A company produces a set of K products at I plants. It then ships these products to J market zones. For k = 1,..., K, i = 1,..., I, and j = 1,..., J, the following data are given:
vik = variable cost of producing one unit of product k at plant i
cijk = cost of shipping one unit of product k from plant i to zone j
fik = fixed cost associated with producing product k at plant i
Mik = maximal quantity of product k produced at plant i
mik = minimal quantity of product k that can be produced at plant i, if plant i produces a nonzero quantity
qik = capacity of plant i used to produce one unit of product k
Qi = capacity of plant i
djk = demand for product k at market zone j
(a) Formulate the problem of minimizing the total cost of production and transportation that the company is facing, as an integer programming problem. Indicate how your model can incorporate the following additional constraints.
(b) No plant may produce more than K1 products.
(c) Every product can be produced in at most I1 plants.
(d) For a particular product k0, plant 3 must produce it if neither plant 1 nor plant 2 produce it.
(e) Each market zone must be sourced by exactly one plant for all products.