Comput. & Qw Res. Vol. 6. PP. X39-223 Pergamon Press Lt., 1979. Printed in Great Britain

ROUTING ELECTRIC METER READERS

HELMAN I. STERN* and M~SHE DROR?

Department of Industrial Engineering and Management, Ben Gurion University of the Negev, Israel

Scope and -The closed tour Chinese postman problem, CPP, is that of finding a postman’s delivery tour through an undirected graph. The tour must begin and end at a common node in the graph and every edge must be traversed at least once such that the total edge distance traveled is minimized. When maximum tour length restrictions are present this problem is known as the M-CPP with applications to the routing of solid waste, snow removal and street cleaning vehicles. In these applications tour distance (time) restrictions are normally large compared to the edge distances. Consequently, tours are comprised of sets of complete edges. In addition, the distances from the origin node of each tour to the service area is comparable to that of the edges that need to be covered.

This research focuses on practical problems that are significantly different in two aspects: (1) open tours are desired since working tour time restrictions do not include the trip between the origin node and the service area, and (2) tours may start and end at intermediate points of an edge since tour restrictions are comparable to edge service times. This type of problem is encountered by electric and water companies that dispatch “meter readers” to measure household consumption for billing purposes. Each reader is transported by auto from a central o5ce to the first house on his work tour, normally of 5 or 6 hr duration. At the end of his time limit he is free to return home or to the office by bus. A heuristic algorithm is employed to reduce the number of required working tours.

This class of problem should be of interest to researchers of edge oriented routing, as well as, managers of delivery (collection) services over street networks.

Abstract-BIectric utility companies employ a crew of workers who periodically visit and read the electric meters of each customer in their service area. Each reader is transported by auto from a central office to the iirst customer on his work list. At the end of his work shift time limit the reader is free to leave the area possibly returning home or to the central office by public bus. Taking a graph that corresponds to the city network of streets meter readers must traverse each street while moving from house to house. It is possible that dead heading may be required-back tracking over a street that has already been covered. A working tour is an open path whose reading time plus deadheading time does not exceed the work limit. The problem is to find the minimum number of working tours. Stating the problem in this manner gives us an optimization problem closely related to the M-Chinese postman problem-an edge oriented routing problem. After presenting some background on this type of problem a heuristic procedure is used to solve an example from the City of Beersheva. The solution provides a 40% reduction in the number of working tours. The paper ends with a discussion of the solution, and provides conditions under which the algorithm should have practical utility.

INTRODUCTION

The problem of decreased productivity in the public sector is one of the major problems faced by municipal public service agencies. This is primarily due to the ever increasing cost of labor. This paper is addressed to the problem of collecting data on household consumption of electricity for billing purposes-a highly labor intensive activity. Electric u&y companies normally employ a crew of workers who perform this task by periodically visiting and reading the electric meters of each customer in the service area. This activity requires the dispatch of readers to widely distributed residential neighborhoods of the community. Normally the crew of readers are scheduled to jointly canvass each neighborhood in ‘sequence so that each neighborhood is revisited over a period of ‘3 or 4 months. Given the desired period, and a rough approximation of the amount of work to be done, the crew size can be approximated on the basis of a manworking day or a given workshift time limit. The size of the crew can be reduced

tMoshe Dror is a doctoral candidate and a lecturer in the Department of Industrial Engineering and Management, Ben Gurion University of the Negev, Beersheva, Israel. He holds an M.Sc. degree in Mathematical Methods in Engineering and an I. E. (Prof.) degree both from the Cohunbii University. Mr. Dror is conducting research in topics related to the Chinese Postman Problem and the Traveling Salesman Problem. He is a member of ORSIS, Operations Research Society of Israel.

*Dr. Hehnan I. Stem is a Senior Lecturer in the Department of Industrial Engineering and Management of fhe Beh Gurion University of the Negev, Beersheva, Israel. He received a B.S. in Ekctrical Bngineering from Drexel University, an M.EA. from George Washington University and an MS. and Ph.D. in Operations Rese.arch.from the University of California at Berkeley. His recent publications have been in Naval Research Logistics Quarterly and Operations Research. He is currently preparing a text on Transport Routing and Scheduling to be published by McGraw-Hi. His primary research interests are in the areas of scheduling and routing in production and transportation systems. He is a member of AIIE and ORSA. He is on the Editorial Advisory Board of Trans. Res. and is an Associate Editor of OPSEARCH.

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210 HELMAN I. STERN and MOSHE DROR

by the efficient routing of the walking tours of each reader through each neighborhood street network. By minimizing the number of workshifts (man-days) for each neighborhood the minimum crew size is easily determined through division by the number of working days in the desired period. The decomposition of the problem on a neighborhood basis is a reasonable approximation in this case as the work rate (number of streets or street segments covered per working day) is relatively slow leading to 10 or 20 working days per neighborhood. This is in contrast to vehicle delivered service (garbage collection, street sweeping) where whole neigh- borhoods can be covered in fractions of working days (for example 0.5 or 3.75 days). For such problems appreciable savings can be made by the more appropriate artificial districting of the entire service area or the splitting of working routes over several neighborhoods.

In the next section a more precise description of the problem is presented after discussing the present method of routing meter readers in the city of Beersheva-the problem that motivated this research. Some background material on arc oriented routing is discussed before presenting our heuristic algorithm. This heuristic is then used to solve the Beersheva example. The paper concludes with a discussion of the solution, and provides conditions under which the algorithm should have practical utility.

PROBLEM DESCRIPTION

This problem was motivated by a desire to improve the dispatch and routing policies of electric meter readers in an urban area. The city of Beersheva was selected as a test city to compare objective routing techniques with the present routing procedure. The city of Beer- sheva is divided into eight neighborhood districts. We have arbitrarily selected neighborhood C as a test case for our investigation. A reduced map of the neighborhood is shown in Fig. 1. In the present method of operation the planning office of the electric company provides each reader with a set of cards where readings are to be recorded. These cards are arranged in the order of streets and households to be visited. The routes were chosen on the basis of intuition and experience of the people actually reading the meters in conjunction with estimates of the time to read meters of various housing complexes and walking times along street segments. Each reader has a maximum, on the street, workshift time limit of 5 hr established by union regulations. Each reader is transported by auto from a central office to the address on his first card where he begins work. At the end of his time limit or after all his cards have been filled in the reader is free to leave the area possibly returning home or to the central office by public bus.

Figure 2 presents a graph that corresponds to the network of streets. The heavy lined edges represent those streets that contain meters and must be covered by the meter readers while moving from house to house. Dotted edges represent streets that contain no meters but may be traversed as connecting streets if required. There are no oneway streets (directed edges) in the graph as readers proceed by foot and walking can be done on sidewalks in any direction. Streets with meters on one side only may be covered in either direction. Narrow streets with meters on both sides may be traversed in one pass in a zig-zag fashion. Streets where crossing is not allowed must be covered twice, again in any direction. All of this justifies the representation of the street network as an undirected graph. We make the further assumption that no u-turns are allowed except at intersections. In a later section we shall describe the process of constructing such a graph from an original street map.

Associated with each edge i is a number Q representing an estimate of the time to traverse i while reading meters. In addition, a non reading walking time (dead heading time), 5, is estimated when it is required to traverse an edge that has already been covered (possibly by another reader or in the case of backtracking or street jumping). Estimates of ti and 6, obtained from the electric company,t are placed above and below the slashed lines adjacent to each edge in Fig. 2. Since Ci in some cases is in the same order of magnitude as T (the workshift time limit) working tours may begin and end at intermediate locations of an edge. A working tour (segment) is defined as an open path on a portion of the graph whose total “length”, comprised of reading time plus dead heading time, does not exceed the working time limit. The primary

tThese estimates include upward adjustments to account for rest and meal breaks.

Routing electric meter readers

Fig. 1. Chigiml street map (neighborhood C, city of Beersheva).

objective is to find the minimum number of working tours needed to cover the required edges in the graph, A secondary objective, given the minimal number of tours, is to find the routes of each tour such that the total length traversed is minimal.

Stating the problem in this manner gives us an optimization problem closely related to the Chinese postman problem, Kwan[l]. The Chinese postman problem is that of finding the shortest closed tour on a connected graph such that each edge is traversed at least once. The basic Chinese postman problem has been extended to the M-Chinese postman problem (see Beltrami and Bodin[2], Christofides [3], Liebling[4], Liebman[5], Marks and Stricker[6], Wicker [7], Orloff [8], where M-Chinese postman closed tours emanating from a single point are required to traverse each edge of the graph. A further variation, studied by Orloff [8], deals

212 HELMAN I. STERN and M~SHE DROR

Routing electric meter readers 213

with a graph in which only a subset of the arcs need be covered by the tour. In the traditional M-Chinese postman problem the working tour distance restrictions are normally large com- pared to the edge distances of the graph. Consequently, tours are comprised of complete sets of edges. In addition the distances from the origin point of each tour to the service area is comparable to the distances of the arcs that need to be covered. Our problem is significantly different in two aspects: (1) open tours are desired in lieu of closed tours since the working time limit does not include the trip from the central office and back, and (2), these tours may start and end at intermediate points of an edge.

Before providing some background on the Chinese postman problem and its variations we make two simple observations concerning bounds on the problem solution. The total reading time of the required edges of the graph provides a lower bound on the minimum total time to service the area. Dividing this number by T and rounding it up to the largest integer gives a lower bound, J4, on the minimal number of working tours, M*. Similarly, solving the Chinese postman problem on the graph of required and nonrequired edges, where the final single closed tour may include multiple traversals of a required edge, gives an upper bound on the minimal total time to service the area. Partitioning this tour into working segments not greater that T provides an upper bound, &f, on M*. Where working segments begin or end on nonrequired edges this total time bound may be further reduced.

BACKGROUND ON EULER TOURS ANDTHE CHINESE POSTMAN PROBLEM

Let G(N,A) be a connected, undirected graph representation of the neighborhood to be serviced. To each edge in the graph is associated a number representing its “length”. An Euler tour on an undirected graph is a continuous path starting and ending at some node, and including each edge exactly once. A necessary and sufficient condition for the existence of an Euler tour (Euler[9]) is that the graph be connected with each of its nodes of even degree (the degree is the number of edges incident to a node). If the graph contains nodes of odd degree an augmented graph of even degree may be constructed through the duplication of edges forming paths between nodes of odd degree. The length of the Euler tour on the augmented graph is then the total edge length of the original graph plus the sum of the lengths of the duplicated paths. Minimizing the length of this Euler tour is equivalent to minimizing the total length of the duplicated paths. This problem has been solved by Edmonds [lo] and consists of determining the shortest paths between each pair of odd nodes and then solving a matching problem. Given mo odd nodes in the graph and a set of mo(mo- 1)/2 shortest paths the matching problem is: select a subset of mo/2 shortest paths such that each odd node appears as an endpoint of only one of selected paths and the total lengths of the selected paths is minimal. The solution of the matching problem indicates which edges are to be added to the original graph to convert it to the required augmented graph of even degree. Finding an Euler tour on the augmented graph amounts to solving the Chinese postman problem, Kwan [ 11. The Chinese postman problem is to find a continuous path on a graph such that every edge in the graph is traversed at least once and the total distance traversed is minimal.

In general it may be required to solve the Chinese postman problem on a graph with the stipulation that only a subset of the edges must be traversed at least once. Thus, one may partition the edges in the original graph into two mutually exclusive and exhaustive subsets: AI- the subset of required edges, and A*- the subset of nonrequired edges. The edges in A2 need not appear in the final Chinese postman tour but may be used if necessary to provide continuity. This restricted Chinese postman problem of finding the minimum length tour which traverses each edge of the required subset of edges has been referred to by Orloff [8] as the Rural Postman Problem and may be solved by the following algorithm.

The rural postman algorithm Step 1. Let Gr(Ni, A,) be the graph obtained from G by removing the nonrequired edges AZ

and all nodes which remain disconnected from A,. We assume the graph G, remains connected. If Gi is an even graph, set G* = G, and Z* = 0 and proceed to step 3. Otherwise, find all shortest paths over the original graph G between all pairs of odd nodes in Gi. Note that the shortest paths may contain nonrequired edges from A2 as well as nodes in N.

214 HELMAN I. STERN and MIXHE DROR

Step 2. Solve the resultant minimal matching problem by Fdmonds algorithm or, if desired, by some heuristic.

For m. odd notes in Gr, mo/2 shortest paths will be selected by the matching procedure. Let A,,, be the set of edges in the selected paths with total length Z*, and N,,, the set of nodes appearing in the selected paths that are in the original node set N but not in Nr. Add A,,, and N,,, to the graph G, to obtain the augmented graph G* which is now even.

Step 3. Construct an Euler tour on G* by using either the isthmus avoiding (Kwan[ l]), cycle building (Harary [ 1 l]), or end pairing/next node sequence (Fdmonds and Johnson[ 121) methods. The total length of this tour is D = L(A1) + Z* where II, is the total length of the set of required edges.

In the case where Gr is not connected we may still apply the above procedure and if the augmented graph G* is connected we have solved the problem. It may happen however that G* is disconnected in which case the cheapest matching solution that results in a connected augmented graph G* should be used. Orloff [8] suggests finding this solution through a branch and bound procedure similar to the subtour elimination algorithm for solving the traveling salesman problem. When G* remains disconnected after applying the rural postman algorithm the problem becomes exceedingly di!IIcult to solve.

A HEURISTIC ALGORITHM FOR SOLVING THE METER READER PROBLEM

The heuristic method employed is of the “route first-cluster second” type. This type of heuristic has been used in other settings such as school bus routing, street sweeper vehicle routing and the multiple truck delivery problem (see for example Beltrami and Bodin[2], Newton and Thomas [13]). The heuristic is a two stage procedure. In the first stage the maximum workshift time limit is relaxed and a single tour is found that covers each of the required edges with an attempt to minimize the total nonreading time (deadheading time). In the second stage the single tour is partitioned into working segments each of length no greater than the maximum workshift limit. A description of the algorithm follows.

Phase I. Determination of an open Euler tour Step 1. Construct an augmented even graph G* by the method described for the rural

postman problem in the last section with the following changes: (i) Determine shortest paths over the original graph, G, using arc nonreading walking times

for distances. Do this for each pair of odd nodes in the graph of required arcs, G,. (ii) Use the following heuristic to solve the resultant matching problem. From the list of

shortest paths extract the shortest path with the minimal walking time. Delete all shortest paths left on the list with endpoints coincident with either of the endpoints of the path just extracted. Continue extracting paths in this manner until the list is exhausted at which point mo/2 paths have been extracted.

(iii) Add the set of extracted paths to the graph Gi to obtain an augmented even graph G*. Continually try to improve the matching on G* as follows: If there exist two extracted paths, say P(a, b) and P(c, d) with common arc (e, f) drop the duplicated arcs on these paths and use the remaining paths P(a, e, c) and P(b, f, d).

Step 2. Remove one of the shortest paths in G*, say, path Pi, so that what remains from G* is the graph G*(i) containing exactly two odd nodes. A good choice may be to remove the shortest path of maximal length.

Step 3. Construct an open Euler tour on the graph G*(i).

Phase II. Partition the open Euler tour into working segments Step 4. Place the edges in the tour obtained in Phase I in a list in the order visited by the

tour. Step 5.. Start at the origin node of the open Euler tour and remove edges from the list

accumulating their traversal times (reading or nonreading time whichever is applicable) until the workshift time limit, T, is exceeded. If the last edge removed from the list is a required edge (real edge) go to (i). If it is a nonrequired or dummy edge go to (ii).

(i) Note the position on this edge where the accumulated time is exactly equal to T and mark this as the end of a workshift segment. Start accumulating traversal times for the next

Routing electric meter readers 215

workshift segment from this point. (ii) Dispose of the nonrequired edge and truncate the last working segment at the end of the

last required edge removed from the list. Start the next working segment with the next required edge on the list. (Note that several non required edges may be discarded in this manner until a real edge is selected.) Continue until the list is empty.

Step 6. Let M be the number of working segments constructed and D the total cumulated time (reading and nonreading time) of all working segments.

If the lower bound on the number of working tours has not been reached or further reductions on total time are desired alternative open Euler tours may be selected in Phase I, step 3 of the algorithm. It is also possible to return to Phase I, step 2, and change the start and end points of the open tour by extracting different shortest paths. We illustrate the algorithm on a small 7 node-10 edge example assuming: (1) identical reading and non readii times, (2) all edges are required edges and (3) a maximum working time limit of 10.

Example Consider the graph G depicted in Fig, 3 with reading times recorded above each edge. The

total work required on the graph is 63.10 units. This then is a lower bound on the minimum time to cover each edge at least once. Taking a maximum workshift time of 10.0 gives a lower bound of 7 on the minimum number of required working tours. The optimal matching of the 4 odd nodes of G yields the shortest paths (1,5) and (3,4,6) of lengths 10.0 and 9.43, respectively. Duplication of the arcs of these paths converts G to the even graph G* comprised of the heavy and dashed edges in Fig. 3. After removal of the largest duplicated path (1,5) a graph G*(l) is obtained of total length 72.53. This number represents an upper bound on the minimum total time, and provides an upper bound of 8 working segments on the minimum number of working tours. Using the odd nodes 1 and 5 of G*(l) as the start and end points of an open Euler tour we construct the first candidate tour E, = (1,2, 3,4, 1,5,4, 3,7,6,4,6,5). Partitioning this tour

/

i

I I

I I I IO.

I I I I I I

\

1

Graph G

1 _--_- Graph G’

Fig. 3. Graphs G and G* for the example.

216 HELMAN 1. STERN and M~SHE DROR

yields 8 working segments of total length 65.98. These segments are designated TLT8 in Fig. 4(a) (beginning and end points of tours are indicated by the straight and arrowed marks, respectively). According to the rule in Phase II, Step 5(ii) of the algorithm we were able to eliminate the duplicated edge (4, 3) by truncating segment T5 at node 4 and commencing segment T6 at node 3 (the X in the figure indicates the point where an accumulated time of 10 is reached). The second duplicated edge (4,6) however remained embedded in segment T7. By partitioning a second candidate tour & = (1,2,3,4, 1,5,4,6,7,3,4,6,5) on G*( 1) (see Fig. 4b) we where able to reduce the number of working segments to its optimal value of 7. The total travel time however remains at 65.98. By partitioning a third candidate tour & = (1,2,3,7,6,5,4,6,4, 3,4, 1,5) on G*( 1) we were again able to obtain 7 working segments with a total working time of 63.10 (see Fig. 4c). As these numbers are identical to the lower bounds we are assured that this solution is optimal. Alternative optimal solutions may also be found for this problem. For example, extracting the duplicated path (3,4,6) from G* to obtain G*(2), and partitioning the open Euler tour (6, 5, 1, 2, 3, 7, 6, 4, 5, 1, 4, 3) yields the optimal number of segments with duplicated edge (5,l) deleted after truncating the first working segment examined.

In this example three Euler paths were constructed until a solution was reached. In general, however, it may be necessary to construct and partition many Euler paths until a reasonable solution is obtained. As this can be a time consuming task it is worthwhile to try to generate Euler paths that promise to have good decompositions. A good strategy might be to: (1) try to avoid absorbing duplicated edges in working tours and (2) have working tour end points land on duplicated edges. One possibility to help achieve these goals is to construct Euler tours in which the edges in duplicated paths retain their original connectedness. This should increase the probability of a tour endpoint landing on a dummy edge thus eliminating the entire set of dummy edges in a duplicated path. To illustrate this point examine candidate tours E, and Ez of the sample problem. In these tours the two edges of the duplicated path (6,4,3) were split, and although a tour endpoint landed on one of them the second remained embedded in a working

10.0

TI

T4

TB ’

E,=(C2,3,4,1,5,4,3,7,6,4,6,5)

M,= 6

D,= 65.96

1.23

Fig. 4(a). Partition of first candidate tour on G*(l) into 8 working segments,

Routing electric meter readers 211

4

E,=(l,2,3,4,1,5,4,6,7,3,4,6,5)

y’7

&= 65.96

Fig. 4(b). Partition of second candidate tour on G*(l) into 7 working segments.

tour. In ES, however, the edges (6,4), (4,3) remained joined, and by landing on edge (6,4) we were able to also remove edge (4,3) before starting a new working tour.

In lieu of the “route first- partition second” procedure a look ahead procedure which constructs and partitions Euler tours in a combined operation may have merit as a one pass heuristic algorithm. In the isthmus avoiding method of constructing Euler tours many choices are usually available in selecting the next edge to be included in the tour (for street networks the number of such edges will be at most three or four). A good strategy would be to select dummy edges if the current work shift endpoint lands on it, and to avoid the selection of dummy edges if they threaten to be absorbed in a working tour. Trying this strategy on our sample graph yields the optimal tour partition. From the partial Euler tour (1,2,3) at node 3 we are in our &cond working tour with 4.23 tmits of accumulated time. The next edge may be selected as dummy edge (3,4), real edge (3,4), or real edge (3,7). As an accumulated time of 10 units will fall on dummy edge (3,4) this is the best “local” choice, followed by dummy edge (4,6) on the same duplicated path.

In all of the above we make no claims that the optimal solution (minimum total time for the minimal number of working tours) lies in partitions of the set of all Euler tours on the minimal augmented even graph, and thus the above procedures are strictly heuristic.

SOLVING THE BEERSHEVA EXAMPLE

The algorithm was applied to the simplified graph G (Fii. 2). The graph G was constructed from the Beersheva street map (Fii. 1) through a process of “cosmeticization”. Through this process we were able to appreciably reduce the number of nodes and edges of the original street map. For example the number of intersections were reduced from 54 to 42 (in Fii. 2 the original numerical node designations appearing in Fig. 1 are retained). The process consists of the absorbtion of appendages such as dead end streets, and the representation of street segments with joint isolated nodes by a single edge. For example, since the appendage attached CAOR Vol. 6 No. 4-D

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HELMAN I. STERN and M~SHE DROR

4.23

1 (5) \

(6-7)

E,.(l,2,3,7,6,5,4,6,4,3,4,1,5)

M,- 7

D,- 63.10

Fig. 4(c). Partition of third candidate tour on G*(I) into 7 working segments and minimum travel time.

to node 26 did not contain any meters it was completely deleted from the graph. We were then able to drop node 26 and represent its two adjacent streets by the single required edge (22,25) with reading and non reading traversal times equal to the sum of its prior components. The situation between nodes 44 and 42 is slightly different since here the dead end streets adjacent to nodes 40, 41 and 43 contained meters. These appendages and their adjacent nodes were absorbed into a single edge from node 44 to 42. The reading time of this edge was established by summing the reading times along each of the original 4 street segments and the 3 dead end streets. The non reading time, however, was taken as the direct walking time of the 4 original segments only (1.25 + 1.9+ 1.75+3.5 = 8.4). We were careful not to apply this absorbtion process to adjacent required and non required street segments. For example, the required streets (39,37) and (36,35) joined by a non required street (37,36) were not absorbed in a single edge (39,35) as we wanted to alford the opportunity of terminating a working tour on edge (37,36) during the partitioning procedure.

The edges of the graph G may be partitioned into the set of required edges A,. (solid edges) and the set of non required edges AZ (dotted edges). After removing the non required edge set A2 from G we obtain the graph Gi (Nr, AI), which remains connected, and affords US the opportunity to construct lower bounds on our solution. Summing the reading traversal times of Gi gives us a lower bound of 4297.52 min. on the total work to be done. Using a workshift time limit of 300 min. provides a lower bound of &f = 15 on the minimum number of required working tours. Application of phase I of the algorithm will provide us with initial upper bounds on the problem. Table 1 is the shortest distance matrix over G. The circled entries indicate the solution to the matching problem using the heuristic of step l(ii). The total value of this matching is 61.32. According to step l(iii) an improved matching of 56.52 was obtained by replacing paths (23-38) and (652) by paths (2244) and (52-38). (The new elements of this matching are indicated by boxed cells in Table 1.) This was possible because the paths (23-28)

,

220 HELMAN I. STERN and MOSHE DROR

and (654) contained a common edge (44,52). The final even graph G* is shown in Fig. 5. The augmented edges are shown as dotted lines and are of two types: duplicated required edges and non required edges. The former are identified by the curved dotted edges. After removing path (23,44)-the shortest path of maximal length, (step 2 of the algorithm) a graph G*(l) was obtained with two odd nodes 23 and 44 between which an open Euler tour was constructed. The length of this tour was 4340.59 comprised of 4297.52 min of reading time over required edges and 43.07 min of walking time over duplicated or non required edges. The upper bounds on the problem are now 4341.59min of 15 for the minimal total time and the minimal number of working tours, respectively.

The results of partitioning the open Euler tour into working segments according to Phase II of the algorithm are shown in Fig. 6. The 15 working segments are identified on the graph as Tl-T15. All tours are of full length (300 min) except for T4 and T 15 of 299.62 and 138.57 min, respectively. This gives the optimal number of working tours M* = 15 and a total working time of D = 4338.19. Through the partition procedure we were able to reduce the total time by only 2.40 min. This was a result of dropping the duplicated edge (10, 11) during the truncation of working segment T4. An attempt was made to further reduce the total working time by partitioning alternative open Euler tours, but unsuccessfully.

DISCUSSION OF THE SOLUTION

We note that the solution to the matching problem gave a rough estimate of the maximum non productive (deadheading time) work to be done in the Beersheva example. As this time was in the order of 40-6Omin only marginal gains could be achieved in trying to minimize dead-heading time. In fact in our final solution only 40.67 of the total of 4338.19 min of on the street time was attributed to deadheading time. Thus, for our problem there was little to be gained by trying to find an exact solution to the matching problem or for that matter trying to eliminate deadheading streets through alternative partitionings. On the other hand, in our small seven node example an exact solution to the matching problem; and successive partitioning of alternative open Euler tours resulted in an appreciable reduction of the number of working tours as well as the total working time. The question may be asked: for what types of problems and under what conditions should the full power of the algorithm be employed in an attempt to achieve improved solutions?

The first step is to compare the value of the matching with the “slack work time” of the set of required edges. In computing the lower bound on the number of working tours [L(A,)/T]+ = M, we note that &f may be divided into &f - 1 “complete working tours” of length T, and 1 working tour with R units of slack non productive work time-the residual of L(A,)/T. If a matching value of Z* is obtained and Z* 5 R then &i = &f and the partition of any open Euler tour will provide a representation of the minimum number of working tours. If Z* > R which

implies ti >&f, then a better matching and repeated application of the partition Phase II to alternative candidate open Euler tours may reduce the number of working tours. Moreover, in the first case if the magnitude of Z* is insigniticant with respect to L(AJ then little can be gained in an attempt to reduce the second objective of minimum total time. This all points to the importance of the magnitude of the potential deadheading time Z*.

Several factors are simultaneously at play in determining the magnitude of Z*. The first has to do with the topology of the graph of required edges. The more odd nodes in this graph and the more widely dispersed they are over the network the more duplicated edges are necessary in order to traverse the network completely. If the number of duplicated edges rises then the occurrences of situations where savings are encountered in the partitioning phase of the algorithm will rise. An additional factor of critical importance is the ratio of edge reading to nonreading time. If the reading times are large in comparison to the nonreading times then the magnitude of Z* is more likely to be small in comparison to L(A,). It is then more likely that reductions in deadheading time will be insignificant. For the Beersheva example we encoun- tered relatively short streets capable of being traversed in strict walking times in the order of 3-4 min. The required streets were heavily loaded with, apartment buildings, some streets with hundreds of meters on them, each requiring approximately 1 min to read. These factors lead to reading/nonreading ratios in the order of 10-45. The seven node example, on the other hand, exhibited ratios of unity.

Routing electric meter readers 221

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Routiq electric meter readers 223

In conclusion the topology of the street network and the ratio of productive vs non- productive traversal times are important considerations in determining how effective the use of increased analytical power (of the algorithm) is in achieving gains in the solution. In cases of low reading/nonreading time ratios, networks of required streets with widely dispersed sections, and many odd degree intersections the fulI power of the algorithm can be appreciated. Conversely, for hi readiinonreading ratios in conjunction with compact required street networks with few odd degree intersections (such as rectangular street networks) the con- struction of a single open Euler tour and its partition should provide a sufficient level of analysis. One practical problem that exhibits the conditions for large improvements is the delivery of newspapers by foot where the productive/non productive ratio is close to unity. It is quite conceivable that other investigators may find additional practical situations exhibiting low ratios and irregular dispersed networks.

CONCLUDING REMARKS

We have presented a methodology based on Chinese postman solution techniques to solve a special type of edge oriented multiple open tour routing problem. This methodology was applied to the problem of routing electric meter readers in one neighborhood of the city of Beersheva. The city of Beersheva currently canvases this neighborhood with 24 working tours each no greater than five hours duration. Our solution has shown that through improved routing the work may be completed in 15 working tours. This corresponds to approximately a 40% savings in manpower or time. If, as expected, these savings can be achieved in the remaining seven neighborhoods of Beersheva this would result in a sign&ant reduction of labor costs. In our application relatively little effort involving a simple construction of an open Euler tour and its partition provided these savings. We have indicated conditions where it is worthwhile to expend additional work in the application of our algorithm in order to exploit the potential savings inherent in the network under consideration.

Acknowledgements-We would lie to give credit to H. Zvi, K. Efraim, G Won, B. Israel and D. Dali, from the Beersheva Branch of the Israeli Electric Company, for providing the detailed data for our Beersheva example.

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