The Decision Sciences Journal of Innovative Education

A Note on the Treatment of the Center-of-Gravity

Method in Operations Management Textbooks

Ching-Chung Kuo

Department of Management

College of Business Administration

P. O. Box 305429

University of North Texas

Denton, TX 76203-5429

(940) 565-4749

KUOC@UNT.EDU

Richard E. White

Department of Management

College of Business Administration

P. O. Box 305429

University of North Texas

Denton, TX 76203-5429

(940) 565-3036

WHITE@UNT.EDU

INTRODUCTION

Location planning is concerned with where to establish a new facility or relocate an existing facility to achieve a certain objective. The facility could be a post office, a sports stadium, a retail store, or a military outpost while the objective could be to minimize distance, minimize cost, or maximize equity. The study of facility location and related issues dates back to the early 17^th century, but location theory was not formally introduced until the beginning of the 20^th century (Love, Morris, and Weslowsky, 1988).  Today, location planning has become a popular research area, with applications in a wide array of industries including manufacturing, transportation, healthcare, education, and airline, among others.

Choosing the location of a facility is a strategic consideration that has profound impact on other aspects of an organization. It has been argued that of all the business decisions made by management, none have more serious ramifications than those regarding where to locate a facility and how to design it (Weiss and Gershon, 1993). This is especially true in recent years when more and more companies have been expanding globally.

The significance of location analysis in the business world is evidenced by the many college textbooks on operations management that devote a chapter or a section to the topic and examine one or more of the following managerial decision aids: center-of-gravity method, factor rating procedure, break-even analysis, load-distance technique, and transportation modeling. It is the unsatisfactory treatment of the center-of-gravity approach in quite a few texts that motivates this teaching brief.

BASIC LOCATION CONCEPTS

Location models arise in a large variety of contexts and differ significantly in their characteristics. Of particular interest here is the mini-sum single facility location problem (MSSFLP), where a new facility is to be established among several existing facilities with known coordinates. The shipping cost is proportional to the distance between facilities as well as the number of trips to be made between them, and the goal is to find the location of the proposed facility that minimizes the total transportation cost.

Three different distance measures are widely used by location planners. Firstly, rectilinear distances are appropriate in metropolitan areas since travel takes place along orthogonal streets. They are also pertinent in many production plants, where materials are moved across aisles in crisscross patterns. Secondly, there exist countless situations in which Euclidean distances are useful. For instance, air flight tends to follow a straight-line path, so does movement of parts on a conveyor. Finally, as an example application of squared Euclidean distances, it has been reported that fire loss generally increases with the square of the Euclidean distance between the disaster site and the fire station.

Location problems can take place in a one-, two-, or three-dimensional space. The one-dimensional model may be applied to such settings as placing an ice cream cart along a stretch of beach and positioning a wrecker on a section of relatively straight highway in response to traffic accidents. The two-dimensional model is helpful in determining where to build a distribution center in a large geographic region or a service hub for an airline on a continent. Finally, the three-dimensional location problem has found interesting applications in configuring electronic components in an integrated circuit and locating fire exits in a multi-story office building.

The three types of distance defined on a plane are illustrated in the figure in Appendix 1. In the remainder of the presentation, a two-dimensional space is used as the primary platform for examining MSSFLPs involving only Euclidean or squared Euclidean distances. Clearly, all of the models and solution methods discussed below are applicable to one- and three-dimensional cases as well.

SQUARED EUCLIDEAN AND EUCLIDEAN MSSFLPS

Let (a_i, b_i) be the location of existing facility i, (x, y) be the location of a new facility to be established, and w_i be the transportation cost per unit of distance between (a_i, b_i) and (x, y), i = 1, 2, …, n. The squared Euclidean MSSFLP is concerned with finding the values of x and y so that the total transportation cost represented by the following function is minimized:

                                                  f₁(x, y) =w_i[(a_i - x)² + (b_i - y)²]                                               (1)

The optimal solution to this problem, (x^*, y^*), is often called the center of gravity (COG), and it may be obtained through these two formulae:

                                                             x^* = (w_ia_i)/(w_i)                                                     (2)

                                                             y^* = (w_ib_i)/(w_i)                                                     (3)

In contrast, the Euclidean MSSFLP seeks the values of x and y for minimizing the following total transportation cost function:

                                                    f₂(x, y) =w_i[(a_i - x)² + (b_i - y)²]^1/2                                          (4)

The problem in (4) is far more difficult to wrestle with than its squared Euclidean counterpart in (1), and the only algorithm available for solving it is the modified gradient procedure (MGP) due to Kuhn (1967). The MGP iteratively develops solutions progressively closer to the optimum until two successive ones are approximately identical:

Step 0: Let k = 0 and (x₀, y₀) = ((w_ia_i)/(w_i), (w_ib_i)/(w_i)).

Step 1: Let k = k + 1 and g_i(x_k-1, y_k-1) = w_i/[(a_i - x_k-1)² + (b_i - y_k-1)²]^1/2, i = 1, 2, …, n, (x_k, y_k) = ([g_i(x_k-1, y_k-1)(a_i)]/[g_i(x_k-1, y_k-1)], [g_i(x_k-1, y_k-1)(b_i)]/[g_i(x_k-1, y_k-1)]).

Step 2: If (x_k, y_k) » (x_k-1, y_k-1), then (x^*, y^*) » (x_k-1, y_k-1) and stop; otherwise, go to Step 1.

NUMERICAL EXAMPLE

As an illustration of the solution methodologies discussed in the previous section, let us consider a location problem adapted from Nahmias (2001). Suppose that XYZ Corporation currently has six plants and is contemplating the construction of a centralized warehouse to better coordinate material flows within the company. The x-y coordinates of the six plants together with the expected number of trips to be made by truck per month between each of them and the new warehouse are given in the table below. Management would like to decide where the warehouse should be located to minimize the total distance traveled on a monthly basis.

Plant                   Coordinates                   No. of trips

-------------------------------------------------------------

1                           (5, 13)                             31

2                           (8, 18)                             28

3                             (0, 0)                             19

4                             (6, 3)                             53

5                         (14, 20)                             32

6                         (10, 12)                             41

-------------------------------------------------------------

If squared Euclidean distances are appropriate in this example, then we have x^* = [31(5) + 28(8) + ... + 41(10)]/(31 + 28 + ... + 41) » 7.62 and y^* = [31(13) + 28(18) + ... + 41(12)]/(31 + 28 + ... + 41) » 10.78. Hence, the optimal location of the new warehouse is at the COG with coordinates (x^*, y^*) = (7.62, 10.78).

Now, suppose that Euclidean distances are pertinent in this problem. Since the center of gravity was previously computed to be at (7.62, 10.78), we let k = 0 and (x₀, y₀) = (7.62, 10.78). In the first iteration with k = 0 + 1 = 1, it can be found that g₁(7.62, 10.78) = 31/[(7.62 - 5)² + (10.78 - 13)²]^1/2 » 9.01, g₂(7.62, 10.78) » 3.87, g₃(7.62, 10.78) » 1.44, g₄(7.62, 10.78) » 6.67, g₅(7.62, 10.78) » 2.85, and g₆(7.62, 10.78) » 15.33. Consequently, x₁ = [9.01(5) + 3.87(8) + ... + 15.33(10)]/(9.01 + 3.87 + ... + 15.33) » 7.90 and y₁ = [9.01(13) + 3.87(18) + ... + 15.33(12)]/(9.01 + 3.87 + ... + 15.33) » 11.43.

As (x₁, y₁) = (7.90, 11.43) ¹ (7.62, 10.78) = (x₀, y₀), we let k = 1 + 1 = 2 and proceed to calculate g₁(x₁, y₁), g₂(x₁, y₁), …, and g₆(x₁, y₁) before assessing the values of x₂ and y₂. Continuing in the same fashion until the 15^th iteration, we obtain the sequences shown in Appendix 2, where (x₀, y₀) » (7.62, 10.78), (x₁, y₁) » (7.90, 11.43), (x₂, y₂) » (8.10, 11.71), …, (x₁₃, y₁₃) » (8.60, 11.91), (x₁₄, y₁₄) » (8.61, 11.91), and (x₁₅, y₁₅) » (8.61, 11.91). As (x₁₅, y₁₅) = (x₁₄, y₁₄), the optimal location of the new warehouse is (x^*, y^*) = (x₁₄, y₁₄) » (8.61, 11.91). Notice that the MGP-based optimal solution is different from the one based on the COG at (x^*, y^*) = (7.62, 10.78).

MISCONCEPTION ABOUT COG

Based on the previous analysis, the COG provides the optimal solution to the squared Euclidean MSSFLP whereas the MGP should be employed to tackle the Euclidean MSSFLP. Considering the fundamental difference between the two distinct distance measures, it is obvious that the COG does not solve the Euclidean MSSFLP. Unfortunately, authors of many operations management (OM) textbooks fail to recognize this when discussing the location planning tool.

A careful review of 35 popular OM texts adopted by many North American universities and colleges reveals that the COG approach is not covered in 14 of them but briefly touched on in nine others with no explicit mention of the use of squared Euclidean distances. Among the remaining 12 books, the method is introduced with the wrong assumption of Euclidean distance by Hanna and Newman (2001), McClain, Thomas, and Mazzola (1992), Russell and Taylor (2003), as well as Vonderembse and White (1996) or rectilinear distance by Reid and Sanders (2002) as well as Monk (1996b). Weiss and Gershon (1993) provide a correct explanation of the method in one-dimensional problems but an incorrect one in two-dimensional models. Moreover, Krajewski and Ritzman (2002) along with Ritzman and Krajewski (2003) caution the non-optimality of the COG under Euclidean and rectilinear distances without indicating its optimality under squared Euclidean distances. In only three of the 35 textbooks surveyed, it is clearly and correctly stated that the COG is where the new facility should be located in an MSSFLP involving squared Euclidean distances. These are Martinich (1997), Nahmias (2001), and Stonebraker and Leong (1994). All of the above observations are summarized in the table in Appendix 3.

In sum, there is no discussion of the COG in 40% (14/35 = 0.40) of the OM texts included in this study and the treatment of the topic is deemed inadequate in 26% (9/35 » 0.26) of them due to lack of specification of the distance measure used. Our main concern, however, stems from the common misconception about the COG in 17% (6/35 » 0.17) of them where an incorrect distance measure is assumed. The materials presented in those textbooks along with the numerical examples given are inaccurate and misleading. These are indicated in the “NC,” “CNE,” and “CIE” columns, respectively, of the table in Appendix 3.

CONCLUSION

The main goal of this paper is to point out an unsatisfactory treatment of the COG approach in a large number of college texts on OM. It is hoped that this problem, which has seemingly existed for a long time, will be fixed in the respective new editions so that future readers have a correct understanding of the methodology.

A secondary goal is to report some new developments in COG-related research. More specifically, several authors in the OM field have noted that while the COG does not solve the Euclidean MSSFLP, it is an excellent starting point for obtaining the optimum (Krajewski and Ritzman, 2002; Martinich, 1997; Ritzman and Krajewski, 2003). Unfortunately, no theoretical or empirical evidence has been offered in support of the statement. In an effort to bridge such a gap, Kuo, White, and Chiang (2002) have shown that locating the new facility at the COG leads to a transportation cost higher than the minimum one by an average of less than 2% over one-, two-, and three-dimensional MSSFLPs involving Euclidean distances. From a pedagogical or a practitioner’s perspective, this finding together with the computational ease makes the COG an attractive substitute for the MGP-based optimal solution. This is because the MGP is a more involved procedure which often calls for much time and effort to implement. In addition, it will fail if the least-cost location sought overlaps with one of the existing facility locations (Francis, McGinnis, and White, 1992).

Appendix 1: Three Different Distance Measures on a Plane

               y

                                                                                    (a₂, b₂)

                     [(a₁ - a₂)² + (b₁ - b₂)²]^1/2                                     

                                                                                             |b₁ - b₂|

                        (a₁, b₁)                                                 (a₂, b₁)

                                                         |a₁ - a₂|

                                                                                                                x

Note:    Rectilinear distance = |a₁ - a₂| + |b₁ - b₂|

Euclidean distance = [(a₁ - a₂)² + (b₁ - b₂)²]^1/2

Squared Euclidean distance = (a₁ - a₂)² + (b₁ - b₂)²

Appendix 2: Results from Implementing MGP

Appendix 3. Comparison of Treatments of the COG Method in OM Textbooks

¹   NC: Not covered.

²   CNE: Covered but not explained.

³   CIE: Covered but incorrectly explained.

⁴   CCE: Covered and correctly explained.

^*   Non-optimality under rectilinear and Euclidean mentioned but optimality under squared Euclidean not indicated.

⁺  CIE in two-dimensional models but CCE in one-dimensional cases.

Appendix 4: References

Adam, E., Jr., & Ebert, R. (1992). Production and operations management: Concepts, models, and behavior (5^th ed.). Englewood Cliffs, NJ: Prentice Hall.

Buffa, E., & Sarin, R. (1987). Modern production/operations management (8^th ed.). New York, NY: Wiley & Sons.

Chase, R., Aquilano, N., & Jacobs, R. (2001). Operations management for competitive advantage (9^th ed.). New York: McGraw-Hill/Irwin.

Davis, M., Aquilano, N., & Chase, R. (1999). Fundamentals of operations management (3^rd ed.). New York, NY: Irwin/McGraw-Hill.

Dilworth, J. (2000). Operations management: Providing value in goods and services (3^rd ed.). Orlando, FL: Dryden.

Evans, J. (1997). Production/operations management: Quality, performance, and value (5^th ed.). Minneapolis/St. Paul, MN: West.

Finch, B., & Luebbe, R. (1995). Operations management: Competing in a changing environment. Orlando, FL: Dryden.

Francis, R., McGinnis, L., Jr., & White, J. (1992). Facility layout and location: An analytical approach (2^nd ed.). Upper Saddle River, NJ: Prentice Hall.

Gaither, N., & Frazier, G. (2002). Operations management (9^th ed.). Cincinnati, OH: South-Western.

Hanna, M., & Newman, R. (2001). Integrated operations management: Adding value for customers. Upper Saddle River, NJ: Prentice-Hall.

Heizer, J., & Render, B. (2004). Operations management (7^th ed.). Upper Saddle River, NJ: Prentice Hall.

Heizer, J., & Render, B. (1996). Production & operations management (4^th ed.). Upper Saddle River, NJ: Prentice Hall.

Knod, E., & Schonberger, R. (2001). Operations management: Meeting customers' demands (7^th ed.). New York, NY: McGraw-Hill/Irwin.

Krajewski, L., & Ritzman, L. (2002). Operations management: Strategy and analysis (6^th ed.). Upper Saddle River, NJ: Prentice Hall.

Kuhn, H. W. On a pair of dual non-linear problems. In Abadie, J. (ed.) (1967). Non-linear programming. New York, NY: Wiley & Sons.

Kuo, C.-C., White, R. E., & Chiang, W. K. (2002). On the treatment of the center-of-gravity method in the existing operations management textbooks. Proceedings of the 2002 Annual Meeting of Decision Sciences Institute, San Diego, CA, November 2002, 1957-1960.

Lee, S., & Schniederjans, M. (1994). Operations management. Boston, MA: Houghton Mifflin.

Love, R., Morris, J., & Wesolowsky, G. (1988). Facilities location: Models & methods. New York, NY: North-Holland.

Markland, R., Vickery, S., & Davis, R. (1995). Operations management: Concepts in manufacturing and services. Minneapolis/St. Paul, MN: West.

Martinich, J. (1997). Production and operations management: An applied modern approach. New York, NY: Wiley & Sons.

McClain, J, Thomas, J., & Mazzola, J. (1992). Operations management: Production of goods and services (3^rd ed.). Englewood Cliffs, NJ: Prentice Hall.

Melnyk, S., & Denzler, D. (1996). Operations management: A value-driven approach. Chicago, IL: Irwin.

Meredith, J. (1992). The management of operations: A conceptual emphasis (4^th ed.). New York, NY: Wiley & Sons.

Meredith, J., & Shafer, S. (2002). Operations management for MBAs (2^nd ed.). New York, NY: Wiley & Sons.

Monk, J. (1996a). Operations management: Theory and problems (3^rd ed.). New York, NY: McGraw-Hill.

Monk, J. (1996b). Theory and problems of operations management (2^nd ed.). New York, NY: McGraw-Hill.

Nahmias, S. (2001). Production and operations analysis (4^th ed.). New York, NY: McGraw-Hill/Irwin.

Noori, H., & Radford, R. (1995). Production and operations management: Total quality and responsiveness. New York, NY: McGraw-Hill.

Reid, D., & Sanders, N. (2002). Operations management. New York, NY: Wiley & Sons.

Ritzman, L., & Krajewski, L. (2003). Foundations of operations management). Upper Saddle River, NJ: Prentice Hall.

Russell, R., & Taylor, B., III. (2003). Operations management (4^th ed.). Upper Saddle River, NJ: Prentice Hall.

Schmenner, R. (1990). Production/operations management: Concepts and situations (4^th ed.). New York, NY: Macmillan.

Schroeder, R. (2000). Operations management: Contemporary concepts and cases. New York, NY: Irwin/McGraw-Hill.

Schroeder, R. (1993). Operations management: Decision making in the operations function. New York, NY: McGraw-Hill.

Starr, M. (1996). Operations management: A systems approach. Danvers, MA: Boyd & Fraser.

Stevenson, W. (2002). Operations management (7^th ed.). New York, NY: McGraw-Hill/Irwin.

Stonebraker, P., & Leong, K. (1994). Operations strategy: Focusing competitive excellence. Boston, MA: Allyn and Bacon.

Vonderembse, M., & White, G. (1996). Operations management: Concepts, methods, and strategies (3^rd ed.). St. Paul, MN: West.

Weiss, H. J., & Gershon, M. E. (1993). Production and operations management (2^nd ed.). Boston, MA: Allyn and Bacon.