the above equations (4) and (5) must be used to ensure maintenance of proper logic. The equations above do not conflict with the traditional CPM calculations (if we think of the DML relationship as a combination SS + FF relationship), but they add more restrictions to keep the lag holding for the entire duration.

DML relationships also work (with the same equations above) in the event that the predecessor and/or the successor is interruptible (noncontiguous).

Conclusion

The DML relationship is a new and useful concept in project scheduling. It should be mainly used for the situation of overlapping activities when the predecessor has to stay ahead of the successor by a certain margin from start to finish. For example, in building construction, this relationship can be used for the series of four activities: framing, drywall, pasting and taping, and wall painting. In infrastructure construction, it can be used for trench excavating and piping operations. There are numerous applications in all types of projects.

The DML relationship eliminates the need for the SS + FF combination relationship and most of the cases where a single SS or FF relationship is used. However, there will be a few cases where an SS or FF relationship is still used. For example, in the case of a developer who is planning to build an apartment complex on a large piece of land (Exhibit 4a/b).

Exhibit 4a: A Case in Which the DML Relationship Cannot Be Used in Lieu of the SS Relationship

In this case, the developer needs to “clear and grub” a certain portion of the land where the building will be erected. No connection exists between the finish “clear and grub” and “excavate.” The scheduler in this case has to make sure the minimum required percentage of “clear and grub” is completed before the excavation can start. A safer way to depict that logic is shown in Exhibit 4b below:

Exhibit 4b: A Better Alternative to Depict the Logic of the Network Shown in Exhibit 4a

For example, assume the following two activities:

We need to maintain at least a 1-day lag between the two activities. Activities are contiguous. Assume Activity B (Install Shingles) to be linear. Using traditional CPM relationships:

1.   At Day 0 (end of Day 0 = start of Day 1):
   
2.   At Day 10 (end of Day 10): Only 40% of activity A (Sheathing) is completed:

   Activity A:	Started on time:
   	Actual duration	= 10 days
   	Remaining duration	= 10 days (productivity will pick up in the last 10 days)
   	At completion duration	= 20 days

   Activity B (Shingles):	Started at end of Day 3 (i.e., beginning of Day 4)
   	Actual duration	= 7 days
   	Remaining duration	= 11 days (assuming the end of activity A + 1 day)
   	Percent complete = 7 / (7+18) = 38.9%

     Using Equation 4 of the DML paper:

   

   Finish date for Activity B ≥ Day 24, not Day 21 as calculated by the FF relationship.

3.   At Day 15: Assume only 70% of Activity A (Sheathing) is completed:

   Activity A:
   	Actual duration	= 15 days
   	Remaining duration	= 6 days (estimated)
   	At completion duration	= 21 days

   Activity B (Shingles):
   	Actual duration	= 12 days
   	Remaining duration	= 7 days (assuming the end of activity A + 1 day)
   	Percent complete = 12 / (12+7) = 63%

     Using Equation 4 of the DML paper:

   

     Finish date for Activity B = Day 22, which is same as calculated by the FF relationship.