3.4 Numerical study
3.4.2 Benchmarking simple priority rules against the optimal policy 40
In this section, we benchmark simple rules against the optimal policy. Tables 3.7-3.10 present the performance of simple rules based on the percentage cost gap. For a simple rule, the percentage cost gap is defined as the percentage increase in the
value of the objective function when the rule is employed instead of the optimal policy (CostRuleCost−CostOpt
Opt ·100%). The simple rule that provides the smallest percentage cost gap is marked in bold in each case for each cost pair.
Percentage cost gap Percentage cost gap
using PA using DC
Table 3.7Effect of TL’s on the percentage cost gap of simple priority rules
The percentage cost gap resulting from the use of the EDD rule is observed to be 0% in Tables 3.7-3.10 whenever the tardiness penalty is proportional to tardiness (fixed cost=0), as shown by Duenyas and Hopp (1995). Thus, in Figures 3.1-3.4, the system performance under EDD is represented by a point that is exactly at the same level on the y-axis as the optimal policy for the (0,1) cost pair. It falls slightly to the left since the on-time probability obtained by employing the EDD rule is slightly worse. Although the resulting long-run average costs are exactly the same, the priority sequencing decisions given by the optimal policy under PA and DC for the (0,1) cost pair are not identical to each other and to the ones given by the EDD rule. In other words, making priority sequencing decisions differently for some states does not result in a different objective value when fixed cost=0.
Suppose that the system is in a state where (r1, r2) = (3,1) and the position of a new arriving order with CRL= 2 is decided. The EDD rule suggestsk = 3, while the optimal policy for (0,1) cost pair under the PA strategy suggests k = 2. The total variable tardiness cost incurred for these three orders is the same under both decisions, no matter the processing times of orders. If each order takes one period to process, incurred total variable tardiness cost is: 0 + 1 + 1 = 2 under k= 3, and 0 + 0 + 2 = 2 underk= 2. On the other hand,k = 2 leads to an on-time completion of the arriving order (with CRL = 2), while the decision k = 3 leads to its late completion. The two decisions are different in terms of the number of late orders they result in, but this is not reflected in the objective value when fixed cost=0.
Whenever a fixed cost of tardiness is involved, the optimal policy deviates from the EDD principle. Adhering to EDD results in a large percentage cost gap
Percentage cost gap Percentage cost gap
Table 3.8Effect of utilization (ρ) on the percentage cost gap of simple priority rules
when the relative value of the fixed cost is high. Furthermore, an increase in the utilization level (ρ) and in the processing time parameter (β), and a decrease in the CRL variance (σ), increase this gap (Tables 3.8-3.10, respectively). Under a high utilization level (ρ= 0.85), the percentage cost gap resulting from the use ofEDD for the pair (1,0) exceeds 56%. This is due to the effects observed in Figures 3.2-3.4.
In other words, the potential benefit provided by policies appropriate for use when there is a fixed cost (e.g. EDDotpP A) is higher when the utilization is higher (Table 3.8), the expected processing time is shorter and the variability in processing times is lower (i.e. β is higher Table 3.9), and CRL variability is lower (Table 3.10).
Percentage cost gap Percentage cost gap
using PA using DC
Table 3.9Effect ofβon the percentage cost gap of simple priority rules
The potential benefit gained by using policies that mitigate the amount of tar-diness, such as EDD, increases with a decreasing utilization level (ρ), increasing processing time parameter β (lower mean and lower variance in processing times) and increasing CRL variance (σ). This can be seen from the increase of the per-centage cost gap resulting from the use of theF CF S discipline under the (0,1) cost pair in Tables 3.8, 3.9 and Table 3.10, respectively.
Percentage cost gap Percentage cost gap
Table 3.10Effects ofCRLmean (µ) and variance (σ) on the percentage cost gap of simple priority rules
When the PA strategy is considered, the simple rule with the smallest percentage cost gap moves from EDD to EDDP Aotp and then to LDD as the relative value of the fixed cost increases. This effect is observed in all instances except the one where the variability in customer-required leadtimes is high (σ = 3). Whenσ = 3,F CF S performs closer to optimal than EDDotpP A, as can be seen from Figure 3.4 as well as from Table 3.10 where the best performing rule for the cost pair (4,1) is F CF S under both strategies.
When the DC strategy is considered, the percentage cost gap resulting from the use of LDD is rarely the smallest. Only for the cost pair (1,0) in instances 6 and 9 in Tables 3.9 and 3.10 this is the case. Thus, as the relative value of the fixed cost increases, the simple rule with the smallest percentage cost gap moves from EDD toEDDotpDC and it stays there for further increases, in all other instances. In the worst case among the ones considered here, the percentage cost gap resulting from the use of the best performing simple rule is 3.9 under the PA strategy and 6.0 under the DC strategy (for instances 8 and 7 in Table 3.10). Thus, results show that the simple rules perform well in comparison to the optimal.
This is an important result from the practical point of view because, in terms of ease of applicability, simple rules are superior to the optimal policy in the following sense.
First, simple rules are given policies, i.e. no computational effort is required to obtain them. Second, as discussed by Hopp and Spearman (2001), they only require
polynomial time sorting algorithms for their implementation. The computational complexity of sorting n orders isO(nlogn) when the queue is maintained unsorted.
The complexity is less in the sorted case. On the other hand, as discussed by Littman (1997), using linear programming, the MDP can be solved in polynomial time in the number of states. However, the number of states increases exponentially in problem size. Thus, the computational time required for obtaining the optimal policy grows exponentially in problem size.
3.4.3. Comparing simple priority rules with each other
In this section, we compare simple rules with each other. Table 3.11 presents the performance of simple rules based on the on-time probability (η) and expected tardiness (E[T]) measures.
When all arriving customers are willing to wait for the same amount of time, i.e.
Dist 1, 2 or 3 applies for theCRLdistribution, the earliest-due-date always belongs to the order that had arrived first. Thus, the simple rules F CF S and EDD are equivalent and the results presented in Table 3.11 show the sameηand E[T] values for EDDand F CF S in instances 7-9. Furthermore, both Figure 3.4 and instances 9-11 in Table 3.11 show that increasing customer-required leadtime variability (σ) increases their performance difference with regard to both measures.
In addition, the following observations can be made from all instances in Table 3.11. The on-time probability is worse under EDD than it is under the basic F CF S wheneverσ >0. This is an interesting result since theEDDrule takes into account the information regarding due-dates while F CF S does not. The reason for this effect is the following: in all instances investigated up-to now (base case and instances 1-11), the maximum customer-required leadtime is 7, on the other hand, the expected production leadtime is 4.937 in the base case (E[W] is obtained using (3.13), equivalently one can consider (3.22) underγ = 0.38, β = 0.5,K = 5).
Thus, a situation in which orders with already reached due-dates are still in the system, may frequently occur. This means that under the EDD rule, the arriving orders with close due-dates are kept waiting for orders that are already late to be completed, which also prevents them from being completed on-time and hurts the on-time probability measure.
On the other hand,EDDotpP AandEDDDCotp treat the queuing orders with no chance of on-time completion differently andLDDgives the highest priority to the order with the highest probability of being completed on-time. Therefore, as also visualized
Instance ρ β µ σ F CF S EDD EDDotpP A EDDDCotp LDD Base 0.7 0.5 4.86 1.88 η 0.596 0.583 0.655 0.709 0.697 (TL 3) E[T] 1.606 1.565 1.742 1.830 2.015 4 0.85 0.49 4.86 1.88 η 0.455 0.435 0.563 0.644 0.635 (TL 3) E[T] 2.461 2.412 2.708 2.837 3.094
Table 3.11The customer-related performance of simple priority rules
in Figures 3.1-3.4, they provide better on-time probability performance with larger expected tardiness thanF CF S. Of these three rules, theEDDotpDC rule provides the best on-time probability in all instances except 6 and 9, in which the best on-time probability is obtained by employing LDD. For instance 5, η under EDDotpDC is slightly better than η under LDD (in fourth digit after the comma).
WhileLDDgives the second best on-time probability whenever it does not give the best, it always results in the largest expected tardiness. This means that a trade-off between two measures is apparent between EDDP Aotp and LDD but EDDDCotp outperformsLDDby providing a better performance based on both, in all instances except 6 and 9.
3.4.4. Comparing simple priority rules with each other in larger