Nondetailed Vehicle Routing Models: Many Vehicle

4.2 The Transportation Operation

4.2.1 Nondetailed Vehicle Routing Models: Many Vehicle

Eilon, et al. (1971), developed simple approximate formulas for the tance of near-optimal vehicle routes in Euclidean square regions. The dis-cussion presented here is based on more recent material extending Eilon et al.'s results to zones of arbitrary shape (Daganzo, 1984a and 1984b), and also incorporating other metrics and the influence of underlying transporta-tion networks (Newell and Daganzo, 1986 and 1986a, and Newell, 1986).

Appendix A summarizes the logic behind some of these results.

In order not to introduce additional notation, we will use N to denote the number of destinations that must be visited. If tours are not being con-structed for all the customers in the region, as occurs later in the chapter, the results can be easily reinterpreted.

We have already mentioned that vehicles should be used to the fullest;

that is, there should be at most one vehicle that makes fewer than C stops, and none if N is an integer multiple of C . Our strategies are of the "clus-ter-first and route-second" type, where the service region is divided into non-overlapping zones of C customers, to be served by separate vehicles.

For a given set of zones, the vehicle routes are easy to construct using some simple rules. To minimize the total distance (and, hence, the cost), these zones should have specific shapes and orientations, dictated by the relative magnitude of N and C² . Two cases need to be considered: (i) when the number of vehicle routes N/C is much greater then the number of stops per route C , N >> C² , and (ii) when only a few vehicle routes are needed N << C².

For case (i), discussed in this subsection, delivery districts (or zones) should have a width comparable with the distance between neighboring points and be as long as necessary to contain C points; see Appendix A.

The formulas are most transparent when expressed in terms of the spatial point density – in points per unit area – evaluated at a point inside the

de-98 One-to-Many Distribution

livery district, x : į(x) = Nf(x) . (Because į(x) varies slowly, just like f(x) , it does not matter which x is used). The factor į(x)^-1/2 , appearing in the formulas, represents a distance close to the average separation between neighboring points in the vicinity of x . For randomly scattered points, it has been found that (see Appendix A):

   



^zonelength



^C/



^6/



^.

/ 6 width

2 / 1 2 / 1

G G

| zone |

These dimensions are close to ideal and relatively independent of the met-ric or underlying network. When į changes over R , distmet-rict dimensions should also change over R, although more slowly. As the solution to the EOQ problem, these expressions are robust; departures from the ideal di-mensions by 20 - 30 percent are largely inconsequential, but larger depar-tures increase distance. This robustness makes it easy to carve out R into delivery districts of satisfactory dimensions.

Zones should also be oriented "toward the depot", but the precise mean-ing of this recipe depends on the underlymean-ing metric. One should build equi-distance contours from the depot and design zones of the right dimensions that are perpendicular to these contours. For the Euclidian metric the con-tours are concentric circles centered at the depot, so that the zones should fan out from the depot in the radial direction. For the L₁ (or "Manhattan") metric,³ the contours are squares centered at the depot, at 45( to the met-ric's preferred directions; in this case the zones should be perpendicular to these contours, so that they don't point exactly toward the depot. Ideal ori-entations can also be defined when the network includes fast/cheap roads.

Because the zones are narrow, it is easy to construct good vehicle routes, once the region has been carved into delivery districts. One simply needs to travel up one side of the zone, visiting the points in order of in-creasing distance to the depot, and then return along the other side visiting the remaining points in the reverse order. The effectiveness of this routing scheme improves with the slenderness of the zones – it is exact if zones are infinitely narrow. This has been verified experimentally albeit indirectly by Robusté et al. (1990).

Before turning our attention to distance formulas, let us show how to partition a region into delivery distances with proper shape and orientation.

We recommend drawing delivery zones around the region's edge away from the depot, and then filling in the remaining space with more delivery

3The L1 distance between two points is the sum of the absolute differences in their coordinates.

routes, always proceeding toward the depot. Figure 4.1 depicts an interme-diate point of this process for an irregular region with an internal depot and a rectangular grid network – note how most districts are perpendicular to the (square) equi-distance (L₁) contours. As we progress toward the depot, it may become necessary to pack a few zones with the "wrong" shape, but most will have the right dimensions and orientation. Because the distance traveled is not overly sensitive to (small) deviations from the ideal design, the distance formulas about to be developed should be accurate. This is confirmed by experiments in Daganzo (1984b), Robusté et al. (1990) and Hall (1993).

Fig. 4.1 Intermediate stage of the delivery district design process

The last of these references considered systems in which the number of stops in a tour depends on the shipment sizes handled at each stop.

The total distance traveled to visit the C points in a given zone contain-ing point x₀ is:

c

100 One-to-Many Distribution

 

^C,

k r distance

Tour »¼º

«¬ª 



|2 G 1/2 x₀ (4.1)

where ¯r is the average distance from the C points to the depot (on the shortest path) and k is a dimensionless constant that depends on the metric (k  0.57 for the Euclidean metric, and k  0.82 for the L1 metric). See Appendix A for more details.

The first term of (4.1) can be interpreted as the line-haul distance needed to reach the center of gravity of the points in the zone, and the sec-ond term as a local distance that must be traveled because the points are not next to one another. Note that each stop contributes toward the total a distance comparable with the separation between neighboring points, kį^-1/2(x0) . This occurs, because the vehicle must be detoured on every leg between successive deliveries. In actuality, because there are only C - 1 such legs, the factor "C" in (4.1) should be replaced by "C - 1" . Thus, a better expression is:

 

^{C- .}

k + r distance

Tour |2 [ G^1/2 x₀ ][ 1] (4.1a)

The improvement afforded by this expression, particularly obvious for C = 1 , fades in importance as C grows. Because (4.1) is more compact, it will be used unless C is small.

Let us now see how the total distance over R can be expressed without regard to the detailed position of points, using a continuum approximation.

Distance (4.1) can be prorated to each one of the points in the zone so that if point i (located at x_i) is r_i distance units away from the depot, then:

   

i / -i

/ -i i

+ k į x C

r + k į x C for x r Distance

2 1

2 0 1

2 2

|

|

(4.2)

where the second approximate equality follows from the slow varying property of į(x).

The total distance traveled in the region is the sum of (4.2) across all points:

 

^.

k + C r

distance

Total ^-1/2

i i i

xi

¦

G

¦

| 2

(4.3)

For large N , the summations can be replaced by integrals independent of

the specific location of all the points:

 

^x

>

^-1/2

 

^x

@  

^x

^{d x ,}

R

i

G G

G ^| ³

¦

i ^-1/2 (4.4a)

and

    ^{d .}

r

x x x

R

³ G

¦ r

i i

|

(4.4b)

Thus, (4.3) can be rewritten as:

   

> ^{2 r /C + k}

^-1/2

@ ^{d .}

³

|

R

x x x

x

G G

distance

Total _(4.4c)

Note that this expression is well suited for continuum approximations be-cause the cost in any given (small) area only depends on the local condi-tions.

An alternative expression for the total distance is obtained after replac-ing į(x)dx by Nf(x)dx in (4.4a and 4.4b); it then becomes clear that these expressions can be interpreted as the product of N and the expectation of r(x) or į^-1/2(x) , when the probability density of position is f(x). Thus, let-ting E(r) and E(į^-1/2) denote these expectations, the total distance can be expressed as:

 

^+kE

 

^N.

C r distance E

Total

¿¾

½

¯®

|­2 ^{ 21/}

G (4.5a)

For a uniform density, E(į^-1/2) = į^-1/2= R| |/ N and we can write:

 

_{N+k | |N,}

C r distance E

Total 2 R

| (4.5b)

where R denotes the surface area of R .

Independent of the specific locations, Eqs. (4.4) and (4.5) are particu-larly useful if cost must be estimated before the point locations are known.

In that instance, it may be reasonable to view the actual locations (x1 , ... ,

102 One-to-Many Distribution

x_N) as outcomes of i.i.d random variables with density f(x) , and interpret Eq. (4.5a) as the average total distance over all possible locations (x₁ , ... , x_N) . In any specific instance there will be some discrepancy between (4.5a) and the actual distance – for large N most of the difference typically will arise from fluctuations in Ȉir_i , which are of order O(N^1/2) and compa-rable to the second term of (4.5a). If more accuracy is desired, one should wait for the point locations to become known. Comparisons made in Hall et. al. (1994) indicate that the approximation formulas just presented are fairly accurate even if the number of stops is not the same for all tours.

That reference examines improved routing methods for problems in which the number of vehicle stops depends primarily on the vehicle's capacity and the shipment sizes handled at each stop.

4.2.2 Non-Detailed Vehicle Routing Models: Few Vehicle