Warren and Williams (W&W) Warehousing is planning to add a new line of automated warehouses for a specialized product. At present, W&W Warehousing has no such warehouses in its service region and does...

Warren and Williams (W&W) Warehousing is planning to add a new line of automated warehouses for a specialized product. At present, W&W Warehousing has no such warehouses in its service region and does not handle this product. Since it is not in the business now, it is, in essence, starting with a “clean sheet of paper.” W&W Warehousing has identified a number of candidate sites for the new warehouses and, for each site, it has identified a number of alternative sizes or capacities for a warehouse that might be built on the site. Clearly, at most one warehouse can be built at each candidate site. In addition, W&W Warehousing has predicted the demand in each of the markets it hopes to serve as well as the unit shipment cost from each candidate site to each market area. W&W Warehousing is interested in identifying the following quantities:

(a) The optimal number of warehouses

(b) The optimal sites for the warehouses

(c) The optimal sizes for the warehouses

(d) The optimal distribution plan (which warehouses supply which markets)

so that the total investment cost (measured on an equivalent daily payment basis) plus the distribution cost is minimized.

Using the notation below and any additional notation that you clearly define, formulate this problem as an integer linear programming problem. Clearly state the objective function and constraints in both words and mathematical notation. Be sure to indicate clearly any indices of summation and for which values of the indices constraints apply.
³

Inputs

hi = demand in market area i
 I

S_jk
= capacity at candidate site j
 J (or size or supply at site j
 J) if the kth sized warehouse is built there (that is, for site j
 J there are K alternative sizes of warehouses that can be built which are indexed by k = 1; ... ; K. This is the size of the kth one on this list.

F_jk
= fixed cost (equivalent daily payment of the investment cost) of building the kth sized warehouse on the list of possible warehouse sizes at site j
 J

c_ij
= unit shipment cost from candidate site j
 J to market area i
 I

³
This problem involves the formulation of a model involving economies of scale. The reader is referred to Osleeb et al. (1986) for a description of such a model applied to investments in ports to be used for exporting coal. Kuby, Ratick, and Osleeb (1991) further describe the application of this model.

b_jk
=unit variable cost of shipping through a warehouse at candidate site j
 J if the kth sized warehouse is built there

Assume that the unit variable costs decrease with the size of the facility (i.e., facilities are ordered so that S_jk
> S_LK-1
and b_jk
Lk-1) but that the fixed costs increase with the size of the facility (i.e., F_jk
> F_Lk-1).

Decision Variables

Y_ij
= number of units to ship from a warehouse at site j
 J to market area i
 I

X_jk
= 1 if we build the kth sized warehouse at candidate site j
 J

0 if not