Qjm = quantity of commodity k consumed in demand region j and month m,

and the parameters ap and ba, are both assumed to be non-negative.

 Let

 X0.m = quantity of commodity k shipped from supply region i to demand region j in

 month m

 c3, = per unit transportation cost from supply region i to demand region j for

 commodity k in month m.

 With these definitions, a quadratic programming model can be written as

 J K M t L
Max E E E (ai.Q 1/2biQ)- E cl, W,
 j=l k=- m-l i.l 1kl


 I K M I J K M
 -E E E c2Z ,,-E E E E
 i-l k*- m-1 i-1 j-l k-I m-1


subject to


U1, b = dr, nW,








E xo. z.
j-1


i~l,...,J; k=l,...,K; m=1,...,M