*Intended learning outcomes: Explain and solve the forward and backward scheduling problems that is calculation of start and completion dates for the order and each operation, as well as the critical path and lead-time margin.*

Here, you will practice some backward and forward scheduling. Figure 13.7.4.1 presents a simple network, including a legend showing the lead- time elements used.

Solve the forward and backward scheduling problems (calculation of start and completion dates for the order and each operation, as well as the critical path and lead-time margin) listed in Figure 13.7.4.2:

**Fig.
13.7.4.1** Scheduled
network.

a. Common forward scheduling.

b. Common backward scheduling.

c. Forward scheduling with a different lead-time-stretching factor, that is, a different order urgency, to accelerate or slow down the order.

d. Forward scheduling with lead-time-stretching factor = 0, which results in the lead time as the sum of operation times plus the technical interoperation times.

**Fig. 13.7.4.2** Various
forward and backward scheduling problems.

Some common problems in the calculation process lead to the following potential errors:

- Calculating incorrect start date and due dates, not respecting interoperation times multiplied by the stretching factor
- Multiplying technical waiting time by the stretching factor
- Incorrectly calculating the longest path in a network
- Not understanding the principle of forward or backward scheduling

*Solutions*:

(ESD stands for earliest start date, ECD for earliest completion date, LSD for latest start date, LCD for latest completion date.)

a. ESD(op10) = 3, ECD(op10) = 3.5; ESD(op20) = 6.5, ECD(op20) = 7; ESD(op30) = 3, ECD(op30) = 3.5; ESD(op40) = 10, ECD(op40) = 11; ESD(order) = 0, ECD(order) = 12. Note the critical path in determining the ESD(op40): The *lower* path is critical. The lead-time margin of the *upper* path is 2.

b. LCD(op40) = 15, LSD(op40) = 14; LCD(op30) = 9.5, LSD(op30) = 9; LCD(op20) = 11, LSD(op20) = 10.5; LCD(op10) = 7.5, LSD(op10)=7; LCD(order) = 16, LSD(order) = 4. Note that — again — the *lower* path is critical. The lead-time margin of the *upper* path is again 2.

c. ESD(op10) = 1.5, ECD(op10) = 2; ESD(op20) = 3.5, ECD(op20) = 4; ESD(op30) = 1.5, ECD(op30) = 2; ESD(op40 = 5.5, ECD(op40) =6.5; ESD(order) = 0, ECD(order) = 7. Note that both paths are critical.

d. ESD(op10) = 0, ECD(op10) = 0.5; ESD(op20) = 0.5, ECD(op20) = 1; ESD(op30) = 0, ECD(op30) = 0.5; ESD(op40) = 3, ECD(op40) = 4; ESD(order) = 0, ECD(order) = 4. Note that the critical path has changed. The *upper* path is now critical. The lead-time margin of the *lower* path is 2.

## Course section 13.7: Subsections and their intended learning outcomes

##### 13.7.1 Exercise: Queues as an Effect of Random Load Fluctuations (1)

Intended learning outcomes: Answer a number of questions using the relevant formulas in queuing theory.

##### 13.7.2 Scenario: Queues as an Effect of Random Load Fluctuations (2)

Intended learning outcomes: Experience average wait time as a function of capacity utilization in a job shop environment with random arrivals, execution of operations in order of arrival as well as operation times subject to a determinate distribution with mean and coefficient of variation.

##### 13.7.3 Exercise: Network Planning

Intended learning outcomes: Calculate a scheduled network with incomplete data for six operations and a start operation (administration time. Determine the critical path.

##### 13.7.4 Exercise: Backward Scheduling and Forward Scheduling

Intended learning outcomes: Explain and solve the forward and backward scheduling problems that is calculation of start and completion dates for the order and each operation, as well as the critical path and lead-time margin.

##### 13.7.5 Exercise: The Lead-Time-Stretching Factor and Probable Scheduling

Intended learning outcomes: Explain and practice the use of the lead-time-stretching factor as well as probable scheduling.

##### 13.7 Scenarios and Exercises

Intended learning outcomes: Assess queues as an effect of random load fluctuations. Calculate examples for network planning, backward scheduling, forward scheduling, the lead-time stretching factor, and probable scheduling.