# 13.3.6 The Lead-Time-Stretching Factor and Probable Scheduling

### Intended learning outcomes: Produce an overview on order urgency and slack time. Differentiate between forward, backward, and probable scheduling. Explain the role of the lead-time-stretching factor in probable scheduling. Describe the equation for recalculation of lead-time-stretching factor.

In practice, the urgency of an order is often more important than an absolute date.

`Order urgency is the urgency of the order’s operations compared with those of other orders.`

A possible measure for order urgency is the lead-time-stretching factor, which is introduced in the following.

`For backward scheduling, slack time is the difference between the latest (possible) start date and the earliest (acceptable) start date; for forward scheduling, it is the difference between the earliest (possible) completion date and the latest (acceptable) completion date.`

Therefore, slack time provides an element of flexibility in planning. Positive slack time allows an increase in lead time, while negative slack time requires that it be shortened.

`In probable scheduling, we take slack time into account to increase or decrease lead time.`

Figure 13.3.6.1 illustrates the principle of probable scheduling using an example with three operations (“op”) and positive slack time. In contrast to forward or backward scheduling, the operations are distributed evenly between the earliest start date and the latest completion date. Then, the start or the completion date of each operation is its probable start date or probable completion date.

Since the technical process itself determines the duration of operations and the technical inter­operation time, we can only modify slack time by increasing or reducing either the nontechnical inter­operation times or the administrative times. All of these time elements are attributes of the product’s master data, its routing sheet, and the work centers. Their values are averages, determined through measuring or estimating.

Fig. 13.3.6.1       Forward, backward, and probable scheduling.

`The lead-time-stretching factor is a numerical factor by which the non-technical inter­operation times and the administrative times are multiplied.`

The choice of the lead-time-stretching factor has the following effects on the scheduling algorithm:

• A factor greater than 1 results in increased lead time.
• A factor equal to 1 results in “normal,” or average, lead time.
• A factor between 0 and 1 results in reduced lead time.
• A factor equal to 0 results in a minimal lead time, in that only the duration of the operations and technical inter­operation times are strung together.
• With a factor of less than 0, the operations overlap.

Probable scheduling takes the latest completion date and the earliest start date as givens and calculates the lead-time-stretching factor. This is the starting point in the cases that follow.

• Customer production orders with a set due date: This due date is the latest acceptable completion date for scheduling. Because delivery dates are often very short term, the earliest start date becomes de facto “today.” The scheduling algorithm calculates the lead-time-stretching factor (less than 1) needed to shorten the inter­operation times so that the order can be completed between “today” and the delivery date. In this case, the lead-time-stretching factor indicates the feasibility of completion of the order cycle (where sufficient capacity is available, of course).
• Orders in process: The earliest start date for the first of all remaining operations is “today.” The latest completion date is generally the date specified when the order is released. Rescheduling calculates the lead-time-stretching factor required for order completion on time. This is very useful if, for example, there are delays after the order is released. A lower lead-time-stretching factor gives this order immediate urgency.
• Early released orders: The earliest start date is provided by the date the order is released; the latest completion date is the date on which warehouse stocks will probably fall below the safety stock level. Again, probable scheduling will calculate the lead-time-stretching factor required for timely order completion. This factor can then serve as a priority rule for queues at the work centers (see also Section 15.3.1).

The lead-time-stretching factor is calculated using an iterative forward or backward scheduling process as follows:

1. Choose a lead-time-stretching factor, such as 1 (randomly) or the last valid factor used (in a previous scheduling process).
2. Schedule forward (or backward) using the chosen lead-time-stretching factor. At the same time, calculate the earliest completion date (or the latest start date) using the lead-time-stretching factor 0, and thus the lead time required for the duration of operations and technical inter­operation times.
3. If the difference between the earliest completion date and the latest completion date in forward scheduling (or the earliest start date and the latest start date in backward scheduling) is approximately zero, then we have found the appropriate lead-time-stretching factor and the process is finished.
4. If the difference is not approximately zero, choose a new lead-time-stretching factor according to the formulas in Figure 13.3.6.3. Begin again with step 2.

Figure 13.3.6.2 shows the result of each iteration in Step 4, in forward scheduling.[note 1307]

Fig. 13.3.6.2       The role of the lead-time-stretching factor in probable scheduling.

Iteration of the forward scheduling algorithm calculates the earliest com­pletion date using the currently valid lead-time-stretching factor. The same iteration of the algorithm calculates the earliest completion date using the lead-time-stretching factor 0. The result yields the minimum load time without an overlapping of the operations. The objective of probable scheduling is, by recalculation of the lead-time-stretching factor, to eliminate the difference, that is, the slack time, between the earliest completion date and the latest completion date. This is shown in Figure 13.3.6.2. Since this involves a multiplication factor, the equation is a proportional relationship, as shown in Figure 13.3.6.3.[note 1308]

Fig. 13.3.6.3       Equation for recalculation of lead-time-stretching factor.

For a production contract with a limited number of serially executed operations, probable scheduling using the formula in Figure 13.3.6.3 usually yields the exact solution after only one iteration subsequent to the initial step. In a network structure, however, there may be a different number of operations with varying inter­operation times in each branch of the network. In any case, there are always situations where one iteration alone does not produce an immediate, exact solution with a slack time of approximately zero. The reasons for this and some suggestions for solving the problem are as follows:

• The lead-time-stretching factor was too inexact. Another iteration of the process will yield a more exact result, namely, a slack time close to zero.
• The calculations were inexact, which we can correct by, for example, calculating to finer units, such as to tenth-days instead of half-days.
• Because of the new lead-time-stretching factor, another path in the network of operations has become time critical; that is, it is now the longest path. A further iteration of the algorithm would yield precise results, provided that the critical path remains the same.
• There is a negative lead-time-stretching factor, and the scheduling algorithm cannot accommodate the operations between the earliest start date and the latest completion date. It is even possible that one of the operations itself is longer than the difference between these two set dates. In both cases, only lengthening the time span will resolve the situation.