Regional Bell telephone operating companies, 18 which buy many products from a much smaller number of suppliers, often solicit discounts based on the total dollar volume of business awarded to particular suppliers. For example, suppliers i = 1,

c, 25 might be required to quote base prices pi,j for needed products j = 1,

c, 200, together with upper limits bi,k, k = 1,

c, 5 for ranges on total dollar volume and corresponding discount fractions di,k that increase with k. Then the actual cost to the telephone company of supplier i’s goods will be 11 - di,12 times the total base price of those goods if that total dollar value falls within the interval [ui,0, ui,1], 11 - di,22 if the total dollar volume falls within [ui,1, ui,2], and so on (assume that ui,0! 0, ui,5 Ú any feasible dollar volume). The company wants to choose bids to equal or exceed all required product quantities rj at least total discounted cost. Formulate an ILP model of this volume discount problem using the decision variables (i = 1,

c, 25; j = 1,

c, 200;

k = 1,

c, 5)

xi,j ! quantity of product j purchased from supplier i wi,k ! dollar volume of goods from supplier i when discount range k applies yi,k ! c 1 if discount range k applies for supplier i 0 otherwise