5. Capítulo 5
5.1 Sistematización plan de intervenciones
When driving time restrictions are imposed, the minimal and maximal cycle time of a distribution pattern have to be expressed as an integer number of days. For the maximal cycle time TMAX, this is done by simply extending the formula for Tmax.
As for multi-tours, the minimal cycle time TMIN is no longer given by a closed expression, but requires solving a mathematical programming problem. The di erent iterations of the di erent tours that a vehicle makes have to be assigned to days in the cycle such that the driving time of the vehicle is less than 8 hours on any given day. The minimum number of days for which this can be done determines the minimal cycle time TMIN. If all tours are between 4 and 8 hours long, no two tours can be made on the same day by the same vehicle without violating the 8-hour constraint and the number of days needed in the cycle is equal to the total number of tours K = Pn
i ki. If tours shorter than 4 hours exist, making more than one tour on a single day may be feasible and the minimal cycle time TMIN may reduce. As such, K is an upper bound on the minimal cycle time.
The mathematical model for determining TMIN is given below. In this model, the parameter TDiagain represents the time needed to nish tour i (i.e. TDi= t+ TT SP (Si+)+P
j2Sitj) and the binary variable Xilt is used to indicate whether the l'th iteration of tour i is made on day t or not (i 2 1::n; l 2 1::ki; t 2 1::n).
Minimize TMIN subject to:
XK t=1
Xilt = 1 i = 1::n; l = 1::ki (2.47)
ki
X
l=1
Xilt 1 i = 1::n; t = 1::K (2.48) Xn
i=1 ki
X
l=1
T DiXilt 8 t = 1::K (2.49) Xn
t=1
t Xilt TMIN i = 1::n (2.50) Xilt 2 f0; 1g i; t = 1::n
Constraint (2.47) ensures that all iterations of each of the tours are made once during the cycle, while Constraint (2.47) imposes that not two iterations of the same tour are on the same day. Constraint (2.49) is the driving time restriction of 8 hours per day. Constraint (2.50), nally, de nes the minimal cycle time, i.e. the minimal number of days needed to nish all iterations of all tours.
Determining the optimal cycle time and cost rate and developing the actual schedule for a distribution pattern under driving time restrictions again re-quires a di erent approach based on whether the customers impose equidistant deliveries or not.
Regular distribution patterns
When driving time restrictions apply, we assume that a tour cannot be made more than once per day. For regular distribution patterns, this means that the time between two iterations of a tour, T=ki has to be an integer number of days. As a result, the cycle time T of the distribution pattern, expressed in number of days, has to be an integer multiple of all frequencies ki. Thus, the cycle time is always an integer multiple of the least common multiple of all frequencies ki. This least common multiple de nes what we call the base cycle time B. E.g. if a distribution pattern makes two tours, one with frequency 2 and another with frequency 3, then the cycle time of the distribution pattern has to be a multiple of B = 6 days if equidistant deliveries are required.
If the vehicle can only drive on weekdays and not during the weekend, the cycle time of the distribution pattern also has to be an integer multiple of 5. Then, the time between consecutive deliveries to any customer is an integer number of weeks, which makes sure that no delivery ever occurs during weekends. In
2.3 Modelling with more complex distribution patterns 43
this case, days 6 and 7 in a schedule are not Saturday and Sunday, but Monday and Tuesday of the next week.
Based on this base cycle time B, the minimal and maximal cycle time can also be adjusted. In the mathematical model for determining the minimal cycle time TMIN, the following constraint can be added.
TMIN mod B = 0 (2.51)
The formula for the maximal cycle time TMAX changes as follows.
TMAX = B
Once a cycle time of a certain number of days is given, the tours in a distribution pattern have to be assigned to days such that (i) the vehicle drives less than a pre-speci ed number of hours (usually 8) per day, and (ii) the number of days between consecutive iterations of the same tour is constant. The scheduling problem arising here is thus an extended version of the scheduling problem for multi-tours. Table 2.13 gives a mathematical model for this problem.
In the model, the binary variable Xilt indicates whether the l'th iteration of tour i is done on day t or not (i 2 1::n; l 2 1::ki; t 2 1::TMAX, while the integer variable Til represents the day on which the l'th iteration of tour i is made (i 2 1::n; l 2 1::ki). TS is the schedule time expressed as a number of days.
The objective is again to construct a schedule with a minimum idle time and thus with a minimal makespan.
Constraint (2.53) xes a starting point in the cycle by stating that the rst iteration of the rst tour has to be made on the rst day. Tour i has to be made kitimes during a cycle, and TMAX is the upper bound on the cycle time. Thus, the rst iteration of tour i should be on day TMAXki at the latest, the second iteration on day 2TMAXki at the latest, and so on. Similarly, because of the lower bound TMIN on the cycle time, the second iteration of tour i can be made on day TMINki +1 at the earliest, the third iteration can be on day 2TMINki +1 at the earliest, and so on. This is imposed by Constraints (2.54), (2.55) and (2.56).
Constraint (2.57) derives the values of the dummy Til variables from the Xilt variables. Constraint (2.58) imposes that deliveries are equidistant, such that the number of days between two iterations of the same tour is always the same, and Constraint (2.59) ensures that the last iteration of tour i is done within the time horizon TS. Note that if TS is smaller than TMAX, all Xilt-variables become zero for t > TSCHED. The actual driving time restriction of 8 hours per day is given in Constraint (2.60). Constraint (2.61), nally, states that the cycle time has to be an integer multiple of the base cycle time B.
In this model, it is assumed that the minimal cycle time is known. However, this requires solving another, related model. To avoid this, the following lower bound on the cycle time TM can be used instead of TMIN.
Table 2.13: Regular scheduling with driving time restrictions Minimize TS
subject to:
X111 = 1; T11= 1 (2.53)
(l 1)XTMINki
t=1
Xilt = 0
i = 1::n
l = 2::ki (2.54)
lTMAXXki
t=(l 1)TMINki +1
Xilt = 1
i = 1::n
l = 1::ki (2.55)
TXMAX
t=lTMAXki +1
Xilt = 0
i = 1::n
l = 1::ki (2.56)
Til=
TMAXX
t=1
t Xilt
i = 1::n
l = 1::ki (2.57)
Ti;l+1= Til+ TS=ki
i = 1::n
l = 1::ki 1 (2.58)
Tiki TS i = 1::n (2.59)
Xn i=1
ki
X
l=1
TDi Xilt
8 t = 1::TMAX (2.60)
TS mod B = 0 (2.61)
Xilt 2 f0; 1g; Til2 1::TMAX
8<
:
i = 1::n l = 1::ki
t = 1::TMAX
2.3 Modelling with more complex distribution patterns 45 Constraints (2.54), (2.55), (2.56) and (2.60) constitute a generalized assign-ment problem, which is known to be NP-hard [10]. This scheduling problem is thus also NP-hard. Therefore, a heuristic solution approach is developed in Section 3.6.
Illustrative example
If we assume that demand only occurs during the 8 working hours of a day, then from all proposed solutions for our illustrative example only two solutions remain valid when the driving time restriction is imposed.
Because of the frequencies of 4, 3, 1 and 2, the cycle time of the full-truckload distribution pattern has to be an integer multiple of 12 days or 96 hours. The only possible cycle time is 120 hours, which is not a multiple of 96 hours, so the full-truckload distribution pattern becomes infeasible.
In the 4-tour distribution pattern with adjusted frequencies, the least common multiples of the frequencies is 6, so the cycle time has to be a multiple of 48 hours. However, there is no multiple of 48 hours between 84 and 90 hours, which is the range of possible cycle times for this distribution pattern, so it becomes infeasible as well.
Next is the powers-of-two distribution pattern. The least common multiple of its frequencies is 2, so the cycle time has to be a multiple of 16 hours. The only feasible cycle time in the possible interval between 44 and 60 hours is at 48 hours, or 6 days. The resulting schedule is shown in Figure 2.10. The tours to customers 2 and 4 have to be made on the same day, which is perfectly possible because they take only 4 hours each. Because the cycle time changes, the cost rate also increases to 77:30 euro per hour, or 618:40 euro per day.
- t (days)
1 3 2 4 1 2 4
1 2 3 4 5 6
Figure 2.10: Schedule for the powers-of-two distribution pattern under driving time restrictions
The optimal solution is the other solution that is still feasible when the driving time restriction is imposed. The cycle time is 72 hours, which is exactly 9 days.
This is an integer multiple of the least common multiple of all frequencies, which is 3. A schedule is shown in Figure 2.11. The tour to customer 1 is done on days 1, 4 and 7, the tour to customers 2 and 4 is done on days 2, 5 and 8 and
the tour to customer 3 is done on day 3. The cost rate of this solution is still the same, 74:24 euro per hour, or 593:9 euro per day. The optimal solution thus remains optimal under the driving time restriction.
- t (days)
1 1 1
2/4 3 2/4 2/4
1 2 3 4 5 6 7 8 9
Figure 2.11: Schedule for the optimal distribution pattern under driving time re-strictions
Irregular distribution patterns
For irregular distribution patterns under driving time restrictions, the cycle time does not have to be an integer multiple of the tour frequencies ki, but merely an integer number of days. In the resulting scheduling problem, all iterations of each of the tours have to be assigned to a day in the cycle with the objective of minimizing the total holding cost at the customers. A mathemat-ical model for this scheduling problem is obtained by discretizing the model of Table 2.11. However, the model is not presented here. As all other ver-sions of the scheduling problem, this is an NP-hard problem and the heuristic procedure presented in Section 2.5 is used for solving it.
The full-truckload distribution pattern of our illustrative example has an opti-mal cycle time of 93:1 hours, which is close to 96 hours or 12 days. A possible schedule is shown in Figure 2.12. In this solution, the deliveries to customers 1, 3 and 4 are equidistant. The resulting stock levels for customer 2, who does not receive perfectly equidistant deliveries, are also shown in Figure 2.12. The cost rate of this full-truckload distribution pattern is 607:7 euro per day. It is thus cheaper than the regular full-truckload distribution pattern without driv-ing time restrictions, which costs 77:67 euro per hour, or 621:4 euro per day, but still more expensive than the optimal, regular distribution pattern shown above.
Another regular distribution pattern solution that becomes infeasible under driving time constraints is the 4-tour distribution pattern with adjusted fre-quencies. When deliveries no longer have to be equidistant, it still gives a feasible irregular distribution pattern under driving time constraints. A pos-sible schedule with a cycle time of 11 days is shown in Table 2.13. In this schedule, customers 3 and 4 still have equidistant deliveries, but customers 1 and 2 no longer have a perfect jigsaw stock level curve. Due to their increased holding cost rates, the cost rate of this distribution pattern increases from 597:4 to 608:6 euro per day.
2.4 Conclusion 47
Figure 2.12: Irregular schedule for the full-truckload distribution pattern
- t (days)
1 4 2 3 1 2 4 1 2
1 2 3 4 5 6 7 8 9 10 11
Figure 2.13: Schedule for another irregular distribution pattern
2.4 Conclusion
When using the concept of tours, it is assumed that a separate vehicle is avail-able for each tour. In other words, the assignment of tours to vehicles is not considered. In long-term routing problems, the vehicle eet becomes variable and a xed cost per vehicle has to be included. As a result, the assignment of tours to vehicles can no longer be neglected. The concept of tours then be-comes inadequate. In this chapter, we introduce a generalized routing concept, to be used in long-term routing problems where a cyclic solution approach is used. This generalized concept is the `distribution pattern', a cyclic routing scheme for a single vehicle consisting of multiple tours that can be repeated with di erent frequencies.
In the concept of distribution patterns, many real-life characteristics and re-strictions can be considered. A maximal cycle time is imposed resulting from the vehicle capacity and the limited customer storage capacities. A minimal cy-cle time is imposed, taking into account driving, loading and unloading times, but also customer visiting frequency restrictions.
Another important feature is the scheduling problem that arises in designing distribution patterns. The nature of this scheduling problem depends on the presence of two other restrictions. The rst extra restriction that can be im-posed is that deliveries have to be equidistant, resulting in so-called `regular' distribution patterns. The second restriction is the driving time restriction, imposing that a vehicle cannot drive more than a certain number of hours
(usually 8) per day, resulting in a discrete version of the scheduling problem.
Although solution approaches for the di erent situations are presented in the next chapter, we will make the realistic assumptions in our computational ex-periments that deliveries have to be equidistant and driving times are restricted to 8 hours per day.
In distribution patterns, vehicle capacity is reused for multiple customer sub-sets, thus reducing the required vehicle eet size. Furthermore, allowing di er-ent tours to be made by the same vehicle, possibly with di erdi er-ent frequencies, gives the opportunity to obtain cycle times closer to the individual optimal tour cycle times, resulting in a better balance between distribution and hold-ing costs. Distribution patterns are therefore the appropriate concept for the long-term, cyclic inventory routing problem in which a three-way cost trade-o has to be found between (i) xed vehicle eet costs, (ii) distribution costs, and (iii) stock holding costs.