The Able Company is a local division of a large public warehousing firm. The management of this company has successfully applied the techniques of scientific management in the past and is currently looking at its layout problem to see if these techniques can indicate whether cost savings can be made in this area. The company has selected a particular warehouse for consideration. This warehouse has two receiving docks (R1 R2) and one shipping dock (S,). The three major products handled by the warehouse are stored in six storage bays.
Management finds that because of order sizes, receiving locations, quantities received, and the like, different times are required to supply and distribute from a storage bay, and these service times depend on the particular product and location of the storage bay in the warehouse. There is a direct relationship between handling costs and handling times for each product and storage bay.

Each storage bay has a certain capacity depending on the product. The following information on storage bay capacity is known:

Management forecasts that it must plan storage space for at least 11,000 units of product 1,4,000 units of product 2, and 12,000 units of product 3 over the next three months. The decision problem is how to allocate the products to the various storage bays (in the proper quantities) so as to minimize the total handling time (cost) required for all products.
Objective function

subject to

And

Gj = capacity of bay for product j
Rj = number of units of product j required to be stored
M = number of storage locations
N = number of products

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HANDLING TIMES (HR.)a PER 1,000 UNITS OF PRODUCT STORED IN VARIOUS BAYS Storage Bay 1 0.90 0.75 0.90 0.80 0.65 0.95 0.60 0.70 0.65 0.70 0.55 0.45 0.50 0.50 0.45 0.40 0.45 0.35 Product Storage Bay Capacity (units) 5,000 3,000 6,000 2 Xys 1.0 for i= 1,2,..,,M XiR for 1,2.., N