In this section we present a modified model with a more flexible execution of the tango movement by changing the assumptions of section 4.1.1. So far, we strictly assume that a tango rearrangement is performed according to the procedure shown in Figure 4.2. However, the time to perform a tango rearrangement can be shortened as shown in Figure 4.14. Consider a tango rearrangement while one of the storage lanes next to the retrieval position is either in state E or H. In this case, the blocking unit may be restored into the respective lane. Consequently, the load handling device does not need to move back to restore the unit to the rear position of the original storage lane (see Figure 4.14, pictures c and d).
a b c
1 1 1
d
1
Figure 4.14: Possibility to shorten tango movement
The blocking unit is always re-stored to the rear position during the execu-tion of a standard tango. When executing the modified tango as explained above, there are two possible scenarios for the storage lane in which the re-storage takes place:
1. A neighboring storage lane is empty (E) and the blocking unit is re-stored to the rear position, as shown in Figure 4.14. In this way, the time needed to move the LHDs sideways from one lane to a neigh-boring lane, td, is saved.
2. A neighboring storage lane is half-filled (H) and the blocking unit is re-stored to the front position. In this way, the saving is td plus the difference between the time needed to access the rear or the front position, i.e., tLH D,r− tLH D, f.
The savings of the modified tango are depending on the state of the neigh-boring storage lanes and are specified in Table 4.2. As the savings are pos-itive in both cases, the modified tango is always beneficial. Consequently, as soon as one of the adjacent storage lanes provides at least one free posi-tion, the modified tango is performed.
State of the adjacent storage lane
LHD time for re-storage
Savings
E tLH D,r td
H tLH D, f td+ (tLH D,r− tLH D, f)
Table 4.2: Possible savings of performing a modified tango depending on the state of the adja-cent lanes compared to performing a standard tango
For an analytical representation of a QC with a modified tango, the tango probability is split into two different terms: First, the probability that a standard tango is performed, and second, the probability that the modified tango is performed. The probability of the regular rearrangement remains unchanged. To determine which tango is performed, we need the proba-bility of performing a tango, P (R5) (see equation 4.12), combined with the allocation probabilities of the neighboring storage lanes. The probability of performing the modified tango is the conditional probability of performing a tango, given that there is at least one neighboring storage lane in state E or H. For the analytical derivation, we ignore effects at the edge of the rack, where storage lanes have only one neighboring lane.
To derive the unknown probabilities, consider the following: Each storage lane chosen for retrieval has two neighboring lanes which both can either be in state E, H or F. Hence, nine possible combinations of storage lane states can occur, as presented in Figure 4.15.
1 2 3
4 5 6
7 8 9
Figure 4.15: Possible combination for the states of the neighboring storage lanes
Eight out of nine states allow to perform a modified tango, while in the re-maining a standard tango is performed. The probability of state nine is the probability that a standard tango is performed:
P (St and ar d | Tang o) = P(F ) · P(F ) = P(F )2 (4.68)
The probability of performing a modified tango is the probability of the complement, which is:
P (Mod i f i ed | Tang o) = 1 − P(F )2 (4.69)
Equations 4.68 and 4.69 show how to weight the tango probability.
We have to consider that the modified tango causes state transitions differ-ent from those of the standard tango. If a modified tango is performed, the state transitions of the regular rearrangements, i.e., R3 and R4, can be ob-served. The transition R5 only occurs when a standard tango is performed.
Figure 4.16 summarizes this. Allowing for the modified tango, makes R3 and R4 more likely compared to the former model without tango modifica-tion and thus affects the probabilities of R3, R4 and R5. A formulamodifica-tion of the changed probabilities is presented in Appendix A. Consequently, the transi-tion probabilities in the Markov Chain change and the statransi-tionary allocatransi-tion
of the storage lanes change too. By changing P (R | H) and P(R | F ) in the transition matrix (see equation 4.23), the derivation of the adjusted station-ary allocation of the storage lanes is possible. But the solution of the system of equations can only be done numerically and the results are neither ap-propriately presentable nor applicable. Therefore, we set-up an analytical model for the modified tango by applying the storage lane allocation of the former model using P(E), P(H) and P(F) from equations 4.29, 4.30 and 4.31.
𝑃(𝐹) 𝑃 𝐻 + 2𝑃(𝐹)
½ P(SSRR) 1 - ½ P(SSRR)
Tango
P(Standard) P(Modified)
Regular Rearrangement
R3 R4 R3 R4
R5
Rearrangement
Changed state transitions with the modified tango
Figure 4.16: Decision tree showing the rearrangement probabilities including both tango variants
The application of the ‘old’ probabilities generates an error in the state probabilities of the storage lanes, as the model overestimates the proba-bility of a storage lane being in state H, and underestimates the probabil-ity of a storage lane being in state E: For the modified tango, a retrieval of type R3, which is H → E, can occur in combination with a tango. In the former model, tango always means a transition of type F → H, which is represented by R4 or R5. When applying the state probabilities of the for-mer model, we therefore automatically ignore R3 transitions that arise due to the modified tango. As a result, the travel time model for the modified tango is slightly inaccurate.
The approximated cycle time for the quadruple command cycle with the modified tango is as follows:
E (QCd d)Mod .Tang o= t0+5 2· (vx
ax+vy
ay) + E(QC )N· L vx
+ P (SSRR) · PRear r ang e· [P (F )2· tTang o+ (1 − P (F )2) · tTang o,mod .] + (1 + (1 − P (SSRR))PRear r ang e· E(RC ))
+ 4 · E(tLH DS ) + 4 · E(tLH DR ) − 2 · tLH D, f
(4.70)
Different from tTang o, the time required to perform the modified tango, tTang o,mod ., is stochastic. To determine tTang o,mod ., we make the following assumption: If both neighboring lanes are available for a modified tango, the storage lane to perform the rearrangement to is randomly chosen. Fig-ure 4.15 illustrates that three cases allow a modified tango to be performed into an empty storage lane (cases 1,3 and 7) and another three cases allow a modified tango to be performed into a half filled storage lane (cases 5,6 and 8). For the remaining cases (2 and 4), both options are possible and thus they are weighted with1/2each.
Re-storage Notation Probability
Rear
position P (Mod , r ear ) P (E )2+2P (E)P (F )+1/22 P (E )P (H )) (1−P(F )2)
Front
position P (Mod , f r ont ) P (H )2+2P (H)P (F )+1/22 P (E )P (H ) (1−P(F )2)
Table 4.3: Probabilities of the re-storage position during a modified tango
Based on the probabilities shown in Table 4.3, the time needed to perform a particular modified tango is the standard tango time, tTang o, minus the savings presented in Table 4.2. Consequently, the mean time to perform the modified tango is
tTang o,mod .= P (Mod, r ear ) · (tTang o− td)
+ P (Mod, f r ont ) · (tTang o− td+ (tLH D,r− tLH D, f)) (4.71)
The idea of the modified tango can be applied to every execution order of the QC. The time of the modified tango, tTang o,mod ., is calculated using the respective stationary allocations of the storage lanes (P(E), P(H) and P(F)).