Descripción global de los instrumentos de evaluación

In a regular distribution pattern, a customer always receives the same quanti-ties with a constant time between the deliveries. As a result, the stock level displays a perfectly regular jigsaw pattern. The formulas for the maximal cy-cle time (2.26), cost rate (2.35) and EOQ cycy-cle time (2.36) are based on this property.

In an irregular distribution pattern, the restriction of equidistant deliveries is dropped. Customers can now receive dierent quantities with diering inter-delivery times, so the stock levels can show irregular jigsaw patterns. Therefore, the formulas for the maximal cycle time, cost rate and EOQ cycle time are no longer valid. Figure 2.8 shows an example of an irregular stock level pattern.

A customer consuming 3 units per hour receives three deliveries in a 20-hour cycle. The delivery quantities q; q⁰; q⁰⁰ are the same, but the time between deliveries varies. As a result, stock builds up and two out of three times, the customer is replenished when there is still stock left (s⁰; s⁰⁰> 0).

- t(h)

@@

@@@

@@

@@@

@@

@@@

@@

@@@

q =20 s=0

q⁰=20

s⁰=5

q⁰⁰=20

s⁰⁰=10

0 5 10 20

Figure 2.8: Irregular stock level pattern

In the cost rate of an irregular distribution pattern, the rst four components are independent of the schedule, but the fth component, the holding cost rate, does depend on the actual schedule and delivery quantities. For the fth component, a closed expression can no longer be found. The holding cost component of Formula (2.35) gives a lower bound for the actual holding cost rate, and thus Formula (2.36) gives an upper bound for the optimal cycle time.

To obtain the actual optimal cycle time and cost rate, an iterative procedure would be needed. Starting with a cycle time given by Formula (2.36), a schedule is constructed. From this schedule, the actual holding cost rate is determined.

Adjusting the formula for the optimal cycle time with this actual holding cost can give a new cycle time. For this adjusted cycle time, a new schedule is constructed, and so on, until the cycle time converges.

For the problem of scheduling the tours in irregular distribution pattern, the mathematical model of Table 2.5 can be recycled. However, in the irregular case the makespan T_sched of the schedule is no longer a variable but a given parameter. The objective is now to minimize holding costs at the customers without violating any of the capacity restrictions. Therefore, the following notations are introduced.

 _i= ^PPj2Sijdj

j2Sidj : the weighted average holding cost rate for the customers in Si.

 q_il: variable that gives the vehicle load to be distributed over the cus-tomers in the l'th iteration of tour i (i 2 1::n; l 2 1::k_i).

 sil: cumulative stock level of the customers in Si at the time of the l'th delivery (i 2 1::n; k 2 1::K).

The mathematical model for the tour scheduling problem in an irregular dis-tribution pattern is shown in Table 2.11.

Constraints (2.41) and (2.42) represent the ow of goods. Constraint (2.43) is the vehicle capacity constraint, and Constraint (2.44) the customer storage capacity constraint. Constraint (2.45), nally, imposes that the total delivered quantity covers the demand during one cycle.

2.3 Modelling with more complex distribution patterns 39

Table 2.11: Mathematical model for irregular distribution pattern scheduling

Minimize 1

The objective of the problem is to choose the delivery timing (t^k_ilvariables) and quantities (qilvariables) such that the total holding cost rate is minimized. The holding cost corresponds to the cumulative surface under the stock level curves (see e.g. Figure 2.8). The objective function is thus quadratic and the compu-tational complexity of this scheduling problem extremely high. Therefore, as for scheduling regular distribution patterns, a heuristic is adopted for solving it (see Section 3.6).

Illustrative example

For our illustrative example, the regular full-truckload distribution pattern has a schedule time that is equal to the maximal cycle time of 120 hours, while the EOQ cycle time is 93:1 hours. If we now allow irregular solutions, this cycle time of 93:1 hours becomes feasible. However, to avoid working with fractional values, we round this cycle time to 96 hours. For the tours to customers 1, 3 and 4, it is possible to keep the deliveries equidistant. For the tour to customer 2, with the incompatible frequency, this is not the case. The second and third delivery have to be started a bit earlier, such that the time between the third iteration in a cycle and the rst iteration in the next cycle is slightly larger than the time between other iterations. Figure 2.9 shows the schedule and the stock level of customer 2 with the cycle time rounded to 96 hours.

- t(h)

1 2 3 1 4 2 1 2 1 4

0 24 48 72 96

- _t(h) HHH

HHHH

HHHHH

93 HH

HHHH

HHH

93

HHHH

HHHH

102

0 8 39 70 96

Figure 2.9: Schedule and stock level for the irregular distribution pattern To have a regular distribution pattern, the vehicle should bring three times 96 units to customer 2, with 32 hours in between. This is infeasible, so the following situation, which is very close to being regular, is obtained. In the rst two visits in a cycle, deliveries of only 93 units are made, so that there is 31 hours until the next delivery. The third and last delivery then has to cover 34 hours and therefore brings 102 units.

Table 2.12 gives an overview of the main characteristics of the irregular distri-bution pattern.

This solution is cheaper than the full-truckload and the powers-of-two distri-bution pattern, but it is still more expensive than the optimal distridistri-bution pattern, which is regular.

2.3 Modelling with more complex distribution patterns 41

Table 2.12: Characteristics of the irregular distribution pattern

T_min 52h

2.3.3 Distribution patterns under driving time