Formulate this lot sizing problem using a combination of

Question

Formulate this lot sizing problem using a combination of variable indices, linear constraints, and stretch constraints.

In each period t, there is a demand dit for product i, and this demand must be met from the stock of product i at the end of period t. At most one product can be manufactured in each period t, represented by variable y_t.

If nothing is manufactured, y_t = 0, where product 0 is a dummy product. The quantity of product i manufactured in any period must be either q_i or zero.

When product i is manufactured, its manufacture must continue no fewer than i and no more than ui periods in a row. The manufacture of product i in any period must be followed in the next period by the manufacture of one of the products in S_i (one may assume 0, i ∈ S_i).

The unit holding cost per period for product i is hi, and the unit manufacturing cost is g_i. The setup cost of making a transition from product i in one period to product j in the next is c_ij (where possibly i and/or j is 0).

Minimize total manufacturing, holding, and setup costs over an n-period horizon while meeting demand.

After formulating the problem, indicate how to replace the variable indices with element constraints.

Hint: let variable x_ij represent the quantity of product i manufactured in period t, and sit the stock at the end of the period. The element(y, x, z) constraint also has a form element(y, z | a). where a is a tuple of constants. It sets z equal to a_y.