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Measuring transport costing error in customer aggregation for facility location.

Computer technology has advanced to the point where much of the practical analysis for locating facilities, such as warehouses, plants, terminals, and ports, is now handled with the aid of location modeling software. In a recent survey of logistics executives, over two thirds indicated that their companies will use a location model or software to assist in their location analyses within the next year.(1) However, the substantial size of practical location problems and the limitations of computer power usually require control of the problem formulation through data aggregation. One common modeling practice is to aggregate individual customer accounts into a limited number of geographic clusters. This reduces the size of the location problem to be formulated and, thus, the amount of data needed for analysis; the size of computer memory required; and the computational time needed to find a solution. According to Magee, Copacino, and Rosenfield, when conducting large-scale location studies:

...(an) important issue in data aggregation for model formulation is the categorization of markets. For many companies the recognition of each individual customer is not feasible.... Generally, some level of aggregation of markets must...be made.(2)

Bender echoed the need for customer aggregation in location analysis but recognized the potential error in doing so when he said:

The most critical step in the (logistics system) analysis and design process is to determine the right level of data aggregation: the more aggregated the data, the greater the potential errors in analysis, but the simpler it is to analyze, and the cheaper it is to assemble....Generally, the most difficult decisions on level of aggregation deal with products, and with demand markets.(3)

Others have also suggested the same practical need for customer clustering.(4)

Sources of Costing Error

There are potentially two types of errors (cost and optimality) due to customer clustering: (1) estimating total transportation cost to aggregated customer centers rather than computing the true transportation cost to each customer, and (2) misallocating customers to source points, and the resulting mislocation of source points, due to the use of clustered demand rather than individual customer demand.(5) As shown in Figure 1, rather than shipping directly from source points to each customer, customers are assumed to be collected into a limited number of groupings. Costs for shipments are computed as if the shipments are made to the center of these clusters. Then, the source points are located so as to minimize these estimated transport costs.

Customer clustering is a broad issue relating to many types of facility location problems, including plant location, warehouse location, and the location of retail and service facilities.(6) The focus of this research is specifically directed at customer clustering as it affects facility location in the domestic distribution of physical products. Therefore, the primary question here is, In the contiguous United States, to what extent is the transport costing error affected by the number of clusters used for data aggregation, the size of the clusters, and the number of source points?

Previous Research

Compared to the substantial research that has been conducted on the methods by which warehouse location problems can be solved, there is a paucity of available research into the data-related questions that can make the location methods effective in practice. Regarding the clustering of customers, various schemes have been employed by analysts, including the use of political boundaries, Standard Metropolitan Statistical Areas,(7) a firm's marketing territories,(8) national geocoding systems,(9) and empirically generated maps.(10) Research conducted on transport costing error due to customer clustering was reported several years ago by House and Jamie.(11) They examined this problem for consumer products distributed to customers from warehouses. Their study drew the following major conclusions:

(1) As the number of market clusters increases, the transport costing error decreases.

(2) The transport costing error cannot be reasonably controlled with less than 100 market clusters.

(3) Transport costing errors are no more than 2 to 3 percent when using at least 150 market clusters.

(4) Very large numbers of market clusters do not significantly reduce costing error.

(5) As the number of warehouses increases relative to the number of market clusters, the transport costing error increases.

These results were based on markets generated from a clustering of zip sectional centers. The approach was to identify the following:

...the principal city within each area and (to allocate) the remaining areas to these cities on the basis of proximity.(12)

This clustering was apparently accomplished by applying a certain amount of judgment to define the clusters, since no precise method was described for determining the principal city within a zip sectional center or for assigning centers by proximity to principal cities.

Selecting warehouses by number and by location was based on the concentration of supermarkets within an area. This gave a variety of warehouse locations ranging from 3 to 26.

Current and Schilling more recently have estimated the costing error for a number of selected problems with up to 70 clusters from 681 demand locations and up to 10 source locations.(13) Their study showed that:

(1) The optimality error (mislocating sources) is small, averaging about 1.5 percent.

(2) Costing error averages about 15 percent.

(3) Both optimality and costing error increase monotonically with the number of sources.

(4) Both optimality and costing errors decrease as the number of demand clusters increase.

(5) There is a bias toward too many source points with fewer demand clusters.

Common Customer Aggregation Practice

Although previous research did not suggest the number of clusters to use in location analysis, practice over the years gives insight into a range of what seems to be typical for consumer products. For example, Bender suggests 80 to 150 clusters for the U.S.(14) Geoffrion and Graves used 127 customer zones for a manufacturer of hospital supplies(15) and Geoffrion used 121 clusters for a large food company.(16) House used 100 to 200 clusters in two studies involving consumer goods.(17) Klincewicz configured a location model with 200 demand clusters.(18) Finally, a survey of 16 of the major commercial-grade location models for facility location showed that 60 percent of them typically configured their models for problems with 100 to 200 customer clusters.(19) It seems reasonable to conclude that customer clustering in the range of 100 to 200 clusters is common practice for warehouse location analysis in the United States.

Contributions of this Research

Against the background of current location modeling practice, this research quantifies the transport costing error associated with customer aggregation in the location modeling process. It attempts to contribute to customer aggregation research in several ways. First, a precisely defined methodology is proposed that can be used to replicate the results of this study and to extend the work to additional settings such as other regions of the world, other methods of transportation, and other geocoding systems than zip codes for geographic demand representation. Second, a data base of customer sales (population is used as a surrogate for sales in this study) by zip codes and class transportation rates are used that are steadily available in most businesses. Third, a larger number of source points (up to 100) is tested than has been the case in previous studies (about 25). Fourth, specific guidelines for selecting the proper number of demand clusters are provided that should be useful for the practitioner who may be conducting a location study. Overall, this research is directed at finding the best number and composition of demand clusters to use when modeling a facility network for the purpose of finding the proper number, size, and location of the facilities.

EXPERIMENTAL DESIGN

The Data Base

This research is directed at consumer products--those sold to the general public. Food and drug items, electronic goods, and housewares are examples. Product distribution is nationwide with sales in proportion to population and dispersed according to the population at large. Shipments made from source points to markets are assumed to vary in size from 500 lb. to full vehicle load.

Population by 3-digit zip codes (zip sectional centers) was selected as the elemental customer cluster. From a practical point of view, firms maintain sales data by customer address and they can easily group customer sales into the 900 three-digit zip codes. (Clustering customers into smaller units of five-digit and nine-digit zip codes was not considered.) These three-digit zip codes provided a first level of demand clustering in a firm's data base. Zip codes allow sales to be easily manipulated into any reduced number of clusters desired. They also can be referenced to coordinate point schemes such as latitude and longitude, which allows for distance computation needed in transport cost determination.

For this study, 1990 population estimates by three-digit zip codes nationwide were obtained from the National Planning Data Corporation. Zip codes outside the contiguous United States (Hawaii, Alaska, Puerto Rico, and APOs) were eliminated from the data base, leaving approximately 900 zip codes with a population of about 248,000,000 persons. To generalize the transport cost computation for a variety of products, population was used as a surrogate for consumer product sales where one person arbitrarily equals 1 lb. of product.

Transportation rates were obtained from the 1990 rate schedules of three of the larger truck common carriers in the U.S. (ABF, Roadway, and Yellow Freight). Rates were found for shipments ranging in size from 500 lb. to 40,000 lb. and for distances originating at a variety of cities throughout the U.S. to other cities from one to 3,000 miles from the origin city. The rate for the various distances and shipment weights were generalized through regression analysis into rate estimating curves. A variety of such rate curves was computed in previous research and is used here.(20) These curves give a reasonable estimation of actual rates since the coefficient of determination is most often above 0.90 for a variety of shipping points and shipment sizes. The purpose of these rate estimating curves was to provide a systematic way of generating rates as distances between source points and demand cluster points changed throughout the clustering process. Although private trucking cost can be converted to similar rate estimating curves, private carriage was not specifically examined in this study. Also, where rates deviate from the class rate structure such as for contract and commodity rates, the results of this research may not apply. It is assumed that the rates generated from these rate estimating curves are accurate. The costing error is measured using these estimated rates.

The Testing Procedure

Clustering customers. Research has not yet shown an optimal way for forming customer clusters. However, Geoffrion has suggested an approach:

A hierarchical clustering approach started with as many clusters as original customers and then combining clusters one at a time...appears to hold promise.(21)

This author agrees with the general nature of the approach and formulates the following specific clustering procedure: Beginning with 900 three-digit zip codes (or clusters), and their associated latitude-longitude coordinates, the great circle distances, corrected for road distances, are computed for all zip code pairs. The pair of zip codes in closest proximity to each other are identified. The populations contained within the two zip codes are combined and placed into one cluster. The coordinates for the replacement cluster are determined by the weighted center of gravity method. That is, combined cluster center coordinates are recomputed as

[Mathematical Expression Omitted]

where

[Mathematical Expression Omitted] = weighted coordinates of the cluster.

[X.sub.i], [Y.sub.i] = coordinates of the zip codes in the cluster.

[V.sub.i] = population of the zip codes within the cluster.

The two clusters, as represented by zip code and associated demand, are replaced by a single one. The total number of clusters is reduced by one. This clustering process is repeated until the desired number of clusters is achieved.

Since clustering is guided only by zip code proximity, cluster size is not directly controlled. Any number of zip codes may by contained within a cluster. Clusters with huge populations may be created, especially in areas where population is great and a large number of zip codes exist relatively close to each other. Alternately, limited cluster size has the effect of increasing the number of clusters in the more densely populated areas, with the potential for reducing transport costing error.

Determining source points. Location modeling practice involves determining the number and location of source points (e.g., warehouses) based on customers aggregated into clusters. Although there are numerous ways in which source point location can be found, including linear programming, integer programming, enumeration, calculus, and various heuristics, the approach that serves this study well is a locational equilibrium (multiple center of gravity) method as outlined by Cooper.(22) This is an iterative heuristic method where, for a specified number of source points to be located among a given number of customer clusters, the best locations are found that minimize transport cost. The assignment of clusters to source points is also noted. Although the method is heuristic, Cooper claims that it gives, on the average, locations within 0.3 percent of optimum.

The costing error. Once the source locations are known for a given number of clusters, the transport costing error can be found. The cost for shipping between sources and assigned clusters is determined from transport rate estimating curves of the form Rate=a+bxDistance, a curve for the particular shipment size being tested. Each rate is then multiplied by the demand represented in each cluster to yield total transportation cost.

The percentage error in transport costing occurs by comparing two conditions: (1) the cost of shipping to cluster centers and (2) the cost of shipping to the customers in all 900 zip codes, given the predetermined source point locations. Regarding the second condition, the 900 zip codes are assigned to the given source point locations on the basis of proximity. Since the source locations are not necessarily optimal for the 900 zip codes, the true transport costs may vary slightly from those calculated in this manner. Recall that Current and Schilling found the error due to mislocation to be quite small. The transport costing error is defined as a ratio of the second condition to the first. For example, suppose the best five source locations have been found based on 200 clusters. The transport cost to the 200 cluster centers is $12,333,581. Assigning the 900 customer clusters to these sources on the basis of proximity, the transport cost is $12,348,425. This gives a costing error of (12,348,425-12,333,581) x 100/12,348,425 = 0.1202%. This type of calculation is repeated for various cluster sizes and for 1, 5, 10, 25, 50, and 100 sources.

The test procedure involved three scenarios. The first was to examine the effect of shipment size (product variation) on the transport costing error. In a transport costing sense, product is represented by different shipment sizes and different class ratings, and the extent of rate discounting. However, different class ratings did little but to change the rates proportionately to the class 100 rates. Rate discounting had the same effect when it was proportionately applied to the tariff. Therefore, by using the class 100 rates at shipment sizes within three weight breaks of 500 to 1,000 lb., 5,000 to 10,000 lb., and 40,000 lb. or greater, the effect of a range of products was tested.

The second scenario was to examine the effect of the number of source points (e.g., warehouses) from which shipments were made to the customer clusters. The number of source points ranged from one to one hundred. Recall that they were selected by a multiple sources locational equilibrium approach. This method gave locations that minimized total transport cost for the network.

The third scenario was to examine different configurations of clusters. Two designs were selected for consideration:

(1) An unrestricted cluster size resulting from clustering 900 zip codes without controlling the population allowed within any one cluster.

(2) A restricted cluster size that limits, where possible, any one cluster from having a population greater than a specified percentage of total demand. Cluster size limits tested were at 0.5, 0.8, 2, and 5 percent.

RESULTS

Product Shipment Size and Classification

Shipment size has a substantial impact on transport costing percentage error. An example of the effect of small (500 to 1000 lb.) and large (full vehicle load) shipment sizes for various numbers of source points is shown in Figure 2. Transport rates vary markedly over this shipment size range. From a previous study of 8,600 rates, the fixed coefficient of a transport rate estimating curve is, on the average, about seven times higher for small shipment sizes versus large ones." The average variable coefficient is about two times higher over the same size range. As the variable coefficient of the rate curve becomes smaller relative to the fixed coefficient, the cost due to distance between source points and cluster centers and, therefore, source point location, is less important to determining total transport cost. Hence, the percentage error is less for small shipment sizes than for large ones. Since the number and location of source points are most affected by the costing error associated with large shipment sizes, subsequent testing was conducted using rates for only the vehicle load shipment size. This gave the maximum costing errors that are likely to be experienced in practice.

Unlimited-Size Clusters

The costing error was examined for 1, 5, 10, 25, 50, and 100 source points for clusters formed by proximity and with no limits as to their sizes. The maximum cluster size was about 7 percent of total demand. The results are shown in Figure 3. Costing error does not uniformly increase as the number of clusters is decreased. However, error does increase sharply after the number of clusters drops below about 100.

There is an increase in costing error as the number of source points in a network increases. This increase error occurs as the number of source points becomes large relative to the number of clusters.

Overall, the costing error seems intolerably high except for relatively few source points, or where the number of clusters is quite high. Limiting the size of clusters seemed an obvious way to reduce the error.

Limited-Size Clusters

Forming demand clusters by proximity and limiting their size to some fraction of the total demand prevented the weighted center of gravity for the clusters from moving too far from the major population centers and, therefore, the centers of high-volume shipments. Cluster size limits of 0.5 percent, 0.8 percent, 2 percent, and 5 percent of total demand (population) were selected for testing and comparison with the case of no limits on cluster size. From Figure 4, it is seen that the costing error is reduced substantially by limiting the maximum allowed cluster size to 0.8 percent of total demand as compared with the unlimited cluster size results shown in Figure 3. The general effect of cluster size on costing error can be seen in Figure 5. It becomes clear that smaller cluster sizes are better. However, as the maximum allowed cluster size is reduced, the minimum number of clusters increases. For example, if size is limited to 0.5 percent of total demand, the minimum number of clusters that can be created is 1/0.005 = 200. Since demand is not uniformly spread over all clusters, the effective number is usually slightly higher than this. Continued reduction of maximum cluster size drives the minimum number of clusters towards the base number of 900 zip codes. Since the number of source points cannot exceed the number of clusters, there is a lower bound on the number of clusters that can be used in location analysis.

Selecting the Number of Clusters

It has been shown that shipment size, number of source points in a network, and the maximum size of customer clusters all affect the level of the transport costing error experienced in location analysis. Although it is not clear how much costing error can be tolerated in any particular analysis, it is possible to provide some guidelines for cluster size selection. Recall that maximum costing error occurs with full vehicle load shipment sizes and that the costing error will be only one-third to one-fifth of the maximum error for 500 lb. shipment sizes. Using the vehicle load shipment size as a case where costing error is greatest, Table 1 can be developed. Given a specified maximum acceptable costing error and a given minimum cluster size, this table shows the recommended minimum number of clusters to use in a location analysis based on an anticipated number of source points. This table is developed from numerous error curves, of the type shown in Figure 3, for various minimum cluster size scenarios. It becomes clear from this table that keeping the costing error low by using only a few clusters is not difficult when there are just a few source points to be located. However, as the number of source points becomes large, the minimum number of clusters that can be used to realize a given costing error must increase, often substantially. If the analyst wishes to limit, perhaps for computational reasons, the number of clusters (given the number of source points), the design of the clusters may be the only decision variable for limiting costing error.

CONCLUSIONS

A number of conclusions can be drawn from this research that are of practical value when selecting and evaluating customer aggregation schemes used for facility location. Recall that location analysis typically involves finding the location of facilities based on transportation costs, among others, computed to the center of customer groups rather than to individual customers. In this study, optimal, or nearly optimal, source locations were found for various customer clusters that were formed by grouping customers according to proximity. Since actual customers were not used in finding source point locations, there was a resulting costing error. The impact of this costing error is to cause an improper balance of the costs of location (production or purchase, transportation, inventory, storage, handling, and fixed costs) that results in mislocation of the facilities. Understating the transportation element of total locational costs puts too few facilities in the network or places them too far from the customers that they are to serve. Minimizing the costing error is in the best interest of accurate location analysis.

The results of this study can enhance our understanding of the transport costing error associated with the common practice of customer clustering. The major findings are as follows:

(1) The common practice of using 100 to 200 customer clusters for consumer products should be challenged. Although the practice is not necessarily inappropriate for all problems, the costing error is sensitive enough to the rates associated with shipment size, number of facilities to be located, and maximum size of a cluster among all potential clusters that good location practice is to select the clustering configuration based on the nature of the location problem at hand. Table 1 is a guide to the appropriate number of clusters, or the proper number of clusters may be generated, depending on the characteristics of a particular location problem. In general, 100 clusters is too few for only a few facilities in a network and/or high costing error levels. The use of 200 clusters is reasonable for locating up to 25 facilities. Above that number, the number of clusters in a location analysis should be increased substantially. Because the number of source points may not initially be known in a location problem, it usually can be approximated to within a reasonable range. Otherwise, the number of clusters will need to be revised as the number of source points is better estimated.
Table 1. Minimum Acceptable Number of Clusters for the Maximum Allowed
Transport Costing Error and for Various Numbers of Network Source Points and
Largest Customer Cluster Sizes

Maximum Largest
Allowed Cluster Approximate Number of Source Points in Network
Error Size 1 5 10 25 50 100

 0.5% 200(c) 325 350 500 650 750
 0.8% 150 150 175 375 450 650
0.5% 2.0% 75 100 300 450 600 650
 5.0% 75 150 250 500 600 750
 Unlimited(b) 50 350 400 500 700 750

 0.5% 200(c) 200(c) 200(c) 200(c) 250 500
 0.8% 150 150 150 175 350 500
1.0% 2.0% 75 75 175 300 500 600
 5.0% 75 100 225 400 500 600
 Unlimited(b) 25 200 250 400 500 600

 0.5% 200(c) 200(c) 200(c) 200(c) 200(c) 350
 0.8% 150 150 150 150 250 450
2.0% 2.0% 75 75 100 250 350 500
 5.0% 75 75 150 300 425 500
 Unlimited(b) 25 75 175 300 450 500

 0.5% 200(c) 200(c) 200(c) 200(c) 200(c)
200(c) 0.8% 150 150 150 150 150
 300 5.0% 2.0% 75 75 75 100 225
 300 5.0% 75 75 75 175 275
 350 Unlimited(b) 25 50 75 200 275
 350

 0.5% 200(c) 200(c) 200(c) 200(c) 200(c)
200(c) 0.8% 150 150 150 150 150
 150 10.0% 2.0% 75 75 75 75 125
 175 5.0% 75 75 75 75 150
 200 Unlimited(b) 25 50 75 100 175
 225

a Largest cluster size among all clusters as a percentage of total demand.

b Cluster size is not specifically limited, but is approximately 7 percent of
total demand.

c Mathematically the minimum number of clusters.


(2) Shipment size, and the associated transport rate, affects the level of the transport costing error. The maximum error occurs when the full vehicle load rate is used. The error is approximately 3 to 5 times higher at full vehicle load than it is for shipments of 500 lb.

(3) Controlling cluster size during the formation of any particular set of clusters can reduce the costing error. Large clusters have centers that are far from major customer concentrations. Limiting the size keeps the center of the clusters close to customer concentrations, but allows cluster size to grow where customer density is lowest. The result is lower total transport costing error. However, reducing cluster size has a limiting effect by increasing the minimum number of clusters that is realized. Minimum cluster sizes that represent 1 percent of total demand, or 100 clusters, is probably a practical lower limit.

(4) The minimum number of clusters used in a location analysis should parallel the number of facilities expected to be located in a network. As the number of facilities in a network increases, the costing error rises. There is no precise error relationship between the number of source points and the number of clusters, but having approximately 5 to 10 times the number of clusters to source points is about what is required so that the maximum costing error does not exceed 1/2 to 1 percent. If a maximum error in the 5 to 10 percent range is acceptable, the multiplier drops to 3 to 5 times the number of source points.

(5) Grouping customers by proximity and controlling cluster size is a reasonable way to form the clusters. Combining customers close to each other keeps transport costs low, thus reducing transport costing error. Recall that House and Jamie found costing error to be between 2 to 3 percent for 150 or more clusters with no more than 26 source points. However, by carefully forming the clusters in the manner suggested in this study, the costing error associated with truckload shipments and 150 clusters does not exceed 1.5 percent. For small shipment sizes, the error can be as low as 0.3 percent.

(6) The results of Current and Schilling are generally supported. That is, costing errors are reduced with increased number of clusters. Also, costing errors increase as the number of facilities increases relative to the number of clusters in a location problem.

The sensitivity of costing error to a number of variables suggests that the conclusions of this study may have only limited transferability to other product types and to other regions of the world. However, it is expected that the general relationships between shipment size, cluster size, and number of source points generally do hold. Therefore, using a single number of clusters for a wide variety of locational problems results in an unacceptably high costing error. The specific number of clusters to use in a location analysis should be determined from the particular dispersion of customer demand in a region, the nature of that product demand, and the characteristics of shipments. It may be that these results can be further generalized, but this conclusion must be left to further research. The methods outlined in this study should facilitate such efforts.

Finally, it should be emphasized that the results of this study are for transportation rates that are reasonably proportional to distance. Where specifically quoted rates or selected discounting of class rates cause rates that are not accurately represented by distance, the effect of demand clustering on transport costing error remains unknown. Since such an analysis is likely to require a case-by-case evaluation, it was considered outside the scope of this study and is left for further research. General discounting of class rates falls within the scope of this study and the results apply.

ENDNOTES

1 Ronald H. Ballou and James M. Masters, "Commercial Software for Locating Warehouses and Other Facilities," Journal of Business Logistics, forthcoming.

2 John F. Magee, William C. Copacino, and Donald B. Rosenfield, Modern Logistics Management: Integrating Marketing, Manufacturing, and Physical Distribution, (New York: John Wiley & Sons, 1985), p. 329.

3 Paul S. Bender, "Logistic System Design" in James F. Robeson and Robert G. House (eds.), The Distribution Handbook (New York: The Free Press, 1985), p. 157.

4 B. M. Khumawala and D.C. Whybark, "A Comparison of Some Recent Warehouse Techniques," The Logistics Review, Vol. 7, No. 31 (1971), p. 6; and Harvey N. Shycon, "The Computer as a Tool in Planning the Distribution System," Proceedings of the Annual Conference, National Council of Physical Distribution Management (Boston, MA: National Council of Physical Distribution Management, 1978), p. 4.

5 E. L. Hillsman and R. Rhoda, "Errors in Measuring Distances from Populations to Service Centers," Annals of the Regional Science Association, Vol. 12 (1978), pp. 74-88.

6 Avijit Ghosh and Sara L. McLafferty, Location Strategies for Retail and Service Firms (Lexington, MA: D.C. Heath and Company, 1987), Chapter 5.

7 Robert G. House, "A Small Scale Facility Location Model," Working paper WPS 78-21 of the College of Administrative Science, The Ohio State University (March 1978).

8 Ibid.

9 Pamela A. Werner, A Survey of National Geocoding Systems, Technical Report No. DOT-TSC-OST-74-26 (Washington, D.C.: Superintendent of Documents, U.S. Government Printing Office, 1974).

10 Ronald H. Ballou, Business Logistics Management, 3rd edition (Englewood Cliffs, NJ: Prentice-Hall, 1992), p. 285.

11 Robert G. House and Kenneth D. Jamie, "Measuring the Impact of Alternative Market Classification System in Distribution Planning," Journal of Business Logistics, Vol. 2, No. 2 (1981), pp. 1-31.

12 Ibid.

13 John R. Current and David A. Schilling, "Elimination of Source A and B Errors in P-Median Location Problems," Geographical Analysis, Vol. 19, No. 2 (April 1987), pp. 95-110.

14 Bender, op.cit., p. 159.

15 A. M. Geoffrion and G. W. Graves, "Multicommodity Distribution System Design by Benders Decomposition," Management Science, Vol. 20, No. 5 (January 1974), p. 842.

16 A. M. Geoffrion, "A Guide to Computer-Assisted Methods for Distribution Systems Planning," Western Management Science Institute, University of California, Los Angeles (Working Paper No. 216; June, 1974).

17 Robert G. House, "A Small Scale Facility Location Model," Working Paper Series WPS 78-21 of the College of Administrative Science, The Ohio State University (March 1978).

18 J. G. Klincewicz, "A Large-Scale Distribution and Location Model," AT&T Technical Journal, Vol. 64, No. 7 (September 1985).

19 Ballou and Masters, "Commercial Software for Locating Warehouses and Other Facilities," op.cit.

20 Ronald H. Ballou, "The Accuracy in Estimating Truck Class Rates for Logistical Planning," Transportation Research-A, Vol. 25A, No. 6 (1991), pp. 327-337.

21 Geoffrion, "Customer Aggregation in Distribution Modeling," op. cit., p. 12.

22 Leon Cooper, "Solutions of Generalized Locational Equilibrium Models," Journal of Regional Science, Vol. 7, No. 1 (1967), pp. 1-18.

23 Ballou, "The Accuracy in Estimating Truck Class Rates for Logistical Planning," op. cit., pp. 334-336.

Mr. Ballou is professor of operations and logistics management, Weatherhead School of Management, Case Western Reserve University, Cleveland, Ohio 44106-7235.
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Date:Mar 22, 1994
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