# 13.4 Order Splitting, Order Overlapping, and Extended Scheduling Algorithms

## 13.4.1 Order Splitting, or Lot Splitting

`Order splitting or lot splitting means distributing the lot to be produced by an operation among two or more machines or employees at a work center for processing. This implies split lots.`

Splitting reduces lead time, but it incurs additional setup costs, since employees must set up multiple machines. Figure 13.4.1.1 shows the situation.

Fig. 13.4.1.1       Reducing lead time for operation i by using a splitting factor > 1.

`The splitting factor for an operation expresses the degree of its potential splitting.`

The initial value of the splitting factor is 1, that is, “no splitting.” Where a splitting factor > 1 is given, run time is divided by this value. To calculate the costs of the operation, however, setup load must be multiplied by the splitting factor.

The split lots may be worked on in parallel or be finished at points that are offset in time.

`A split offset factor expresses the possible temporal shift of the split lots, according to the principle illustrated in Figure 13.4.1.2.`

The split offset factor is expressed as a percentage of the operation time after splitting. The initial value of this factor is zero, that is, “no split offset.”

Fig. 13.4.1.2       The split offset factor offsets the split lots in time.

## 13.4.2 Overlapping

`We speak of overlapping within an operation when the individual units of a lot are not produced sequentially, or one after the other, but rather overlap one another.`

Consider the example of an assembly operation for machines. The operation may comprise several partial operations. Figure 13.4.2.1 shows the situation for the lot as a whole.

Fig. 13.4.2.1       The principle of overlapping within an operation.

A later partial operation on the first machine of the lot may be worked on parallel to the first partial operation on a subsequent machine of the lot.

`The run time offset, or offset of the next run time, is a measure for the overlapping within an operation.`

Run time offset is expressed as a percentage of run time. The standard value for run time offset is 100%, or “no overlapping.”

For some production processes, you can overlap entire operations.

`In an operation overlapping or an overlapped schedule, we begin the next operation on a portion of the lot before the entire lot is completed with the previous operation.`

Figure 13.4.2.2 shows an example. Schedulers can use operation overlapping to accelerate a production order.

Fig. 13.4.2.2       The principle of operation overlapping.

`The maximum offset of the next operation is a measure of operation overlapping. It is based on one operation and shows the maximum lapse of time before the next operation begins.`

In practice, the next operation begins immediately after the setup time and run time for the first unit (or first units) of the order lot. (See, for example, near-to-line production in Figure 6.2.2.2).

The initial value of the maximum offset of the next operation is infinite, that is, “no overlapping.” If the time we calculate (based on operation time and inter­operation times) until beginning the next operation is smaller than the actual value, we take the smaller time as the new offset time.

## 13.4.3 An Extended Formula for Manufacturing Lead Time (*)

The following lists the definitions set out in Section 13.3.2 for the components of operation time. Here, we have added the following abbreviations for the elements defined above.

LOTSIZE          :=  lot size ordered
ST[i]                    :=  setup time for operation i
RT[i]                  :=  run time per unit produced for operation i
STREFAC         :=  lead-time-stretching factor
SPLFAC[i]        :=  splitting factor for operation i
SPLOFST[i]      :=  split offset factor expressed as a percentage
RTOFST[i]        :=  run time offset for operation i expressed as a percentage
MAXOFST[i]   :=  maximum offset of the operation immediately following operation i (a duration)

We can express the operation time for an operation i, OT[i], by the formula shown in Figure 13.4.3.1. This formula is much more complex than the one in Section 13.3.2.

Fig. 13.4.3.1       Extended operation lead time.

For a sequence of operations as the order of the operations, the formula in Figure 13.4.3.2 yields the lead time for the order.

Fig. 13.4.3.2       Extended lead time formula (first version).

LTI represents the lead time for LOTSIZE and will vary when lot sizes are different. In Figure 13.4.3.3, we attempt to define partial sums to express lead time as a linear function of lot size.

As in Figure 13.3.2.3, we can store the partial sums in the lead time formula as attributes of the product and recalculate them after each modification of the routing sheet. Correspondingly, the formula according to Figure 13.3.2.4 holds.

Fig. 13.4.3.3       Extended partial sums for the lead time formula.

Fig. 13.4.3.4       Extended lead time formula (second version).

Because of the overlapping of operations, which is expressed in the formula for LTI in Figure 13.4.3.2 as a minimization, LTI is not equivalent to LTI': For either one or the other operation, the maximum offset of the next operation is smaller than the sum of the other time elements (the “normal” time period until the beginning of the next operation).

Figure 13.4.3.5 shows a possible plotting of the two lead times as functions of lot size.

Fig. 13.4.3.5       Influence of overlapping of operations upon lead time.

In most circumstances LTI' is precise enough and certainly suffices for rough-cut planning. If necessary, we can set a lot size limit for the lead time formula.If the lot size is less than or equal to this quantity, we calculate lead time according to the “quick” lead time formula (the second version in Figure 13.4.3.4). Other­wise, we apply the more involved, “slow” formula in Figure 13.4.3.2.

In a directed network of operations as the order of the operations, similar considerations to those examined in Section 13.3.2 apply.

## 13.4.4 Extended Scheduling Algorithms (*)

We can now extend the scheduling algorithms presented in Section 13.3.3 to include the definitions introduced in the subsections above. These include:

• The introduction of a lead-time-stretching factor that multiplies inter­operation times
• The introduction of splitting and overlapping and an expanded formula for lead time
• The inclusion of multiple partial orders for each production order
• The inclusion of divergent product structures, as — for example — the case of temporary assembly
• Ongoing planning for released orders with work remaining to be done

We can derive a generalized algorithm from the algorithm presented in Section 13.3.3, for both a sequence of operations and for a directed network of operations. This would complicate the algorithm further, and we will not present it here in detail.

The extensions introduced thus far may not be sufficient for lead time scheduling in every potential scenario. A first case is the undirected network of operations with a repetition of operations. During a chemical process or in the production of electronic components, for example, production has to repeat certain operations. This may be because inspection has uncovered defects in quality. Here, the number of iterations and the individual operations to be repeated become evident only during the course of work and cannot be planned in advance. In this case, it is not possible to calculate lead time precisely. Instead, we have to use expected mean values for the number of iterations and accompanying deviation. However, we have to take into account that each calculation of lead time itself is based on estimations of the time elements, particularly wait time in front of the work center.

Another case arises in process industries. The processor-oriented concept implemented in these industries may require sequencing or, more precisely, the planning of optimum sequences of operations, as early as the phase of long- and medium-term planning. Because of the extremely high setup costs, planners should establish suitable lots even prior to order release to keep changeover costs at a minimum. To this category belongs, for example, cut optimizations for glass, sheet metals, or other materials. The scheduling of an individual order will depend on whether it may be combined with other orders and with what orders, to achieve optimal usage of the raw material, the reactors, or processing containers.

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