Study Results

Table 4.4.2 summarises strategy performance measure, Iπ, defined in Equation (4.4.1),

averaged across 100 simulations with associated standard errors given in parentheses. The boxplots in Figure 4.4.2 support the summary statistics in Table 4.4.2, clearly demonstrating the value of accounting for the carryover of incomplete work in Aggregate Planning. For example, we see from theCaboxplot that capturing the flow of incomplete work over the horizon can provide the opportunity to resolve on average 78.5% of the incomplete work resulting from a primary skill allocation for each period independently (Ba).

The baseline model reflects the use of workers’ highest efficiency skills only, with work constrained to the day it is initially planned for. Work left incomplete after allocation will require completion via expensive outsourcing or overtime options. We can therefore interpretICaas an upper bound on the proportion of savings that can be made by using

the existing workforce to resolve carryover instead of paying for extra resources.

With incomplete work carrying over until there are the spare resources to resolve it, a longer planning horizon naturally provides more opportunity to resolve all demand. This

● ● ● ● ● ● ● ● 10 20 30 40 50 60 70 80 90 100 CT Ea CT+Ea Ca Ca+CT Ca+Ea Ca+CT+Ea Strategies % Reduction in ter minal cum ulativ e carr y o v er relativ e to str ategy Ba

Early completion limit

0 1 3

Figure 4.4.2: Case Study: boxplots of percentage reduction in terminal cumulative demand carryover for strategiesπ ∈Π\Ba, relative to strategy Ba, by early completion limit

relationship between the length of the planning horizon,|T|, and the value of modelling carryover is summarised in Figure 4.4.3. The solid line in this plot illustrates how the mean improvement from modelling carryover increases with the length of planning horizon considered. Across an 84-day horizon, the benefit of incorporating carryover measures in allocation (using strategyCainstead ofBa) approaches 80%. It can be seen, however, that much of the value of widening the planning horizon beyond independent single-period allocations can be gained from planning across a much shorter 21-period window. This is illustrated by the solid curve increasing steeply from zero as the length of the planning horizon is increased. Modelling the carryover of incomplete work across just one day (by solving multiple two-day allocation problems with strategy Ca) can lead to resolving on average 25.1% of excess work, with the benefit rising to 58.8% with a window of 3 days.

0 20 40 60 80 0 20 40 60 80

Mean % reduction in ter

minal cum ulativ e carr y o v er relativ e to str ategy Ba

Planning Horizon Length Ca

Ea

Figure 4.4.3: Mean percentage reduction in terminal cumulative demand carryover for strate- gies Caand Ea, relative to strategy Ba, as a function of planning horizon length. The solid line represents the value of using strategyCa. The dashed line represents the value of using strategyEawith an early-completion limit, lj, of 1 day

influenced by the length of the planning horizon however. This is illustrated by the almost stationary dashed curve in Figure 4.4.3. Indeed, allocating using anEa strategy over a window of only 2 days results in almost all the value that an 84-day horizon might provide. More generally, to reap the benefits of an Ea allocation strategy with early-completion limit lj we need only plan over horizons which accommodate lj, i.e.

horizons of length |T|≥lj + 1.

It is important to note the trade-off that exists between utilising spare capacity for the early completion of work versus the picking up of late running work. Comparing the boxplots for strategiesCa and Ca+Ea, we see the marginal benefit of the model allowing the early completion of work is small in comparison to its added value in a non-carryover setting. Late running work takes priority over completing some work early, a property we encouraged by settingaj < cj. Late running work occupies some of

value ofaj (in relation to cj) has on the quantity of work that is advanced is discussed

in more depth in Section 4.5.3.

The day-by-day approach reflected by the baseline model Ba (and any strategy not featuringEa or Ca) is more appropriate for latter stages of planning with short notice before the start of operations. At this higher level of planning, we argue that it is bene- ficial for the planner to open up their horizon of consideration and exploit opportunities for resolving excess work with previous and future spare capacity. Solutions only resort to an alteration in the timing of demand when supply at their intended period is ex- hausted. That said, the reported benefits of these temporal demand flexibility measures should be viewed as upper bounds in application. The early completion or delay of some work will likely not be feasible for all types of demand or during all phases of the planning horizon, with the negative impact on the customer increasing as the notice before operations decreases.

An additional benefit of capturing carryover in modelling is that it allows us to mon- itor the evolution of excess work throughout the planning horizon. The plots in Figure 4.4.4 give an example of the evolution of total cumulative carryover, Hπ

t, across the

horizon (for t ∈ {1, . . . ,|T|}) for a particular time series realisation of demand. Here, early completion of demand was restricted to 1 day. Strategies in isolation and in combi- nation are plotted separately in Figures 4.4(a) and 4.4(b) respectively, with the trace for baseline strategyBa appearing in both as a reference point. Figure 4.4(a) demonstrates the unique flexibility afforded by carryover strategies to contain the amount of cumula- tive carryover over time. This results in excess work being diminished in periods with spare capacity, with the count of excess work maintained at a level below 2000 hours for the majority of periods. Figure 4.4(b) highlights the additional benefit of strategiesCT andEa in combination withCa, with all three such combinations mitigating cumulative incomplete work across the horizon very well.

0 20 40 60 80 0 2000 4000 6000 8000 10000 Ba Ea CT Ca Cum ulativ e carr y o v er Time (days)

(a) Strategiesπ∈ {Ea, CT, Ca}

0 20 40 60 80 0 2000 4000 6000 8000 10000 Ba Ea+CT Ca+CT Ca+Ea Ca+CT+Ea Cum ulativ e carr y o v er Time (days)

(b) Combination strategies π∈ {Ea+CT, Ca+CT, Ca+Ea, Ca+

CT +Ea}

Figure 4.4.4: The evolution of cumulative demand carryover throughout a planning horizon for a single problem instance and demand realisation

The up-shift in cumulative carryover at day 73, seen most clearly in the solid line plot for baseline model Ba, highlights a particularly higher than average level of de- mand pushing incomplete work up significantly. These plots of cumulative incomplete work over time, are of particular use in assessing when these spikes in demand can be absorbed by the existing workforce (using temporal flexibility and cross-training) and how long it may take to restore cumulative carryover to 0. In this case, it takes one week (until day 80) to resolve the impact of this spike in demand so that there are 7 successive days of excess work which could result in some work running 7 days late. This high- lights this period in the horizon as one for which we may consider an injection of extra resources through outsourcing or overtime. Although a similar sized jump in cumulative carryover can be seen at day 36, it is quickly resolved using temporal flexibility and/or cross-training. Solving an Aggregate Planning model which incorporates these flexible strategies aids the identification of problem periods which cannot be easily identified from the time series of demand, or the baseline strategy cumulative carryover alone. It is the balance between supply and demand which dictates a period to be problematic and so identification of such periods relies on the output of an Aggregate Planning model which quantifies the carryover of incomplete work after supply allocation.

The final key observation we draw from Table 4.4.2 concerns the potential gains of considering the utilisation of the cross-trained workforce early in the planning process. The allocation solution provided by the model is designed to provide the scheduler with richer information upon which to make informed decisions about the proportion of time that individual workers should aim to spend on their different skills. Secondary and tertiary skills are more commonly omitted from these early stages of planning and de- ployed as emergency efforts to balance supply and demand at the Operational Planning stage. By planning the utilisation of cross-training early in the horizon we see that on average 34.9% of the incomplete work resulting from a primary-skill only alloca-

tion can be resolved by also considering secondary and tertiary skills in allocation. In combination with the above discussed temporal demand flexibility, we reach a powerful

Ca+CT +Ea planning strategy which sees, on average, up to 91.8% of the terminal cumulative demand carryover resulting from the baseline strategy being resolved.

The marginal gains of incorporating cross-training are rather less in the temporal demand flexibility domain however, adding an additional 4.3% mean improvement to the Ca+Ea model (with lj = 3) compared to the the 58.5% marginal benefit when

whenCa and Ea are not available. Since incomplete work remains in the system when the timing of demand is not totally fixed as, opposed to exiting the system under theBa strategy, the system experiences a greater level of demand, reducing opportunities to exploit secondary and tertiary skills. The spare capacity required to benefit from cross- training is more frequently soaked up in resolving late running work or to accommodate the early completion of work, to the detriment of the utilisation of secondary and tertiary skills. This highlights the strength of cross-training to be in cases where there is limited flexibility to alter the timing of demand delivery. It is therefore important that, when evaluating the benefits of cross-training, the extent to which there is some flexibility to complete work early and the extent to which carryover is a real and present feature of the planning problem should be carefully considered.

The later in the planning horizon that we have the flexibility to amend the timing of demand, the closer the additional benefit of cross-training will be to 4.3%. In or- ganisations that must commit to the timing of demand early in the horizon when it is likely to be subject to further significant change ahead of operations, the value of cross- training will approach the fixed-timing value (strategy CT) of 58.5%. Existing studies into the value of cross-training are universally conducted in the latter-described domain, with restrictions typical of Operational Planning. We argue that organisations should consider modelling early completion of work and carryover to obtain a more accurate

evaluation of the potential value of cross-training.