In this section, our heuristic solution approach is compared to the results of Viswanathan and Mathur (1997) [30], and Sindhuchao et al.(2005) [28].
4.2 Comparison to other heuristics 101
Figure 4.12: Interaction between the factors AREA and CCAP Table 4.17: E ect of the service area size when CCAP is inactive AREA Total Fleet Transport Dispatch Deliver Holding
[75; 100] 270.74 112.13 82.65 15.11 18.19 42.66
[150; 175] 395.53 164.88 143.86 15.70 15.45 55.65
Di 46.09% 47.05% 74.07% 3.86% -15.05% 30.43%
AREA CPU-time nrVeh nrTours Utilization Stock Cust/tour
[75; 100] 2778.77 2.96 36.89 0.79 1173.49 2.52
[150; 175] 1793.48 4.39 44.70 0.77 1441.40 1.98
Di -35.46% 48.10% 21.18% -2.14% 22.83% -21.70%
Table 4.18: E ect of the service area size when CCAP is active AREA Total Fleet Transport Dispatch Deliver Holding [75; 100] 333.43 150.13 106.37 15.45 30.46 31.03 [150; 175] 529.64 235.13 211.06 17.42 29.73 36.31
Di 58.85% 56.62% 98.42% 12.79% -2.39% 17.02%
AREA CPU-time nrVeh nrTours Utilization Stock Cust/tour
[75; 100] 300.67 3.91 22.03 0.79 657.49 3.88
[150; 175] 242.98 6.06 25.76 0.83 724.70 3.15
Di -19.19% 54.95% 16.97% 4.45% 10.22% -18.62%
Viswanathan and Mathur
Viswanathan and Mathur [30] developed a heuristic that generates a so-called stationary nested joint replenishment policy (SNJRP). They term a policy to
be stationary if replenishments are made at equally spaced points in time. This is the equivalent of what we have called regular. A nested policy means that if the replenishment interval of a given customer is larger than that of another customer served by the same vehicle, the former is a multiple of the latter. Thus, the nestedness corresponds to our delivery frequencies, because not all customers visited by the same vehicle have the same replenishment interval. As in most papers found in the literature on cyclic inventory routing, a xed vehicle cost is not considered in the cost structure of Viswanathan and Mathur. As a result, the assignment of tours to vehicles is not taken into account and the assumption is made that there is a separate vehicle per tour.
For the cycle times of the tours, only vehicle capacity is used as a constraining element. This means that, compared to our model, the restrictions follow- ing from (i) travelling, loading and unloading times, (ii) customer imposed maximal delivery frequencies, and (iii) limited customer storage capacities, are discarded. In other words, a minimal cycle time for the (nested) tours is not being considered.
To compare our heuristic method to that of Viswanathan and Mathur, the following adaptations were made to our model as presented in Chapter 2:
The xed vehicle cost is discarded: = 0.
The vehicle speed is very high ( = 1), such that the minimal cycle time is always negligible.
The problem instances that Viswanathan and Mathur used in their compu- tational testing were not available from the authors, so we reused the set of problem instances that were generated for our design of experiments (see Section 4.1), excluding the instances with customer storage capacity restric- tions. Because a common base cycle time is used in the solution approach of Viswanathan and Mathur, we impose a 1 day common base cycle time for replenishment in these tests by activating the driving time restriction in our solution approach.
Figure 4.13 and Table 4.19 show the average improvement of the solutions obtained with our solution method using distribution patterns (DP) over those obtained with the stationary nested joint replenishment heuristic (SNJRH). For all problem instances, solutions were found that are cheaper than those proposed by Viswanathan and Mathur. On average, there is a decrease in cost of no less than 7:4%. This is mostly due to the more general routing concept that we are using. These results show that allowing vehicles to make multiple tours gives the opportunity to use vehicle capacity much more eciently and nd much better cost trade-o s, even without considering xed vehicle costs. Our solution approach is thus capable of nding better trade-o s between dis- tribution and customer holding costs due to the generalized model. When con- sidering xed vehicle costs, the model of Viswanathan and Mathur becomes
4.2 Comparison to other heuristics 103
Figure 4.13: Comparing our solutions to that of the stationary nested joint replen- ishment heuristic
Table 4.19: Improvement of our heuristic over SNJRH
VCAP HC NR AREA DP SNJR 100u 10c [30,70] [150,175] 178.82 161.49 10.7% 50u 10c [30,70] [150,175] 211.74 193.52 9.4% 100u 1c [30,70] [150,175] 88.90 85.57 3.9% 50u 1c [30,70] [150,175] 128.62 127.35 1.0% 100u 10c [80,120] [150,175] 384.88 354.06 8.7% 50u 10c [80,120] [150,175] 466.82 425.05 9.8% 100u 1c [80,120] [150,175] 197.75 191.02 3.5% 50u 1c [80,120] [150,175] 287.83 283.82 1.4% 100u 10c [30,70] [75,100] 141.30 127.15 11.1% 50u 10c [30,70] [75,100] 162.35 145.46 11.6% 100u 1c [30,70] [75,100] 65.08 61.43 5.9% 50u 1c [30,70] [75,100] 88.39 86.87 1.7% 100u 10c [80,120] [75,100] 276.59 253.39 9.2% 50u 10c [80,120] [75,100] 327.33 291.23 12.4% 100u 1c [80,120] [75,100] 130.67 124.47 5.0% 50u 1c [80,120] [75,100] 179.58 176.88 1.5% Average 207.29 193.05 7.4%
obsolete, while our approach is still valid and is capable of nding three-way cost trade-o s between distribution, customer holding and xed vehicle costs.
Sindhuchao et al.
Sindhuchao et al. [28] develop a branch-and-price algorithm that nds the op- timal solution for a set of very small problem instances. Again, xed vehicle costs are not considered in the cost structure, such that only solutions with a single tour per vehicle are obtained.
In the literature on cyclic inventory routing, Sindhuchao et al. are the only ones that consider a lower bound on tour cycle times. However, this minimal cycle time is not the natural one arising due to the travelling, loading and unloading times. Instead, they impose a vehicle frequency constraint F , stating that a vehicle cannot make more than F tours per period.
To compare our heuristic method to the results of Sindhuchao et al., the fol- lowing adaptations had to be made:
The xed vehicle cost is discarded: = 0.
The vehicle speed is very high ( = 1), such that the original minimal cycle time is negligible.
The new minimal cycle time due to the vehicle frequency constraint is imposed as follows: Pi=1::nki< F T , with F the maximum number of
tours per period. This gives Tmin=Pi=1::nki=F .
Our adjusted heuristic solution approach was then applied to the set of very small problem instances for which Sindhuchao et al. found the optimum. The results are shown in Table 4.20.
Table 4.20: Optimal cost vs. obtained solutions
Optimum Solution Gap
1 2778.1 2856.0 2.80% 2 2645.8 2682.7 1.40% 3 2598.6 2629.5 1.19% 4 2761.2 2804.9 1.58% 5 2726.0 2726.0 0.00% 6 2699.3 2707.2 0.29% 7 2526.7 2564.8 1.51% 8 2426.2 2449.1 0.94% 9 2577.2 2606.9 1.15% 10 2825.5 2865.4 1.41% Avg 2656.5 2689.2 1.23%
It can be seen that our heuristic performs very well on these 10 instances. The average gap between our results and the optimal solutions is only 1:23% and the largest gap is 2:80%. This again con rms the power of our approach. While it can handle much more general versions of the cyclic inventory routing problem, it still manages to get close to the optimum for this speci c version.
4.3 Performance of the scheduling heuristic 105