Determining tour frequencies

Once the subset of customers assigned to a vehicle is partitioned into sub-subsets and tours are constructed for each sub-subset, the frequencies for these tours in the distribution pattern have to be determined. These frequencies should be such that the resulting cost rate (and thus also the resulting reduced cost rate) is minimal.

Every tour in a distribution pattern has its individual optimal cycle time (see Section 2.1). Thus, the frequencies should be chosen such that the resulting cycle times for the tours are as close as possible to their respective optimal

cycle times. If the optimal cycle time of tour i is denoted T_i^, then we are looking for a distribution pattern cycle time T and frequencies ki such that T=ki T_i^.

The smallest minimal cycle time that can be obtained for a distribution pattern is when all frequencies ki are 1 (see Formula (2.25)). This is in fact the multi-tour solution. If the optimal cycle times of the multi-tours are all smaller than this minimal cycle, the multi-tour gives the best result. Increasing frequencies would result in an increase of the distribution pattern cycle time and take the resulting tour cycle times further away from their optimal values, making the distribution pattern more expensive. For these cases, the tour frequency determining procedure introduced below need not be run.

If the optimal cycle times of the tours are not smaller than the cycle time of the multi-tour solution, it does make sense to have frequencies higher than 1. To determine the appropriate tour frequencies, we propose an iterative procedure that will evaluate a number of alternatives and keep the best one. In this procedure, we start from initial frequencies chosen such that there is at least one tour with frequency 1, for simplicity. The frequencies are then iteratively increased one by one until a stopping criterion is met. The obvious way to determine initial frequencies is the following: k_i =j

maxk(T_k^) T_i^

k; 8i. The obvious

stopping criterion is then to increase frequencies until kil

maxk(T_k^) T_i^

m; 8i.

As explained in Section 2.3, idle time is necessary in the schedule of a regular distribution pattern to make sure that deliveries are equidistant. For `incom-patible' frequencies (e.g. k1 = 2 and k2= 3), this idle time can be very high, leading to a schedule time Tsched that is higher than the maximal cycle time T_max, which makes the distribution pattern infeasible. For distribution pat-terns with powers-of-two frequencies, much less idle time is needed, and the chance of having an infeasibility is much smaller. Therefore, we decided that the initial frequencies of the tours should be powers-of-two. Instead of the smallest integer smaller than ^max_T^k^(Tk)

i , the frequency is initialized with the smallest power-of-two smaller than ^max_T^k^(Tk)

i . The stopping rule also changes:

instead of stopping at the next integer value, we now stop at the next power-of-two value.

1. The frequencies are initialized by rounding down to power-of-two values:

ki = 2



log₂(^{maxk(T }_{T } ^{k )}

i )

. Cycle times and cost rates are calculated and a schedule is constructed for these initial frequencies.

2. The tour in the distribution pattern for which k_iT_i^ is minimal, has its frequency increased by one. If this frequency is now more than double of the initial value (k_i > 2¹⁺



log₂(^{maxk(T }_{T } ^{k )}

i )

), the process stops. If not, changes in cycle times and cost rates are calculated.

3.5 Determining tour frequencies 59

3. A schedule is constructed. If it is feasible and the cost rate has decreased, these frequencies are kept. The process returns to Step 2.

The frequency determining procedure as described above is somewhat ine-cient. In Step 2, a schedule is being constructed for every frequency combina-tion that is encountered. This is not always necessary, because in some cases, we can already conclude that the considered frequency combination will not give a better result, without actually constructing the schedule. We have found four situations for which this is true, based on the following considerations.

 Increasing a frequency always leads to an increase of the minimal cycle time Tmin, but not necessarily in an increase of the maximal cycle time Tmax. The schedule time Tsched may even decrease, if the frequencies become more compatible. A useful measure of frequency compatibility (FC) is the least common multiple (lcm) of all frequencies k_i: FC = lcm(k_i). If this least common multiple decreases by increasing one of the frequencies, it means that the frequencies have become more compatible.

E.g. the combination (1; 2; 3) is less compatible than (1; 2; 4), because lcm(1; 2; 3) = 6 is bigger than lcm(1; 2; 4) = 4.

 We are looking for frequencies k_i and a cycle time T such that the result-ing tour cycle times are close to the optimal tour cycle times: T=k_i T_i^. A useful measure of frequency optimality (FO) is the sum of the (absolute values of the) dierences between T=k_i and T_i^: FO =P

ijT=k_i T_i^j.

The four situations for which constructing the schedule can be skipped, are the following:

(I) If T_min has become bigger than T_max, the frequency combination is in-feasible.

(II) If all k_i are even, the frequency combination gives the same result as the combination of which it is the double.

(III) If (i) the previous frequency combination was infeasible, (ii) T_max has not increased and (iii) the frequency compatibility measure FC has not decreased, then the current frequency combination will also be infeasible.

(IV) If the frequency optimality measure FO (with T the optimal cycle time in the interval [T_min; T_max]) has increased and the previous combination was feasible, then the current frequency combination cannot be better than the previous one.

The procedure to determine the right tour frequencies taking into account these considerations is then as follows.

1. The frequencies are initialized by rounding down to power-of-two values:

k_i = 2^{blog2(maxk(T T i k ))c}. Cycle times and cost rates are calculated and a schedule is constructed for these initial frequencies.

2. The tour in the distribution pattern for which kiT_i^ is minimal, has its frequency increased by one. If this frequency is now more than double of the initial value (k_i > 2^{1+blog2(maxk(Tk)Ti )c}), the process stops. If not, changes in cycle times and cost rates are calculated. If none of the four situations (I), (II), (III) and (IV) described above occurs, the process goes to Step 3. Else, this step (2) is repeated without going to Step 3.

3. A schedule is constructed. If it is feasible and the cost rate has decreased, these frequencies are kept. The process returns to Step 2.

Illustrative example

To illustrate the frequency determining procedure, we apply it to our small 4-customer example that was introduced in the previous chapter. In the dis-tribution pattern that has a separate tour for each of the customers, the tours have the following optimal cycle times: T₁^ = 30h, T₂^ = 29:81h, T₃^ = 73:03h and T₄^ = 36:51h. Table 3.1 shows the dierent iterations when determining the frequencies of these four tours.

Table 3.1: Determining the frequencies for the illustrative example k_i FC FO (arg)min_i(k_iT_i^) Schedule Cost rate

1 2; 2; 1; 2 2 19:73 59:63 (2) yes 74:75

2 2; 3; 1; 2 6 29:35 60:00 (1) no (IV)

-3 -3; -3; 1; 2 6 20:27 73:03 (3) yes 74:68

4 3; 3; 2; 2 6 36:71 73:03 (4) no (IV)

-5 3; 3; 2; 3 6 34:73 89:44 (2) yes infeasible 6 3; 4; 2; 3 12 41:85 90:00 (1) no (III) -7 4; 4; 2; 3 12 16:71 109:54 (4) yes infeasible 8 4; 4; 2; 4 4 19:73 119:26 (2) no (II) -9 4; 5; 2; 4 20 25:35 120:00 (1) no, STOP

From iteration 5 on, the solutions are infeasible, because the schedule times become higher than the maximal cycle times. Iteration 8 is the double of the rst iteration and is therefore skipped. In iteration 9, a frequency of 5 appears, which is more than the double of the initial frequency 2, so the process stops there.