The Cost of Completing Work Early

We conclude the study with a discussion on the impact that the early completion cost

aj has on the quantity of work recommended for early completion; and hence on the

performance of the Ea strategy. Table 4.5.9 demonstrates the leverage that aj has on

the performance of this strategy alone, with aj = 0.1 leading to an additional 35.7%

reduction in terminal cumulative carryover relative to the aj = 0.9 case. The more

between carryover and early completion of demand in the discussion around Table 4.5.3 and Figure 4.5.1. The presence of carryover reduces the spare capacity opportunities that the Ea strategy is able to capitalise on when used in isolation. It is clear from these studies that so long as the timely spare capacity exists, early completion of work will be exploited to positive effect. This is true, whether carryover is modelled or not, whenaj < cj.

Early Completion Cost Strategy (π) aj = 0.1 aj = 0.9

Ea 64.4 (0.110) 28.7 (0.066) CT+Ea 81.5 (0.139) 64.3 (0.140) Ca+Ea 85.0 (0.128) 82.0 (0.121) Ca+CT+Ea 92.3 (0.128) 90.9 (0.123)

Table 4.5.9: Mean (standard error) percentage reduction in terminal cumulative demand car- ryover forEa-type strategies, relative to strategyBa, by early completion cost

This condition is not required for the utilisation of the Ea strategy however. Recall from Section 4.3.2 that the definition for the quantity of incomplete work,δj,t,t+1, differs for models which include or exclude carryover. This means that the cost trade-off between using spare capacity to resolve late running work versus completing work early differs for π∈ΠCa and π∈Π\ΠCa.

Consider a quantity of incomplete work,gj, for skill j in period t and suppose there

exists sufficient spare capacity in periodt−1 to completeall of this work early. In models that do not capture carryover, constraints (4.3.9) define δj,t,t+1 such that incomplete work incurs a one-off cost before being removed from the system. We will therefore take up the opportunity to complete this work early when the one-off cost of moving it to the previous period is less than the cost of letting it run on as incomplete work, i.e. if ajδj,t,t−1 ≤ cjδj,t,t−1. Since we assume we have enough capacity in period t−1 to accommodate gj we would therefore take the opportunity to move it all so that this

inequality becomesajgj ≤cjgj. The exploitation of early completion as a strategy then

reduces to the one-to-one comparison of costs aj ≤ cj so that non-carryover strategies

would be costed out if aj > cj.

Consider now the same situation in a strategy that does incorporate carryover. Con- straints (4.3.3) define δj,t,t+1 such that incomplete work continues to incur a cost for every subsequent period that experiences demand which outstrips supply. That is, until we reach a future period of spare capacity which we can use to resolve late running work, we continue to incur the cost cjgj. Suppose that there is no spare capacity for

skill j work in the p periods that follow t, i.e. until period t+ (p+ 1). We will utilise early completion of demand ifajgj ≤(p+ 1)cjgj, i.e. ifaj ≤(p+ 1)cj. This means that

the active range for aj is greater for strategies which model carryover.

This property is illustrated in Figure 4.5.2 for a fixed problem instance. For a single demand simulation the proportion of total demand identified for early completion is defined by P|T| t=1 Pt−1 τ=(t−lj)+δjtτ P|T| t=1djt .

For a range of early-completion costs,aj, we average this value over 100 simulations to

give plots of the mean percentage of demand completed early. When carryover is not modelled, we see the quantity of demand moved to an earlier period falls to 0 when

aj > cj. When carryover is captured in allocation, a non-zero quantity of demand is

completed early even when aj > cj. Note that the inclusion of carryover reduces the

volume of demand completed early.

The value of p, the number of successive under-supplied days which follow a given period, clearly differs for each skill and at each period in the planning horizon. It is a quantity hidden within the optimisation process. This strong relationship between the number of periods for which an hour of carryover continues to incur a cost and the cost

0.0 0.5 1.0 1.5 2.0 0 5 10 15 20 ● ●●● ●●●●● ●_●● ●●●●● ●●● ●●●●● ●● ● ● ● ● ● ● ● ● ●_● ●● ● ●●_{●●●●● ●●●} ●_{●●●● ●● ● ● ●}

Mean % demand completed ear

ly

Cost of early completion

Early−completion limit 1 Day 3 Days

(a) Non-carryover strategiesπ∈Π \ΠCa

0.0 0.5 1.0 1.5 2.0 0 2 4 6 8 ● ●●●●●●●● ●●●●●●●● ●●●● ●●●● ●● ● ● _● ● ●●_● ●●●_● ● ●●●●●●●● ●●● ● ●●●● ●● ● ● ●

Mean % demand completed ear

ly

Cost of early completion

Early−completion limit 1 Day 3 Days

(b) Carryover strategiesπ∈ΠCa

of moving work to an earlier period can be utilised in practice. When the cost aj is

not directly measurable it can be used as a lever within the optimisation model. For example, we might chooseaj to reflect the number of days, q, we can conceivably allow

work to run late without violating service level agreements. This would simply involve settingaj ≤(q+ 1)cj.

All studies presented in this chapter have featuredaj ≤cj, reflecting a belief that we

would always rather look to move work to an earlier period than let it run evenq+ 1 = 1 day late. Note that model 4.3.2 is not restricted to this case however.