Read the mini-case Hornby Products Company (reproduced from Baker's book and attached below; the table exhibits

are available electronically on Canvas).

Construct an (mixed-)integer linear programing optimization model in Python using PuLP to examine the problem of

determining the optimal number and locations of warehouses to lease. Produce a recommendation of the optimal

number and locations of warehouses to lease.

EXHIBIT 7.5 Unit Costs for Distribution Atlanta Buffalo Chicago Cincinnati Detroit Pittsburgh Richmond St Louis Atlanta 13 16 11 15 1 1 12 Birmingham 15 12 10 15 16 13 11 Buffalo 13 1 10 8 4 4 13 8 10 Charleston 6 11 6 Charlotte 4 15 13 Chattanooga Chicago 16 Cincinnati 9 Cleveland 13 Columbus 10 Detroit 15 Evansville A -UMONDO NO-NONANAA Ft. Wayne 14 JA D N - NO NANJOONN- A Indianapolis 12 Knoxville Louisville Memphis Nashville Peoria 16 Pittsburgh 11 Richmond St Louis 12Case: Homby Products Company* The Hornby Products Company, headquartered in Denver, Colorado, markets a broad line of handcrafted home furnishings that are produced either in its own plants or by local artisans working under contract with the company. Hominy Products has established markets throughout most of the area west of the Mississippi. Its products are distributed to these markets mainly through a series of specialized manufactureris representatives. In a few areas, the company uti lizes architectural rms and interior decorators as distributors. The company has been so successful in western markets that management has decided to expand its market area to the east. The most recent expansion has been into a region east of the Mississippi from Illinois to western New York and as far south as Alabama and South Carolina. The company is currently serving this new region from its warehouse in Denver and a regional warehouse in St Paul. Saies in the eastern region have grown to such a level that management has decided to establish a system of distribution warehouses to serve this market. The company is now asking, how many additional warehouses are needed. and where should they be located\"? History The Hornby product line contains the full gamut of home furnishings. from heavy pieces of handcrafted furniture to delicate pottery and statues. The company's management has always insisted that the workmanship in its products meet the highest standard. Because ofthis insis tence. the company has attained an industrywide reputation for outstanding quality. In addition to product quality. management has also concentrated on the quality of its customer service. As its reputation for quality and service has spread, the company has begun to experience very rapid growth. With a vigorous management team, it appears possible to sustain this rapid growth rate wid'tout much difculty. The company has several policies that have permitted this growth without major capital additions. Subcontracting of production has reduced the necessary investment in plant and equipment. Leasing rather than buying warehouses has practically eliminated investment in this area. The company has. however. deviated from its minimum investment policy in order to maintain a high level of customer service. A full line of products is stocked at every distri- bution warehouse and every effort is made to provide delivery within 24 hours. The company considers this policy critical to expanding the market for its products. Homby Products has contracted with 22 different manufacturer's representatives and architectural rms in the expansion east of the Mississippi. As the company's product line has become established in that area, the representatives have been placing larger Orders. necessitating more and more shipments from Denver rather than from St Paul. Denver shipments have been required in order not to drastically deplete the stock at St Paul. This has made it dif cult to adhere to the 24hour delivery policy. The situation has worsened to the point where management has concluded that the sales growth will he stifled unless distribution warehouses *Adapted from Berry, WL. and D. Clay Whybark, Computer Augmented Cases in Operations and Logistics Management (19H) SouthWestern Publishing. An edited version was graciously provided by Alan Neebe. 292 Chapter 7 Integer Programming: Logical Constraints are set up to serve the eastern region. The final impetus to establish these warehouses was pro- vided by the representative in Indianapolis, who complained that late delivery of samples had caused the loss of a sizable contract for the furnishings of a major hotel chain. Hornby management recognized that this would be an increasing problem if better delivery service couldn't be arranged. They went to work on establishing a new distribution system for the east- ern region. EXHIBIT 7.1 Potential Warehouse Locations for the New Region MN WI NY MI Buffalo Detroit IA Chicago PA Ft. Cleveland NJ Wayne OH Pittsburgh Peoria . Columbus MD Indianapolis IL DE Cincinnati MO IN WV VA St. Louis Charleston Richmond Louisville (Evansville KY . Willow Springs Knoxville NC TN . Nashville Charlotte Chattanooga AR Memphis Birmingham Atlanta SC Little Rock MS AL. GA Key " Potential warehouse locations and manufacturer's representatives. . Manufacturer's representatives only.Case: Hornby Products Company 293 EXHIBIT 7.2 Cost Structure at a Typical Warehouse Annual cost F U Volume Alternatives As a first step in establishing the system, company officers visited each of the cities in which they had manufacturing representatives. These 22 cities are shown on the map in Exhibit 7.1. (After initially drawing the map, the company officers decided that Denver should continue to serve Little Rock and Willow Springs.) Eight of these cities were judged to have sufficient transpor tation services and warehouse facilities for distribution warehouses. Preliminary work was done on estimating the costs and determining the availability of specific facilities. Since company policy was to lease distribution warehouses, little difficulty was encountered in finding at least some warehouse space in each of the eight possible cities. The typical warehouse facility offers a leasing arrangement that follows the cost structure diagrammed in Exhibit 7.2. The cost structure consists of a fixed cost (F) and a variable cost (V). In addition, there is a lower limit (L) and an upper limit (U) associated with annual volume. The EXHIBIT 7.3 Warehouse Cost Data Fixed Variable Min. Max. W/H Location cost cost (L) (U) Atlanta 2700 200 1750 Buffalo 2900 150 1250 Chicago 3500 250 2000 Cincinnati 2200 200 1500 DO YOUAWN- Detroit 3300 200 1750 Pittsburgh 3000 200 1500 Richmond 2000 150 1000 St. Louis 1800 200 1500294 Chapter 7 Integer Programming: Logical Constraints upper limit is a volume ceiling dictated by material handling equipment: no more than this amount of product can be moved through the warehouse. The lower limit functions as a guar- antee of sorts to the leasing company. In each city, lease arrangements require a certain mini- mum capacity to be leased if a warehouse is established. Specifically, Hornby can choose to operate at a volume lower than L, but the fixed cost F would be charged in any event. Only volumes in excess of L incur additional costs, and here the charge is equal to the variable cost V multiplied by the excess volume. Exhibit 7.3 presents the detailed cost and capacity estimates. For the purposes of evaluating expansion alternatives, sales levels have been forecast two years into the future. Detailed sales forecasts for each representative have been converted to hundredweights (cwt) sold annually at each location. (The term hundredweight refers to 100 pounds of product.) At Hornby Products Company, the use of hundredweights as a unit of measure turns out to be a relatively accurate means of aggregating demand, and it is convenient in working out transportation costs. The sales forecasts are given in Exhibit 7.4. EXHIBIT 7.4 Forecast for Annual Demands Rep. no. Location Volume Atlanta 275 Birmingham 160 Buffalo 240 Charleston 260 Charlotte 135 Chattanooga 160 Chicago 400 Cincinnati 200 Cleveland 320 Columbus 220 Detroit 190 Evansville 100 Ft Wayne 140 Indianapolis 310 Knoxville 125 Louisville 340 17 Memphis 240 18 Nashville 210 19 Peoria 150 20 Pittsburgh 340 21 Richmond 300 22 St Louis 260