Route-based approach

In the route-based approach, each combination of routes is investigated to find the node that comes close to at least two routes.

Consider the scenario i n Figure 3 . 1 5 in which the purple dashed line indicates the near radius for each route in a rectilinear setting . Every point inside the g reen region, is close to both routes and represents a collection of pOints of close a pproach . O ne node is close to this region and becomes a candidate for coordination site.

3 Coordination 55

Fig u re 3 . 1 5 : Route-based approach .

Step One Step Two Step Th ree Step Fou r Step Five Route-based approach

Find the list of close approaches for each orientation option . Order these close approaches i n increasing order.

Look at each close approach . Is the point of close a pproach near a node?

If not, reject that point; repeat Step Th ree with the next on the list.

Otherwise, go to Step Five.

If no more pOints of close a pproach, repeat from Step One with another route order / orientation com bination . If no more com binations, STOP.

The node is a possible coordination site.

Proced u re 3 . 2 : Route-based approach for determining a

coordination site at a node.

Occasionally, the closest node to a close approach lies on a route that is not involved i n formi ng the close approach, as shown in Fig u re 3 . 1 6 . By dividing the work cycle into discretised time steps and determining the vehicle positions and system status at each time, the path traversed by each vehicle is represented in the model by a series of discrete positions, and, because of this step-wise natu re of creating the path from one node to the next, a close a pproach might be represented as a different point on each route.

56

Close approach paints

3 Coordination

Figure 3 . 16 : The situation prior to exchange. The closest node lies on a third route.

At the coordination site, six part-routes exist. They are the part-routes representing the routes before- and after-coordination-site. The outcome depends on two factors :

1 . _{the vehicle that carries the demand for the coordination site, and} 2. the way in which the part-routes re-combi ne,

and the combination of these results in several options, shown in Table 3 .4.

Carrier of demand

Option Part-route com bination

for coord ination point

Cl Vehicle A Original (except Vehicle C) . C2 Veh icle B Original (except Vehicle C ) .

C3 Vehicle A Part-routes swap vehicles.

C4 Vehicle B Part-routes swap vehicles.

CS Vehicle C O riginal.

C6 Vehicle C Part-routes swap vehicles. Table 3 . 4 : Combination options.

In options Cl - C4, the coordination site is removed from its original route and the demand is carried by Vehicle A or Vehicle B ; Veh icle C has a reduced payload .

3 Coordination

Vehicle A

Veh icle C Figure 3 . 1 7 : Options C l a n d C2.

57

Options Cl and C2 differ only in the carrier of the demand for the coordination site. In option C l , the demand is carried by Vehicle A; in C2, it is carried by Vehicle B.

Vehicle C

Figu re 3 . 1 8 : O ptions C 3 and C4.

Options C3 and C4 differ only in the carrier of the demand for the coordination site. In option C3, the demand is carried by Vehicle A; in C4, it is carried by Vehicle B. Options Cl and C3 differ in the post-coordination routing. In C l , the original route is adhered to; in C3, Vehicle A continues with the latter part of Vehicle B's original route . Options C2 and C4 differ in a similar fashion .

I n options CS and C6, the demand for the coordination site is carried by Vehicle C, which would have carried it had coordination not been applied . Vehicles A and B still visit the coordination site because that is the node closest to each of

58 3 Coordination

their routes. Thus, the coordination site is attended by th ree vehicles, not necessarily at the same time.

Vehicle A

Figure 3 . 1 9 : O ption CS.

Options CS and C6 differ only in the post-coordination parts of their routes.

In option CS, the origi nal routes are adhered to; in C6, the latter parts a re swapped .

3 Coordi nation 59

3.9.3 Intersection-based approach

I n the intersection-based a pproach, the aim is to find a node close to an intersectio n . This approach differs from the route-based approach in that the orientation and route order are determined a posteriori. The routing scheme such as that shown in Figu re 3 . 2 1 is investigated to determine whether i ntersections exist without regard to orientation or to route order. If any intersections exist, each is considered in tu rn . When a suitable node has been located, the orientation and route order combinations that allow that intersection to occu r are found. The resulting part-routes are the same as those in the route-based fig ures above.

Figure 3 . 2 1 : Intersection-based approach .

I n the scheme shown in Figure 3 . 2 1 , the purple dotted line indicates the near radius of an intersection. Two nodes lie within this l imit and thus, are close to the intersection . One of the nodes is on a route that forms the intersection. The other node lies on a third route.

3.9.4 Lim itations

It shou ld not be assumed that a node suitable for acting as a coordination site actually exists. In each of the approaches a bove, there is no guarantee that a node will be capable of hosting a coordination site.

This study considers coordination in both the Euclidean (Chapter 4) and rectilinear grid (Chapter 5) scenarios, each of which presents slightly different characteristics for the identification of su itable nodes. In the Euclidean plane, coordination sites are restricted to existing only at the locations of existing nodes, and the paths between nodes are unique straight lines, thus limiting the number of interactions that can exist between routes to one location per intersection . In contrast, the rectilinear g rid allows paths between nodes to take the form of a

60 3 Coordination contiguous series of small links in the grid, thus a llowing much g reater choice in the actual path taken by a vehicle. With the increased choice of path there is a concomitant increase in the number of potential interactions, each of which is considered when i nvestigating the existence of suitable locations for coordination . This is illustrated in Figure 3 . 18 which demonstrates the effect of having just two possible paths for each of two part-routes. Two of the instances contain no intersections; one of the instances has vehicles sharing a common section .

Figure 3 . 2 2 : Paths in the rectilinear g rid . Even with only two possible paths for each of the two routes, the environment for coordination is altered .

Also, because of the nature of the g rid which allows nearby paths to contain para l lel sections, i nteractions may occu r over a region rather than at a point as in the Eucl idean plane.

Consequently, the existence and identity of coordination sites varies depending on whether the scenario occu rs in a E uclidean plane or rectilinear g rid. For this reason, a discussion of the types of coordination that can be employed is dealt with in the appropriate chapters below.

4 Coordi nation i n

CHAPTER 4

COORDINATION I N THE EUCLlDEAN PLANE

The aim of this chapter i s to investigate the effect o f coordination as a route improvement method i n the Euclidean plane. This coordinate system has been chosen because it presents no limit on the location of pOints or paths, and to provide a basis for com parison with Euclidean vehicle routing problem schemes that do not use coordination. By a llowing com plete freedom in the l ocation of nodes and paths, coord ination sites can be created anywhere within the plane . On the other hand, the lack of restriction on vehicle paths g reatly limits the number of direct path interactions since paths are less likely to have coincident sections, reducing chances of path overlap. Consequently, regions a round paths are i nvestigated ( once path locations are known) for potential interactions with other paths. The next chapter deals with the rectilinear g rid which forces path loci i nto restricted configurations and enables path interactions to be investigated more fully.

4. 1 Node location

Nodes a re located randomly a t integer positions within a plane measuring 1 00 by 1 00 distance u nits with the exception that

• a l l coordinates a re integ ral,

• no two nodes occupy the same location, and • no node is co-located with a depot.

Restricting the coordinates to integers permits them to be represented relatively simply by matrices, and allows for straightforward i m plementation in the MATLAB programming language.

6 2 4 Coord ination i n the Euclidean

4.2 Euclidean d istances

The standard Euclidean metric has been used to calcu late symmetric inter-nodal distances. The distance between node A, located at (XA, YA) , and node B, located at (XB, YB) is g iven by

4.3 Depot location

The depot can be located at any point within the E uclidean framework but it has been restricted i nitially to a constan t position for consistency. Unless otherwise stated, systems being com pared have the same depot location in order to prevent such location from posing an obfuscatory influ ence on the outcome of a routing i mprovement scheme.

To investigate the effect of depot location on route structu re, twenty datasets were examined, each at five depot locations ( results a re shown in Table 4 . 1 ) . When the depot is located at or near the centroid of nodes, routes are even ly balanced typically with one or two single-noded routes, severa l routes of fou r to six nodes, and with the fleet containing two to fou r multi-routed vehicles. When the depot is located outside the convex h u l l defi ned by the nodes, the distance between the depot and the first node on a route increases, resulting in reduced manoeuvrability for coordination . When the depot is located in a corner, well away from almost all nodes, the routes are u n balanced with few nodes per route and few (if any) multi-routed vehicles; most of the d istance allowance is used just to travel between the depot and the first node of a route.

Depot location Nodes per route Fleet size No. m ulti-routed vehicles

Centroid [ 1 , 6] [ 5 , 6] [ 2,4 ]

Near centroid [ 1 , 6] [ 5 , 6] [ 2,4 ]

Just outside hull [ 1 , 5] [6 ,9 ] [ 1 , 2 ] Near edge of plane [ 1 , 3 ] [8, 1 7] [ 0, 1 ] Near corner of plane [ 1 , 3 ] [8, 20] [ 0, 1 ]

Table 4 . 1 : Effect of depot location on routes for a single depot system of 25 nodes located randomly across the plane .

Since some coordination requires the existence of m ul ti-routed veh icles, the placement of a depot at an extreme position (edge or corner) away from the nodes

4 Coordination i n the

was not pursued. A default location was adopted near the centrold to a llow m ulti­ rou ted vehicles to participate i n coordination. As a bonus, this adds an air of realism since, i n practice, it is not u ncom mon for depots to be situated slightly off­ centre with respect to their clientele.

4.4 Customer demand

Each customer is assigned an i nteger-valued demand generated from a normal random distribution, which has mean 0 and variance 1, by the equation

Demand == floor(a bs(Random x 1 00 » .

Where the demand is g reater than the maxim u m vehicle capacity and load splitting is not permitted, the demand is reduced to the value of the maximum vehicle capacity. This allows a feasible solution to be reached and allows slightly more loads of larger value to be generated. An alternative scheme of generating only demands of value less than or equal to the vehicle capacity would a lso have been acceptable . A normal distribution is chosen, in preference to a u niform distribution, to simul ate m ore closely the actua l demands received by a courier com pany. Demands belong to the same com modity class a nd have equal priority.

4.5 Del ivery times

The time required to deliver a customer's demand is a l i near function of the demand and the d istance travelled to reach the customer's node. To deliver x u nits of demand made u p of y different items, the delivery time is

Delivery Time == A + Bx + Cy

where A represents the fixed time to make a delivery, and B and C are user-defined constants, typical ly of the order 0 . 1 and 0 . 0 5 . This function acknow ledges both the longer u nloading time of loads that i nvolve more items and the economies of scale of u nloading bulky items, and the values of B and C have been chosen to reflect that.

64 4 Coordi nation in the Euclidean

4.6 Coordi nation opportunities in the Euclidean plane

The structure of a Euclidean plane does not limit the position ing of coord ination sites . They can occu r at existing nodes, on existi ng non-coordinated routes between nodes and at pOints away from such routes . Travel can occur anywhere withi n the plane, and in a non -coordi nated system vehicles travel along edges joining pairs of nodes because this results in shortest paths. In a coord inated system, the smallest total distance travell ed may include some edges that do not form the shortest paths between a pair of nodes, since veh icles may have to detour. The i ntersection will be within a convex h u l l defined by the depot and node locations as shown in Figure 4. 1 .

Convex hull

Possible coordination site

Figure 4 . 1 : Euclidean convex h u l l . On the left, two pairs of nodes with con necti ng edges. On the ri g ht, a possi ble coordi nation site within the convex h u l l created by the fou r nodes.

Coordi nation sites at nodes have the advantage that thei r locations are known before coordination is atte m pted . An existing node can act as a coord ination site in two ways :

1 . The demand for that node can be ca rri ed by more than one vehicle. This is known as split delivery.

2. The node ca n act as a tem porary distribution centre, where loads ca n be deposited for later uplift to other nodes.

The two mechanisms are quite different and not mutually excl usive . Beca use both tem pora l and spatial coordi nation between vehicles is i m portant, both the routi ng and schedul ing phases are affected . For the pu rposes of this study, any node can fu nction as either (or both ) type(s) of coord ination site.

The assu m ption is that nodes already have the faci lity to accept goods, whether those intended for them or for other nodes. Access a n d avai labil ity are not

4 Coordination i n the Euclidean

limited ; there i s a fin ite number o f existing nodes. In contrast, there i s a n i nfinite number of possible coordination sites on existing routes.

Normal ly, the function of a node in a pure pick u p or delivery system is to provide a vehicle or vehicles with goods or to accept goods from a vehicle; such goods a re to be delive red to or from the d epot. It is not the standard function of a node to store vehicles or to store goods for delivery to other nodes. In contrast, the node in a coordinated system can act as a temporary depot. The nodal zone of influence is the area defined by a minimum d istance from a node within which coordination has no significant effect. That is, the region in which the act of coord ination m ig ht as easily occur at the node. The value of a "significant effect" is determined as being within one u nit in the smallest decimal place in the measurement of total distance . For example, a tra nsfer of goods d irectly adjacent to a node cou l d have occurred at that node with no l oss of d istance advantage. W ithout zones of infl uence, nodes would exist merely as infinitely smal l pOints in a plane and no path would intersect them other than to service the m . The use of zones of influence confers fin ite areas on nodes and enables continuous edges to be considered as objects, which can be divided i nto discrete quanta .

The nodal zone of i nfluence can b e represented a s a spatial and a temporal measure; each can be fractional or absolute. As a fractional spatial measure, the rad i us of the zone is a fraction of the length of the edge u nder consideration. As a fractional temporal measure, the radius is a fraction of the travel time along a n edge. Two i ntersecting edges will have d ifferent fractional measures but the same a bsolute values. The spatia l zone of influence is referred to as a n horizo n ; the temporal zone is a threshol d . During coord ination, both horizon and threshold are i mportant.

1 . 3 u n its

3.9 units

Fig u re 4 . 2 : Zones of influence. Each node has a spatial zone of i nfluence of (say) 0 . 3 u nit radius. For the longer edge, the zone of i nfluence has fractional value 0 . 08, and for the shorter edge 0 . 23. Activities occurring withi n the zone of i nfluence can be considered as occurring at the node itself.

4 Coordination i n the Euclidean

Similarly, i n m u ltiple depot systems, there is a depot zone of i nfluence . However, inter-depot coordi nation a n d transfers a re not considered i n this study since it deals with single depot systems.

There is no loss of generality in considering coordination at nodes and coordination on existing routes together since routes are defi ned by the nodes throug h which they pass. Any position on an edge of an existing route can be defined by a value combining

• the nodes at either end of that edge, and • the d istance from one end of the edge.