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Designing an Integrated Distribution System at DowBrands, Inc.

E. POWELL ROBINSON, JR.

Li-LiAN G A O

STANLEY D . MUGGENBORG

Department of Business Analysis and Research College of Business Administration Texas A&M University College Station, Texas 77843-4217

Department of Management School of Business Hofstra University Hempstead, New York 11550

Logistics Planning DowBrands, Inc. PO Box 68511 Indianapolis, Indiana 46268

Merging two independent distribution systems into an inte- grated whole poses significant challenges and opportunities for managers. Not only must they analyze the trade-offs among facility, inventory, and transportation costs, but they must also consider customer-service issues. We developed an optimiza- tion-based decision support system (DSS) for designing two- echelon, multi-product distribution systems and applied it to a problem facing DowBrands, Inc. The DSS provided valuable insight into the problem's cost and service trade-offs. Savings in logistics arising from its application are conservatively estimated to be $1.5 million per year.

In 1985 Dow Consumer Products, Inc., amanufacturer of food-care products {for example, Saran Wrap and Ziploc storage bags), acquired the Texize home-care product lines of Morton Thiokol, Inc. and formed DowBrands, Inc., subsidiary of Dow Chemical Company. The combina- tion gave Dow much needed marketing muscle and the potential for increased dis- tribution efficiency if it could successfully

merge the two complementary product lines.

By early 1986 the sales organizations were integrated, but the distribution orga- nizations remained, for the most part, sep- arate. Each organization employed an echelon of full-service central distribution centers (CDCs) that received products from their respective manufacturing plants, performed product mixing operations.

Ciipyrifiht tc 1993. The [nslitute o( Management Sc O091-21O2/93/2303/01O7$O1.25 This paper was referred

FACILITIES/ EQUIPMENT PLANNING—LOCATION INDUSTRIES—CONSUMER PRODUCTS

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stored inventories, and provided truckload (TL) and less-than-truckload (LTL) deliv- ery to customers. In addition, the food-care distribution network maintained a second echelon of satellite or regional distribution centers (RDCs) that received product from CDCs and provided only LTL delivery to customers.

Each organization felt that its distribu- tion system was best for its products. But to achieve the anticipated economic bene- fits of the merger, they needed an inte- grated distribution system. This propelled management to explore alternative consoli- dation strategies for the product lines.

Management was particularly interested in finding out whether a two-echelon sys- tem consisting of CDCs and RDCs or a single-echelon of CDCs was preferable. In addition, it wanted to know (1) the best number and location of facilities at each distribution echelon, (2) the best service assignments by product and shipment size for each facility, (3) the best assignment of customer demand to facilities, and (4) the best shipment routings by product and shipment size through the distribution system.

DowBrands, Inc., asked us to discuss these issues and to suggest a DSS that could analyze its problem. In reviewing the literature, we uncovered several applica- tions of optimization procedures for locat- ing a single-echelon of facilities [Erienkotter 1978; Fitzsimmons and Allen 1983; and Geoffrion and Graves 1974J. hi addition, Kaufman, van den Eede, and Hansen [1977], Ro and Tcha [1984], and Tcha and Lee [1984 [ discuss mathematical models and optimization procedures for two-echelon distribution system design.

However, we did not find any procedure capable of solving a two echelon design problem of the size and complexity of DowBrands.

During the next two years, we devel- oped an efficient optimization-based DSS for solving two echelon distribution system design problems. Concurrently, Dow- Brands used scenario evaluation to guide integration of the home-care product line into the two echelon food care distribution system. The Operations and Competitive Environment

DowBrands, Inc., manufactures and dis- tributes over 80 products nationwide with additional sales in international markets. Based on consumer application, and manu- facturing and distribution characteristics, it classifies the products into food-care and home care product lines. Both product lines are sold to retailers whose order size ranges from 100 pounds for a mom-and- pop operation to multiple truckloads for national supermarket chains. The sales price includes the product price plus trans- portation costs to the customer's dock.

DowBrands operates multiple manufac- turing plants for both product lines. Each plant focuses on a single product line. The distribution system consists of two-eche- lons of facilities, CDCs and RDCs, which are hierarchically linked. CDCs receive TL shipments from the manufacturing plants; perform product mixing operations; and maintain seasonal, order cycle, and safety- stock inventories. The CDCs ship TL quan- tities to replenish RDC inventories and ser- vice large-volume customers and LTL quantities to supply small-volume customers.

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RDCs operate as CDC satellites. Each RDC is supplied from a single CDC and maintains regional cycle and safety-stock inventories but only minimal seasonal in- ventories. RDCs provide only LTL delivery.

Public warehousing is used for product storage and order processing. Costs vary by location and consist of a handling cost per unit throughput and a storage cost per unit per unit time. Leases are flexible to permit the addition or deletion of capacity on a three-month notice.

All of DowBrands' freight movements are by common carrier. Freight rates are negotiated based on a Class 80 rating for food-care products and a Class 55 rating for home-care products. Most shipments to customers are in TL quantities. LTL ship- ment sizes range from 100 to 20,000 lbs. Several customers maintain private fleets and provide their own transportation ser- vices. These customers are credited at TL rates for picking up their own orders. Con- sequently, DowBrands bears all transporta- tion costs from the manufacturing plants to the customer's door.

Competition in the consumer products industry is intense; several regional and national manufacturers offer rival products and vie for limited retailer shelf space. Al- though product features, product quality, promotions, and price are important fac- tors in the customer's selection process, lo- gistics performance is key to a firm's success.

Retailers often employ just-in-time in- ventory concepts requiring distributors to bear the bulk of inventory maintenance costs and provide quick and reliable deliv- eries. Poor logistics performance leads to

lost sales due to stockouts and can result in lower allocations of retail shelf space. In addition, TL deliveries are frequently as- signed delivery time windows of two hours at the customer's dock; missing a time window means the delivery must be rescheduled for the following day. A missed time window yields not only poor

DowBrands manufactures and distributes over 80 products nationwide.

customer service and a potential loss of sales, but ties up a truck and driver for an extra day, adding to the cost of delivery.

Industry norms require that LTL deliver- ies in each customer zone originate from a single facility. TL shipments may originate at different facilities for each product line. An average order cycle lead time of seven to 10 days is standard for the industry as long as delivery occurs when promised. As a rule of thumb, the maximum delivery distance for LTL shipments is set at 500 miles to avoid consolidation or breakbulk terminals, which add to the mean and variance of delivery lead time. TL ship- ment distances are not constrained since they bypass consolidation centers, and their transit lead times are more predict- able. Fixed-Charge Network Programming Model

We modeled the problem as a fixed- charge network programming model in which nodes represent customer zones and candidate facility locations, and arcs repre- sent potential shipment paths. The graphi- cal aspects of the network model provide a

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convenient, comfortable, and effective in- strument for communicating and structur- ing the problem.

A unique feature of the network model is the representation of RDCs by both their location and the supplying CDC. This maintains the structural and cost relation- ships of DowBrands's problem and is a critical component of the model. Other two-echeion facility location problem for- mulations do not have this characteristic.

Figure 1 shows the network model's structure and draws on the notation in

Jensen and Barnes [1980]. The first two echelons of arcs are fixed-charge arcs which correspond to facility open and close decisions at potential CDC and RDC locations, respectively. If any product is shipped into a facility node, then the facil- ity must be opened and the supplying arc's fixed-charge (that is, annual facility over- head costs) is incurred. Otherwise, the fa- cility is closed and the fixed-charge is not applied.

We aggregated customers into three de- mand classes (LTL demand for both prod-

Source Node

CDC Node

RDC Node (U)

Demand Node (k)

Demand Node Description (zone, item)

[6] (F,, 0, Z,)

[-1] d.LTL)

[-1] (1,TLFood)

h i ] (i.TLHome)

[-1] (2. LTL)

[-1] (2, TLFood)

[-1] (2. TLHome)

Figure 1: The network model with two CDCs, two RDCs, and two customer zones. The first two echelons of fixed-charge arcs correspond to facility open and close decisions. The third echelon of arcs represents potential shipment paths to customers. The parameters above the arcs are the fixed-charge, the unit flow cost, and the decision variable associated with the arc The objective is to find the least-cost routing of the six units of supply at the source node such that each demand node receives exactly one unit.

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uct lines, TL demand for food-care prod- ucts, and TL demand for home-care prod- ucts), and identified these with a unique node for each demand class in each cus-̂ tomer zone.

The third echelon of arcs connects the potential facilities to the demand nodes. Each arc represents a specific shipment path through the two echelons of facilities.

Logistics performance is key to a firm's success.

The unit arc cost is the total variable trans- portation and facility throughput costs for using the replenishment path.

A dummy RDC j is defined for each CDC ), where f,y = 0 when / = y. These fa- cilities maintain a strict hierarchical linkage of nodes and arcs for algorithm implemen- tation and provide a mechanism for serv- ing TL and LTL demand from CDCs. CDCs can serve (through their dummy RDCs) both TL and LTL demand nodes, while satellite RDCs (that is, i ¥" j) are re- stricted to serving only LTL demand nodes.

We describe a mathematically equivalent mixed integer programming formulation of the network model in the appendix. Gao and Robinson [1992] describe the solution algorithm.

Because public warehousing is abundant at all potential locations, we assumed facil- ities are uncapacitated and allowed the model to determine what the facility ca- pacities should be at each location. This as- sumption also guarantees that each facility and demand node is served from a single shipment path in the optimal solution.

Hence, each opened RDC is linked to one CDC, such that the costs for establishing an RDC are not duplicated in the optimal solution. In addition, each customer de- mand node is served from a single facility as set forth in the firm's customer-service policy.

A single node represents LTL demand in each customer zone. Since the food-care and home-care plants are focused by prod- uct line and geographically dispersed, it is sometimes more efficient to supply TL de- mand through different CDCs for each product line. Hence, we modeled TL food- care and TL home-care demand using sep- arate nodes. This level of aggregation satis- fies the single-sourcing customer-service policy at minimum cost.

A prespecified maximum distance for LTL shipments serves as a surrogate mea- sure for in-transit delivery lead time. In calculating the cost parameters for the re- plenishment paths into the LTL demand nodes, we checked the distance from each customer zone to each potential facility lo- cation. If the distance exceeded the maxi- mum permissible shipment distance, we eliminated the shipment path from consideration. Data Collection and Parameter Estimation

The network model requires parametric estimates for facility operating costs, TL and LTL transportation costs for each product line over all shipment paths, and demand forecasts by product line and shipment size for each customer zone.

Facility operating costs include the cost of storing inventory and processing orders at the CDCs and RDCs. We calculated RDC inventory storage costs using regres-

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sion analysis to predict facility inventory level IROC given facility throughput VRPC . The RDC regression equation in thousand units is Unc = 78.544 + 0.09636 (VKDC) with an K-squared of 0.775. Given man- agement's minimum acceptable facility throughput level V /̂.v and the per-unit holding cost S, the fixed cost for opening the minimum-size facility is f,, = S((78.544 + 0.09636 (VM,,V)).

The slope of a second regression equa- tion with the y-intercept set equal to VMIN provides an estimate of the incremental in- ventory required to service each unit of throughput at the RDC (R-squared = 0.705). This slope, multiplied by the storage rate, yields the unit throughput cost for RDC inventory storage. We added unit processing and handling costs to unit storage costs to derive the total unit vari- able cost for RDC throughput.

Due to the confounding effects of sea- sonal inventories, regression analysis of CDC inventory level and CDC throughput failed to provide a strong causal relation- ship between the two variables. Using available data and management's experi- ence, we estimated the minimum inven- tory level excluding seasonal inventory for opening a CDC to be five times that of the minimum-sized RDC. Given this estimate, we calculated the CDC operating costs as we did those of the RDCs.

For the study, we aggregated demand into 93 customer zones and stated it in LTL and TL shipment sizes by product line. A single transportation rate structure applies for all TL movements. Separate LTL rate structures apply to food-care and home- care products, reflecting differences in the freight classifications of the products. At

this level of aggregation, the model re- quires approximately 6,700 freight charges.

Although actual transportation charges are based on specific origin-destination pairs, regression analysis indicates that over 80 percent of the variation in trans- portation rates for each shipment size and commodity are distance-related. Conse- quently, we used individual regression equations for each shipment size and prod- uct line to calculate mileage-based freight rates per hundred weight (cwt). We multi- plied these rates by the appropriate ship- ment weight and distance to determine the freight charge. We used transportation dis- tances from the Standard Highway Mileage Guide [1985] in the study. Computer Implementation and Software Performance

The computer code was written in FOR- TRAN and implemented on an IBM 4381 mainframe computer at Indiana University. We set the problem dimensions at 13 CDC locations, 23 RDC locations, and 93 market zones with three demand classes in each zone. This equated to 13 fixed-charge arcs for CDCs, 299 fixed-charge arcs for RDCs, and 83,421 shipment arcs. The total num- ber of feasible distribution configurations exceeded 68.7 biilion, from which we de- termined the optimal system design. The central processing unit (CPU) times for problem solution and report generation ranged from 0.33 to 28.55 seconds with an average time of 7.7 seconds. During the study, we evaluated over 60 different cost and customer-service scenarios. Performing the System Design Study

The system design study consisted of four parts: verification of the model and data set, sensitivity analysis, analysis of

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cost-service trade-offs, and what-if evaluation. Verification

We solved several problems to verify the accuracy of the data set and computer code. We checked the internal validity of the algorithm by solving identical small test problems with both the specialized computer code and a general purpose mixed-integer programming computer code. Both computer codes found identical solutions, which gave us confidence in the numerical accuracy of the solution procedure.

Next, we solved the problem using the complete data set and studied the output reports to verify the external validity of the network model and the data set. We fo- cused on establishing whether the model accurately reflected the assumptions of the problem and produced reasonable solu- tions. We identified and corrected several minor inconsistencies in the data set dur- ing this process. Sensitivity Analysis

Once we had accepted the validity of the model and data set, we investigated the sensitivity of the optimal solution to changes in cost parameters. We were par- ticularly interested in the potential impact of errors in the estimation of facility fixed- cost structures. Using the estimates from the regression analysis as the base case, we evaluated eight different cost scenarios with the fixed costs ranging from 50 to 200 percent of the base case. Since the purpose of this evaluation was to study cost inter- actions, we imposed no limits on the maxi- mum distance for LTL product shipments.

This analysis provided three major ob- servations. First, the optimal system design

was relatively insensitive to errors in fixed- cost estimation. The number of opened CDCs and RDCs ranged from four to five and from zero to two, respectively. All of the eight optimal solutions included com- binations of the same five CDCs indicating their strong dominance over the other eight candidate CDCs that did not appear in any of the optimal system designs. Sec- ond, the solutions did not utilize RDCs un- til the RDC fixed costs were 45 percent of the base case. This suggested that, consid- ering only cost, a single-echelon of CDCs would be more efficient than the two-eche- lon system of CDCs and RDCs. Finally, the analysis provided insight into how the distribution system would evolve given changes in facility cost structures.

Sensitivity analysis for different CDC/ RDC fixed-cost ratios, transportation cost structures, and demand rates confirmed the relative insensitivity of the optimal sys- tem design to changes in these problem parameters. Customer-Service Trade-offs

In the next phase of the study, we inves- tigated the relationship between the least- cost system design and customer service, as defined by the maximum shipment dis- tance for LTL deliveries. We evaluated eight different levels of customer service ranging from a distance of 500 miles to a distance of 2,000 miles. The maximum LTL distance of 500 miles reflected manage- ment's existing service policy. The service constraints were not binding in the 2,000- mile scenario.

Figure 2 shows the results of our analy- sis. The cost-service curve follows the el- bow shape typically associated with these curves [Rosenfield, Shapiro, and Bohn

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1985]. The graph helps management under-

stand the cost-service trade-offs and pro- vides evidence to support decisions affect- ing customer service. For example, total costs decrease by only $100,000 when the maximum LTL transit distance is increased from 500 to 600 miles. This may support management's preference for a 500-mile service level. However, costs are reduced approximately $700,000 if the maximum service distance is increased from 500 to 700 miles. Management must decide whether the 500-mile service policy is ap- propriate. Do customers perceive a suffi- cient difference between a 500-mile ship- ment distance and a 700-mile distance to warrant the $700,000 cost increase?

The cost-service analysis also clarifies

Cost (In thousands)

23,500

23,300 _

23,100 _

22,900 _

22,700 _

22,500 _

22.300 _

the roles of the two facility types. Without customer service restrictions, the company handles all demand from CDCs. It adds RDCs as it decreases the maximum LTL shipment distance. RDCs thus provide the least-cost alternative for providing higher LTL customer-service levels. Consequently, for DowBrand's problem, the justification for a two-echelon distribution system rests on a differentiated distribution strategy for TL and LTL movements.

The customer-service analysis provided additional insights into what constitutes a good distribution system design for Dow- Brands. Of the possible 13 CDC sites, only six entered into the optimal solutions. Of these five were at plant locations. Of the 23 possible RDCs, the model suggested opening only seven. These RDCs were lo-̂

2000 (4.0)

Miles (CDC,RDC)

Figure 2: The cost-service curve indicates the relationship between total system costs and max- imum LTL service distances. The two numbers in parentheses below the X-axis are the optimal open number of CDCs and RDCs for the specified maximum LTL shipment distance.

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cated primarily in the Midwest and North- west. The specific RDCs the model sug- gested varied considerably depending upon the customer-service policy. For ex- ample, service policies of 700 and 800 miles were most economically served by five CDCs and two RDCs. However, RDCs in Kansas City and Seattle were associated with the 700-mile service policy, whereas the model opened facilities in Minneapolis and Portland to meet an 800-mile service policy. Whal-If Evaluation

In the final phase of the study, we merged the most attractive distribution system configurations we identified in the sensitivity and cost-service analyses with management's experience to generate sev- eral what-if configurations for evaluation. These what-if configurations typically in- volved forcing an existing facility that was not recommended in the optimal solutions into the set of opened facilities. We then solved the model to optimality and evalu- ated the what-if scenario.

An example illustrates the benefits of the what-if analysis. An existing CDC in Chi- cago was not recommended in any of the computer-generated solutions. This con- cerned the management team since Chi- cago was a high demand market area. Fur- thermore, Denver, which was recom- mended as a satellite RDC, could be served from Chicago with frequent and reliable LTL service, eliminating the need for an RDC in Denver. Based on these observa- tions, we evaluated a what-if scenario.

In the scenario, we forced Chicago open and solved the resulting problem. In this solution, Chicago replaced an existing CDC in Ohio, for an annual cost increase

of $36,000 over the previous solution. Given the relatively small cost difference, management could make the choice be- tween keeping the Chicago CDC open or keeping the Ohio CDC open primarily on the basis of customer service.

However, management was uncomfort- able closing either one of the existing CDCs. We conducted a second what-if analysis, forcing both CDCs open. This so- lution increased costs by approximately $400,000 annually over the scenarios in which either the Chicago or the Ohio CDC was open and increased service only mod- erately. These what-if analyses allowed management to study the cost and service trade-offs associated with a nonoptimal system design. Implications, Benefits, and Conclusions

DowBrands, Inc., began consolidating its distribution system in 1986 right after it acquired Texize, Inc. It reduced the initial configuration of five CDCs and 19 RDCs to six CDCs and seven RDCs by 1988. However, management felt uncertain as it moved into the final stages of the design process.

It needed a better understanding of the roles of CDCs and RDCs, whether it should use a single-echelon or a two-eche- lon system, and what type, number, and location of facilities would correspond to the least-cost system design for current (or forecast) cost and demand parameters. In addition, management did not understand the impact of alternative customer-service policies on the optimal distribution system configuration and on cost.

The optimization procedures gave man- agement the analytical support it needed to eliminate these uncertainties and develop

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guidelines for change. The sensitivity anal- ysis, the cost-service trade-off analysis, and the optimization-based what-if analysis clarified all the major cost and service trade-offs, and gave managers confidence that the distribution strategy they proposed was economically sound.

The savings projected for the recom- mended system configuration with a maxi- mum 500-mile LTL distance constraint over the one in operation in 1988 were $762,400 per year. When this constraint was relaxed to 700 miles, as recommended by the management team, the projected savings were $1,466,800 per year. These savings were primarily due to a reduction in the number of RDCs.

The academics involved in the project gained considerable benefits as well. Our preliminary discussions about the problem provided a meaningful research topic that culminated in a PhD dissertation [Gao 1988] and two research papers [Robinson and Gao forthcoming; and Gao and Robinson 1992]. The solution algorithms performed well using hypothetical data, but the actual data and research support DowBrands provided helped us to docu- ment the true applicability of the new so- lution procedures. In essence, we had a living laboratory in which to test our re- search findings. The lessons we learned will serve well both in the classroom and in research. Acknowledgments

Special thanks go to Robert Larson, vice- president of material flow and Sharon Gillie, manager of analysis, at DowBrands, Inc. The study benefited immensely from their encouragement, insight, and suggestions.

APPENDIX

We use the following notation in the mathematical problem statement: m = the number of candidate central dis-

tribution centers (CDCs), n ^ the number of candidate regional

distribution centers (RDCs), and q = the number of customer zones.

Each customer zone represents the demand in a specific geographic area, for a particular product and ship- ment size. For each / G {1, 2, . . . , m } , 7 e {1, 2 » } , andfrE j l , 2, ...,q}

Z, = the binary decision variable for opening CDC (,

y,, = the binary decision variable for opening RDC / and supplying it from CDC /,

X,,t = the continuous decision variable for the fraction of zone k's demand that is served through CDC ( and RDC f,

F, = the annualized fixed cost for opening CDC /,

Fij ^ the annualized fixed cost for opening RDC / and supplying it from CDC i,

C,,i ^ the cost of serving zone k's demand through CDC / and RDC j where

l,,i, = the unit throughput cost at CDC / and RDC ;, and transportation costs from the plant to CDC ( and from CDC ( through RDC / to zone k, and

(4 ^ the annual demand at zone k. The mixed-integer programming model

of the two-echelon uncapacitated facility location problem is

min Z = 1=1 ;=1

(1) Z 2^ 2/ C,-,iX,,v, 1=1 i=i k^i

subject to

q, (2)

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-Z, + r,, < 0, / = 1, 2 m / - 1 , 2 tt,

- y , , + X,,, < 0 , / - 1 , 2 m

j = \,2 n A: - 1, 2 q,

0 < Z, < 1 and integer i

= 1, 2 m, 0 < Y;, < 1 and integer i = I, 2, . . . , m

/ - 1, 2 n,

0 < X,,̂ < 1, / = 1, 2 m

y - 1, 2 n k = \,2 q.

(3)

(4)

(5)

(6)

(7)

The objective function represents the fixed costs for establishing CDCs and RDCs and the variable costs for serving customers. Constraint set (2) requires that all demand be served. Constraint set (3) prevents a CDC from supplying an RDC unless the CDC is opened. Constraint set (4) prevents an RDC from supplying a customer zone unless the RDC is opened.

References Erienkotter, D. 1978, "A dual-based procedure

for uncapacitated facility location," Opera- tions Research, Vol. 26, No. 6, pp. 992-1009.

Fitzsimmons, J. A. and Allen, L. A. 1983, "A warehouse location model helps Texas comp- troller select out-of-state audit offices," Inter- faces, Vol. 13, No. 5, pp. 40-46.

Gao, L. 1988, ' A revised formulation and two complementary optimization procedures for the two-echelon uncapacitated facility loca- tion problem," PhD diss., Indiana University, Bloomington, Indiana.

Gao, L and Robinson, E. P. 1992, "A dual- based optimization procedure for the two- echeion uncapacitated facility location prob- lem," Naval Research Logistics, Vol. 39, No. 2, p p . 191-212.

Geoffrion, A. M. and Graves, G. W. 1974, "Muiticommodity distribution system design by benders decomposition," Management Sci- ence, Vol. 20, No. 5, pp. 822-844.

Jensen, P. A. and Barnes, J. W., eds. 1980, Net- work Programming, John Wiley and Sons, New York, New York.

Kaufman, L.; van den Eede, M.; and Hansen, P. 1977, "A plant and warehouse location prob- lem," Operational Research Quartertx/, Vol. 28, No, 3, pp. 547-554.

Ro, H. and Tcha, D. 1984, "A branch-and- bound algorithm for the two-level uncapaci- tated facility location problem with some side constraints," European Journal of Operational Research, Vol. 18, pp. 349-358.

Robinson, E. P. and Gao, L. forthcoming, "A new formulation and linear programming based optimization procedure for the two- echelon uncapacitated facility location prob- lem," The Annals of the Society of logistics En- gineers. Vol. 2, No, 2,

RosenBeld, D. B.; Shapiro, R. D.; and Bohn, R. E. 1985, "Implications of cost-service trade-offs on industry logistics structures," In- terfaces, Vol. 15, No. 6, pp. 47-59.

Standard Highway Mileage Guide 1985, Rand McNally and Co., Chicago, Illinois.

Tcha, D. and Lee, B. 1984, "A branch-and- bound algorithm for the multi-level uncapaci- tated facility location problem," European journal of Operational Research, Vol. 18, pp. 35-43.

Robert E. Larson, Vice-President, Cus- tomer and Business Services, DowBrands L. P., 9550 North Zionsville Road, PO Box 68511, Indianapolis, Indiana 46268-0511, writes, "The warehousing model gave us the insight and confidence in our decision concerning where to locate our warehouses while assimilating the purchase of Texize from Morton Thiokol. The model helped us look at cost trade-offs of various loca- tions and the associated service levels to our customers. The projected savings of $1.4MM a year is substantial to our business.

"We are implementing changes to our warehouse system to match the results of this study. The model has provided us with the best of both worlds—less ware- houses to operate and a cost savings."

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