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Operations Research Models and Methods
Problems Section
Nonlinear Programming Models
 - Multi-item Warehouse Stocking

A firm stocks n different items in the same warehouse. Item i requires a(i) square feet of warehouse space per unit. The total warehouse space available is A square feet. All n items are replenished independently in batches of size Q(i). Holding costs assessed on the average inventory level amount to h dollars per dollar invested per year. The value of item i is v(i) dollars per unit. The fixed ordering cost for item i is K(i) per batch, and the annual demand is d(i).

a. Develop a mathematical program to minimize the annual total inventory holding and ordering costs for all n items subject to warehouse availability. Assume that each item is allotted a space in the warehouse required to store Q(i).

b. Given that the warehouse space available is A and all will be used, form the Lagrangian function and determine expressions to find the optimal order quantities Q*(i) (i.e., set partial derivatives to zero).

c. Use A = 2000 sq ft, h = 0.2, and the following data to find the optimal Q*(i) and the minimum cost. How much of the total cost can be attributed to the warehouse restriction?





d(i) (units)




v(i) ($/unit)




K(i) ($/order)




a(i) (sq ft)





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Operations Research Models and Methods
by Paul A. Jensen
Copyright 2001 - All rights reserved