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Methods for solving combined two level location-routing problems of realistic dimensions
Methods for solving combined two level location-routing problems of realistic dimensions Oli B.G. MADSEN IMSOR, The Institute of Mathematical Research. The Technrcal Unioerslt.y Lp~g~y, Dervnark
Statistics and Operatmns of Denmark, DK 2800
Received August 198 1 Revised September 1982
Very few distribution studies have dealt with combined location-routing problems, the problems of locating depots from which customers are served by tours rather than individual trips. This paper gives a survey of methods solving combined location-routing problems. Some methods are analyzed and three new heuristic methods are developed, implemented and compared. A newspaper delivery system consisting of 4500 customers is solved. The results seem to indicate that an alternate location-allocation-savings procedure and a savings-drop procedure are promising.
LEGEND (‘
-
Tour Customer
0
0 Fig. 1. Combined
Depot
location-routing
structure
with
no factory.
1. Introduction
In many distribution systems a hybrid location-routing’problem occurs. For instance in a two level system commodities are sent from factories via depots to customers. Several depots are served on one route outgoing from a factory and several customers are served on one route outgoing from a depot. In this problem three types of decisions are to be made: (a) the number and location of depots, (b) the design of tours originating at the factory to serve the depots, and, (c) the design of tours emanating from depots to serve customers. In this paper we will focus on the problem with either no factory (as shown in Fig. 1) or one factory having a given location (as shown in Fig. 2),
LEGEND +
Primary
-+
Secondary
0 U
North-Holland Publishing Company European Journal of Operational Research
0377-2217/83/0000-0000/$03.00
12 ( 1983)
295-301
@ 1983 North-Holland
Tour Tour
(PiI (ST)
Customer Depot
F1g. 2 Combined location-routing and primary tours.
structure
uith
one factor!
296
0. B. G. Madsen
/
Two level location-routing
several depots, and customers with a tour structure between depots and customers.
2. Review of location routing studies The location and routing literature have increased greatly during the last two decades (see e.g. Lea [16], Francis and Goldstein [9], Golden, Magnanti and Nguyen [ 111, and Francis, McGinnis and White [lo]). Unfortunately very few papers have tried to combine the two subject areas. Location theory deals with direct deliveries to customers and Routing theory deals with fixed locations. In this section a few papers dealing with combined location-routing problems will be briefly reviewed. Christofides and Eilon [5] were among the first to consider the problem of locating a depot from which customers are served by tours rather than individual trips. The basic result is an approximation formula giving the total length of the tours as a function of the average number of customers per vehicle, the size of the area served by the depot and the sum-of depot-customer distances. Using the approximation formula, the location problem becomes a standard weberian type of problem. Furthermore they point out that the position of the depot does not affect appreciably the expected tour length. The approximation formula is valid under the assumption that customers are uniformly distributed in the area, which very often is not the case. Webb [22] demonstrates how the use of simple cost functions in a wholesale distribution depot location problem may produce misleading results. Marks and Stricter [17] point out that routing models should become part of a complete system of models for distribution and location problems when efficient solution techniques have been developed. Burness and White [4] suggest an iterative procedure for the solution of the travelling salesman location problem, wheie one new facility is to be inserted on a travelling salesman tour through m existing facilities. In one iteration. a single facility location problem is solved using suitably adjusted weights on hyperbolically approximated distances, and a travelling salesman problem is solved by standard procedures. The literature contains only a few papers on ‘location-routing’ problems. Six examples are
problems
Watson-Gandy and Dohrn [21], TFD [20], Jacobsen and Madsen [ 12,131, Or and Pierskalla [ 181, Bednar and Strohmeier [3] and Laporte, Nobert and Pelletier [ 14,151. In the first and second of those papers the problem is solved by transforming its location part into an ‘ordinary’ location problem using the Christofides-Eilon-like approximation mentioned above. The location-part is then solved by a standard location-allocation technique producing depot locations and customer allocations (Cooper [7] and Rapp [19]). Next, the routing-part is solved by standard vehicle scheduling techniques, such as the Savings Method (Clarke and Wright [6]). The Watson-Gandy and Dohrn paper is concerned with a company operating in the food and drink industry in the southern part of England and Wales. In addition to distribution and depots costs, sales revenue (on the average, sales decline with distance from a depot) were included in the model. The solution recommended to the company involved the closing of 3 out of 11 existing depots, and the opening of 8 new depots, resulting in an increase of an estimated f60000 per annum in profits. This solution was later implemented. The TFD-paper is concerned with the delivery of goods to approximately 100 shops in Sweden, and the location of depots that could be placed in 13 possible locations. The results are claimed to be good, but very few details have been reported. The Jacobsen and Madsen papers will be discussed in more details in a subsequent section. The problem may be described as a newspaper-delivery case from the western part of Denmark. From the printing office with a given production rate the newspapers are loaded on to primary vehicles (small trucks). Each vehicle serves from one to three transfer points, where the newspapers are transferred to secondary vehicles (vans and ordinary cars) that make the final delivery to 4510 customers (e.g., retailers, canteens). The transfer points that are to be located can be located anywhere at virtually no cost. The routing costs consist of a fixed and a distance dependant cost. Each sales point has a specified latest delivery time, and some additional constraints are included in the model such as a production rate constraint, capacity constraints. and a secondary tour duration constraint. The objective is to minimize the total transportation costs. Jacobsen and Madsen solved the problem using three different heuristics:
0. B. G. Madsen
/ Two level location
(1) a tour construction method with implicit transfer point locations, (2) an alternate location-allocation procedure, and (3) a savings procedure for secondary tour routing, followed by a drop procedure for locations followed by a savings procedure for primary tour routing. The derived solution showed slightly better results than the existing solution (2.9%5.6%) and suggested a decrease was needed from 42 to 12 points. Or and Pierskalla have suggested a transportation location-allocation model for the location of bloodbanks from which vehicles are sent on tours to deliver blood to hospitals. The model is called BTAP (Blood Transportation Allocation Problem). It requires a given number of bloodbanks and a number of potential locations where these banks can be located. In addition, the model takes vehicle capacity and maximum travel distance for a vehicle into consideration. The model is formulated as a large zero-one programming problem. This problem can be solved with a two-stage heuristic procedure. In one stage a number of vehicle dispatching problems (VDP) are solved. Based on these results a kind of savings is calculated and the assignment of hospitals to bloodbanks is changed. Then a new group of VDP is solved. When all reasonable exchanges are completed (i.e., no positive exchange savings are found) the procedure is terminated. The actual data for 117 hospitals in the Greater Metropolitan Chicago area was used as test data. A near optimal solution was found in 40-70 seconds on a CDC 6400 computer. Bednar and Strohmeier constructed a model for distributing consumer goods to a large number of outlets. One goal was to find warehouse locations, customer to warehouse assignments and routes minimizing the warehouse and transportation costs. The problem was solved by a two-stage procedure. First the grouping of customers in zones and thereby also the location of warehouses was found by cluster analysis. The cluster analysis was performed by the standard code CLUSTAN 1A. A minimum variance criterion also called the kmeans-method (Anderberg [I]) was used. When the districting had been done a straightforward standard code for vehicle scheduling, ROOT (Route Organization and Optimization Technique), for UNIVAC computers was used in stage two. This algorithm is based on the savings procedure.
-routing
problems
291
To strengthen the results from the two-stage procedure a supporting mixed-integer-programming model was used for comparison. Here the standard code was FMPS (Functional Mathematical Programming System) for UNIVAC computers. The study was implemented by an Austrian manufacturer of consumer goods. First the customers were aggregated into 1500 customer groups. One group for each zipcode number. Then an extensive analysis of the road network was performed in order to have the necessary data for calculating transportation costs. Very little has been reported about the computer runs and the results. In the Laporte and Nobert paper [ 141 and algorithm for finding an exact solution to a combined location-routing problem is discussed. One depot has to be located and the distance matrix has to be symmetric, but not necessarily Euclidean. The objective is to minimize the fixed depot costs plus the travel and operating costs. A mixed-integer-programming model is formulated and the model is solved by relaxation. The algorithm is applied to ‘random problems ranging from 20 to 50 points. In the case of Euclidean distances the computation time seems to be exponentially increasing with a doubling of the computational time with each increment of 5-6 points. This work has been continued in the paper of Laporte, Nobert and Pelletier [ 151. This time the algorithm is extended to solve multi-depot problems, but still gives exact solutions. Both the problem involving fixed costs on the depots, and the problems of no fixed costs are examined. Also, an upper bound on the number of depots is introduced. The algorithm is applied to random problems ranging from 20 to 50 points and solution times range from 1 to 100 seconds on a CYBER 173 computer. In summary, it appears that only a few methods are available for solving location-routing problems. The Bednar and Strohmeier (BS) approach is very similar to the second approach of Jacobsen and Madsen (JM2). The clustering in BS uses the minimum variance criterion as its objective, while JM2 uses the Euclidean distance criterion in the location-allocation stage. Ihis means that BS penalizes customers far from the depot-locations a little more. The Or and Pierskalla (OP) approach is also similar to BS and JM2 except that OP is repeating the two stages more than one time. Recently algorithms finding exact solutions to com-
0. B. G. Madsen
298
/
Two level locatron
bined location-routing problems have been suggested (Laporte, Nobert and Pelletier [ 151).
3. The tree-tour heuristic (TIM) The tour construction method by Jacobsen and Madsen [ 121 and [ 131 is called a tree tour heuristic (TTH). If we consider a tour pattern with no return arcs (i.e. the secondary tours stop at the last sales point), the graph becomes a spanning tree with the characteristic that only the factory and the transfer points have multiple successors (as shown in fig. 3). Thus the task at hand may be viewed as one of constructing a spanning tree satisfying timing and capacity constraints at minimum cost. Due to the timing constraints the tree is constructed starting with the customers with tightest time constraints. The customers are sorted according to urgency of delivery. Then the customers are added one at a time to the tour structure according to some priority rules in which cost minimization is indirectly taken into consideration. by the predefined sequencing of customers, computational efficiency is gained over a mddified Savings-mgthod, where search is required for the next sales point to be added. In other words, the
LEG END =+
Primary
--+
Secondary
0
Customer
0 Fig. 3. Tour
pattern
Tour
(Pi)
Tour
ISTi
Transfer
Point
ITPI
represented
as a spanning
tree
-routing
problems
Tree Tour Heuristic is a ‘one shot’ feasibility seeking construction method. To store and update a large-scale tour structure in a convenient way the Extended Threaded Indexing method was adapted from network optimization (Barr, Glover an Klingman [2]). The solution of the newspaper delivery problem showed disappointing results with total costs exceeding the present costs by 25%. It was concluded that the TT’H probably would be useful in cases with extremely tight time constraints so that the problem was more to find a feasible solution than an optimal solution. To solve the problem with 4510 customers, the code plus arrays areas used some 200 K bytes of core storage. The computing times ranged around 12 minutes on an IBM 370/165 computer using FORTRAN G. Table 1 in Section 5 gives a summary of the performance characteristics of the TTH.
4. The ALA-SAV heuristic The second method of Jacobsen and Madsen [ 131 briefly mentioned in Section 2 was ALA-SAV, a three stage heuristic procedure composed from the Alternate Location Allocation method (ALA) by Rapp [19] and Cooper [7], and the Savings method (SAV) by Clarke and Wright [6]. It involved (i) a preselected number of transfer points located by a standard ALA-procedure and customers allocated to specific transfer points; (ii) secondary tours that are formed applying a SAV-procedure sequentially to each transfer point and it’s allocated secondary tours (attention is paid here to the Secondary Tour Duration Constraints and the Delivery Time Constraints); and, (iii) primary tours that are formed used a standard SAV-procedure with the factory as the depot and the transfer points as customers (feasibility checks are made here on the primary Vehicle Capacity Constraints and the Delivery Time Constraints). The transfer point-locations in (i) are determined with regard to the customers (each having weight one); that is, the incoming primary tour from the factory exercises no pull on the transfer point. The transfer point-locations are rectilinear distance median location that are determined substantially faster than the Euclidean median loca-
0. B. G. Madsen
/ Two level location
tions. The former are determined essentially by a single pass procedure. Experience shows that the two medians are virtually coincident, especially when all points have unit weight. Between (i) and (ii) a check on the feasibility of the Delivery Time Constraints is inserted. Then (ii) requires solution of some lo-50 constrained routing problems, each with 450-90 customers. Thus, it was necessary to speed up the SAV-method by using a ‘row maximum entry’-method instead of the standard ‘matrix maximum entry’-method. Because of this slight alteration the number of savings evaluated per connection established is reduced dramatically (approximately by a factor of 100 in the present case). In (iii) the only non-standard feature is that the order in which the primary vehicles leave the factory may have to be changed during primary tour formation, in order to create the best match between the Production Rate Constraint and the Delivery Time Constraint. In order to cut down the number of secondary tours (ii) was modified. This modification consisted of the application of another savings concept. The saving is simply taken to be the negative of the distance between the two sales points being considered for connection. In other words, (ii) attempts to establish the shortest possible distances between sales points when constructing the secondary tours, and thus utilizes the secondary tourduration as economically as possible. This modification worked reasonably well in the sense that the solutions obtained are comparable to reality. Computational results from the newspaper case showed that it was possible to get 2.9% better solutions than the existing one. The ALA-SAV procedure was coded in some 500 FORTRAN IV statements. The core storage demand is 164 K bytes. One computer run requires some 2 CPU-minutes on an IBM 370,/165 (Gcompiler): more than 1 CPU-minute is spent on secondary tour-routing. Table 1 in Section 5 gives a summary of the performance characteristics of ALA-SAV.
5. The SAV-DROP
heuristic
The third method of Jacobsen and Madsen [ 131 briefly mentioned in Section 2 was SAV-DROP, a
-routing
problems
299
three stage heuristic procedure composed from the Savings method (SAV) by Clarke and Wright [6] and the Drop method (DROP) by Feldman, Lehrer and Ray [8]. Its three stages are: (i) secondary tours are formed by a SAV-procedure regarding the factory as the depot (feasibility checks include only the secondary tour Duration Constraints); (ii) transfer points are located by a DROPprocedure (initially, a transfer point is located at the first customer of each secondary tour; feasibility checks are made on the Delivery Time Constraints assuming each transfer point is served by a primary tour coming directly from the factory); and, (iii) primary tours are formed by the same SAV-procedure as in (iii) of ALA-SAV. To speed up (i), the savings matrix includes only the 14 largest entries in each row. That is, a salespoint may only be connected to one of its 14 nearest neighbours. In addition, the ‘row maximum entry’-method explained in the preceding section is used. ) (i) produces secondary tours without orientation. Before entering (ii), the secodary tours are oriented so that delivery time slack is maximized. (ii) is by far the most time consuming since reallocation of secondary tours out of a transfer point to be dropped may change the order in which the primary tours leave the factory. All transfer points should be considered for dropping in order to find the best solution. The SAV-DROP heuristic is in some sense the most ‘natural’ of the three procedures described in Section 3, 4 and 5 as it reduces the volume of data sequentially. (i) reduces 4500 salespoints to some 200 secondary tours. (ii) reduces 200 secondary tours to some 15 transfer points. Finally, (iii) attempts to reduce the 15 transfer points to a smaller number of primary tours. SAV-DROP produces too many secondary tours relative to reality. As a remedy, (i) was modified in rhe same way as (ii) of ALA-SAV: that is. the having was redefined to be the negative of the distance between the two salespoints considered tar connection. Computational resulta from the newspaper case showed that it was possible to get .5.6’%better solutions than the existing one. The SAV-DROP procedure was coded in some 600 FORTRAN IV statements. The core storage demand is 380 K bytes. One computer run requires
0. B. G. Madsen
300 Table 1 Summary
Total No. No. No. Total Total CPU Core,
of performance
/ Two leoel locaiion
-routing
problems
characteristics
cost (Dkr/day) of PT’s ’ of TP’s = of ST’s’ PT mileage (km) ST mileage (km) minutes b K-bytes
A One run, more runs are necessary. b On an IBM 370/165 (FORTRAN ’ See Fig. 2.
Benchmark
TTM
ALA-SAV3
SAV-DROP3
35 404 26 42 190 2461 10421
45 505 21 45 231 3496 18882 12 “ 200
34375 12 12 213 1182 13728 2” 164
33 409 13 13 202 1353 12323 12 380
-
-
G-Compiler).
some 10 CPU-minutes on an IBM 370/165 (Gcompiler); secondary tour-routing requires only a couple of seconds, whereas most of the time is spent in the DROP-procedure. The generation and row-wise sorting of the savings file required some 3 CPU-minutes. (For a summary of performance characteristics see Table 1.)
6. Conclusion In conclusion, we have at hand the following four types of solution methods for the combined location-routing problem: (1) (a) Christofides-Eilon like transformations, then alternate location-allocation (ALA), then routing (SAV); (b) First ALA, then SAV and then perhaps SAV; (2) First SAV, then DROP and then perhaps SAV; (3) The Tree Tour Heuristic; and, (4) Exact relaxation algorithm. (4) can be used for problems containing not more than 50-60 points. The last SAV in (lb) and (2) is only necessary if primary tours are included in the model. Watson-Gandy and Dohrn, TFD belongs to (la), Jacobsen and Madsen (from Section 4), Or and Pierskalla, Bednar and Strohmeier belong to (1 b). Jacobsen and Madsen (from Section 5) belongs to (2) and Jacobsen and Madsen (from Section 3) belongs to (3). Apart from problems with very tight time constraints, (3) cannot be recommended. Both (1) and (2) have shown their usefulness. It seems perfectly possible to obtain
good solutions to location-routing problems by combining well known heuristics for location and routing.
References [ 1] M.R. Anderberg, Cluster Analysis for Applications (Academic Press. “w York, 1973). [2] R. Barr, F. Clover and D. Klingman, Enchancements of spanning tree labelling procedures for network optimization, Research Report CCS262, The University of Texas, Austin, TX (1976). ]3] L. von Bednar and E. Strohmeier, Lagerstandortoptimierung und Fuhrparkeinsatzplanung in der Konsumgiiter-Industrie,.Z. Operations Res. 23 (1979) B89-B104. [4] R.C. Burness and J.A. White, The travelling salesman location problem, Transportation Sci. 10 (1976) 348-360. ]5] N. Christofides and S. Eilon, Expected distances in distribution problems, Operational Res. Quart. 20 (1969) 437-443. [6] G. Clarke and J.W. Wright, Scheduling of vehicles from a central depot to a number of delivery points, Operations Res. 12 (1964) 568-581. [7] L. Cooper, Heuristic Methods for location-allocation problems, SIAM Rec. 6 (1964) 37-53. [8] E. Feldman, F.A. Lehrer and T.L. Ray, Warehouse location under continuous economies of scale, Manugement Sci. 12 (1966) 670-684. [Y] R.L. Francis and J.M. Goldstein, Location theory. a selective bibliography, Operations Re.r. 22 (1974) 400-411. [IO] R.L. Francis, L.F. McGinnis and J.A. White, Locational analysis, European J. Operational Res. 12 (1983) 220-252. this issue. i 1 I ] B.L. Golden, T.L. Magnanti and H.Q. Nguyen. Implementing vehicle routing algorithms, Networks 7 (1977) 113.- 148. [ 121 SK. Jacobsen and O.B.G. Madsen, On the location of transfer points in a two-level newspaper delivery system A case study. Research Report 5/197X, IMSOR, The Institute of Mathematical Statistic and Operations Re-
0. B. G. Madsen
[13]
Two level locairon
search, The Technical University of Denmark, Lyngby. Denmark (1978). S.K. Jacobsen and O.B.G. Madsen, A comparative study of heuristics for a two-level routing-location problem, European
[14]
/
J. Operational
Res. 5 (1980)
G. Laporte and Y. Nobert. An exact miring routing and operating costs European
J. Operational
378-387.
algorithm in depot
for minilocation,
Res. 6 (198 I) 224-226.
[ 151 G. Laporte, Y. Nobert and P. Pelletier, Hamiltonian location problems, European J. Operational Rex 12 (1983) 80-87. [ 161 A.C. Lea, Location-allocation systems: An annotated bibliography, Discussion Paper No. 13, Department of Geography, University of Toronto, Toronto, Canada (1973). ]17] D.H. Marks and R. Stricter, Routing for public service vehicles, Amer. Sot. Civil Engrg.. J. Urban Planning and Development Divrsion 97 (1971) 165- 178.
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problems
301
[ 181 I. Or and W.P. Pierskalla, A transportation location-allocation model for regional blood banking, AIIE Trans. II (1979) 86-95. [ 191 Y. Rapp, Planning of exchange locations and boundaries, Ericsson Technics 2 (1962) l-22. [20] TDF (Transportforskningsdelegationen), Distribution planning using mathematical methods (in Swedish), Report 1977:1, Contract Research Group for Applied Mathematics, Royal Institute of Technology, S-10044 Stockholm, Sweden (1977). [21] C.T.D. Watson-Gandy and P.J. Dohrn. Depot location with van salesman - A practical approach, Omega I (1973) 321-329. (221 M.H.J. Webb, Cost functions in the location of depots for multiple-delivery journeys. f?prmfronal Rc? Qunrt !O (1968) 311-320.
Statistics and Operatmns of Denmark, DK 2800
Received August 198 1 Revised September 1982
Very few distribution studies have dealt with combined location-routing problems, the problems of locating depots from which customers are served by tours rather than individual trips. This paper gives a survey of methods solving combined location-routing problems. Some methods are analyzed and three new heuristic methods are developed, implemented and compared. A newspaper delivery system consisting of 4500 customers is solved. The results seem to indicate that an alternate location-allocation-savings procedure and a savings-drop procedure are promising.
LEGEND (‘
-
Tour Customer
0
0 Fig. 1. Combined
Depot
location-routing
structure
with
no factory.
1. Introduction
In many distribution systems a hybrid location-routing’problem occurs. For instance in a two level system commodities are sent from factories via depots to customers. Several depots are served on one route outgoing from a factory and several customers are served on one route outgoing from a depot. In this problem three types of decisions are to be made: (a) the number and location of depots, (b) the design of tours originating at the factory to serve the depots, and, (c) the design of tours emanating from depots to serve customers. In this paper we will focus on the problem with either no factory (as shown in Fig. 1) or one factory having a given location (as shown in Fig. 2),
LEGEND +
Primary
-+
Secondary
0 U
North-Holland Publishing Company European Journal of Operational Research
0377-2217/83/0000-0000/$03.00
12 ( 1983)
295-301
@ 1983 North-Holland
Tour Tour
(PiI (ST)
Customer Depot
F1g. 2 Combined location-routing and primary tours.
structure
uith
one factor!
296
0. B. G. Madsen
/
Two level location-routing
several depots, and customers with a tour structure between depots and customers.
2. Review of location routing studies The location and routing literature have increased greatly during the last two decades (see e.g. Lea [16], Francis and Goldstein [9], Golden, Magnanti and Nguyen [ 111, and Francis, McGinnis and White [lo]). Unfortunately very few papers have tried to combine the two subject areas. Location theory deals with direct deliveries to customers and Routing theory deals with fixed locations. In this section a few papers dealing with combined location-routing problems will be briefly reviewed. Christofides and Eilon [5] were among the first to consider the problem of locating a depot from which customers are served by tours rather than individual trips. The basic result is an approximation formula giving the total length of the tours as a function of the average number of customers per vehicle, the size of the area served by the depot and the sum-of depot-customer distances. Using the approximation formula, the location problem becomes a standard weberian type of problem. Furthermore they point out that the position of the depot does not affect appreciably the expected tour length. The approximation formula is valid under the assumption that customers are uniformly distributed in the area, which very often is not the case. Webb [22] demonstrates how the use of simple cost functions in a wholesale distribution depot location problem may produce misleading results. Marks and Stricter [17] point out that routing models should become part of a complete system of models for distribution and location problems when efficient solution techniques have been developed. Burness and White [4] suggest an iterative procedure for the solution of the travelling salesman location problem, wheie one new facility is to be inserted on a travelling salesman tour through m existing facilities. In one iteration. a single facility location problem is solved using suitably adjusted weights on hyperbolically approximated distances, and a travelling salesman problem is solved by standard procedures. The literature contains only a few papers on ‘location-routing’ problems. Six examples are
problems
Watson-Gandy and Dohrn [21], TFD [20], Jacobsen and Madsen [ 12,131, Or and Pierskalla [ 181, Bednar and Strohmeier [3] and Laporte, Nobert and Pelletier [ 14,151. In the first and second of those papers the problem is solved by transforming its location part into an ‘ordinary’ location problem using the Christofides-Eilon-like approximation mentioned above. The location-part is then solved by a standard location-allocation technique producing depot locations and customer allocations (Cooper [7] and Rapp [19]). Next, the routing-part is solved by standard vehicle scheduling techniques, such as the Savings Method (Clarke and Wright [6]). The Watson-Gandy and Dohrn paper is concerned with a company operating in the food and drink industry in the southern part of England and Wales. In addition to distribution and depots costs, sales revenue (on the average, sales decline with distance from a depot) were included in the model. The solution recommended to the company involved the closing of 3 out of 11 existing depots, and the opening of 8 new depots, resulting in an increase of an estimated f60000 per annum in profits. This solution was later implemented. The TFD-paper is concerned with the delivery of goods to approximately 100 shops in Sweden, and the location of depots that could be placed in 13 possible locations. The results are claimed to be good, but very few details have been reported. The Jacobsen and Madsen papers will be discussed in more details in a subsequent section. The problem may be described as a newspaper-delivery case from the western part of Denmark. From the printing office with a given production rate the newspapers are loaded on to primary vehicles (small trucks). Each vehicle serves from one to three transfer points, where the newspapers are transferred to secondary vehicles (vans and ordinary cars) that make the final delivery to 4510 customers (e.g., retailers, canteens). The transfer points that are to be located can be located anywhere at virtually no cost. The routing costs consist of a fixed and a distance dependant cost. Each sales point has a specified latest delivery time, and some additional constraints are included in the model such as a production rate constraint, capacity constraints. and a secondary tour duration constraint. The objective is to minimize the total transportation costs. Jacobsen and Madsen solved the problem using three different heuristics:
0. B. G. Madsen
/ Two level location
(1) a tour construction method with implicit transfer point locations, (2) an alternate location-allocation procedure, and (3) a savings procedure for secondary tour routing, followed by a drop procedure for locations followed by a savings procedure for primary tour routing. The derived solution showed slightly better results than the existing solution (2.9%5.6%) and suggested a decrease was needed from 42 to 12 points. Or and Pierskalla have suggested a transportation location-allocation model for the location of bloodbanks from which vehicles are sent on tours to deliver blood to hospitals. The model is called BTAP (Blood Transportation Allocation Problem). It requires a given number of bloodbanks and a number of potential locations where these banks can be located. In addition, the model takes vehicle capacity and maximum travel distance for a vehicle into consideration. The model is formulated as a large zero-one programming problem. This problem can be solved with a two-stage heuristic procedure. In one stage a number of vehicle dispatching problems (VDP) are solved. Based on these results a kind of savings is calculated and the assignment of hospitals to bloodbanks is changed. Then a new group of VDP is solved. When all reasonable exchanges are completed (i.e., no positive exchange savings are found) the procedure is terminated. The actual data for 117 hospitals in the Greater Metropolitan Chicago area was used as test data. A near optimal solution was found in 40-70 seconds on a CDC 6400 computer. Bednar and Strohmeier constructed a model for distributing consumer goods to a large number of outlets. One goal was to find warehouse locations, customer to warehouse assignments and routes minimizing the warehouse and transportation costs. The problem was solved by a two-stage procedure. First the grouping of customers in zones and thereby also the location of warehouses was found by cluster analysis. The cluster analysis was performed by the standard code CLUSTAN 1A. A minimum variance criterion also called the kmeans-method (Anderberg [I]) was used. When the districting had been done a straightforward standard code for vehicle scheduling, ROOT (Route Organization and Optimization Technique), for UNIVAC computers was used in stage two. This algorithm is based on the savings procedure.
-routing
problems
291
To strengthen the results from the two-stage procedure a supporting mixed-integer-programming model was used for comparison. Here the standard code was FMPS (Functional Mathematical Programming System) for UNIVAC computers. The study was implemented by an Austrian manufacturer of consumer goods. First the customers were aggregated into 1500 customer groups. One group for each zipcode number. Then an extensive analysis of the road network was performed in order to have the necessary data for calculating transportation costs. Very little has been reported about the computer runs and the results. In the Laporte and Nobert paper [ 141 and algorithm for finding an exact solution to a combined location-routing problem is discussed. One depot has to be located and the distance matrix has to be symmetric, but not necessarily Euclidean. The objective is to minimize the fixed depot costs plus the travel and operating costs. A mixed-integer-programming model is formulated and the model is solved by relaxation. The algorithm is applied to ‘random problems ranging from 20 to 50 points. In the case of Euclidean distances the computation time seems to be exponentially increasing with a doubling of the computational time with each increment of 5-6 points. This work has been continued in the paper of Laporte, Nobert and Pelletier [ 151. This time the algorithm is extended to solve multi-depot problems, but still gives exact solutions. Both the problem involving fixed costs on the depots, and the problems of no fixed costs are examined. Also, an upper bound on the number of depots is introduced. The algorithm is applied to random problems ranging from 20 to 50 points and solution times range from 1 to 100 seconds on a CYBER 173 computer. In summary, it appears that only a few methods are available for solving location-routing problems. The Bednar and Strohmeier (BS) approach is very similar to the second approach of Jacobsen and Madsen (JM2). The clustering in BS uses the minimum variance criterion as its objective, while JM2 uses the Euclidean distance criterion in the location-allocation stage. Ihis means that BS penalizes customers far from the depot-locations a little more. The Or and Pierskalla (OP) approach is also similar to BS and JM2 except that OP is repeating the two stages more than one time. Recently algorithms finding exact solutions to com-
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298
/
Two level locatron
bined location-routing problems have been suggested (Laporte, Nobert and Pelletier [ 151).
3. The tree-tour heuristic (TIM) The tour construction method by Jacobsen and Madsen [ 121 and [ 131 is called a tree tour heuristic (TTH). If we consider a tour pattern with no return arcs (i.e. the secondary tours stop at the last sales point), the graph becomes a spanning tree with the characteristic that only the factory and the transfer points have multiple successors (as shown in fig. 3). Thus the task at hand may be viewed as one of constructing a spanning tree satisfying timing and capacity constraints at minimum cost. Due to the timing constraints the tree is constructed starting with the customers with tightest time constraints. The customers are sorted according to urgency of delivery. Then the customers are added one at a time to the tour structure according to some priority rules in which cost minimization is indirectly taken into consideration. by the predefined sequencing of customers, computational efficiency is gained over a mddified Savings-mgthod, where search is required for the next sales point to be added. In other words, the
LEG END =+
Primary
--+
Secondary
0
Customer
0 Fig. 3. Tour
pattern
Tour
(Pi)
Tour
ISTi
Transfer
Point
ITPI
represented
as a spanning
tree
-routing
problems
Tree Tour Heuristic is a ‘one shot’ feasibility seeking construction method. To store and update a large-scale tour structure in a convenient way the Extended Threaded Indexing method was adapted from network optimization (Barr, Glover an Klingman [2]). The solution of the newspaper delivery problem showed disappointing results with total costs exceeding the present costs by 25%. It was concluded that the TT’H probably would be useful in cases with extremely tight time constraints so that the problem was more to find a feasible solution than an optimal solution. To solve the problem with 4510 customers, the code plus arrays areas used some 200 K bytes of core storage. The computing times ranged around 12 minutes on an IBM 370/165 computer using FORTRAN G. Table 1 in Section 5 gives a summary of the performance characteristics of the TTH.
4. The ALA-SAV heuristic The second method of Jacobsen and Madsen [ 131 briefly mentioned in Section 2 was ALA-SAV, a three stage heuristic procedure composed from the Alternate Location Allocation method (ALA) by Rapp [19] and Cooper [7], and the Savings method (SAV) by Clarke and Wright [6]. It involved (i) a preselected number of transfer points located by a standard ALA-procedure and customers allocated to specific transfer points; (ii) secondary tours that are formed applying a SAV-procedure sequentially to each transfer point and it’s allocated secondary tours (attention is paid here to the Secondary Tour Duration Constraints and the Delivery Time Constraints); and, (iii) primary tours that are formed used a standard SAV-procedure with the factory as the depot and the transfer points as customers (feasibility checks are made here on the primary Vehicle Capacity Constraints and the Delivery Time Constraints). The transfer point-locations in (i) are determined with regard to the customers (each having weight one); that is, the incoming primary tour from the factory exercises no pull on the transfer point. The transfer point-locations are rectilinear distance median location that are determined substantially faster than the Euclidean median loca-
0. B. G. Madsen
/ Two level location
tions. The former are determined essentially by a single pass procedure. Experience shows that the two medians are virtually coincident, especially when all points have unit weight. Between (i) and (ii) a check on the feasibility of the Delivery Time Constraints is inserted. Then (ii) requires solution of some lo-50 constrained routing problems, each with 450-90 customers. Thus, it was necessary to speed up the SAV-method by using a ‘row maximum entry’-method instead of the standard ‘matrix maximum entry’-method. Because of this slight alteration the number of savings evaluated per connection established is reduced dramatically (approximately by a factor of 100 in the present case). In (iii) the only non-standard feature is that the order in which the primary vehicles leave the factory may have to be changed during primary tour formation, in order to create the best match between the Production Rate Constraint and the Delivery Time Constraint. In order to cut down the number of secondary tours (ii) was modified. This modification consisted of the application of another savings concept. The saving is simply taken to be the negative of the distance between the two sales points being considered for connection. In other words, (ii) attempts to establish the shortest possible distances between sales points when constructing the secondary tours, and thus utilizes the secondary tourduration as economically as possible. This modification worked reasonably well in the sense that the solutions obtained are comparable to reality. Computational results from the newspaper case showed that it was possible to get 2.9% better solutions than the existing one. The ALA-SAV procedure was coded in some 500 FORTRAN IV statements. The core storage demand is 164 K bytes. One computer run requires some 2 CPU-minutes on an IBM 370,/165 (Gcompiler): more than 1 CPU-minute is spent on secondary tour-routing. Table 1 in Section 5 gives a summary of the performance characteristics of ALA-SAV.
5. The SAV-DROP
heuristic
The third method of Jacobsen and Madsen [ 131 briefly mentioned in Section 2 was SAV-DROP, a
-routing
problems
299
three stage heuristic procedure composed from the Savings method (SAV) by Clarke and Wright [6] and the Drop method (DROP) by Feldman, Lehrer and Ray [8]. Its three stages are: (i) secondary tours are formed by a SAV-procedure regarding the factory as the depot (feasibility checks include only the secondary tour Duration Constraints); (ii) transfer points are located by a DROPprocedure (initially, a transfer point is located at the first customer of each secondary tour; feasibility checks are made on the Delivery Time Constraints assuming each transfer point is served by a primary tour coming directly from the factory); and, (iii) primary tours are formed by the same SAV-procedure as in (iii) of ALA-SAV. To speed up (i), the savings matrix includes only the 14 largest entries in each row. That is, a salespoint may only be connected to one of its 14 nearest neighbours. In addition, the ‘row maximum entry’-method explained in the preceding section is used. ) (i) produces secondary tours without orientation. Before entering (ii), the secodary tours are oriented so that delivery time slack is maximized. (ii) is by far the most time consuming since reallocation of secondary tours out of a transfer point to be dropped may change the order in which the primary tours leave the factory. All transfer points should be considered for dropping in order to find the best solution. The SAV-DROP heuristic is in some sense the most ‘natural’ of the three procedures described in Section 3, 4 and 5 as it reduces the volume of data sequentially. (i) reduces 4500 salespoints to some 200 secondary tours. (ii) reduces 200 secondary tours to some 15 transfer points. Finally, (iii) attempts to reduce the 15 transfer points to a smaller number of primary tours. SAV-DROP produces too many secondary tours relative to reality. As a remedy, (i) was modified in rhe same way as (ii) of ALA-SAV: that is. the having was redefined to be the negative of the distance between the two salespoints considered tar connection. Computational resulta from the newspaper case showed that it was possible to get .5.6’%better solutions than the existing one. The SAV-DROP procedure was coded in some 600 FORTRAN IV statements. The core storage demand is 380 K bytes. One computer run requires
0. B. G. Madsen
300 Table 1 Summary
Total No. No. No. Total Total CPU Core,
of performance
/ Two leoel locaiion
-routing
problems
characteristics
cost (Dkr/day) of PT’s ’ of TP’s = of ST’s’ PT mileage (km) ST mileage (km) minutes b K-bytes
A One run, more runs are necessary. b On an IBM 370/165 (FORTRAN ’ See Fig. 2.
Benchmark
TTM
ALA-SAV3
SAV-DROP3
35 404 26 42 190 2461 10421
45 505 21 45 231 3496 18882 12 “ 200
34375 12 12 213 1182 13728 2” 164
33 409 13 13 202 1353 12323 12 380
-
-
G-Compiler).
some 10 CPU-minutes on an IBM 370/165 (Gcompiler); secondary tour-routing requires only a couple of seconds, whereas most of the time is spent in the DROP-procedure. The generation and row-wise sorting of the savings file required some 3 CPU-minutes. (For a summary of performance characteristics see Table 1.)
6. Conclusion In conclusion, we have at hand the following four types of solution methods for the combined location-routing problem: (1) (a) Christofides-Eilon like transformations, then alternate location-allocation (ALA), then routing (SAV); (b) First ALA, then SAV and then perhaps SAV; (2) First SAV, then DROP and then perhaps SAV; (3) The Tree Tour Heuristic; and, (4) Exact relaxation algorithm. (4) can be used for problems containing not more than 50-60 points. The last SAV in (lb) and (2) is only necessary if primary tours are included in the model. Watson-Gandy and Dohrn, TFD belongs to (la), Jacobsen and Madsen (from Section 4), Or and Pierskalla, Bednar and Strohmeier belong to (1 b). Jacobsen and Madsen (from Section 5) belongs to (2) and Jacobsen and Madsen (from Section 3) belongs to (3). Apart from problems with very tight time constraints, (3) cannot be recommended. Both (1) and (2) have shown their usefulness. It seems perfectly possible to obtain
good solutions to location-routing problems by combining well known heuristics for location and routing.
References [ 1] M.R. Anderberg, Cluster Analysis for Applications (Academic Press. “w York, 1973). [2] R. Barr, F. Clover and D. Klingman, Enchancements of spanning tree labelling procedures for network optimization, Research Report CCS262, The University of Texas, Austin, TX (1976). ]3] L. von Bednar and E. Strohmeier, Lagerstandortoptimierung und Fuhrparkeinsatzplanung in der Konsumgiiter-Industrie,.Z. Operations Res. 23 (1979) B89-B104. [4] R.C. Burness and J.A. White, The travelling salesman location problem, Transportation Sci. 10 (1976) 348-360. ]5] N. Christofides and S. Eilon, Expected distances in distribution problems, Operational Res. Quart. 20 (1969) 437-443. [6] G. Clarke and J.W. Wright, Scheduling of vehicles from a central depot to a number of delivery points, Operations Res. 12 (1964) 568-581. [7] L. Cooper, Heuristic Methods for location-allocation problems, SIAM Rec. 6 (1964) 37-53. [8] E. Feldman, F.A. Lehrer and T.L. Ray, Warehouse location under continuous economies of scale, Manugement Sci. 12 (1966) 670-684. [Y] R.L. Francis and J.M. Goldstein, Location theory. a selective bibliography, Operations Re.r. 22 (1974) 400-411. [IO] R.L. Francis, L.F. McGinnis and J.A. White, Locational analysis, European J. Operational Res. 12 (1983) 220-252. this issue. i 1 I ] B.L. Golden, T.L. Magnanti and H.Q. Nguyen. Implementing vehicle routing algorithms, Networks 7 (1977) 113.- 148. [ 121 SK. Jacobsen and O.B.G. Madsen, On the location of transfer points in a two-level newspaper delivery system A case study. Research Report 5/197X, IMSOR, The Institute of Mathematical Statistic and Operations Re-
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[13]
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search, The Technical University of Denmark, Lyngby. Denmark (1978). S.K. Jacobsen and O.B.G. Madsen, A comparative study of heuristics for a two-level routing-location problem, European
[14]
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G. Laporte and Y. Nobert. An exact miring routing and operating costs European
J. Operational
378-387.
algorithm in depot
for minilocation,
Res. 6 (198 I) 224-226.
[ 151 G. Laporte, Y. Nobert and P. Pelletier, Hamiltonian location problems, European J. Operational Rex 12 (1983) 80-87. [ 161 A.C. Lea, Location-allocation systems: An annotated bibliography, Discussion Paper No. 13, Department of Geography, University of Toronto, Toronto, Canada (1973). ]17] D.H. Marks and R. Stricter, Routing for public service vehicles, Amer. Sot. Civil Engrg.. J. Urban Planning and Development Divrsion 97 (1971) 165- 178.
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[ 181 I. Or and W.P. Pierskalla, A transportation location-allocation model for regional blood banking, AIIE Trans. II (1979) 86-95. [ 191 Y. Rapp, Planning of exchange locations and boundaries, Ericsson Technics 2 (1962) l-22. [20] TDF (Transportforskningsdelegationen), Distribution planning using mathematical methods (in Swedish), Report 1977:1, Contract Research Group for Applied Mathematics, Royal Institute of Technology, S-10044 Stockholm, Sweden (1977). [21] C.T.D. Watson-Gandy and P.J. Dohrn. Depot location with van salesman - A practical approach, Omega I (1973) 321-329. (221 M.H.J. Webb, Cost functions in the location of depots for multiple-delivery journeys. f?prmfronal Rc? Qunrt !O (1968) 311-320.