*Intended learning outcomes: Describe the determination of the lead-time-stretching factor. Explain the equation for recalculation of lead-time-stretching factor.*

*Continuation from previous subsection (13.3.6)*

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

- Choose a lead-time-stretching factor, such as 1 (randomly) or the last valid factor used (in a previous scheduling process).
- 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.
- 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.
- 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.

## Course section 13.3: Subsections and their intended learning outcomes

##### 13.3 Scheduling of Orders and Scheduling Algorithms

Intended learning outcomes: Describe the manufacturing calendar and the calculation of the manufacturing lead time. Differentiate between Backward Scheduling and Forward Scheduling. Explain network planning, central point scheduling, the lead-time stretching factor, and probable scheduling. Present scheduling of process trains.

##### 13.3.1 The Manufacturing Calendar, or Shop Calendar

Intended learning outcomes: Present characteristics of the manufacturing calendar, or shop calendar. Explain an example of a manufacturing calendar.

##### 13.3.2 Lead Time Scheduling: Calculating the Manufacturing Lead Time

Intended learning outcomes: Produce an overview on lead time scheduling. Identify definitions for the elements of operation time. Present the lead time formula and the start date as a function of completion date. Differentiate between manufacturing lead time, cycle time and throughput time.

##### 13.3.3 Backward Scheduling and Forward Scheduling

Intended learning outcomes: Produce an overview on lead time scheduling. Explain forward scheduling and backward scheduling. Describe a simple algorithm for backward scheduling.

##### 13.3.4 Network Planning and CPM — Critical Path Method

Intended learning outcomes: Explain network planning and the critical path method (CPM). Present an example of a scheduled network. Describe a network algorithm for backward scheduling.

##### 13.3.5 Central Point Scheduling

Intended learning outcomes: Explain central point scheduling. Describe several possible solutions in a directed network of operations.

##### 13.3.6 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.

##### 13.3.6b Calculating the Lead-Time-Stretching Factor

Intended learning outcomes: Describe the determination of the lead-time-stretching factor. Explain the equation for recalculation of lead-time-stretching factor.

##### 13.3.7 Scheduling Process Trains

Intended learning outcomes: Differentiate between reverse flow scheduling, forward flow scheduling, and mixed flow scheduling.